agglomeration in scale-free random graphs - universidade federal de … · 2019. 11. 15. ·...
TRANSCRIPT
Universidade Federal de Minas Gerais
Instituto de Ciências Exatas
Departamento de Matemática
Agglomeration in scale-free random graphs
por
Rodrigo Botelho Ribeiro
Belo Horizonte
2016
Universidade Federal de Minas GeraisInstituto de Ciências ExatasDepartamento de matemática
Agglomeration in scale-free random
graphs
por
Rodrigo Botelho RibeiroOrientador: Remy Sanchis
⇤
Tese apresentada ao Departamento de Matemática da Universidade Federal de Minas Gerais, como parte
dos requisitos para obtenção do grau de
DOUTOR EM MATEMÁTICA
Belo Horizonte, 18 de julho de 2016
⇤O autor foi bolsista da CAPES durante a elaboração deste trabalho.
"Young man, in mathematics you do
not understand things. You just get
used to them."
John von Neumann
Agradecimentos
Aos meus pais e a todas as balas e chocolates que eles venderam para financiar minha educação básica.
Ao meu orientador e à estrutura pública da UFMG que me mostraram o fantástico mundo da ciência. E
por fim, mas não menos importante, ao lindo broto que nasceu em meu jardim.
ii
Resumo
Neste trabalho investigamos três modelos de grafos aleatórios que geram grafos livres de escala. Nosso
interese reside na formação de subgrafos completos, coeficientes de aglomeração, distribuição dos graus
e diâmetro. Nossos principais resultados mostram a existência assintoticamente quase certamente de
um subgrafo completo cuja ordem vai para infinito, além de cota superior para o diâmetro em um dos
modelos. Mostramos também que em um dos modelos, conhecido como Holme-Kim, os coeficientes de
aglomeração local e global possuem comportamentos bastante diferentes. Enquanto o primeiro permanece
longe do zero, o segundo tende a zero à medida que o tempo vai para infinito.
Palavras-Chaves: grafos, cliques, livres de escala, aglomeração, diâmetro, lei de potência.
iii
Abstract
In this work we investigate three random graph models capable of generating scale-free graphs. Our
main interest relies on formation of cliques, calculating clustering coefficients, the degree distribution and
diameters. The main results we have proven show the existence asymptotically almost surely the existence
of a clique whose order goes to infinity as the graph’s order goes to infinity, concentration inequalities for
the degrees and upper bound for the diameter in one of theses models. We also show that in the model
known as Holme-Kim’s model the clustering coefficients local and global present quite distinct behavior.
Whereas the former is bounded away from zero a.a.s, the latter goes to zero as the time goes to infinity.
Key-Words: graphs, cliques, scale-free, agglomeration, diameters, power-law
iv
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 Basic notation and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Clustering coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Brief discussion of our results . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. The Holme-Kim model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Definition of the process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Heuristics and a seemingly general phenomenon . . . . . . . . . . . . 13
2.1.3 Main technical ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.4 Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Formal definition of the process . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Technical estimates for vertex degrees . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Degree increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Upper bounds on vertex degrees . . . . . . . . . . . . . . . . . . . . . 19
2.4 Positive local clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Vanishing global clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Preliminary estimates for number of cherries . . . . . . . . . . . . . . 24
2.5.2 The bootstrap argument . . . . . . . . . . . . . . . . . . . . . . . . . 29
Contents 3
2.5.3 Wrapping up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Final comments on clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Power law degree distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3. The p-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 Bounds for the vertices’ degree . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 The Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Small-World Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. The p(t)-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1 Continuity of the parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 The case p(t) = 1/t� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Power-law distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 The expected maximum degree . . . . . . . . . . . . . . . . . . . . . 55
4.3 The case p(t) = 1/ log(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Power-law distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Lower bounds for the diameter . . . . . . . . . . . . . . . . . . . . . 57
5. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Concentration Inequalities for Martingales . . . . . . . . . . . . . . . . . . . 63
1. INTRODUCTION
Before we overview the area and introduce the problems we treat in this thesis, we beginby the basic notation and definitions needed to understand our results. We believe this willmake the reading easier for those who are not familiar enough with all the graph theoreticalnotations. Those who are already familiar with graph theory may start theirs by the overviewand go back to the basic notation when it is necessary.
1.1 Basic notation and definitions
In this section, we give all the general notations and definitions needed to state and proveour results. We start from the most basic and general graph theory notation.
Recall that a graph G = (VG
, EG
) consists of a set VG
of vertices and a set EG
⇢�
VG
2
�
of edges. Given G and v, w 2 VG
, we say v and w are neighbors, and write v $ w, if{v, w} 2 E
G
. We also write �G
(v) = {w 2 V : w $ v} for the neighborhood of v 2 VG
and e(�G
(v)) for the number of edges between the neighbors of v. In this case, we define thedegree of a vertex v in G by the number of neighbors it has in G and we denote this quantityby d
G
(v). However, some of our models may generate multiple edges, i.e., two vertices maybe connected by more than one edge or we may have loops, edges whose end points are thesame vertex. In these cases, we let e
G
(v, w) be the number of edges connecting v and w anddefine d
G
(v) in terms of eG
(v, w) writing it as dG
(w) =P
w2�G(v)
eG
(v, w), i.e, the total ofedges whose one of its ends is v.
Another quantity we have interest of is the maximum degree of a graph, or multigraph. It isthe value of the largest degree in G and we denote it by d
max
(G).
A complete subgraph in G is a subgraph S ⇢ G whose vertices are all connected, by atleast one edge, to each other. The order of the complete subgraph is simply the number ofvertices it has. Usually, we will call a complete subgraph in G clique or community.
A triangle in a graph G = (VG
, EG
) is a subset {u, v, w} 2�
VG
3
�
with u $ v $ w $ u.We denote the number of triangles formed by a set of vertices S by �(S). Thus, the total
1. Introduction 5
number of triangles in G is �(G).
A path of length k in G is an alternating sequence of vertices and edges v0
, e0
, ..., ek�1
, vk
in which vi
6= vj
for all i, j k and the edge ej
connects the vertex j to the vertex j + 1.The diameter of G, denoted by diam(G) is the length of the largest path in G. We alsohave particular interest of paths of length 2, which we call cherries. Observe that
�
dG(v)
2
�
counts the number of cherries which middle vertex is v. Thus,P
v2V�
dG(v)
2
�
counts the totalnumber of cherries in G, quantity we denote by C
G
. For a fixed vertex v 2 VG
, we denotethe number of triangles sharing at least the common vertex v by �
G
(v).
We will often consider the sequence of graphs {Gt
}t
, which is indexed by a discrete timeparameter t � 0. When considering this sequence, we replace the subscript G
t
with t in ourgraph notation, so that V
t
:= VGt , Et
:= EGt , etc. Given a sequence of numerical values
{xt
}t�0
depending on t, we will let � xt
:= xt+1
� xt
.
Finally, we will use the Landau o/O/⇥ notation at several points of our discussion. Thisalways pressuposes some asymptotics as a parameter n or t of interest goes to +1. Justwhich parameter this is will always be clear from the context.
1.1.1 Clustering coe�cients
We now define the clustering coe�cients that appear in our work and latter we point outwhy their behavior must be clarified to avoid the confusion found in some works.
Definition 1 (Local clustering coe�cient in v). Given a vertex v 2 G of degree at least two,the local clustering coe�cient at v, is
CG
(v) =�
G
(v)�
dG(v)
2
�
.
For vertices of degree one, we put its local clustering coe�cient as zero.
Notice that 0 CG
(v) 1 always, since there can be at most one triangle in {v} [ �G
(v)for each pair of neighbors of v. In probabilistic terms, C
G
(v) measures the probability thata pair of random neighbors of v for an edge, ie. how likely it is that “two friends of v arealso each other’s friends.”
The two coe�cients for the graph G are as follows.
Definition 2 (Local and Global Clustering Coe�cients). The local clustering coe�cient
of G is defined as
Cloc
G
:=
P
v2VGCG
(v)
|VG
| ,
1. Introduction 6
whereas the global coe�cient is
Cglo
G
:= 3⇥ �(G)P
v2VG
�
dG(v)
2
�
.
Bollobas and Riordan [1, page 18] have observed that Cloc
G
and Cglo
G
are used interchangeablyin the non-rigorous literature. They warned that:
In some papers it is not clear which of the two definitions above is intended;when it is clear, sometimes Cloc is used, and sometimes Cglo; In more balancedgraphs the definitions will give more similar values, but they will still di↵er by atleast a constant factor much of the time.
In fact, more extreme di↵erences are possible for non-regular graphs.
Build a graph G consisting of an edge e and n � 2 other vertices connected to the twoendpoints of e, it is easy to see that Cloc
G
= 1� 2
n
. On the other hand, it is straightforward
to see that Cglo
G
= 3(n�2)
n�2+(n�1)(n�2)
= 3
n
..
One of our results shows that we have a random graph model wich obeys a power-lawdistribution and satisfies the less extreme bound Cloc
Gt> 0 and Cglo
Gt! 0 for t large. This
contradicts the numerical findings of [26], where the Holme-Kim model is cited as a modelthat exhibits positive “clustering coe�cient”, but the definition given is Cglo
G
and also thestatement made in [25] which a�rms that both definition are the same, less than constantfactors.
1.2 Overview
Since the 2000’s, the scientific community has made e↵orts to understand the structure oflarge complex networks – e.g., Twitter [15], Scientific coauthorship networks [23]– and also topropose and investigate random models capable of capturing properties pointed out by em-pirical results. Some models have shown to reproduce some of these properties. Three goodexamples are: the model proposed by Strogatz and Watts [29], which captures a phenomenonknown as small-world ; the one proposed by Albert and Barabasi [2], whose rule of evolution– known as preferential attachment –, is capable of producing graphs obeying a power-lawdistribution; and another one proposed by Holme and Kim [18], which captures the tendencythat some networks have of closing triangles. However, the same models have failed to cap-ture others properties also seen in real world networks – the Strogatz-Watts’ model does notobey power laws and the Albert-Barabasi’s model does not have agglomeration [1].
1. Introduction 7
Over the years, variations of the classic models have appeared, mainly of Albert-Barabasi’s,followed by papers that have investigated whether these random graph models had certainproperties beyond the power-law. In [14], the authors determine, asymptotically, the ex-pected value of the global clustering coe�cient of a random graph process which is a slightmodification of Albert-Barbsi’s model. Although this modification still obeys a power-law,it doesn’t have large cliques nor a high clustering coe�cient. In [13], the diameter of a largeclass of models with preferential attachment rules is investigated and all cases consideredhave diameters tending to infinity as the graph’s order increases. In [4, 9, 12], a modelcombining preferential attachment rules and vertices fitness is investigated; however, littleis known about its structure.
Other important related questions are about the spread of diseases in scale-free graphs andtheir vulnerability to deliberate attack – see [6] for an example – which are related somehow tothe existence of complete subgraphs. In the context of vulnerability, cliques play a significantrole. When the attack is completely random, large cliques have high probability of remainingconnected. On the other hand, deliberate attacks directed towards them represent a threatto network’s connectedness. Still in the practical context, the presence of certain subgraphsin biological networks, called motifs, is related to functional properties selected by evolution[21].
The importance of cliques goes beyond the examples cited above. They are useful in GraphTheory, too. The clique-number of G provides a lower bound to the chromatic number of G,i.e., the minimum number of colors such that G has a proper coloring. Furthermore, it alsoprovides a lower bound to the number of triangles in G, a fundamental quantity to study theso-called global clustering coe�cient of G (see e.g. [1, 14, 26, 28]). For more works relatedto cliques in scale-free random graphs, see also [5, 16, 17].
In short, we may say all these models wanted to achieve the following desiderata.
• Power-law degree distribution. This means that the fraction of nodes with degree kshould be roughly of the order k�� for some � > 0. This contrasts starkly with thesituation in Erdos-Renyi graphs with a similar edge density, where p
k
would have aPoisson shape. Power-law degree distributions are generally believed to exist in a widevariety of real-life complex networks [24]. The Barabasi/Albert model is rigorouslyknown to satisfy this property, with � = 3 [8].
• Clustering. In addition to the power law, social and metabolic networks have thefeature that “friends of friends tend to be friends”, ie. that if a node a is connectedto both b and c, the latter have higher propensity of being connected. This trait isnot captured by the Barabasi/Albert model. The Small World model of Strogatz andWatts [29], which is also extremely well-known, does have this property (but has nopower-law).
1. Introduction 8
• Small distances. A third characteristic of real-life networks is that, unlike e.g. pathsor patches of regular lattices, they tend to have diameter that is logarithmic in thenumber of nodes. This property is shared by the two aforementioned models and infact by many others, including some homogeneous ones.
• Large communities. Real-life networks, mainly social-networks, contain large completesubgraphs, also called communities in this context, which are explained by the tendencypeople have of forming large groups where everyone knows everyone.
Unfortunately, these models have failed in two aspects:
1. the graphs generated by them do not capture simultaneously all the properties seen inempirical results;
2. their dynamics do not take into account connections made by a�nity among the ver-tices, i.e., theirs dynamics ignore a phenomenon observed empirically which says thatthe vertices also tend to connect to other vertices of similar age or sharing some othercharacteristic.
Beyond questions about the model’s capability of capturing desired properties, we haveobserved that even the set of properties is not well defined, in the sense that some of theseproperties still have definition problems, as is the case of the concept of the clusteringcoe�cient, which we have discussed in Section 1.1.1.
To conclude this brief overview, we observe that despite many works of investigation ofrandom graph models with preferential attachment rules, such as [22, 14, 13], we believe thescientific community has not yet explored the subject in its full range. There are relevantmodels in the computer science literature whose properties lack mathematical rigor and somestructures of the graphs generated by these models, such as large cliques, have not receivedenough attention. The present thesis is a contribution to the matters above. We investigatemodels which may capture simultaneously all the properties aforementioned and also proposeanother one we believe is more realistic than those we have studied.
1. Introduction 9
1.3 Brief discussion of our results
We have investigated three random graph models, which will be formally defined in separatedchapters. For now we limit ourselves to informally state the main results we have concerningeach of them.
Holme-Kim model:
• The degree distribution follows a power-law distribution with exponent � = 3. We alsoprove sharp concentrations for this power law;
• With high probability, the model has a local clustering coe�cient bounded away fromzero.
• With high probability, the model’s global clustering coe�cient goes to zero as thenumber of nodes in the graph goes to infinity: we also compute its decay speed;
• We give a formal explanation for the di↵erent behavior among the two coe�cients;
• We also prove the Weak Law of Large Numbers for the numbers of paths of size twoin the graphs generated by this model;
The p-model:
• The degree distribution follows a power law distribution with exponent which is afunction of its parameter p given by � = 2 + p
2�p
. Actually, this results may be foundin [10], but we cite it here for completeness;
• The existence, with high probability, of a community whose order goes to infinity asthe graph’s size increases. We specify the community’s order in function of the model’sparameter p. This result may be found in the preprint at http://arxiv.org/abs/
1509.04650
• By a coupling with the classical Albert-Barabasi’s model we show that the modelexhibits the small-world phenomenon;
The p(t)-model:
• The degree distribution follows a power law distribution with exponent which dependson how fast the function p(t) goes to zero;
• A lower bound for the diameter for a specific function p(t).
