supplementary materials: successful strategies for ......1.2 basic properties of dynamical processes...

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Supplementary Materials:

Successful strategies for competing networks

Contents

1 Interplay between the eigenvector centrality and the dynamical properties of complexnetworks 21.1 Definition of eigenvector centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Basic properties of dynamical processes on complex networks . . . . . . . . . . . . . . . . 2

1.2.1 Dynamics towards equilibrium: the role of the transition matrix, its associatedeigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Competition time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Applications to population dynamics: RNA neutral networks . . . . . . . . . . . . . . . . 31.4 Applications to epidemics: The susceptible-infected model . . . . . . . . . . . . . . . . . . 41.5 Applications to rumor spreading: The Maki-Thompson model . . . . . . . . . . . . . . . . 5

2 Analytical approach to the phenomenology of competing networks and definition ofsuccessful strategies 72.1 Analytical derivation of the eigenvector uT,1, the eigenvalue λT,1, and the competition time

tc,T associated to a network T formed by two connected networks A and B . . . . . . . . 72.2 Study of a numerical example: application of the analytical results . . . . . . . . . . . . . 92.3 Summary of successful strategies in competing networks . . . . . . . . . . . . . . . . . . . 11

3 Application of the results to generic competing networks 123.1 Competition between two directed networks . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Competition between N networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Study of competition on large networks through exactly solvable examples 17

5 Competition strategies based on increasing the maximum eigenvalue associated tothe competing networks 215.1 Successful strategies based on increasing the number of nodes or links . . . . . . . . . . . 215.2 Successful strategies based on local network rewiring . . . . . . . . . . . . . . . . . . . . . 21

6 Competition in real networks 23

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1 Interplay between the eigenvector centrality and the dynam-ical properties of complex networks

1.1 Definition of eigenvector centrality

The eigenvector centrality has been traditionally used to quantify the importance of a node in a network.The main contribution of the eigenvector centrality is that it takes into account the importance of thenode’s neighbours by giving a score to the node that is proportional to the scores of all its connectednodes. In this way, the mathematical definition of the eigenvector centrality xk of a node k is related toan iterative process were the centrality is calculated as the sum of the centralities of its neighbours:

xk = γ−1∑j

Gkjxj , (1)

where γ is a constant, xk is the eigenvector centrality of node k and Gkj are the components of theadjacency matrix. In matrix notation Eq. S1 reads γx = Gx so that x can be expressed as a linearcombination of the eigenvectors vk of the adjacency matrix G, being γk the set of the correspondingeigenvalues. Note that Eq. S1 can be regarded as an iterative process that begins at t = 0 with aset of initial conditions x0. No matter what the values of x0 are, the value of xk(t) at t → ∞ will beproportional to the eigenvector v1 associated to the dominant eigenvalue γ1. Therefore, the eigenvectorcentrality is directly obtained from the eigenvector v1 of the adjacency matrix G.

The concept of eigenvector centrality can be easily extended to weighted connectivity matrices, wherethe weight of the connections wkj is proportional to the amount of interaction between two nodes. In thecase of a weighted connectivity matrix W, we have that λx = Wx, and the eigenvector centrality willbe, in this case, obtained from eigenvector u1 of the weighted matrix W [1].

1.2 Basic properties of dynamical processes on complex networks

For the sake of clarity and completeness, we briefly present and prove the main properties of a genericdynamical process that takes place on a complex network (adapted from [2]).

1.2.1 Dynamics towards equilibrium: the role of the transition matrix, its associated eigen-values and eigenvectors

A variety of dynamical processes occurring in a network can be mathematically described as

n(t+ 1) = Mn(t) , (2)

where n(t) is a vector whose components are the state of each node at time t (for example, the populationof individuals at each node), and M, with Mij ≥ 0, is a transition matrix that contains the peculiaritiesof the dynamical process.

M is a primitive matrix. For this reason, its largest eigenvalue is positive, it verifies that λ1 > |λi|,∀ i > 1, and its associated eigenvector is also positive (i.e., all its elements are positive). Therefore, thedynamics becomes

n(t) = Mtn(0) =m∑i=1

(n(0)ui)λtiui , (3)

where n(0) is the initial condition and ui the i−th eigenvectors of M.From Eq. S3 we obtain that the system evolves towards an asymptotic state independent of the initial

condition and proportional to the first eigenvector u1

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limt→∞

(n(t)

(n(0)u1)λt1

)= u1 , (4)

while its associated eigenvalue λ1 yields the growth rate at the asymptotic equilibrium. If n(t) is normal-ized such that |n(t)| = 1 after each iteration, n(t) → u1 when t → ∞. This way, there is a correspondencebetween the eigenvector centrality and the asymptotic state of the system at equilibrium: both quantitiesare proportional to the eigenvector associated to the largest eigenvalue of the transition matrix M.

