Hierarchy in networksHierarchy in networks

Peter Náther, Mária Markošová, Boris RudolfVyjde : Physica A, dec. 2009

OutlineOutline

Networks in Nature and their properties

Ravasz – Barabási hierarchical network

Vásquez modelHierarchy in the growing scale free

networks with local rules

Properties of natural Properties of natural networksnetworks1. Scale free property2.Small world property3.Hierarchy of nodes

Small world property

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21

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ii

k

kE

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Average shortest distance is shortened due to the few shortcuts, while preserving local structure expressed in high clustering coefficient.

Scale free property

1.Network has rich self simillar complex structure.

2.Network has power law degree distribution.

log (k)

log(P(k))=log (N(k)/N)

kN

kNkP

Gamma is tangens of the angle A.

A

N(k) : number of nodes having degree k.

Degree distribution

Structure of the network is a result of its dynamics:

Preferential attachment of new nodes is responsible for the scale free structure of the network.

Can by shown analytically : Barabasi – Albert model

Is some type of dynamics responsible for the hierarchy of nodes in the network?

Sign of hierarchy: power law Sign of hierarchy: power law clustering coefficient distribution clustering coefficient distribution

log (k)

kClog kkC

kC - Average clustering coefficient of nodes with degree k.

RavRaváász - Barabsz - Barabáási hierarchical si hierarchical networknetwork

1. Scale free property coexists with hierarchy of nodes in many real networks (metabolic networks, protein interaction networks, www network, social network).

2. There should be a simple mechanism of creating hierarchical network while maintaining its scale free property.

3. Hierarchy in networks is expressed in the power law scaling of the average clustering coefficient for the nodes with degree k:

4. Hierarchy appears, if certain pattern is added each time unit into the network.

kkC

R-B process of net creation – deterministic version

R-B process of net creation – deterministic version

If the process runs sufficiently long, numerical analysis shows that:

1.Network has scale free character :

2.Network is hierarchical and has power law distribution of clustering coefficients :

3.Average clustering coefficient is constant

161.2, kkP

1, kkC

734.0C

Why ?1

a) Node in the centre of 5 – node modulus has clustering coefficient one and k=4. There are 20 such nodes.

b) Nodes at the centre of 25 node modulus have clustering coefficient 3/19 , k=20, and there are four such nodes.

c) One node at the centre of 125 node modulus has clustering coefficient k=84 and clustering coefficient 3/83.

R – B process, stochastic variant

Several real networks (actor network, semantic word web, www network, internet on the domain level) have hierarchical structure fulfilling approximately the law:

Is this an universal property, or scaling exponent differs from case to case?

Are scaling exponents for the clustering coefficient distribution and the degree distribution in hierarchical scale free network functionally dependant?

1, kkC

Stochastic variant of R-B hierarchical scale free

model.

Pick a p fraction of newly added nodes, and connect each of them independently and preferentially to the nodes of central module .

Preferential attachment means, that the probability of linking new node is proportional to the degree of the central module node .

What shows the numerical analysis of this stochastic model?

Changing p influences both exponents of clustering coefficient and degreedistribution:

decreases with increasing p.

decreases with increasing p.

Network preserves scale free and hierarchical property simultaneously.

, kkC

, kkP

Is an attachment of some regular, or at least of some virtually regular pattern responsible for the hierarchy in growing networks?

VVásquez modelásquez model

Vásques: Hierarchy and scale free property in networks emerges due to the local attachment rules.

Local rules: rules involving node and its nearest neighborhood

Exploring real networks (www, citation network)

Example A : Exploring www 1. One finds new www page using hyperlinks given on already known page. 2. One finds new www page using a search engine.

Example B: Exploring citation network 1. One finds a new paper by following citation list of known paper. 2. One finds a new paper randomly by searching.

Random walking, surfing on graph

Searching the net

Probability that certain node in the network will be visited if we start at randomly chosen node:

ij

outj

ej

outj

jije

ei

J

k

qk

vJq

N

qv ,

1

probability that the surfer decides to follow one link from the node

node out degree

adjacency matrix

Probability of node i being visited from node j by random walk

Probability of node i being visited by random jump from somewhere

inie

ei kq

N

qv

1 Mean field approximation of the

same formula.

Average probability, that a vertex pointing to vertex i is visited

How to calculate ?

average probability, that vertex having certain out degree and pointing to i node is visited.

average probability to leave this node through one link pointing out of the node i.

outk

v

outk

1

Visiting a new node means sometimes adding a link to it. Therefore, when exploring network by moving in it vertices are visited and links are added in average; being a probability that visited vertex increases its degree by one. (e.g. hyperlink is created to web page in www).

Nv Nvqvvq

Nvqt

Et

N

vs

a

---------- number of added nodes per time

unit,

---------- number of added edges per time unit, where is the number of surfers walking on net.

s

number of edges added by one surfer

a

vsoutin

vs

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v

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tvN

outk

vFrom equations

we get, that

a

vsoutin

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a

Number of added nodes per time unit

Number of surfersProbability that the

visited vertex increases its degree by one

Number of nodes in the network

Probability of increasing in degree of vertex Probability of increasing in degree of vertex with bywith byoneone

ink

ins

aeev

in

inie

eini

inv

in

kqqqN

kA

kqN

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1

probability that a visited vertex increases its in degree by one

probability that a surfer follows one of the outgoing links

number of added nodes per time unit

number of surfers

probability that the visited vertex increases its degree by one

Nq sv

a

How does the network look? What tells us the How does the network look? What tells us the walk about the network structure?walk about the network structure?

The structure is expressed in the degree distribution and clustering coefficient distribution.

Degree distribution: How many nodes has degree k?