2. THE HOLME-KIM MODEL
In this Chapter we provide a rigorous analysis of a specific non-homogeneous random graphmodel, whose motivation was to combine scale-freeness and clustering. This model wasintroduced in 2001 by Holme and Kim [19]. The Holme-Kim or HK model describes arandom sequence of graphs {G
t
}t2N that will be formally defined in Section 2.2. Here is
an informal description of this evolution. Fix two parameters p 2 [0, 1] and a positiveinteger m > 1, and start with a graph G
1
. For t > 1, the evolution from Gt�1
to Gt
consistsof the addition of a new vertex v
t
and m new edges between vt
and vertices already in Gt�1
.These m edges are added sequentially in the following fashion.
1. The first edge is always attached following the preferential attachment (PA)mechanism,that is, it connects to a previously existing node w with probability proportional tothe degree of w in G
t�1
.
2. Each of the m� 1 remaining edges makes a random decision on how to attach.
(a) With probability p, the edge is attached according to the triad formation (TF)mechanism. Let w0 be the node of G
t�1
to which the previous edge was attached.Then the current edge connects to a neighbor w of w0 chosen with probabilityproportional to number of edges between w and w0.
(b) With probability 1� p (p 2 [0, 1] fixed) the edge follows the same PA mechanismas the first edge (with fresh random choices).
The case p = 0 of this process, where only preferential attachment steps are performed, isessentially the Barabasi-Albert model [3]. The triad formation steps, on the other hand, arereminiscent of the copying model by Kumar et al. [20]. Holme and Kim argued on the basisof simulations and non-rigorous analysis that their model has the properties of scale-freenessand positive clustering.
Our rigorous results partly confirm their findings. The degree power law can be checkedby known methods. On the other hand, we show that there were aspects of the clusteringphenomenon (or lack thereof) that were not made evident in [19] or in other papers in thelarge networks literature (e.g. [27]). We will see that the question of whether the Holme-Kim
2. The Holme-Kim model 11
model has clustering admits two di↵erent answers depending on how we define clustering.
2.1 Definition of the process
In this section we give a more formal definition of the Holme-Kim process.
The model has two parameters: a positive integer numberm � 2 and a real number p 2 [0, 1].It produces a graph sequence {G
t
}t�1
which is obtained inductively according to the growthrule we describe below.
Initial state. The initial graph G1
, which will be taken as the graph with vertex setV1
= {1} and a single edge, which is a self-loop.
Evolution. For t > 1, obtain Gt+1
from Gt
adding to it a vertex t+ 1 and m edges betweent+ 1 and vertices Y (i)
t+1
2 Vt
, 1 i m. These vertices are chosen as follows. Let Ft
be the�-field generated by all random choices made in our construction up to time t. Assume weare given i.i.d. random variables (⇠(i)
t+1
) independent from Ft
. We define:
P⇣
Y(1)
t+1
= u�
�
�
Ft
⌘
=dt
(u)
2mt,
which means the first choice of vertex is always made using the preferential attachmentmechanism. The next m � 1 choices Y
(i)
t+1
, 2 i m, are made as follows: let F (i�1)
t
be
the �-field generated by Ft
and all subsequent random choices made in choosing Y(j)
t+1
for1 j i� 1. Then:
P⇣
Y(i)
t+1
= u�
�
�
⇠(i)
t+1
= x,F (i�1)
t
⌘
=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
dt(u)
2mt
if x = 0,
et(Y(i�1)t+1 ,u)
dt(Y(i�1)t+1 )
if x = 1 and u 2 �t
(Y (i�1)
t+1
),
0 otherwise.
In words, for each choice for the m�1 end points, we flip an independent coin of parameter pand decide according to it which mechanism we use to choose the end point. With probabilityp use the triad formation mechanism, i.e., we choose the end point among the neighbors ofthe previously chosen vertex Y
(i�1)
t+1
. With probability 1� p, we make a fresh choice from Vt
using the preferential attachment mechanism. In this sense, if ⇠(i)t
= 1 we say we have takena TF-step. Otherwise, we say PA-step.
2. The Holme-Kim model 12
2.1.1 Our results
The main results in this Chapter show that such a disparity between local and global cluster-ing does indeed occur in the specific case of the Holme-Kim model, albeit in a less extremeform than what the Example suggests.
Theorem 2.1.1 (Positive local clustering for HK). Let {Gt
}t�0
be the sequence of graphsgenerated by the Holme-Kim model with parameters m � 2 and p 2 (0, 1). Then there existsc > 0 depending only on m and p such that the local clustering coe�cients Cloc
Gtof the graphs
Gt
satisfy:lim
t!+1P�
Cloc
Gt� c
�
= 1.
Theorem 2.1.2 (Vanishing glocal clustering for HK). Let {Gt
}t�0
be as in Theorem 2.1.1.Then there exist constants b
1
, b2
> 0 depending only on m and p such that the global clusteringcoe�cients Cglo
Gtsatisfy:
limt!+1
P✓
b1
log t Cglo
Gt b
2
log t
◆
= 1.
Thus for large t, one of the two clustering coe�cients is typically far from 0, whereas theother one goes to 0 in probability, albeit at a small rate. This shows that the remark byBollobas and Riordan is very relevant in the analysis of at least one network model. Ourresults contradict the numerical findings of [27], where the Holme-Kim model is cited asa model that exhibits positive “clustering coe�cient”, but the definition of clustering usedcorresponds to the global coe�cient1.
For completeness, we will also check in the Appendix that the HK model is scale-free withpower-law exponent � = 3. The proof follows from standard methods in the literature.
Theorem 2.1.3 (The power-law for HK). Let {Gt
}t�0
be as in the previous theorem. Alsolet N
t
(d) be the number of vertices of degree d in Gt
and set
Dt
(d) :=EN
t
(d)
t.
Then
limt!1
Dt
(d) =2(m+ 3)(m+ 1)
(d+ 3)(d+ 2)(d+ 1).
andP⇣
|Nt
(d)�Dt
(d) t| � 16d · c ·pt⌘
(t+ 1)d�me�c
2.
1Their error may be partly explained by the slow decay of Cglo
Gt.
2. The Holme-Kim model 13
2.1.2 Heuristics and a seemingly general phenomenon
The disparity between Cloc
G
and Cglo
G
should be a general phenomenon for large scale-free graphmodels with many (but not too many triangles). This will transpire from the follwowingheuristic analysis of the Holme-Kim case with p 2 (0, 1).
To begin, it is not hard to understand why Theorem 2.1.1 should be true. By Theorem2.1.3, there is a positive fraction of nodes with degree m. Moreover, a positive fraction ofthese vertices are contained in at least one triangle because of TF steps. A more generalobservation may be made.
Reason for positive local clustering: if a positive fraction of nodes havedegree d (assumed constant), and a positive fraction of these nodes are con-tained in at least one triangle, then the local clustering coe�cient Cloc
Gtmust be
bounded away from zero.
We now argue that the vanishing of Cglo
Gtshould be a consequence of the power law degree
distribution. The global clustering coe�cient Cglo
Gtis essentially the ratio of the number of
triangles to the number of cherries in Gt
, the latter being denoted by Ct
. Now, one can easilyshow that the number of triangles in G
t
grows linearly in t with high probability, so:
Cglo
Gt⇡ # of triangles in G
t
Ct
⇡ t
Ct
.
To estimate Ct
, we note that each vertex v of degree d in Gt
is the “middle vertex” of exactlyd(d� 1)/2 ⇡ d2 cherries. This means
Ct
t⇡
t
X
d=1
Nt
(d)
td2
⇣
Nt(d)
t
⇠ Dt
(d) ⇡ d�3 by Theorem 2.1.3⌘
⇡t
X
d=1
1
d⇡ log t.
Our reasoning is not rigorous because it requires bounds on Nt
(d) for very large d. However,we feel our argument is compelling enough to be true for many models. In fact, consideringthe case where N
t
(d)/t ⇡ d�� for 0 < � 3, one is led to the following.
Heuristic reason for large number of cherries: if the fraction of nodesof degree d in G
t
is ⇡ d�� for some 0 < � 3, the number of cherries Ct
is superlinear in t. More precisely, we expect Ct
/t ⇡ t3�� for 0 < � < 3 andC
t
/t ⇡ log t for � = 3.
2. The Holme-Kim model 14
The power law range 0 < � 3 corresponds to most models of large networks in theliterature. Likewise, we believe that the disparity between Cglo
Gtand Cloc
Gtshould hold for all
“natural” random graph sequences with many triangles and power law degree distributionwith exponent 0 < � 3. The general message is this.
Heuristic disparity between local and global clustering: achievingpositive local clustering is “easy”: just introduce a density of triangles in sparseareas of the graph. On the other hand, if the number of triangles in G
t
growslinearly with time, and the fraction of nodes of degree d in G
t
is ⇡ d�� for some0 < � 3, then one expects a vanishingly small global clustering coe�cient.
2.1.3 Main technical ideas
At a high level, our proofs follow standard ideas from previous rigorous papers on complexnetworks. For instance, suppose one wants to keep track of the number of nodes of degreed at time t, for d = m,m + 1, . . . , D. Letting N
t
= (Nt
(m), . . . , Nt
(D)), the basic strategyadopted in several previous papers is to find a deterministic matrix M
t�1
and a deterministicvector r
t�1
, both measurable with respect to G0
, . . . , Gt�1
, such that:
Nt
= Mt�1
Nt
+ rt�1
+ ✏t
, where E(✏t
| G0
, . . . , Gt
) ⇡ 0.
This can be seen as “noisy version” of the deterministic recursion Nt
= Mt�1
Nt�1
+ rt�1
with ✏t
the “noise” term. One then studies the recursion and uses martingale techniques(especially the Azuma-Ho↵ding inequality) to prove that N
t
concentrates around the solutionof the deterministic recursion. Our own proof of the degree power law follows this outline,and is only slightly di↵erent from the one in [11].
Once the degree sequence is analyzed, Theorem 2.1.1 is then a matter of observing that adensity of vertices of degree m will be contained in at least one triangle, due to a TF step.On the other hand, the analysis of global clustering is harder due to the need to estimate thenumber of cherries C
t
. Justifying the heuristic calculation above would require strong controlof the degree distribution up to very large values of d. We opt instead to write a “noisyrecursion” for C
t
itself. However, the increments in this noisy recursion can be quite large,and the Azuma-Ho↵ding inequality is not enough to control the process. We use insteadFreedman’s concentration inequality, which involves the quadratic variation, but even thatis delicate because the variation might “blow up” in certain unlikely events. In the end, weuse a kind of “bootstrap” argument, whereby a preliminary estimate of C
t
is fed back intothe martingale calculation to give sharper control of the predictable terms and the variation.The upshot is a Weak Law of Large Numbers for C
t
, given in Theorem 2.1.4 below:
2. The Holme-Kim model 15
Theorem 2.1.4 (The Weak Law of large Numbers for Ct
). Let Ct
be the number of cherriesin G
t
, thenC
t
t log tP�!
✓
m+ 1
2
◆
.
Overall, we believe our martingale analysis of Ct
is our main technical contribution.
2.1.4 Organization.
The remainder of this Chapter is organized as follows. Section 2.2 presents a formal definitionof the model. In Section 2.3 we prove technical estimates for the degree which will be usefulthroughout the Chapter. Sections 2.4 and 2.5 are devoted to prove the bounds for thelocal and the global clustering coe�cients, respectively. Section 2.6 presents a comparativeexplanation for the so distinct behavior of the clustering coe�cients. The proof of thepower-law distribution is done at the end of the Chapter, in Section 2.7 because it followswell known martingale arguments.
2.2 Formal definition of the process
In this section we give a more formal definition of the Holme-Kim process (compare withthe beginning of this Chapter).
The model has two parameters: a positive integer numberm � 2 and a real number p 2 [0, 1].It produces a graph sequence {G
t
}t�1
which is obtained inductively according to the growthrule we describe below.
Initial state. The initial graph G1
, which will be taken as the graph with vertex setV1
= {1} and a single edge, which is a self-loop.
Evolution. For t > 1, obtain Gt+1
from Gt
adding to it a vertex t+ 1 and m edges betweent+ 1 and vertices Y (i)
t+1
2 Vt
, 1 i m. These vertices are chosen as follows. Let Ft
be the�-field generated by all random choices made in our construction up to time t. Assume weare given i.i.d. random variables (⇠(i)
t+1
) independent from Ft
. We define:
P⇣
Y(1)
t+1
= u�
�
�
Ft
⌘
=dt
(u)
2mt,
which means the first choice of vertex is always made using the preferential attachmentmechanism. The next m � 1 choices Y
(i)
t+1
, 2 i m, are made as follows: let F (i�1)
t
be
2. The Holme-Kim model 16
the �-field generated by Ft
and all subsequent random choices made in choosing Y(j)
t+1
for1 j i� 1. Then:
P⇣
Y(i)
t+1
= u�
�
�
⇠(i)
t+1
= x,F (i�1)
t
⌘
=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
dt(u)
2mt
if x = 0,
et(Y(i�1)t+1 ,u)
dt(Y(i�1)t+1 )
if x = 1 and u 2 �t
(Y (i�1)
t+1
),
0 otherwise.
In words, for each choice for the m�1 end points, we flip an independent coin of parameter pand decide according to it which mechanism we use to choose the end point. With probabilityp use the triad formation mechanism, i.e., we choose the end point among the neighbors ofthe previously chosen vertex Y
(i�1)
t+1
. With probability 1� p, we make a fresh choice from Vt
using the preferential attachment mechanism. In this sense, if ⇠(i)t
= 1 we say we have takena TF-step. Otherwise, we say a PA-step was performed.
2.3 Technical estimates for vertex degrees
In this section we collect several results on vertex degrees. Subsection 2.3.1 describes theprobability of degree increments in a single step. In Subsection 2.3.2 we obtain upper boundson all degrees. Some of these results are fairly technical and may be skipped in a first reading.
2.3.1 Degree increments
We begin the following simple lemma.
Lemma 2.3.1. For all k 2 {1, ..,m}, there exists positive constants cm,p,k
such that
P (�dt
(v) = k|Gt
) = cm,p,k
dkt
(v)
t+O
✓
dk+1
t
(v)
tk+1
◆
.
In particular, for k = 1 we have cm,p,1
= 1/2.
Proof. We begin proving the following claim involving the random variables Y (i)
t
Claim: For all i 2 {0, 1, 2, . . . ,m},
P⇣
Y(i)
t+1
= v�
�
�
Gt
⌘
=dt
(v)
2mt. (2.3.2)
2. The Holme-Kim model 17
Proof of the claim: The proof follows by induction on i. For i = 1 we have nothing to do.So, suppose the claim holds for all choices before i� 1. Then,
P⇣
Y(i)
t+1
= v�
�
�
Gt
⌘
= P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
= v�
�
�
Gt
⌘
+ P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
6= v�
�
�
Gt
⌘
=(1� p)
4m2
d2t
(v)
t2+ P
⇣
Y(i)
t+1
= v, Y(i�1)
t+1
6= v�
�
�
Gt
⌘
.(2.3.3)
For the first term on the r.h.s the only way we can choose v again is following a PA-stepand then choosing v according to preferential attachment rule. This means
P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
= v�
�
�
Gt
⌘
=(1� p)
4m2
d2t
(v)
t2. (2.3.4)
For the second term, we divide it in two sets, whether the vertex chosen at the previouschoice is a neighbor of v or not.