1.2.2 Competition time

The distance ∆(t) to the equilibrium state at time t starting with an initial condition n(0) can be writtenas

∆(t) =

∣∣∣∣Mtn(0)

(n(0)u1)λt1

− u1

∣∣∣∣ =∣∣∣∣∣

m∑i=2

n(0)ui

n(0)u1

(λi

λ1

)t

ui

∣∣∣∣∣ . (5)

In order to estimate the time to reach the equilibrium state, we fix a tolerance ξ, and define thecompetition time tc as the number of iterations required for ∆(tc) < ξ. For generic cases, tc can beextracted from Eq. S5 and approximated by

tc ∼ln |n(0)u2

n(0)u1| − ln ξ

ln λ1

λ2

(6)

because of the exponentially fast suppression of the contributions due to higher-order terms (λi ≥ λi+1,∀i). Therefore, we finally obtain

tc ∝ (lnλ1/λ2)−1. (7)

1.3 Applications to population dynamics: RNA neutral networks

The results of this paper are applicable to generic processes n(t + 1) = Mn(t) evolving on weightednetworks. However, for the sake of clarity, and without any loss of generality, we suppose the followingtransition matrix throughout the simulations of the paper:

M = (R− µ)I+µ

SG , (8)

where I is the identity matrix and G is the adjacency matrix of a connected graph, whose elementsare Gij = 1 if nodes i and j are connected and Gij = 0 otherwise. M describes a process modelling apopulation that, each time step: a) replicates at each node with a growing rate R > 1, b) its daughterindividuals leave the node with probability µ, being 0 < µ ≤ 1, and c) the probability of remaining aliveafter leaving a node of degree ki is ki/S [2]. In particular, the competition times shown in Fig. 4 of themain text and Fig. S4 were obtained making use of Eqs. S7 and S8.

Equation S8 can be directly applied to describe, for instance, the dynamics of a population of RNAmolecules evolving through point mutations on a neutral network. An RNA sequence is composed of thecombination of four nucleotides (guanine, adenine, uracil and cytosine) and a neutral network representsall nodes with different primary structures (genotypes) that fold into the same secondary structure (as aproxy for the phenotype). Two nodes, or genotypes, are connected when their sequences differ in only onenucleotide, which allows a molecule to evolve by one point mutation over the space of genotypes. If themutation pushes the molecule out of the neutral network, it is discarded. In particular, the parametersof Eq. S8 are translated to RNA evolutionary dynamics as follows: R stands for the growing rate of eachmolecule per generation, µ for the mutation rate and µ/S = µ/(3L) for the probability of mutating to aviable neighbour. Finally, L is the number of nucleotides per sequence [2, 3].




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1.4 Applications to epidemics: The susceptible-infected model

In the framework of epidemic dynamics, one example of the dynamical implications of the eigenvectorcentrality is represented by the susceptible-infected (SI) model [4]. In this model, a disease propagatesthrough a network of individuals whose dynamical state can be either susceptible or infected by thedisease. In this section we show that in this model the probability of being infected at short times(t << ∞) is proportional to the eigenvector centrality ( [1]).

In the SI model, the probability that a node k becomes infected is given by Ik(t), where Sk(t) =1 − Ik(t) is the probability of being susceptible (i.e., not infected). Considering β as the infection ratebetween susceptible and infected individuals, the probability that a node k becomes infected betweentimes t and t + dt is proportional to the number of neighbours that are already infected β

∑j GkjIj ,

being G the adjacency matrix. Since only susceptible individuals can get infected, the dynamics of Sk(t)and Ik(t) can be described by a set of N differential equations, N being the total number of individuals(nodes):

dSk

dt= −βSk

∑j

GkjIj = −βSk

∑j

Gkj(1− Sj), (9)

dIkdt

= βSk

∑j

GkjIj = β(1− Ik)∑j

GkjIj , (10)

with Sk + Ik = 1. If the disease starts at a small number of nodes, in the limit of large system size Nand ignoring quadratic terms, Eq. S10 becomes:

dIkdt

= β∑j

GkjIj , (11)

which in matrix form readsdI

dt= βGI, (12)