0,11 ininininin

in

kakkskksk nAnAt

n

probability, that node having degreegains a link.

1inkprobability, that node having degreegains a link.

ink this is zero for 0inkHow the amount of nodes having degree changes with time?

ink

inkk

sininkkkk

sk

inkkskksk

k

nA

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equationthereorganizewe

knAnAt

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ininininininin

inininin

in

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0,

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inkk

sk

k

nA

t

n ininin

Thus to get degree distribution, our aim is to solve this differential equation, which gives us a good asymptotic for great networks and long walking times.

inkk

sk

k

nA

t

n ininin

Let , where is a stationary probability of having node with

degree . Incorporating this into previous equation and integrating in we get

inin kkNpn ink

p

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aeevk

kqqqN

A in

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kevska

kins

aeevins

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Solvable differential equation

e

qk

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inkinkp ein

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Network is scale free

Solution of the differential equation

Clustering coefficient distribution

ini

outii

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kkk

inodeofneighborsamong

edgesofnumberEinodeoftcoefficienclusteringkE

C

,

2

How changes with time by random walk ?iE

ieiniev

i vqkqqt

E

Probability, that a visited vertex increases its degree by one

Probility of folowing one link

Average probability that a vertex pointing to vertex i is visited

Probability that vertex i is visited

ieiniev

i vqkqqt

E

First term denotes the probability, that a

vertex pointing to vertex i is visited, the second one denotes the probability, that a vertex i is visited and the walk follows an out link to its neighbor.

Incorporating and into this equation and using one finds

And by integrating this equation and using the basic formula for the clustering coefficient one finds

iv ink

ini At

k

inike

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1

1 kC Network is hierarchical.

Our model with local rulesOur model with local rulesNáther, Markošová, Rudolf, to appear in Physica A

Network dynamics:

1.We start from the small network, from three totaly interconected nodes.2.Each time unit one node comes from the universe. If it comes at time s, it has a label s. The node brings m>1 new edges to the system.3.One new edge is linked to an old node by clustering driven preference, that means the linking probability is proportional to the clustering coefficient of the old link. The other m-1 edges is randomly distributed among neighbors of the old node.

Clustering coefficients of all nodes in starting graph is one.

1 2

3

4

Now clustering coefficients are 1,3

2,3

2,1 4321 CCCC

5

Clustering driven dynamics (CD model)

Numerical studies of CD model shows, that the network is scale free and hierarchical:

The most simple variant of CD model (SCD – model), in which each new node brings exactly two new edges into the network, is analytically tractable.

1

3

kkC

kkP

1. We can show analytically, that the clustering driven node addition is not responsible for the scale free final network structure.

2. It is possible to map SCD model to the model of Vásquez and to calculate analytically P(k) and C(k) distributions to show scale free and hierarchical property.

Clustering driven node addition is Clustering driven node addition is not responsible for the scale free not responsible for the scale free net structure.net structure.

In the SCD model each step creates a triangle of connected nodes Therefore for each node s the number of edges among its neighbors is k(s)-1, and its clustering coefficient is.

Let us therefore solve the model, where each time unit a node comes and links to the older node with probability proportional to .

sksksk

sksksk

sC2

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1k

We should solve an equation

t

dstsk

tsk

t

tsk

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,,

tgsfbtsk ,Let us look for the solution in a form:

Giving this into main integro - differential equation we have:

And from the initial conditions k(t,t)=1, we get f(s)=1/2-g(s). Seeking g(t) in a form we finally have:

which does not lead to the power law degree distributionm, and clustering driven network is not scale free.

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Remapping of the SCD model to the Remapping of the SCD model to the VVásquez modelásquez model

1. SCD model can be mapped to the Vásquez model with only one surfer and the probability ( prob. that each visited vertex increases its degree by one). Let us call this V- model.

2. Therefore probability, that vertex with degree gains a new link is:

1vq

ink

in

s

aeevk

kqqqN

A in

11 Basic formula

inaeekkvqq

NA in 1

1 V - model formula

1s

3. In our SCD model each time unit two new edges are added. It is therefore comparable to the V – model with .

4. We also have clustering driven preference for finding a node by jump and edges are undirected.

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1eq

kvq

CkqN

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Applying to the rate equation

and using

we get

And after the integration

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kqCk

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kpt

kp

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eq

kkp1

1, Because as was measured for our network.

3,2

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sksksk

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And what about hierarchy?

All nodes s with degree k in the SCD model have clustering coefficient given as:

That means, that

1, kkC

CD model, in which more then two nodes are added in one time unit, has been studied numerically.

CD modelCD model

SCD model is a simplified variant of CD model, in which one node and more then two edges are added in one time unit. One edge is linked by clustering driven preference to the certain old node s, other edges are distributed randomly among neighbors of the node s. Numerical studies of such model shows that the network has scale free property and hierarchy:

m is a number of edges added in each time unit.

mkkC

kkP

,

3,

SummarySummary

1. Local rules are responsible for the hierarchy in the networks. They lead to the fact, that in average certain pattern is added each time unit.

2. The fact, that the network is clustering driven is not responsible for the scale free structure, nor for hierarchy of nodes.

3. Virtual preferential attachment is responsible for the scale free property. If we link a node with clustering driven preference (or even with random preference ) the addition of another edges to the neighbor nodes ids in fact preferential. The higher degree has the neighbor, the higher probability it has to catch a new link.

ReferencesReferences1. Ravasz, Barabasi: Hierarchical organization in complex netwoks,

cond – mat 026130v2, 20022. Vásquez: Growing network with local rules…Phys. Rev. E 67 (2003)

0561043. Náther, Markošová, Rudolf: Hierarchy in the growing scale free

network with local rules, Physica A 388 (2009) 5036

Thank you for the attention