P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
6= v�
�
�
Gt
⌘
=X
u/2�Gt (v)
P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
= u�
�
�
Gt
⌘
+X
u2�Gt (v)
P⇣
Y(i)
t+1
= v, Y(i�1)
t+1
= u�
�
�
Gt
⌘
= (1� p)dt
(v)
2mt
0
@
X
u/2�Gt (v)
dt
(u)
2mt
1
A
+ p
0
@
X
u2�Gt (v)
eGt(u, v)
dt
(u)
dt
(u)
2mt
1
A
+(1� p)
2m
dt
(v)
t
0
@
X
u2�Gt (v)
dt
(u)
2mt
1
A
=dt
(v)
2mt� (1� p)
4m2
d2t
(v)
t2.
We used our inductive hypothesis and that
X
u2�Gt (v)
dt
(u)
2mt+
X
u/2�Gt (v)
dt
(u)
2mt= 1� d
t
(v)
2mt
Returning to (2.3.3) we prove the claim. ⌅
We will show the particular case k = 1 since we have particular interest in the value of cm,p,1
and point out how to obtain the other cases. We begin noticing that the process of choices
2. The Holme-Kim model 18
is by definition a homogeneous Markovian process. This means to evaluate the probabilityof a vertex increasing its degree by exactly one, the case k = 1, we just need to know theprobabilities of transition. In this way, use the notation P
t
to denote the measure conditionedon G
t
and go to the computation of the probabilities of transition. We start by the hardestone.
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
6= v⌘
=X
u2�Gt (v)
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= u⌘
Pt
⇣
Y(i)
t+1
= u�
�
�
Y(i)
t+1
6= v⌘
+X
u/2�Gt (v)
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= u⌘
Pt
⇣
Y(i)
t+1
= u�
�
�
Y(i)
t+1
6= v⌘
(2.3.5)
When u 2 �Gt(v), we can choose v taking any of the two steps. This implies the equation
below
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= u⌘
= (1� p)dt
(v)
2mt+ p
et
(u, v)
dt
(u), (2.3.6)
but when u /2 �Gt(v) the only way we can choose v is following a PA-step, which implies
that
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= u⌘
= (1� p)dt
(v)
2mt. (2.3.7)
We also notice that the equation below holds, since u 6= v and our claim is true
Pt
⇣
Y(i)
t+1
= u�
�
�
Y(i)
t+1
6= v⌘
=Pt
⇣
Y(i)
t+1
= u⌘
Pt
⇣
Y(i)
t+1
6= v⌘
Claim
=dt
(u)
2mtPt
⇣
Y(i)
t+1
6= v⌘ . (2.3.8)
And the same Claim also implies that
1
Pt
⇣
Y(i)
t+1
6= v⌘ =
1
1� dt(v)
2mt
= 1 +1X
n=1
✓
dt
(v)
2mt
◆
n
. (2.3.9)
Combining the last three equations to (2.3.5), we are able to get
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
6= v⌘
= (1� p)dt
(v)
2mt+ p
dt
(v)
2mt
1 +1X
n=1
✓
dt
(v)
2mt
◆
n
!
=dt
(v)
2mt+O
✓
d2t
(v)
t2
◆
.
(2.3.10)
2. The Holme-Kim model 19
If we chose v at the previous choice, the only way we select it again is following a PA-step,this means that
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= v⌘
= (1� p)dt
(v)
2mt. (2.3.11)
From these two probabilities of transition we may obtain the other ones.
To compute the probability of {�dt
(v) = 1} given Gt
we may split it in m possible ways toincrease d
t
(v) by exactly one. Each possible way has an index i 2 {1, ...,m} meaning thestep we chose v and then we must avoid it at the other m� 1 choices. This means that eachof these m ways has a probability similar to the expression below
✓
1� dt
(v)
2mt�O
✓
d2t
(v)
t2
◆◆
i�1
✓
dt
(v)
2mt+O
✓
d2t
(v)
t2
◆◆✓
1� dt
(v)
2mt�O
✓
d2t
(v)
t2
◆◆
m�i
which implies that
P (�dt
(v) = 1|Gt
) =dt
(v)
2t+O
✓
d2t
(v)
t2
◆
.
The cases k > 1 are obtained in the same way, considering the�
m
k
�
ways of increase dt
(v) byk.
Observation 1. We must notice that term O⇣
d
k+1t (v)
t
k+1
⌘
given by the Lemma actually is
a statement stronger than the real big-O notation. Since the Lemma’s proof implies theexistence of a positive constant c
m,p,k
such that
O
✓
dk+1
t
(v)
tk+1
◆
cm,p,k
dk+1
t
(v)
tk+1
,
for all vertex v and time t.
2.3.2 Upper bounds on vertex degrees
To control the number of cherries, Ct
, we will need upper bounds on vertex degrees. Thebound is obtained applying the Azuma-Ho↵ding inequality to the degree of each vertex,which is a martingale after normalizing by the quantity defined below.
�(t) :=t�1
Y
s=1
✓
1 +1
2s
◆
. (2.3.12)
A fact about �(t) will be useful: there exists positive constants b1
and b2
such that
b1
pt �(t) b
2
pt,
for all t.
2. The Holme-Kim model 20
Proposition 2.3.13. For each vertex j, the sequence (X(j)
t
)t�j
defined as
X(j)
t
:=dt
(j)
�(t)
is a martingale.
Proof. Since the vertex j will remain fixed throughout the proof, we will write simply Xt
instead of X(j)
t
.
Observe that we can write dt+1
(j) as follows
dt+1
(j) = dt
(j) +m
X
k=1
1n
Y(k)
t+1
= jo
.
In addition, we proved in Lemma 2.3.1 that, for all k 2 1, ...,m, we have
P⇣
Y(k)
t+1
= j�
�
�
Gt
⌘
=dt
(j)
2mt. (2.3.14)
Thus, the follow equivalence relation is true
E [dt+1
(j)|Gt
] =
✓
1 +1
2t
◆
dt
(j). (2.3.15)
Then, dividing the above equation by �(t+ 1) the desired result follows.
Once we have Proposition 2.3.13 we are able to obtain an upper bound for dt
(j).
Theorem 2.3.16. There is a positive constant b3
such that, for all vertex j
P⇣
dt
(j) � b3
pt log(t)
⌘
t�100.
Proof. The proof is essentially applying Azuma’s inequality to the martingale we obtainedin Proposition 2.3.13. Again we will write it as X
t
.
Applying Azuma’s inequality demands controlling Xt
’s variation, which satisfies the upperbound below
|�Xs
| =
�
�
�
�
�
ds+1
(j)��
1 + 1
2s
�
ds
(j)
�(s+ 1)
�
�
�
�
�
2m
�(s+ 1) b
4ps. (2.3.17)
Thus,t
X
s=j+1
|�Xs
|2 b5
log(t).
2. The Holme-Kim model 21
We must notice that none of the above constants depend on j. Then, Azuma’s inequalitygives us that
P (|Xt
�X0
| > �) 2 exp
⇢
� �2
b5
log(t)
�
. (2.3.18)
Choosing � = 10pb5
log(t) and recalling Xt
= dt
(j)/�(t) we obtain
P✓
�
�
�
�
dt
(j)� m�(t)
�(j)
�
�
�
�
> 10p
b5
�(t) log(t)
◆
t�100.
Finally, using that b1
pt �(t+ 1) b
2
pt, comes
P✓
�
�
�
�
dt
(j)� m�(t)
�(j)
�
�
�
�
> 10p
b5
b2
pt log(t)
◆
t�100.
Implying the desired result.
An immediate consequence of Theorem 2.3.16 is an upper bound for the maximum degreeof G
t
Corollary 2.3.19 (Upper Bound to the maximum degree). There exists a positive constantb1
such that
P⇣
dmax
(Gt
) � b1
pt log(t)
⌘
t�99.
Proof. The event involving dmax
(Gt
) may be seen as follows
n
dmax
(Gt
) � b1
pt log(t)
o
=[
jt
n
dt
(j) � b1
pt log(t)
o
.
Using union bound and applying Theorem 2.3.16 we prove the Corollary.
The next three lemmas are of a technical nature. Their statements will become clearer inthe proof of the upper bound for C
t
.
Lemma 2.3.20. There are positive constants b6
and b7
, such that, for all vertex j and alltime t
0
t, we have
P
dt0(j) > b
4
r
t0
tdt
(j) + b5
pt0
log(t)
!
t�100.
Proof. For each vertex j and t0
t, consider the sequence of random variables (Zs
)s�0
defined as Zs
= Xt0+s
, which is adaptable to the filtration Fs
:= Gt0+t
.
2. The Holme-Kim model 22
Concerning Zs
’s variation, using (2.3.17), we have the following upper bound
|�Zs
| = |�Xt0+s
| b4p
t0
+ s.
Thus,t�t0X
s=0
|�Zs
|2 b5
log(t).
Applying Azuma’s inequality, we obtain
P (|Zt�t0 � Z
0
| � �) 2 exp
⇢
� �2
b5
log(t)
�
. (2.3.21)
But, the definition of Zs
and the fact that �(t) = ⇥�p
t�
the inclusion of events below istrue
⇢
dt0(j) > b
2
pt0
�+b2
pt0
b1
ptdt
(j)
�
⇢ {|Zt
� Z0
| � �} ,
which, combined to (2.3.21), proves the lemma if we choose � = 10pb5
log(t).
Lemma 2.3.22. There is a positive constant b8
such that
P
t
[
j=1
t
[
t0=j
�
dt0(j) > b
8
pt0
log(t)
!
2t�98.
Proof. This Lemma is consequence of Theorem 2.3.16 and Lemma 2.3.20, which state, re-spectively
P⇣
dt
(j) � b3
pt log(t)
⌘
t�100, (2.3.23)
P
dt0(j) > b
4
r
t0
tdt
(j) + b5
pt0
log(t)
!
t�100. (2.3.24)
In which the constants b3
, b4
and b5
don’t depend on the vertex j neither the times t0
and t.
Now, for each t0
t and vertex j, consider the events below
At0,j :=
�
dt0(j) > b
8
pt0
log(t)
,
Bt0,j :=
(
dt0(j) > b
4
r
t0
tdt
(j) + b5
pt0
log(t)
)
,
eC
t,j
:=n
dt
(j) � b3
pt log(t)
o
.
2. The Holme-Kim model 23
Now we obtain an upper bound for P (At0,j) using the bounds we have obtained for the
probabilities of Bt0,j and C
t,j
.
P (At0,j) = P (A
t0,j \ Bt0,j) + P
�
At0,j \ Bc
t0,j
�
P (Bt0,j) + P
�
At0,j \Bc
t0,j\ C
t,j
�
+ P�
At0,j \Bc
t0,j\ Cc
t,j
�
P (Bt0,j) + P (C
t0,j) + P�
At0,j \ Bc
t0,j\ Cc
t,j
�
.
(2.3.25)
However, notice we have the following inclusion of events
Bc
t0,j\ Cc
t,j
⇢�
dt0(j) (b
4
b3
+ b5
)pt0
log(t)
.
Thus, choosing b8
= 2(b4
b3
+ b5
) we have At0,j \Bc
t0,j\Cc
t,j
= ;, which allows us to concludethat
P (At0,j) 2t�100.
Finally, an union bound over t0
followed by a union bound over j implies the desired result.
2.4 Positive local clustering
In this section we prove Theorem 2.1.1, which says that the local clustering coe�cient isbounded away from 0 with high probability.
Proof of Theorem 2.1.1. We must find a lower bound for
Cloc
Gt:=
1
t
X
v2Gt
CGt(v).
Let vm
be a vertex in Gt
whose degree is m. Observe that each TF-step we took when vm
was added increase e (�Gt(vm)) by one. So, denote by T
v
the number of TF-steps taken atthe moment of creation of vertex v. Since all the choices of steps are made independently,Tv
follows a binomial distribution with parameters m � 1 and p. Now, for every vertex weadd to the graph, put a blue label on it if T
v
� 1. The probability of labeling a vertex isbounded away from zero and we denote it by p
b
.
By Theorem 2.1.3, with probability at least 1� t�100, we have
Nt
(m) � b1
t� b2
p
t log(t).
Thus, the number of vertices in Gt
of degree m which were labeled, N (b)
t
(m), is boundedfrom below by a binomial random variable, B
t
, with parameters b1
t � b2
p
t log(t) and pb
.But, about B
t
we have, for all � > 0,
P✓
Bt
E[Bt
]
4
◆
�
1� pb
+ pb
e��
�
b1t�b2
pt log(t)
expn
�pb
⇣
b1
t� b2
p
t log(t)⌘
/4o
2. The Holme-Kim model 24
and choosing � properly we conclude that, w.h.p,
N(b)
t
(m) � pb
⇣
b1
t� b2
p
t log(t)⌘
/4. (2.4.1)
Finally, note that each blue vertex of degree m has CGt(v) > 2m�2. Combining this with
(2.4.1) we have
Cloc
Gt>
1
t
X
v2N(b)t (m)
CGt(v) > t�1N
(b)
t
(m)2m�2
> t�12m�2pb
⇣
b1
t� b2
p
t log(t)⌘
/4
! 2m�2pb
b1
> 0 as t ! +1,
proving the theorem.
2.5 Vanishing global clustering
This section is devoted to the proof of Theorem 2.1.2 which states that the global clusteringof G
t
goes to zero at 1/ log t speed. Since the proof depends on estimates for the number ofcherries, C
t
, we first derive the necessary bounds and finally put all the pieces together atthe end of this section.
2.5.1 Preliminary estimates for number of cherries
Let
Ct
:=t
X
j=1
d2t
(j)
denote the sum of the squares of degrees in Gt
. We will to prove bounds for Ct
insteadof proving them directly for C
t
. Since Ct
= Ct
/2 �mt the results obtained for Ct
directlyextend to C
t
.
Lemma 2.5.1. There is a positive constant B3
such that
E
⇣
Cs+1
� Cs
⌘
2
�
�
�
�
Gs
�
B3
dmax
(Gs
)Cs
s.
2. The Holme-Kim model 25
Proof. We start the proof noticing that for all vertex j we have d2s+1
(j)�d2s
(j) 2mds
(j)+m2
deterministically From this remark the inequality below follows.
Cs+1
� Cs
2ms
X
j=1
ds
(j)1 {�ds
(j) � 1}+ 2m2. (2.5.2)
Since all vertices have degree at least m, we have m2 mP
s
j=1
ds
(j)1 {�ds
(j) � 1}, thus
Cs+1
� Cs
4ms
X
j=1
ds
(j)1 {�ds
(j) � 1} . (2.5.3)
Applying Cauchy-Schwarz to the above inequality, we obtain
⇣
Cs+1
� Cs
⌘
2
16m2
s
X
j=1
ds
(j)1 {�ds
(j) � 1} · 1 {�ds
(j) � 1}!
2
16m2
s
X
j=1
d2s
(j)1 {�ds
(j) � 1}!
s
X
j=1
1 {�ds
(j) � 1}!
16m3
s
X
j=1
d2s
(j)1 {�ds
(j) � 1} .
(2.5.4)
Recalling that
P (�ds
(j) � 1|Gs
) ds
(j)
2swe have
E✓
⇣
Cs+1
� Cs
⌘
2
�
�
�
�
Gs
◆
B3
s
X
j=1
d3s
(j)
s B
3
dmax
(Gs
)Cs
s,
concluding the proof.
Theorem 2.5.5 (Upper bound for Ct
). There is a positive constant B1
such that
P�
Ct
� B1
t log2(t)�
t�98.
Proof. We show the result for Ct
, which is greater than Ct
. To do this, we need to deter-mine E[d2
t+1
(j)|Gt
].
As in proof of Proposition 2.3.13, write
dt+1
(j) = dt
(j) +m
X
k=1
1n
Y(k)
t+1
= jo
2. The Holme-Kim model 26
and denoteP
m
k=1
1n
Y(k)
t+1
= jo
by �dt
(j). Thus,
d2t+1
(j) = d2t
(j)
✓
1 +�d
t
(j)
dt
(j)
◆
2
= d2t
(j) + 2dt
(j)�dt
(j) + (�dt
(j))2.
Combining the above equation with (2.3.14) and (2.3.15), we get
E⇥
d2t+1
(j)�
�Gt
⇤
= d2t
(j) +d2t
(j)
t+ E
⇥
(�dt
(j))2�
�Gt
⇤
. (2.5.6)
Dividing the above equation by t+ 1, we get
E
d2t+1
(j)
t+ 1
�
�
�
�
Gt
�
=d2t
(j)
t+
E [(�dt
(j))2|Gt
]
t+ 1, (2.5.7)
which implies
E
"
Ct+1
t+ 1
�
�
�
�
�
Gt
#
=C
t
t+
m2
t+ 1+
t
X
j=1
E [(�dt
(j))2|Gt
]
t+ 1. (2.5.8)
It is straightforward to see that �dt
(j) (�dt
(j))2 m�dt
(j), which implies
E⇥
(�dt
(j))2�
�Gt
⇤
= ⇥
✓
dt
(j)
t
◆
.
Thus, (2.5.8) may be written as
E
"
Ct+1
t+ 1
�
�
�
�
�
Gt
#
=C
t
t+⇥
✓
1
t
◆
. (2.5.9)
Now, define
Xt
:=C
t+1
t+ 1.
Equation (2.5.9) states that Xt
is a martingale up to a term of magnitude ⇥�
1
t
�
. In order toapply martingale concentration inequalities, we decompose X
t
as in Doob’s Decompositiontheorem. X
t
can be written as Xt
= Mt
+ At
, in which Mt
is a martingale and At
is apredictable process. By Equation (2.5.9), we have
At
=t
X
s=2
E [Xs
|Gs�1
]�Xs�1
=t
X
s=2
⇥
✓
1
s
◆
. (2.5.10)
I.e., At
= ⇥ (log(t)) almost surely.
2. The Holme-Kim model 27
The remainder of the proof is devoted to controlling Xt
’s martingale component using Freed-man’s Inequality. Once again, by Doob’s Decomposition Theorem, we have
Mt
:= X0
+t
X
s=2
Xs
� E [Xs
|Gs�1
] .
Observe that Mt+1
= Mt
+Xt+1
� E [Xt+1
|Gt
], thus
|�Ms
| = |Xs+1
� E [Xs+1
|Gs
]| |Xs+1
�Xs
|+ b9
s
�
�
�
�
�
Cs+1
��
1 + 1
s
�
Cs
s+ 1
�
�
�
�
�
+b9
s
b10
dmax
(Gs
)
s+ b
11
Cs
s2+
b9
s.
(2.5.11)
Since �Cs
attains its maximum when the vertices of maximum degree in Gs
receive at leasta new edge at time s+ 1. Furthermore, since d
max
(Gs
) ms and Cs
m2s2, there exists aconstant b
12
such that maxst
|�Ms
| b12
almost surely.
Combining
|�Ms
|
�
�
�
�
�
Cs+1
��
1 + 1
s
�
Cs
s+ 1
�
�
�
�
�
+b9
s
with Cauchy-Schwarz and Lemma 2.5.1, we obtain positive constants b13
, b14
and b15
suchthat
E⇥
(�Ms
)2�
�Gs
⇤
CS
b13
Eh
(�Cs
)2�
�
�
Gs
i
s2+ b
14
C2
s
s4+
b15
s2
Lemma 2.5.1
b16
dmax
(Gs
)Cs
s3+ b
14
C2
s
s4+
b15
s2.
(2.5.12)
Now, define Vt
as
Vt
:=t
X
s=2
E⇥
(�Ms
)2�
�Gs
⇤
and call bad set the event below
Bt
:=t
[
j=1
t
[
t0=j
�
dt0(j) > b
8
pt0
log(t)
,
observe that Lemma 2.3.22 guarantees P(Bt
) 2t�98.
2. The Holme-Kim model 28
Also notice that Cs
b17
dmax
(Gs
)s almost surely and in Bc
t
we have dmax
(Gs
) b8
ps log(t)
for all s t. Then, outside Bt
we have
Vt
(2.5.12)
t
X
s=2
b16
dmax
(Gs
)Cs
s3+ b
14
C2
s
s4+
b15
s2
t
X
s=2
b16
b28
b17
s2 log2(t)
s3+
b14
b217
s3 log2(t)
s4+
b15
s2
b18
log3(t).
(2.5.13)
So, by Freedman’s inequality, we obtain
P�
Mt
> �, Vt
b18
log3(t)�
exp
⇢
� �2
2b18
log3(t) + 2b12
�/3
�
.
Therefore, if � = b19
log2(t) with b19
large enough, we get
P�
Mt
> b19
log2(t), Vt
b18
log3(t)�
t�100. (2.5.14)
The inequality (2.5.13) guarantees the following inclusion of events
Bc
t
⇢�
Vt
b18
log3(t)
. (2.5.15)
And also,�
Xt
� b21
log2(t)
⇢�
Mt
� (b21
� b20
) log2(t)
. (2.5.16)
since At
b20
log(t) and Mt
� Xt
� b20
log(t).
Finally,
P�
Mt
> b19
log2(t)�
= P�
Mt
> b19
log2(t), Vt
b18
log3(t)�
+ P�
Mt
> b19
log2(t), Vt
> b18
log3(t)�
t�100 + P (Bt
)
P�
Mt
> b19
log2(t)�
3t�98,
proving the Theorem..
We notice that from equation (2.5.8) we may extract the recurrence below
Eh
Ct
i
=
✓
1 +1
t� 1
◆
Eh
Ct�1
i
+ c0
,
in which c0
is a positive constant depending on m and p only. Expanding it, we obtain
Eh
Ct
i
=t�1
Y
s=1
✓
1 +1
s
◆
Eh
C1
i
+ c0
t�1
X
s=1
t�1
Y
r=s
✓
1 +1
r
◆
,
which implies E[Ct
] = ⇥(t log t). This means the upper bound for Ct
given by Theorem 2.5.5is exactly E[C
t
] log(t).
2. The Holme-Kim model 29
2.5.2 The bootstrap argument
Obtaining bounds for Ct
requires some control of its quadratic variation, which requiresbounds for the maximum degree and C
t
, as in Lemma 2.5.1. Applying some deterministicbounds and upper bounds on the maximum degree we were able to derive an upper boundfor C
t
, which is of order E[Ct
] log t. To improve this bound and obtain the right order,we proceed as in proof of Theorem 2.5.5, but making use of the preliminary estimatejustdiscussed. This is what we call the bootstrap argument.
The result we obtain is enunciated in Theorem 2.1.4 and consist of a Weak Law of LargeNumbers, which states that C
t
divided by t log t actually converges in probability to a con-stant depending only on m.
Proof of Theorem 2.1.4. In proof of Theorem 2.5.5, we decomposed the process Xt
= Ct
/tin two components: M
t
and At
. The first part of the proof will be dedicated to showingthat M
t
= o(log(t)), w.h.p. Then we show that At
= (m2 +m) log(t) also w.h.p.
We repeat the proof given for Theorem 2.5.5, but this time we change our definition of badset to
Bt
=t
[
s=log
1/2(t)
n
Cs
� b20
s log2(s)o
.
By Theorem 2.5.5 and union bound, P(Bt
) log�97/2(t). Observe that an upper bound forC
s
gives an upper bound for dmax
(Gs
), since
d2max
(Gs
) Cs
=) dmax
(Gs
) q
Cs
=) dmax
(Gs
) ps log(s),
when Cs
s log2(s).
Using (2.5.12) we have, in Bc
t
,
Vt
t�1
X
s=1
b16
dmax
(Gs
)Cs
s3+ b
14
Cs
2
s4+
b14
s2
log
1/2(t)�1
X
s=1
b016
+t�1
X
s=log
1/2(t)
b17
ps log(s)s log2(s)
s3+ b
18
s2 log4(s)
s4+
b14
s2
b19
log1/2(t),
(2.5.17)
since dmax
(Gs
) m · s and Cs
2m2 · s2 for all s and, in Bc
t
, dmax
(Gs
) pb20
s log(s) andC
s
b20
s log2(s) for all s � log1/2(t). Then, by Freedman’s inequality,
P⇣
Mt
� log1/4+�(t), Vt
b19
log1/2(t)⌘
= o(1). (2.5.18)
2. The Holme-Kim model 30
Recall equation (2.5.8)
E
"
Ct+1
t+ 1
�
�
�
�
�
Gt
#
=C
t
t+
m2
t+ 1+
t
X
j=1
E [(�dt
(j))2|Gt
]
t+ 1.
Now, we recall from Lemma 2.3.1 that for all k 2 {1, ...,m}
P⇣
Y(k)
t+1
= v�
�
�
Gt
⌘
=dt
(v)
2mt.
Furthermore,
P⇣
Y(k)
t+1
= v, Y(j)
t+1
= v�
�
�
Gt
⌘
= O
✓
d2t
(v)
t2
◆
.
Thus,
E⇥
(�dt
(v))2�
�Gt
⇤
=dt
(v)
2t+O
✓
d2t
(v)
t2
◆
, (2.5.19)
which implies that
E
"
Ct+1
t+ 1
�
�
�
�
�
Gt
#
=C
t
t+
m2 +m
t+ 1+O
Ct
t3
!
,
and consequently
At
=t
X
s=2
m2 +m
s+ 1+O
Cs
s3
!
.
As we have already noticed before, all the constants involved in the big-O notation do notdepend on the time or the vertex. From the above equation we deduce that, in Bc
t
,
t
X
s=2
O
Cs
s3
!
b11
log
1/2(t)
X
s=2
Cs
s3+
t
X
s=log
1/2(t)
s log2(s)
s3 b
12
log(log(t)).
Thus, in Bc
t
,A
t
= (m2 +m) log(t) + o(log(t)).
Finally, fix a small positive "
P
�
�
�
�
�
Ct
t log(t)�m2 +m
�
�
�
�
�
> "
!
= P✓
�
�
�
�
Mt
+ At
log(t)�m2 +m
�
�
�
�
> "
◆
P✓
�
�
�
�
Mt
+ At
log(t)�m2 +m
�
�
�
�
> ", Bc
t
◆
+ P (Bt
) .
(2.5.20)
2. The Holme-Kim model 31
We also have that Bc
t
⇢ {Vt
b19
log1/2(t)}, which, combined to (2.5.18), implies
P✓
�
�
�
�
Mt
+ At
log(t)�m2 +m
�
�
�
�
> ",Mt
� log1/4+�(t), Bc
t
◆
= o(1).
And recall that, in Bc
t
, Mt
is at most log1/4+�(t) and At
= (m2 +m) log(t) + o(log(t)), thus
P✓
�
�
�
�
Mt
+ At
log(t)�m2 +m
�
�
�
�
> ",Mt
< log1/4+�(t), Bc
t
◆
= 0
for large enough t.
Recalling that Ct
= Ct
/2�mt we obtain the desired result.
2.5.3 Wrapping up
Until here we devoted our e↵orts to properly control the number of cherries in Gt
. Now, wecombine these results with simple bounds for the number of triangles in G
t
to finally obtainthe exact order of the global clustering.
Proof of Theorem 2.1.2. By Theorem 2.1.4 we have
Ct
= ⇥ (t log(t)) ,w.h.p.
But, observe that number of triangles in Gt
, �Gt , is bounded from above by
�
m
2
�
t. And notethat every TF-step we take increases �
Gt by one. Then,
�Gt � Z
t
=t
X
s=1
Ts
where Ts
is the number of TF-steps we took at time s. Since all the choices concerning whichkind of step we follow are independent, T
s
⇠ bin(m � 1, p) and Zt
⇠ bin((m � 1)t, p). ByCherno↵ Bounds, Z
t
� �(m� 1)pt, for a small �, w.h.p. Thus,
�Gt = ⇥(t),w.h.p,
which conclude the proof.
2.6 Final comments on clustering
We end our discussion about clustering by comparing the two clustering coe�cients from adi↵erent perspective than in Section 2.1.2. Recall that Cloc
Gtis an unweighted average of local
2. The Holme-Kim model 32
clustering coe�cients.
Cloc
Gt:=
1
t
X
v2Gt
CGt(v).
On the other hand, Cglo
Gtis a weighted average, where the weight of vertex v is the number
of cherries that it belongs to,
Cglo
Gt= 3⇥
P
v2GtCGt(v)
�
dt(v)
2
�
P
v2Gt
�
dt(v)
2
�
(2.6.1)
Thus the weight of v in Cglo
Gtis basically proportional to the square of the degree. This skews
the distribution of weights towards high-degree nodes. The clustering of the high degreevertices is the reason why the two coe�cients present so distinct behavior.
We will show below that CGt(v) for a vertex v of high degree d is of order d�1, which explains
why Cglo
Gtgoes to zero. Recall that the r.v. e
t
(�v
) counts the number of edges between theneighbors of v. Due to the definition of our model, one can only increse e
t
(�v
) by one if dt
(v)is also increased by at least one unit. Since e
t
(�v
) can only increase by m units in each timestep, we have:
et
(�v
) mdt
(v),
which implies an upper bound for CGt(v) ⇡ e
t
(�v
)/dt
(v)2 of order d�1
t
(v). The next propo-sition gives a lower bound of the same order.
Proposition 2.6.2. Let v be a vertex of Gt
. Then, there are positive constants, b1
and b2
,such that
P✓
CGt(v)
b1
dt
(v)
�
�
�
�
dt
(v) � b2
log(t)
◆
t�100.
This proposition does not prove our clustering estimates, but seems interesting in any case.
Proof of the Proposition. Observe that if we choose v and take a TF-step thereafter, weincrease e
t
(�v
) by one. Then, if we look only at times in which this occurs, et
(�v
) must begreater than a binomial random variable with parameters: number of times we choose v atthe first choice and p. Since all the choices concerning the kind of step we take are madeindependently of the whole process, we just need to prove that number of times we choose vat the first choice, denoted by d
(1)
t
(v), is proportional to dt
(v) w.h.p.
Recall that Y(1)
s
indicates the vertex chosen at time s at the first of our m choices. Therandom variable d
(1)
t
(v) can be written in terms of Y (1)’s just as follows
d(1)
t
(v) =t
X
s=v+1
1�
Y (1)
s
= v
. (2.6.3)
2. The Holme-Kim model 33
We first claim that if dt
(v) is large enough, a positive fraction of its value must come
from d(1)
t
(v).
Claim: There exists positive constants b1
and b2
such that
P⇣
d(1)
t
(v) b1
dt
(v)�
�
�
dt
(v) � b2
log(t)⌘
t�100.
Proof of the claim: To prove the claim we condition on all possible trajectories of dt
(v). Inthis direction, let ! be an event describing when v was chosen and how many times at eachstep. We have to notice that ! does not record whether v was chosen by a PA-step or aTF-step. The event ! can be regarded as a vector in {0, 1, ...,m}t�v�1 such that !(s) = kmeans we chose v k-times at time s. For each !, let d
!
(v) be the degree of v obtained bythe sequence of choices given by !.
Recall the Equations (2.3.10) and (2.3.11) which states that
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
6= v⌘
=dt
(v)
2mt+O
✓
d2t
(v)
t2
◆
and
Pt
⇣
Y(i+1)
t+1
= v�
�
�
Y(i)
t+1
= v⌘
= (1� p)dt
(v)
2mt.
For any ! such that !(s) = k � 1, we may show, using (2.3.10) and (2.3.11), that thereexists a positive constant � depending only on m and p, such that
P�
Y (1)
s
= v�
�!�
� �. (2.6.4)
Furthermore, given !, the random variables 1{Y (1)
s
= v} are independent. This implies
that, given !, the random variable d(1)t
(v) dominates stochastically another random variablefollowing a binomial distribution of parameters d
!
(v)/m and �. Thus, by Cherno↵ bounds,we can choose a small b
1
such that
P⇣
d(1)
t
(v) b1
d!
(v)�
�
�
!⌘
exp (�d!
(v)) .
Since we are on the event Dt
:= {dt
(v) � b2
log(t)}, all d!
(v) � b2
log(t) for some b2
thatcan be chosen in a way such that
P⇣
d(1)
t
b1
d!
(v)�
�
�
!⌘
t�100
for all ! compatible with Dt
. Finally, to estimaten
d(1)
t
(v) b1
dt
(v)o
, we condition on all
possible history of choices !
P⇣
d(1)
t
(v) b1
dt
(v)�
�
�
Dt
⌘
=X
!
P⇣
d(1)
t
(v) b1
dt
(v)�
�
�
!, Dt
⌘
P (!|Dt
) t�100
(2.6.5)
2. The Holme-Kim model 34
and this proves the claim. ⌅
As we observed at the beginning, et
(�v
) dominates a random variable bin(d(1)t
(v), p). And
by the claim, d(1)t
(v) is proportional to dt
(v), w.h.p. Using Cherno↵ Bounds, we obtain theresult.
2.7 Power law degree distribution
We end this Chapter by proving the power law distribution.
Lemma 2.7.1 (Lemma 3.1 [11]). Let at
be a sequence of positive real numbers satisfying thefollowing recurrence relation
at+1
=
✓
1� bt
t
◆
at
+ ct
.
Furthermore, suppose bt
! b > 0 and ct
! c, then
limt!1
at
t=
c
1 + b.
Proof of Theorem 2.1.3. We divide the proof into two parts. Part (a) is the power law forthe expected value of the proportion of vertices with degree d. Part (b) is the concentrationinequalities N
t
(d).
Proof of part (a).The is essentially the same gave in Section 3.2 of [11]. The key step isobtain a recurrence relation involving E[N
t
(d)] which has the same form of that requiredby Lemma 5.1.1. To obtain the recurrence relation, observe that N
t+1
(d) can be written asfollow
Nt+1
(d) =X
v2Nt(d)
{�dt(v)=0} +X
v2Nt(d�1)
{�dt(v)=1} + . . .+X
v2Nt(d�m)
{�dt(v)=m}. (2.7.2)
Taking the conditional expected value with respect to Gt
on the above equation, applyingLemma 2.3.1 and recalling that N
t
(d) t, we obtain
E[Nt+1
(d)|Gt
] = Nt
(d)
1� d
2t+O
✓
d2
t2
◆�
+Nt
(d�1)
(d� 1)
2t+O
✓
(d� 1)2
t2
◆�
+O
✓
1
t
◆
.
Finally, taking the expected value on both sides, denoting ENt
(d) by a(d)
t
, we have
a(d)
t+1
=
2
41�d
2
+O⇣
d
2
t
⌘
t
3
5 a(d)
t
+ a(d�1)
t
2
4
d�1
2
+O⇣
(d�1)
2
t
⌘
t
3
5+O
✓
1
t
◆
. (2.7.3)
2. The Holme-Kim model 35
From here, the proof follows by an induction on d � m and application of Lemma 5.1.1,assuming ENt(d�1)
t
�! Dd�1
, which gives us
at
(d)
t�!
Dd�1
(d�1)
2
1 + d/2= D
d�1
d� 1
2 + d=: D
d
and this gives us that
Dd
=2
2 +m
d
Y
k=m+1
(k � 1)
k + 2=
2(m+ 3)(m+ 1)
(d+ 3)(d+ 2)(d+ 1)
which proves the part (a).
Proof of part (b). The proof is in line with proof of Theorem 3.2 in [11]. For this reason weshow only that following process
X(d)
t
:=N
t
(d)�Dd
t+ 16d · c ·pt
d
(t),
in which d
(t) is defined as
d
(t) :=t�1
Y
s=d
✓
1� d
2s
◆
is a submartingale and we give an upper bound for its variation.
As in Theorem 3.2 of [11], the proof follows by induction on d.Inductive step: Suppose that for all d0 d� 1 we have
P⇣
Nt
(d0) Dd
0t� 16d0 · c ·pt⌘
(t+ 1)d0�me�c
2. (2.7.4)
Recalling that
E [Nt+1
(d)|Gt
] =
✓
1� d
2t+O
✓
d2
t2
◆◆
Nt
(d) +(d� 1)N
t
(d� 1)
2t+O
�
t�1
�
we have the following recurrence relation
Eh
d
(t+ 1)X(d)
t
�
�
�
Gt
i
�✓
1� d
2t
◆
Nt
(d) +(d� 1)N
t
(d� 1)
2t
+O�
t�1
�
+ 16d · c ·pt�D
d
(t+ 1).
(2.7.5)
The inductive hypothesis assure us that
Nt
(d� 1) � Dd�1
t� 16(d� 1) · c ·pt,
2. The Holme-Kim model 36
with probability at least 1� (t+ 1)d�1�me�c
2. Thus, returning on (2.7.5),
Eh
d
(t+ 1)X(d)
t
�
�
�
Gt
i
�✓
1� d
2t
◆
Nt
(d)
+(d� 1)D
d�1
2�D
d
(t+ 1) + 16d · c ·pt+O
�
t�1
�
.
(2.7.6)
But, observe that about the r.h.s of the above inequality, we have
(d� 1)Dd�1
2�D
d
(t+ 1) + 16d · c ·pt+O
�
t�1
�
�✓
1� d
2t
◆
⇣
�Dd
t+ 16d · c ·pt⌘
()(d� 1)D
d�1
2�D
d
+O�
t�1
�
� dDd
2t� 8d2cp
t
()(d� 1)D
d�1
2+
8d2cpt+O
�
t�1
�
� Dd
+dD
d
2t(2.7.7)
but the last inequality is true since we have (d�1)Dd�1
= (d+2)Dd
and Dd
= 2(m+3)(m+1)
(d+3)(d+2)(d+1)
.
Returning to (2.7.6), we just proved that X(d)
t+1
is a submartingale with fail probability
bounded from above by (t + 1)d�me�c
2. And its variation �X
(d)
t
satisfies the upper boundbelow
�
��X(d)
s
�
� �Ns
(d) +Dd
+ 16dcs�1/2 + dNs
(d)(2s)�1
d
(s+ 1)
m+ 2/(d+ 2) + 17dcs�1/2 + d/2
d
(s+ 1)
2d
d
(s+ 1)+
17dcps
d
(s+ 1),
(2.7.8)
since �Ns
(d) m, Ns
(d) s and Dd
2/(d + 2) for all s and d. Thus, there is a positiveconstant M , such that
�
��X(d)
s
�
�
2 16d2
2
d
(s+ 1)+
Md2c2
s 2
d
(s+ 1). (2.7.9)
The lower bound for Nt
(d) is proven applying Theorem 2.36 of [11] on X(d)
t
, setting
� = 2c ·
v
u
u
t
t+1
X
s=d
�
�
�
�X(d)
s
�
�
�
2
.
2. The Holme-Kim model 37
The upper bound is obtained the same way, but considering the process
�X(d)
t
:=N
t
(d)�Dd
t� 16d · c ·pt
d
(t),
which is a supermartingale.
3. THE P-MODEL
In this chapter, we investigate a random graph model in a class known as GLP in theliterature of Computer Science and Physics. Informally, it is a modification of the modelproposed by Barabasi - Albert in [2], in which links among existing vertices are allowed. Thee↵ect of this alteration has positive consequences in the sense that this model outperformsothers popular models when the task is predicting or mimicking real-world complex networks(In [30], the authors have a quantitative version of this statement).
Let us briefly describe the process. This model has two parameters: a real number p 2 [0, 1]and an initial graph G
1
. For the sake of simplicity we will consider G1
to be the graph withone vertex and one loop. We consider the following two stochastic operations that can beperformed on the graph G:
• Vertex-step - Add a new vertex v, and add an edge {u, v} by choosing u 2 G withprobability proportional to its degree.
• Edge-step - Add a new edge {u1
, u2
} by independently choosing vertices u1
, u2
2 Gwith probability proportional to their degrees. We note that we allow loops to beadded, and we also allow a new connection to be added between vertices that alreadyshared an edge.
For a more formal definition, consider a sequence (Xt
)t�1
of i.i.d’s random variables, such
that Xt
d
= Ber(p). We define inductively a random graph process (Gt
)t�1
as follows: startwith G
1
, the graph with one vertex and one loop. Given Gt
, form Gt+1
by performing avertex-step on G
t
when Xt
= 1, and performing an edge-step on Gt
when Xt
= 0. Theresulting process is the object of study of this chapter.
The chapter’s goal is to investigate the existence of large complete subgraphs in Gt
, whichwe may refer as communities. We are interested in communities whose vertex set cardinality,which we also call order, goes to infinity as the process evolves.
This chapter is organized as follows: in Section 3.1.1 we prove an upper bound for the degreeof all vertices at all times using Azuma’s inequality. This upper bound is used in the proof ofthe lower bound for the degrees. Section 3.1.2 is dedicated to lower bounds for the degrees.
3. The p-model 39
Theses bounds require more work and some technical lemmas. The proof we present heredoes not use martingales inequalities. In Section 3.2, we rigorously prove the existence of alarge community applying the bounds obtained in Section 3.1.2. In Section 3.3 we presenta upper bound for the diameter of the graph generated by this model coupling it with thatgenerated by Barabasi-Albert’s model.
3.1 Bounds for the vertices’ degree
3.1.1 Upper bounds
In this section we will establish a simple bound on the probability that a given vertexhas a large degree. The result will follow from a application of Azuma’s inequality (seeTheorem 2.19 from [10]).
We will let i and j denote the i-th and j-th vertices to enter the random graph processrespectively. The random time in which the j-th vertex is added to the graph will bedenoted by T
j,1
.
Definition 3. Since the constant 1 � p/2 is going to appear many times throughout thisChapter, it deserves a special notation. We write
cp
:= 1� p/2.
Proposition 3.1.1. For each vertex j and each j t0
t the sequence of random vari-ables (Z
t
)t�t0
defined as
Zt
:=dt
(j)1{Tj,1=t0}Q
t�1
s=1
�
1 + cp
s
� (3.1.2)
is a martingale starting from t0
.
Proof. We consider the process (Gt
)t�0
to be adapted to a filtration (Ft
)t�1
. Recall �dt
(j) =dt+1
(j)� dt
(j). It is clear that �dt
(j) 2 {0, 1, 2}. Furthermore, conditioned on Ft
, we knowthe probability that �d
t
(j) takes each of these values. So that, assuming that j alreadyexists at time t, we have
E [�dt
(j)|Ft
] = 1 · pdt(j)2t
+ 1 · (1� p)2dt
(j)
2t
✓
1� dt
(j)
2t
◆
+ 2 · (1� p)(d
t
(j))2
4t2
= cp
dt
(j)
t.
3. The p-model 40
The information “vertex j exists at time t” can be introduced in the equation using therandom variable 1{Tj,1=t0}. Using the fact that 1{Tj,1=t0} is F
t0 measurable, we gain
E⇥
dt+1
(j)1{Tj,1=t0}�
�Ft
⇤
=⇣
1 +cp
t
⌘
dt
(j)1{Tj,1=t0}. (3.1.3)
Dividing the above equation byQ
t
s=1
�
1 + cp
s
�
we obtain the desired result.
Now we prove the main result of the section.
Proposition 3.1.4 (Upper bound for the degree). There exists a universal positive con-stant C
1
, such that for every vertex j we have that
P
dt
(j) � C1
tcp
s
log(t)
j1�p
!
1
t100.
Proof. We define �(t) :=Q
t�1
s=1
�
1 + cp
s
�
. Note that we can write
dt
(j)
�(t)=
t
X
t0=j
dt
(j)
�(t)1{Tj,1=t0},
and by Proposition 3.1.1 each term in the sum is a martingale. We want to apply Azuma’sinequality for each summand, but first we need some bounds on �. In this direction, Lemma5.1.2 assures us the existence of a positive constant c such that �(t) > ctcp .
In order to apply Azuma’s inequality, we must bound the variation of the random variable Zt
defined in (4.2.5), which satisfies the following upper bound
at
:=
�
�
�
�
dt+1
(j)1{Tj,1=t0}
�(t+ 1)�
dt
(j)1{Tj,1=t0}
�(t)
�
�
�
�
�
�
�
�
�
dt+1
(j)��
1 + cp
t
�
dt
(j)
�(t+ 1)
�
�
�
�
�
2 + cp
�(t+ 1), (3.1.5)
since �dt
(j) 2 and dt
(j) t. By the above discussion about � we have that a2t
< (2+cp)2
c
21t
2�p ,
which impliesP
t
s=t0a2s
< c0t�(1�p)
0
for some constant c0 > 0. Then, applying Azuma’sinequality, we obtain
P✓
�
�
�
�
dt
(j)1{Tj,1=t0}
�(t)� E
dt0(j)1{Tj,1=t0}
�(t0
)
�
�
�
�
�
> �
◆
2 exp�
��2t1�p
0
/2c0�
, (3.1.6)
Note that
E
dt0(j)1{Tj,1=t0}
�(t0
)
�
= �(t0
)�1P(Tj,1
= t0
) �(t0
)�1.
3. The p-model 41
By choosing � = c2
q
tp�1
0
log(t), where c2
is a su�ciently large positive constant dependingonly on p, we gain
P
dt
(j)1{Tj,1=t0} � c2
tcp
s
log(t)
t1�p
0
+�(t)
�(t0
)
!
1
t101.
Using the union bound and the asymptotic behaviour of �, we obtain the result for allpossibles times t
0
� j: there exists a positive constant C1
depending only on p such that
P
t
[
t0=j
(
dt
(j)1{Tj,1=t0} � C1
tcp
s
log(t)
j1�p
)!
1
t100.
This finishes the proof of the Proposition.
Via a union bound over all vertices j in Gt
, we obtain the following result:
Corollary 3.1.7 (Upper bound to the maximum degree). Let dmax
(Gt
) denote the maximumdegree among all the vertices of G
t
. Then, there exists a universal positive constant C2
suchthat
P⇣
dmax
(Gt
) � C2
tcpp
log(t)⌘
1
t99.
3.1.2 Lower bounds
This section is devoted to proving the results needed to state a useful lower bound on thedegree of the vertices that entered the random graph early in the history of the process. Weprove two lemmas that let us control the tail of the random times Tm
j,k
, defined bellow, beforeproving the main result, Propositions 3.1.17. First we need a new notation:
Definition 4. Fix a vertex j and two integers m, k � 1. We define the random time
Tm
j,k
:= inft�1
8
<
:
jm
X
i=(j�1)m+1
dt
(i) = k
9
=
;
.
We also write Tj,k
:= T 1
j,k
. In other words, Tj,k
is the first time that the j-th vertex hasdegree at least k. Tm
j,k
can then be explained in the following way: assume that we identifyall the vertices 1 through m, then identify all the vertices m + 1 through 2m and so on.We let d
t,m
(j) denote the degree of the j-th block of m vertices. Then Tm
j,k
is the first timethat d
t,m
(j) is larger than, or equals to, k.
3. The p-model 42
Lemma 3.1.8. Let 0 < � < (c�1
p
� 1). For every k � 1, every large enough m 2 Z andj � (m2/(1�p) + 1), we can construct a sequence ⌘
m
, .., ⌘k
of independent random variables,with
⌘i
d
= Exp⇣
cp
⇣
1� 1� p
2(2� p)i�
⌘
i⌘
, for i = m, . . . , k, (3.1.9)
such that the whole sequence is independent of Tm
j,m
, .., Tm
j,k+1
, and such that
P�
Tm
j,k+1
> t�
P
Tm
j,m
exp
k
X
i=m
⌘i
!
> t
!
+m
[(j � 1)m]99
Proof. We follow the idea of the proof of Lemma 3.1 in [22]. But in our context the existenceof the edge-step demands more attention and prevents a straightforward application of thislemma.
We begin by constructing the k + 1 � m independent random variables ⌘m
, .., ⌘k
with dis-tribution given by (3.1.9), the whole sequence being independent of the random timesTm
j,m
, .., Tm
j,k+1
. Observe that
P�
Tm
j,k+1
> t�
=1X
s=k
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
=k
1+�X
s=k
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
+1X
s=k
1+�
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
P�
Tm
j,k
k1+�
�
+1X
s=k
1+�
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
.
(3.1.10)
We obtain an upper bound for the term P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
in the following way: once thevertex (block) j reaches degree k at time s, we must avoid choosing j at all the subsequentsteps until time t. We note that there exists the possibility that Tm
j,k+1
= Tm
j,k
, in the casethat we add a loop to j at time Tm
j,k
, but in this case P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
is equal to 0,and our calculations remain the same. Noting that at each step s+1 we choose the vertex jwith probability
cp
ds
(j)
s� (1� p)d2
s
(j)
4s2,
3. The p-model 43
and recalling that cp
:= 1� p/2, we obtain, for s � k1+�,
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
t�1
Y
r=s
✓
1� cp
k
r+
(1� p)k2
4r2
◆
=t�1
Y
r=s
1� cp
k
r
✓
1� (1� p)k
2(2� p)r
◆�
t�1
Y
r=s
1� cp
k
r
✓
1� (1� p)
2(2� p)k�
◆�
.
(3.1.11)
We introduce the notation
�k
:=(1� p)
2(2� p)k�
. (3.1.12)
Observe that
1� cp
k(1� �k
)
s exp
✓
1
s
◆�cpk(1��k)
✓
1 +1
s
◆�cpk(1��k)
=
✓
s
s+ 1
◆
cpk(1��k)
.
Plugging the above inequality into (3.1.11), noting that this results in a telescopic product,and recalling the definition of ⌘
i
, we get
P�
Tm
j,k+1
> t�
�Tm
j,k
= s�
⇣s
t
⌘
cp(1��k)k
= P�
Tm
j,k
e⌘k > t�
�Tm
j,k
= s�
.
Combining the above inequality with (3.1.10), we obtain
P�
Tm
j,k+1
> t�
P�
Tm
j,k
k1+�
�
+1X
s=k
1+�
P�
Tm
j,k
e⌘k > t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
P�
Tm
j,k
k1+�
�
+1X
s=k
P�
Tm
j,k
e⌘k > t�
�Tm
j,k
= s�
P�
Tm
j,k
= s�
= P�
Tm
j,k
k1+�
�
+ P�
Tm
j,k
e⌘k > t�
.
(3.1.13)
Writeerr(k) := P
�
Tm
j,k
k1+�
�
.
By the above equation, we also have, recalling that ⌘k
is independent from both Tm
j,k�1
and ⌘k�1
,
P�
Tm
j,k
e⌘k > t�
=
Z 1
0
P✓
Tm
j,k
>t
s
◆
P (e⌘k = ds)
Z 1
0
P✓
Tm
j,k�1
e⌘k�1 >t
s
◆
+ err (k � 1)
�
P (e⌘k = ds)
P�
Tm
j,k�1
e(⌘k+⌘k�1) > t�
+ err(k � 1),
(3.1.14)
3. The p-model 44
where P (e⌘k = ds) denotes the measure in R induced by the random variable e⌘k . Proceedingin this way, we obtain
P�
Tm
j,k+1
> t�
P
Tm
j,m
exp
k
X
i=m
⌘i
!
> t
!
+k
X
n=m
err(n).
It remains to be shown that the sum of errors is su�ciently small. First we note that:
�
Tm
j,n
n1+�
=
8
<
:
mj
X
i=m(j�1)+1
dn
1+� (i) � n
9
=
;
.
But for small n and large j the above event is actually empty, since none of the m verticesin the j-th block has enough time to be added by the process. In order to the above eventto be non-empty, we need at least one of the random variables d
n
1+� (i), for i 2 {(j � 1)m+1, . . . , jm} to be possibly not identically null. For this, n and j must satisfy the inequalitybelow:
n1+� � (j � 1)m+ 1 () n � [(j � 1)m+ 1]1
1+� .
A straightforward application of Dirichlet’s pigeon-hole principle shows that
�
Tm
j,n
n1+�
⇢jm
[
i=(j�1)m+1
n
dn
1+� (i) � n
m
o
.
We note that, ifn
m� C
1
n(1+�)cpp
(1 + �) log n
(j � 1)1�p2
, (3.1.15)
thenn
dn
1+� (i) � n
m
o
⇢(
dn
1+� (i) � C1
n(1+�)cpp
(1 + �) log n
(j � 1)1�p2
)
.
But (3.1.15) is always valid for large n, since (1 + �)cp
< 1 and j � m2/(1�p) + 1. Sincei > (j � 1)m, Proposition 3.1.1 implies
P�
Tm
j,n
n1+�
�
mn�100(1+�).
Consequentlyk
X
n=1
err(n) =k
X
n=[(j�1)m+1]
11+�
err(n) m
[(j � 1)m]99,
which concludes the proof.
3. The p-model 45
Lemma 3.1.16. For all vertex j and all m,R 2 N, there exists a positive constant c =cm,R,p
> 0 such thatE[TR
jm,1
] cjR.
Proof. Note that we can write
Tjm,1
= 1 +jm�1
X
i=1
(Ti+1,1
� Ti,1
),
so that Tjm,1
is distributed as 1 plus a sum of jm�1 independent geometric random variablesof parameter p. Recall that a random variable which follows a negative binomial distributionof parameters jm� 1 and p has moment generating function
G(s) = (1� p)jm�1
(1� pes)jm�1
.
By taking the R-th derivative of G(s) and evaluating it at 0, one can conclude the Lemma’sstatement.
Now we state and prove the main theorem of this section.
Proposition 3.1.17 (Lower bound for the degree). Fix m su�ciently large, and let
1 < R < mcp
(1� �m
).
Then there exists a positive constant c = c(m,R, p) such that, for
� 2 (0, cp
(1� �m
)),
we have:
P�
dt,m
(j) < t��
cjR
tR��R/cp(1��m)
+m
[(j � 1)m]99.
Proof. Fix m su�ciently large. By Lemma 3.1.8,
P(dt,m
(j) < t�) = P(Tm
j,t
� > t) P
0
@Tm
j,m
exp
0
@
t
�X
i=m
⌘i
1
A > t
1
A+m
[(j � 1)m]99. (3.1.18)
We need to control the first term of the inequality’s right hand side. We have that
P
0
@Tm
j,m
exp
0
@
t
�X
i=m
⌘i
1
A > t
1
A =1X
n=1
P
0
@exp
0
@
t
�X
i=m
⌘i
1
A > t/n
1
AP�
Tm
j,m
= n�
, (3.1.19)
3. The p-model 46
because Tm
j,m
is independent of ⌘i
for all possible values of i. Since ⌘i
d
= Exp(icp
(1� �i
)), wehave that its moment generating function is given by
Gi
(s) =1
1� s
icp(1��i)
,
for s < icp
(1� �i
). Then, for 1 < R < mcp
(1� �m
), Markov’s Inequality implies
P
0
@exp
0
@
t
�X
i=m
⌘i
1
A > t/n
1
A nR
tR
t
�Y
i=m
✓
1� R/[cp
(1� �i
)]
i
◆�1
nR
tR
t
�Y
i=m
✓
1� R/[cp
(1� �m
)]
i
◆�1
.
(3.1.20)
The last product can be written in terms of the Gamma Function, using its multiplicativeproperty. Note that
t
�Y
i=m
✓
1� R/cp
(1� �m
)
i
◆
=�(m)�
�
t� + 1�R/cp
(1� �m
)�
� (m�R/cp
(1� �m
))�(t� + 1).
This in turn implies the existence of a constant b = bm,R,p
> 0 such that
t
�Y
i=m
✓
1� R/cp
(1� �m
)
i
◆
> bt��R(cp(1��m))
�1.
Then, combining this bound with inequality (3.1.20), we obtain
P
0
@exp
0
@
t
�X
i=m
⌘i
1
A > t/n
1
A bnRt�R(cp(1��m))
�1
tR. (3.1.21)
Notice that, by the time that the jm-th vertex enters the graph, the j-th block of m ver-tices has total degree at least m, that is, Tm
j,m
Tjm,1
. This fact, together with (3.1.19),Lemma 3.1.16, and the above inequality, implies
P
0
@Tm
j,m
exp
0
@
t
�X
i=m
⌘i
1
A > t
1
A b
tR��R(cp(1��m))
�1
1X
n=1
nRP�
Tm
j,m
= n�
b
tR��R(cp(1��m))
�1
1X
n=1
nRP (Tjm,1
= n)
=b
tR��R(cp(1��m))
�1 E⇥
TR
jm,1
⇤
Lemma 3.1.16
cjR
tR��R(cp(1��m))
�1 ,
(3.1.22)
3. The p-model 47
for some constant c = cm,p,R
> 0. Combining (3.1.22) with (3.1.18) gives the desired result.
Recall that in Corollary 2.3.19 we proved an upper bound for the maximum degree, Ie, weshowed that, w.h.p,
dmax
(Gt
) C1
t1�p/2
p
log(t).
The Corollary below states that we may have dmax
(Gt
) � t1�p/2�", in which " is small as wewant.
Corollary 3.1.23 (Lower bound for the maximum degree). For all " > 0, there exists apositive constant C
2
= c("), such that
P�
dmax
(Gt
) C2
t1�p/2�"
�
= o(1).
Proof. It is enough to show that there exists at least one vertex j with degree large enough,w.h.p. For this, in Theorem 3.1.17, choose m large enough so that �
m
", � = (1� ")cp
(1��m
) and let j be the t"R/2-th block of m vertices. With all these choices, Theorem 3.1.17gives us
P�
dt,m
(t"R/2) < t(1�")(1�p/2)
�
c
t"R/2
.
The above inequality means, w.h.p, that at least one vertex in the t"R/2-th block of verticeshas degree greater than t(1�")(1�p/2)/m, which concludes the proof.
3.2 The Community
This section is devoted to prove the existence asymptotically almost surely of a large com-munity – clique, complete subgraph – in the graph generated by the p-model’s dynamics.I.e, the existence of a community in G
t
whose size grows to infinity polynomially in t. Thestatement is formalized in the theorem below.
Theorem 3.2.1 (The Community). Let " > 0, then, G2t
has a complete subgraph of or-
der t(1�")
(1�p)2�p , asymptotically almost surely.
Proof. Fix " > 0. Let ↵ = (1 � ") (1�p)
2�p
, fix � = 1+"2
�1(1�p)
2
and choose "0 > 0 such that"0 < ↵. By Proposition 3.1.17 and the union bound we have that
P
0
@
t
↵[
j=t
"0
�
dt,m
(j) < t�
1
A C
tR� �R
cp(1��m)�↵(R+1)
+m
(mt)98"0. (3.2.2)
3. The p-model 48
For our choice of ↵, �, and for su�ciently large m and R, the right hand side of the aboveinequality goes to 0 as t goes to infinity. The immediate consequence of this fact is thatinside each one of those t↵ � t"
0blocks of vertices of size m there exists at least one vertex
whose degree is larger than t�/m, with high probability. We denote by Lj
= Lj,m,p
the vertexof largest degree among the vertices of the j-th block, and we call L
j
the leader of its block.As we noted, d
t
(Lj
) � t�/m.
We now prove that these vertices of large degree are connected with high probability. Wewrite i = j to denote the fact that there are no edges between the vertices i and j. Wedefine the event
Bt
=t
↵[
j=t
"0
�
dt,m
(j) < t�
.
Let gs
(i, j) be the indicator function of the event where we add an edge between Li
and Lj
in an edge-step at time s. We define the random variable
Y ij
2t
=2t
Y
s=t+1
(1� gs
(i, j)).
In other words, Y ij
2t
is the indicator function of the event where we don’t connect Li
and Lj
in any of the edge-steps between times t+ 1 and 2t. We have that
E⇥
1B
ct(1� g
2t
(i, j))�
�F2t�1
⇤
=
✓
1� 2(1� p)d2t�1
(Li
)d2t�1
(Lj
)
4(2t� 1)2
◆
1B
ct
✓
1� (1� p)t2�
8t2m2
◆
1B
ct.
(3.2.3)
Now, observe that Y ij
2t
= Y ij
2t�1
(1� g2t
(i, j)). So, by using (3.2.3), we obtain
E⇥
1B
ctY ij
2t
�
�F2t�1
⇤
✓
1� (1� p)t2�
8t2m2
◆
Y ij
2t�1
1B
ct. (3.2.4)
Proceeding inductively, taking the conditional expectation with respect to Fs�1
at eachstep s, we gain the following inequality
E⇥
1B
ctY ij
2t
�
�Ft
⇤
✓
1� (1� p)t2�
8t2m2
◆
t
1B
ct exp
�
�c1
t2��1
�
1B
ct, (3.2.5)
where c1
is a positive constant depending on both p and m. Since 1{Li=Lj in G2t} Y ij
2t
, theabove inequality implies
P (Li
= Lj
in G2t
, Bc
t
) exp�
�c1
t2��1
�
P (Bc
t
) .
3. The p-model 49
Now, by the union bound, we have
P
0
@
[
t
"0i,jt
↵
{Li
= Lj
in G2t
} , Bc
t
1
A t2↵ exp�
�c1
t2��1
�
P (Bc
t
) .
And finally,
P
0
@
[
t
"0i,jt
↵
{Li
= Lj
in G2t
}
1
A = P
0
@
[
t
"0i,jt
↵
{Li
= Lj
in G2t
} , Bc
t
1
A
+ P
0
@
[
t
"0i,jt
↵
{Li
= Lj
in G2t
} , Bt
1
A
t2↵ exp�
�c1
t2��1
�
P (Bc
t
) + P (Bt
) .
The above inequality, together with our choice of ↵, �, and inequality (3.2.2), imply theexistence of a subgraph of G
t
with order t↵(1� o(1)) asymptotically almost surely.
3.3 Small-World Phenomenon
In this section we obtain an upper bound for diam(Gt
) of order log(t). The bound is aimmediate consequence of the next Theorem.
Theorem 3.3.1. Let G(p)
t
be the graph generated by the p-model and G(1)
t
the graph generatedby the Barabasi-Albert model. Then there exists a coupling of the models which gives
P⇣
diam⇣
G(p)
t
⌘
diam⇣
G(1)
t
⌘⌘
= 1.
Proof. We construct the coupling explicitly.
1. Start with G1
the graph with one vertex and one loop attached to it.
2. Obtain Gt+1
from Gt
in the following way
• With probability p add a solid vertex and connect it with u 2 Gt
chosen withprobability dGt(u)/2t;
• With probability 1 � p add a ghost vertex and let two edges leaves from it.Choose u
1
, u2
2 Gt
to be the tails of each edge independently and again withprobability dGt(u)/2t.
3. The p-model 50
We will call the first edge by attractive edge and it will be represent by the dashed edge on thedrawings below. Whereas the second one will be called as connective edge and representedby a solid edge. The ghost vertices will be represent by a circle whereas the solid verticeswill be the black disks.
Fig. 3.1: A ghost-vertex with its attractive edge and connective edge
Before we continue, we must make an important observation.
Observation. The degree of a vertex does not count attractive edges.
From the dynamics we have just described above we may conclude that the graph G(p)
t
maybe obtained from G
t
simply identifying the ghost vertices with the tail of their attractiveedges. Whereas the graph G
(1)
t
is obtained from Gt
just declaring all vertices equals andremoving the attractive edges.
To see why the first statement is true, observe that the presence of the attractive edge is justanother way to add a new edge to G
(p)
t
. The attractive edge tells us where goes the head ofthe new edge, whereas the connective tells where goes its tail. For the second statement, ifwe always ignore attractive edges and all vertices are truly vertices, does not matter whichstep we take, we always add a new vertex and connect it to a existing vertex with probabilityproportional to its degree.
Now, observe that for each path �G
(1)t
in G(1)
t
we have two options: it uses ghost vertices or
it does not. A path � that uses only solid vertices is also contained in G(p)
t
, which means �has the same, or smaller, size in G
(p)
t
. On the other hand, if �G
(1)t
does use at least one ghost
vertex, then we have two options: the attractive edge of this vertex points to another vertexoutside of �
G
(1)t
or it points to another in �G
(1)t.
Fig. 3.2: A path with attractive edge pointing outside
3. The p-model 51
Observe that when the attractive edge points outside �G
(1)t
we do not increase the size of the
path when we regard it as a path in G(1)
t
. Whereas if the attractive points to a vertex in�G
(1)t
when we identify the vertices to regard � as a path in G(p)
t
we may decrease its size.
Fig. 3.3: A path with attractive edge pointing inside
This proves that all paths in G(p)
t
are smaller than the paths in G(1)
t
, which gives us thedesired result.
The above Theorem combined to the upper bound for diam⇣
G(1)
t
⌘
given in [7] gives us the
following Corollary.
Corollary 3.3.2 (Small-world). There exists a positive constant C such that
P⇣
diam⇣
G(p)
t
⌘
� C log(t)⌘
= o(1).
4. THE P (T )-MODEL
This chapter has as main objective introduces a model we proposed and shows our progressinvestigating it. We intend to show a some results which may clarify why the the model maybe a good model to study.
The p(t)-model is a modification of the p-model in which the parameter p is taken to be adecreasing function of the time t. We also demands that this function p(t) goes to zero as tgoes to infinity and that it is bounded from above by one. Recall that in the p-model, theparameter p is the probability of adding a new vertex to the graph. So, in the p(t)-modelthis probability goes to zero as the time goes to infinity. This property has as main objectivebe more realistic, since in many complex networks, as the social ones, the probability of newindividuals become part of the network decreases as the network’s order increases.
Since this model is very similar to the p-model, we briefly describe its dynamics. The modelhas two parameters: a decreasing function of the time p(t) and an initial graph G
1
, which wewill consider to be the graph with one vertex and one loop. As in the p-model, we considerthe following two stochastic operations that can be performed on the graph G:
• Vertex-step - Add a new vertex v, and add an edge {u, v} by choosing u 2 G withprobability proportional to its degree.
• Edge-step - Add a new edge {u1
, u2
} by independently choosing vertices u1
, u2
2 Gwith probability proportional to their degrees. We note that we allow loops to beadded, and we also allow a new connection to be added between vertices that alreadyshared an edge.
Then, we obtain Gt+1
performing a vertex-step on Gt
with probability p(t) or an edge-stepwith probability 1� p(t).
4. The p(t)-model 53
Power-Law exponent Expected Max. Degree Diameter
p(t) = p 2 + p
2�p
⇥⇣
t2�p2
⌘
O(log(t))
p(t) = 1/ log(t) 2 ⇥⇣
t
log(t)
⌘
� log(t)
log(log(t))
p(t) = 1/t� 2� � ⇥ (t) constant
Tab. 4.1: Comparison table among the di↵erent values for p(t).
4.1 Continuity of the parameter
From [10] and Proposition 3.1.1 two properties of the p-model are known:
1. It obeys a power-law distribution of exponent � = 2 + p/(2� p);
2. The expected value of the maximum degree satisfies:
E [dmax
(Gt
)] = ⇥�
t1�p/2
�
.
Looking to these two properties, we may note that � tends to 2 and the expected value of themaximum degree tends to t as p goes to zero. Of course, taking the limit on p is meaningless,since p is fixed on the p-model. Even if p is very close to zero, we still have � > 2 and theexpected value of the maximum degree far from being of order t. In this way, the p(t)-modelcomes to give a proper sense to this limit since its parameter goes to zero. So, we wouldexpect the following two properties of it:
1. It obeys a power-law distribution of exponent � = 2;
2. The expected value of the maximum degree satisfies:
E [dmax
(Gt
)] = ⇥ (t) .
Curiously, that is not the case. The p(t)-model may behave very di↵erently depending onhow fast p(t) goes to zero. The Table 4.1 summarizes this brief discussion.
4. The p(t)-model 54
4.2 The case p(t) = 1/t�
In this section we denote by G�
t
the graph generated by the p(t)-model choosing p(t) = t��,where � 2 (0, 1). We begin noticing that E[|V (G�
t
)|] = ⇥(t1��) since we may write
|V (G�
t
)| =t
X
s=2
Ber(p(s)),
in which (Ber(p(s)))s
are independent random variables following a Bernoulli’s distributionof parameter p(s). And by Cherno↵ bounds we may assure that the number of vertices inG�
t
is close to its mean w.h.p.
4.2.1 Power-law distribution
The proof of the power-law distribution is essentially the same we gave in Chapter 2 forHolme-Kim’s model, which is very similar to that given in [10] for the p-model. For thisreason we will just point out the main di↵erences for this case.
We keep the notation used in Chapter 2, denoting the number of vertices of degree d byN
t,�
(d). We are interested of evaluate the limit of E [Nt,�
(d)] /t1�� when t goes to infinity.Notice the term t1�� which represent the order of total number of vertices in G�
t
instead ofthe usual t as in Theorem 2.1.3.
Theorem 4.2.1 (Power-law distribution). There exists a positive constant c(�) such that
E [Nt,�
(d)]
t1��
�! c(�)
d2��
Proof. Let at
(d) be E [Nt,�
(d)]. Proceeding as in proof of Theorem 2.1.3 we may derive thefollowing recurrence relation for a
t
at+1
(d) =
✓
1� (2� p(t))d
2t
◆
at
(d) +(d� 1)(2� p(t))
2tat
(d� 1) +O�
t�1
�
. (4.2.2)
So, supposingat
(k)
t1��
�! Mk
,
for all k d and writing a0t
(d) = t�at
(d) we have
a0t
(d)
t=
at
(d)
t1��
4. The p(t)-model 55
and the that a0t
(d) satisfies the following recurrence relation
a0t+1
(d) =
✓
1� (2� p(t))d
2t
◆✓
1 +1
t
◆
�
a0t
(d) +(d� 1)(2� p(t))
2t1��
at
(d� 1) +O�
t�1+�
�
.
(4.2.3)Now, observe that as t gets large we may write
✓
1 +1
t
◆
�
= 1 +�
t+O
�
t�2
�
.
Returning to (4.2.3) we have
a0t+1
(d) =
✓
1� (2� p(t))d� 2�
2t
◆
a0t
(d) +(d� 1)(2� p(t))
2t1��
at
(d� 1) +O�
t�1+�
�
.
Finally, using Lemma 5.1.1 and the inductive hypothesis we obtain
E [Nt,�
(d)]
t1��
=a0t
(d)
t�! (d� 1)
d+ 1� �M
d�1
which is enough to conclude the desired result.
4.2.2 The expected maximum degree
The result is an immediate consequence of the Proposition below, which is essentially theProposition 3.1.1 we proved for the case p(t) = p.
Proposition 4.2.4. For each vertex j and each j t0
t the sequence of random vari-ables (Z
t
)t�t0
defined as
Zt
:=dt
(j)1{Tj,1=t0}Q
t�1
s=1
⇣
1 + 1�p(s)/2
s
⌘ (4.2.5)
is a martingale starting from t0
.
Proof. As in proof of Proposition 3.1.1, define �dt
(j) := dt+1
(j)� dt
(j). So, assuming thatj already exists at time t, we have
E [�dt
(j)|Ft
] = 1 · p(t)dt(j)2t
+ 1 · (1� p(t))2dt
(j)
2t
✓
1� dt
(j)
2t
◆
+ 2 · (1� p(t))(d
t
(j))2
4t2
=
✓
1� p(t)
2
◆
dt
(j)
t.
(4.2.6)
Dividing the above equation byQ
t
s=1
⇣
1 + 1�p(s)/2
s
⌘
we obtain the desired result.
4. The p(t)-model 56
From the Proposition, specifically from equation (4.2.6), we have that
E [dt
(j)] =t
Y
s=j
✓
1 +1� s��/2
s
◆
= ⇥
✓
t
j
◆
which implies the desired result.
Observation. We observe that from the above equation we also deduce the expected degreewhen p(t) = 1/ log(t).
4.3 The case p(t) = 1/ log(t)
4.3.1 Power-law distribution
This time we denote the random graph by Gt
and we observe that |V (Gt
)| = ⇥ (t/ log(t)).
Theorem 4.3.1 (Power-law distribution). There exists a positive constant c1
such that
E [Nt
(d)]
t/ log(t)�! c
1
(d+ 1)d
Proof. The proof is essentially the same we gave for Theorem 4.2.1 but here we let
a0t
(d) = at
(d) log(t).
This leads to the following recurrence relation involving a0t
(d)
a0t+1
(d) =
✓
1� (2� p(t))d
2t
◆
log(t+ 1)
log(t)a0t
(d)+log(t+ 1)
log(t)
(d� 1)(2� p(t))
2ta0t
(d� 1). (4.3.2)
Now, observe that as t gets large we may write
log(t+ 1)
log(t)= 1 +
1
t log(t)+O
�
t�2
�
.
Which yields
a0t+1
(d) =
✓
1� (2� p(t))d
2t+O
✓
1
t log(t)
◆◆
a0t
(d) +log(t+ 1)
log(t)
(d� 1)(2� p(t))
2ta0t
(d� 1).
Finally, using Lemma 5.1.1 and the inductive hypothesis we obtain
E [Nt
(d)]
t/ log(t)=
a0t
(d)
t�! (d� 1)
d+ 1M
d�1
which is enough to conclude the desired result.
4. The p(t)-model 57
4.3.2 Lower bounds for the diameter
Here we prove a lower bound for the diameter of Gt
showing that we have a large numberof paths of size log(t)/ log(log(t)), which we call snakes. The main result is the following
Theorem 4.3.3. There exists a positive constant c such that
P✓
diam (Gt
) � clog(t)
log(log(t))
◆
= o(1).
We begin by the main object needed for the proof of the above theorem.
Definition 5 (Snake). Given a sequence of steps of size l, ~t = (t1
, .., tl
), a path of Gt
whosevertices are those created at times t
1
, t2
, .., tl
plus the neighbor of the vertex created at timet1
and that doesn’t have edges created at other times will be called a snake of size l anddenoted by s
~
t
.
In other words, a snake s~
t
is a path created exactly as vector of times ~t = (t1
, .., tl
) instructs:at time t
1
we follow a vertex-step and connect the vertex created to another vertex, at timet2
we again take a vertex-step connecting the new vertex to the vertex created at time t1
and so on. Beyond the sequence of vertex-step, we require that no other edge is connectedto the path.
t0 t1 tl
...
Fig. 4.1: A snake-path of size l
Let Sl
(t) be the set containing all snakes of size l in Gt
and denote by N l
ct,t
the numberof snakes whose vertices were created at times between ct and t. We will need some lowerbounds for E
⇥
N l
ct,t
⇤
which will be expressed in the following two lemmas below.
Lemma 4.3.4 (Snake Lemma). For any 0 < c < 1 and any integer l min{(1 � c)t, ct},the following lower bound to E
⇥
N l
ct,t
⇤
holds
E⇥
N l
ct,t
⇤
�✓
(1� c)t
l
◆
1
(2t)l�1 logl(t)
✓
1� 2l
ct
◆
t
.
Proof. The random variable N l
ct,t
can be written as
N l
ct,t
=X
t1<t2<...<tlti2[ct,t]
1 {s~
t
2 Sl
(t)} (4.3.5)
4. The p(t)-model 58
Then, its expected value is
E⇥
N l
ct,t
⇤
=X
t1<t2<...<tlti2[ct,t]
P (s~
t
2 Sl
(t))
So, it su�ces to obtain a proper lower bound to P (s~
t
2 Sl
(t)). We claim that
P (s~
t
2 Sl
(t)) � 1
(2t)l�1 logl(t)
✓
1� 2l
ct
◆
t
.
To see why the claim above is true, observe that the snake requires the birth of l vertices attimes greater than ct. This explains the factor of log�l(t). When the vertex born, it mustconnects to the previous vertex in the snake, this is done with probability at least (2t)�1.Letting the vertex created at time t
1
connects to any vertex in Gt1�1
, we have the factorof (2t)l�1. We also must protect the snake from each possible edge created at times not in ~t.This is made with probability at least
�
1� 2l
ct
�
, because the probability of to connect to thesnake at each step is at most 2l/ct.
Finally we have the following lower bound to E⇥
N l
ct,t
⇤
E⇥
N l
ct,t
⇤
�✓
(1� c)t
l
◆
1
(2t)l�1 logl(t)
✓
1� 2l
ct
◆
t
. (4.3.6)
Lemma 4.3.7. There exists positives constants c1
, c2
and c3
such that, for l = c2
log(t)
log(log(t))
,
E⇥
N l
c1t,t
⇤
� tc3 .
Proof. We use the lower bound provided by Lemma 4.3.4 substituting l = c2
log(t)
log(log(t))
. In this
way, we must obtain lower bound for�
(1�c1)t
l
�
. But, since l << t, Stirling’s formula assuresthat there exists c
3
such that✓
(1� c1
)t
l
◆
� c3
(1� c1
)ltlell�l� 12 .
Choosing c1
= 1� e�1, we have✓
(1� c1
)t
l
◆
� c3
tll�l� 12 � c
3
exp {l log(t)� 2l log(l)} .
Combining the above inequality and (4.3.5) we have, for t large enough,
E⇥
N l
c1t,t
⇤
� c4
exp�
l log(t)� 2l log(l)� (l � 1) log(2t)� l log(log(t))� 2c�1
1
l
.
4. The p(t)-model 59
Finally, substituting l = c2
log(t)
log(log(t))
in the above inequality, we are able to chose c2
such that
E⇥
N l
c1t,t
⇤
� tc5
and this proves the lemma.
Some notations and definitions will be useful throughout the proof of Theorem 4.3.3. Wewrite {e
s
9 s~
t
} for the event in which the edge created at the s-th step doesn’t connect toany vertex of s
~
t
.
Definition 6 (Snake’s degree). Given a snake s~
t
, we denote by dr
(s~
t
) the sum of the degreesof each snake’s vertex. I.e.,
dr
(s~
t
) =X
ti2~t
dr
(vertex added at time ti
)
If r > t and s~
t
has size l, then dr
(s~
t
) = 2l � 1. And more
P (es+1
9 s~
t
| s~
t
⇢ Gs
) = 1�✓
1� p(s)
2
◆
ds
(s~
t
)
s. (4.3.8)
We write
cp
(s) := 1� p(s)
2.
Finally we are able to prove the main section’s theorem.
Proof of Theorem 4.3.3. Paley-Zigmund’s inequality assures that, for any 0 ✓ 1
P�
N l
ct,t
> ✓E⇥
N l
ct,t
⇤�
� (1� ✓)2�
E⇥
N l
ct,t
⇤�
2
Eh
�
N l
ct,t
�
2
i . (4.3.9)
If we guarantee that
1. E⇥
N l
ct,t
⇤
! 1;
2.�
E⇥
N l
ct,t
⇤�
2
= (1� o(1))Eh
�
N l
ct,t
�
2
i
,
choosing ✓ = ✓(t) such that ✓(t) goes to zero slower than E⇥
N l
ct,t
⇤
goes to infinity the Theoremis proven.
4. The p(t)-model 60
Lemma 4.3.7 provides a option for the constant c. We let c be the constant c2
in thelemma’s statement. Moreover, it also guarantees that E
⇥
N l
ct,t
⇤
goes to infinity. So, the proofis dedicated to show item 2.
To create the snake s~
t
we must add new vertex at time t1
then we avoid the snake from timet1
+ 1 to time t2
� 1. At time t2
, we add a new vertex and connect it to t1
, next we againavoid the snake in all steps between t
2
+ 1 and t3
� 1. We keep following theses rules untiltime t. We are saying that
P (s~
t
2 Sl
(t)) = p(t1
)t2�1
Y
r1=t1+1
✓
1� cp
(r1
)
r1
◆
p(t2
)1
2(t2
� 1)...
t
Y
rl=tl+1
✓
1� cp
(rl
)(2l � 1)
rl
◆
(4.3.10)
Note that If we have two snakes s~
t
and s~r
, by the definition of snakes, they can’t oc-cur simultaneously, unless they have disjoints vertex sets. With this in mind we compareP (s
~
t
, s~r
2 Sl
(t)) with P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t)). We would like to show the following
Claim: For two snakes with disjoints vertices set, we have
P (s~
t
, s~r
2 Sl
(t)) = (1 + o(1))P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t)) .
Proof of the claim: We separate in cases.
Case 1: s 2 ~t but s /2 ~r (s /2 ~t but s 2 ~r.).
To calculate P (s~
t
, s~r
2 Sl
(t)), we must add a new vertex at time s and connect it to the lastvertex added of s
~
t
. We do this with probability equal p(s) 1
2(s�1)
. On the other hand, to deter-
mine P (s~r
2 Sl
(t)), we must avoid s~r
and we do this with probability 1� cp(s�1)ds�1(s~r)
s�1
. Whilein P (s
~
t
2 Sl
(t)) we see exactly p(s) 1
2(s�1)
. This means in P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t)) we see⇣
1� cp(s�1)ds�1(s~r)
s�1
⌘
p(s) 1
2(s�1)
while in P (s~
t
, s~r
2 Sl
(t)), the term p(s) 1
2(s�1)
. This case occurs
2l times since the snakes are disjoints. Thus, recalling s 2 [ct, t], l = c log(t)/ log(log(t)) andthat the degree of each snake is at most 2l � 1, we can bound the product of all the terms
of the form⇣
1� cp(s�1)ds�1(s~r)
s�1
⌘
from above by
⇣
1� c3
t
⌘
2l
and from bellow by✓
1� c4
l
t
◆
2l
.
4. The p(t)-model 61
Observe that both product goes to 1 as t goes to infinity.
Case 2: s /2 ~t and s /2 ~r.
In P (s~r
2 Sl
(t)) as in P (s~
t
2 Sl
(t)) we see terms like 1� cp(s�1)ds�1(s~r)
s�1
, since we must avoidthe snakes in each event. But, in P (s
~
t
, s~r
2 Sl
(t)) we observe
1� cp
(s� 1)(ds�1
(s~
t
) + ds�1
(s~r
))
s� 1
since we must avoid both snakes at same time. Now, observe that
⇣
1� a
s
⌘⇣
1� c
s
⌘
=
✓
1� (a+ c)
s
◆✓
1 +ac
s2(1� o(1))
◆
.
We have ⇥(t) terms likes⇣
1 + ac
s
2(1�o(1))
⌘
. But again, as in Case 1, we still have their product
going to 1. This proves the claim. ⌅
Now, observe that
Eh
�
N l
ct,t
�
2
i
= E
2
4
0
@
X
~
t
1 {s~
t
2 Sl
(t)}
1
A
X
~r
1 {s~r
2 Sl
(t)}!
3
5
=X
~
t,~r
disjoints
P (s~
t
, s~r
2 Sl
(t)) + E⇥
N l
ct,t
⇤
.(4.3.11)
And that
�
E⇥
N l
ct,t
⇤�
2
=X
~
t,~r
disjoints
P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t)) +X
~
t,~r
~r\~t 6=;
P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t)) .
Thus, using the equations above, Lemma 4.3.7 and Jensen’s Inequality, we are able to obtain
1 Eh
�
N l
ct,t
�
2
i
�
E⇥
N l
ct,t
⇤�
2
P
~
t,~r
disjoints
P (s~
t
, s~r
2 Sl
(t))
P
~
t,~r
disjoints
P (s~
t
2 Sl
(t))P (s~r
2 Sl
(t))+
1
E⇥
N l
ct,t
⇤ = 1 + o(1)
which proves the desired result.
5. APPENDIX
5.1 Auxiliary Lemmas
Lemma 5.1.1 (Lemma 3.1 [10]). Let at
be a sequence of positive real numbers satisfying thefollowing recurrence relation
at+1
=
✓
1� bt
t
◆
at
+ ct
.
Furthermore, suppose bt
! b > 0 and ct
! c, then
limt!1
at
t=
c
1 + b.
Lemma 5.1.2. Let c > 0 be a fixed constant and �(t) be a real function defined as
�(t) :=t�1
Y
s=1
⇣
1 +c
s
⌘
, for t � 1.
Then, there exists positive constants b1
and b2
depending on c only such that
b1
tc �(t+ 1) b2
tc, for all t � 1.
Proof. The proof is essentially controlling of Taylor Series. Observe that, for all 0 < x < 1,we have
ex = (1 + x)
1 +x2
1 + x
1X
k=2
xk�2
k!
!
(1 + x)
1 + x2
1X
k=2
1
k!
!
(1 + x)�
1 + ex2
�
.
(5.1.3)
Since 1 + x ex, we also have
t�1
Y
s=1
✓
1 +(ce)2
s2
◆
exp
(
t�1
X
s=1
(ce)2
s2
)
eb, (5.1.4)
5. Appendix 63
in which, b = expn
P1s=1
(ce)
2
s
2
o
. Finally, combining the inequality above we have
�(t) �exp
�
P
t�1
s=1
c
s
Q
t�1
s=1
⇣
1 + (ce)
2
s
2
⌘ � b1
tc.
The upper bound comes from the bound 1 + x ex.
5.2 Concentration Inequalities for Martingales
Theorem 5.2.1 (Azuma’s Inequality - [10]). Let (Mn
)n�1
be a (super)martingale satisfying
|Mi+1
�Mi
| ai
Then, for all � > 0 we have
P (Mn
�M0
> �) exp
✓
� �2P
n
i=1
a2i
◆
.
Theorem 5.2.2 (Freedman’s Inequality). Let (Mn
,Fn
)n�1
be a (super)martingale. Write
Vn
:=n�1
X
k=1
E⇥
(Mk+1
�Mk
)2�
�Fk
⇤
and suppose that M0
= 0 and
|Mk+1
�Mk
| R, for all k.
Then, for all � > 0 we have
P�
Mn
� �, Vn
�2, for some n�
exp
✓
� �2
2�2 +R�
◆
.
Proof. Without lost of generality, we assume R = 1. If it is not, another martingale can bedefined just dividing by R. To simply our writing, denote by D
k
the di↵erence Mk
�Mk�1
and put ⌧ = inf{n : Mn
> �}. The proof will require the inequality below, which we juststate since it may be derived via straightforward calculus arguments.
esx 1 + sx+ (es � 1� s)x2, (5.2.3)
for all s > 0 and x 1. The proof is essentially the follow
5. Appendix 64
Claim: For all s > 0,
E⇥
esDk+1�
�Fk
⇤
e(es�1�s)E[D2
k+1|Fk].
Proof of the claim: By hypothesis |Mk+1
�Mk
| 1. Thus, (5.2.3) gives us
esDk+1 1 + sDk+1
+ (es � 1� s)D2
k+1
.
Taking the conditional expectation on both sides and using that Mn
is a (super)martingal,we have
E⇥
esDk+1�
�Fk
⇤
1 + (es � 1� s)E⇥
D2
k
�
�Fk�1
⇤
e(es�1�s)E[D2
k+1|Fk]
proving the claim. ⌅
Now, let Sn
be the following
Sn
:= esMn�(e
s�1�s)Vn = Sn�1
esDn�(e
s�1�s)E[D2n|Fn�1]
and observe that our claim guarantees (Sn
)n
is a positive supermartingal. Thus, by theOptional Stopping Theorem,
E [S⌧^n] 1, for all n
and by Fatou’s Lemma,Z
{⌧<1}S⌧
lim infn!1
Z
{⌧<1}S⌧^n 1.
Now, let A denotes the event we are interest in and observe
S⌧
1A
� 1A
es��(e
s�1�s)�
2,
which impliesP(A) e�s�+(e
s�1�s)�
2. (5.2.4)
Minimizing on s the above inequality, we obtain
s = log
✓
�+ �2
�2
◆
.
Returning on (5.2.4), we have
P(A) ✓
�2
�+ �2
◆
�+�
2
e� exp
⇢
� ��2
2�2 + �
�
,
since log(1 + x) � 2x/(2 + x), which concludes the proof.
BIBLIOGRAPHY
[1] B. Bollobas. Mathematical results on scale-free random graphs. In Handbook of Graphsand Networks, pages 1–37, 2003.
[2] A. Barabasi and R. Albert. Emergence of scaling in random networks. Science,(5439):509–512, 1999.
[3] A-L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 1999.
[4] G. Bianconi and A. L. Barabasi. Bose-Einstein Condensation in Complex Networks.Physical Review Letters, 86:5632–5635, 2001.
[5] G. Bianconi and M. Marsili. Emergence of large cliques in random scale-free networks.EPL (Europhysics Letters), 74(4):740, 2006.
[6] B. Bollobas and O. Riordan. Robustness and vulnerability of scale-free random graphs.Internet Math., 1(1):1–35, 2003.
[7] B. Bollobas and O. Riordan. The diameter of a scale-free random graph. Combinatorica,24(1):5–34, 2004.
[8] Bla Bollobs, Oliver Riordan, Joel Spencer, and Gbor Tusndy. The degree sequence ofa scale-free random graph process. RANDOM STRUCTURES AND ALGORITHMS,18:279–290, 2001.
[9] C. Borgs, J. T. Chayes, C. Daskalakis, and S. Roch. First to market is not everything:an analysis of preferential attachment with fitness. In STOC, pages 135–144. ACM,2007.
[10] F. Chung and L. Lu. Complex Graphs and Networks (Cbms Regional Conference Seriesin Mathematics). American Mathematical Society, Boston, MA, USA, 2006.
[11] F. Chung and L. Lu. Complex Graphs and Networks (Cbms Regional Conference Seriesin Mathematics). American Mathematical Society, Boston, MA, USA, 2006.
[12] S. Dereich and M. Ortgiese. Robust analysis of preferential attachment models withfitness. Combinatorics, Probability and Computing, 23:386–411, 2014.
Bibliography 66
[13] S. Dommers, R. van der Hofstad, and G. Hooghiemstra. Diameters in preferentialattachment models. Journal of Statistical Physics, (1):72–107, 2010.
[14] N. Eggemann and S.D. Noble. The clustering coe�cient of a scale-free random graph.Discrete Applied Mathematics, 2011.
[15] M. Gabielkov and A. Legout. The Complete Picture of the Twitter Social Graph. InACM CoNEXT 2012 Student Workshop, Nice, France, December 2012.
[16] M. Girvan and M.E.J Newman. Community structure in social and biological net-works. Proceedings of the National Academy of Sciences of the United States of America,99(12):7821–7826, 2002.
[17] S. Janson, T. Luczak, and I. Norros. Large cliques in a power-law random graph. J.App. Prob., 2010.
[18] B. Jim and P. Holme. Growing scale-free networks with tunable clustering. Phys. Rev.E, 2002.
[19] B. Jim and P. Holme. Growing scale-free networks with tunable clustering. Phys. Rev.E, 2002.
[20] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal.Random graph models for the web graph. In FOCS, pages 57–65, 2000.
[21] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Networkmotifs: simple building blocks of complex networks. Science, 298(5594):824–827, Octo-ber 2002.
[22] T. Mori. The maximum degree of the barabasi-albert random tree. Comb. Probab.Computing, 2005.
[23] M. E. J. Newman. Who is the best connected scientist? A Study of scientific coauthor-ship networks. Phys. Rev., page 016131, 2001.
[24] M. E. J. Newman. Networks: An Introduction. Oxford University Press, Inc., NewYork, NY, USA, 2010.
[25] M. E. J. Newman, D. J Watts, and S. Strogatz. Random graph models of social net-works. Proceedings of the National Academy of Sciences of the United States of America,99:2566–2572, 2002.
[26] L. Ostroumova, A. Ryabchenko, and E. Samosvat. Generalized preferential attachment:Tunable power-law degree distribution and clustering coe�cient. Lectures Notes inComputer Science, 2013.
Bibliography 67
[27] L. Ostroumova, A. Ryabchenko, and E. Samosvat. Generalized preferential attachment:Tunable power-law degree distribution and clustering coe�cient. Lectures Notes inComputer Science, 2013.
[28] L. Ostroumova and E. Samosvat. Global clustering coe�cient in scale-free networks.Lectures Notes in Computer Science, 2014.
[29] S.H Strogatz and D. J. Watts. Collective dynamics of ’small-world’ networks. Nature,1998.
[30] W.-Q. Wang, Q.-M. Zhang, and T. Zhou. Evaluating network models: A likelihoodanalysis. EPL (Europhysics Letters), 98(2):28004, 2012.