I being a vector of components Ik. The temporal evolution of I can be expressed as a linear combinationof the eigenvectors uk of the transition matrix, which, in this case, coincides with the adjacency matrixG:

I(t) =N∑

k=1

ak(t)uk, (13)

where uk are the eigenvectors of the λk eigenvalues of G. Then

dI(t)

dt=

N∑k=1

dak(t)

dtuk = βG

N∑k=1

ak(t)uk = βN∑

k=1

λkak(t)uk, (14)

Comparing terms multiplying uk we obtain:

dakdt

= βλkak, (15)

which has the solutionak(t) = ak(0)e

βλkt. (16)

If we substitute Eq. S16 into Eq. S13 we arrive to the final expression on I(t):

I(t) =N∑

k=1

ak(0)eβλktuk. (17)




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Since the largest eigenvalue λ1 dominates over the others, we can approximate the infected populationas:

I(t) ∼ eβλktu1. (18)

Thus, for t → ∞ the exponential term leads to I → 1, i.e. the whole population gets infected at thefinal state. Nevertheless, for low to intermediate time scales (t << ∞), the term u1, i.e. precisely theeigenvector centrality of the nodes, controls the distribution of probabilities of getting infected.

1.5 Applications to rumor spreading: The Maki-Thompson model

The diffusion of rumors is strongly related to epidemic processes. The idea of treating the diffusion ofrumor/ideas in a similar way as a disease spreads over a population was first introduced by Daley &Kendal [5]. In this case, a rumor “infects” individuals that exchange information across its network ofconnections leading to a final state where the rumor attains some parts of the population but not others.In this section, we show that the probability of hearing a rumor can be shown to be strongly related tothe eigenvector centrality.

This point is illustrated using the Maki-Thompson (MK) model [6], a variation of the initial modelintroduced by Daley and Kendal (where similar results are obtained). The MK model considers threedifferent kinds of actors in a population of N individuals: a) ignorants I, who never heard about therumor, b) spreaders S, who spread the rumor to their contacts and c) stiflers R, who have heard therumor but decide to stop its diffusion. Three kinds of interactions can be defined when two individualsare in contact:

S + Iα→ 2S, (19)

indicating that when a spreader meets an ignorant, the ignorant becomes a spreader of the rumor witha probability α;

S + Sβ→ S +R, (20)

saying that when two spreaders meet, one of them realizes that the rumor does not have novelty andbecomes a stifler with a probability β; and

S +Rβ→ 2R, (21)

indicating that when a spreader meets a stifler, the former becomes a stifler with a probability β. Anyother kind of contact, such as I+I or R+R, does no introduce any change in the state of the individuals.Note that N = I + S + R since the total population is fixed. Given the interaction rules, the incrementof each population for short time intervals will be:

∆I ≈ −∆tαSI/N, (22)

∆S ≈ −∆t(αSI

N− βS2

N− βSR

N), (23)

∆R ≈ −∆t(βS2

N+

βSR

N). (24)

If we define the percentages of ignorants, spreaders and stiflers as x = I/N , y = S/N and z = R/N ,we can express Eqs. S22-S24 as a set of three differential equations:

dx

dt= −αxy, (25)




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dy

dt= yαx− βy2 − βy(1− y − x) = (α+ β)xy − βy, (26)

dz

dt= βy(1− x). (27)

We can extend Eqs. S25-S27 into a network of contacts just by considering that interactions betweenindividuals are defined by an adjacency matrix G:

dxk

dt= −αxk

∑j

Gkjyj , (28)

dykdt

= (α+ β)xk

∑j

Gkjyj − βyk, (29)

dzkdt

= β(1− xk)∑j

Gkjyj , (30)

where now, xk, yk and zk are the probabilities that node k is, respectively, ignorant, spreader orstifler. Assuming that at t = 0 the number of individuals that are prone to transmit the rumor is low, inthe limit of large number of nodes N the probability of a node to be a spreader is:

dykdt

= (α+ β)∑j

Gkjyj − βyk =∑j

[(α+ β)Gkj − βδkj ]yj , (31)

δkj being the Kronecker delta. In matrix form, Eq. S31 reads

dy

dt= (α+ β)My, (32)

where M is a transition matrix given by M = G − βα+β I, where I is the identity matrix. Note that M

and G only differ by a term that is multiple to the identity matrix. Therefore, it is possible to expressy as a linear combination of the eigenvectors uk of M, which are also the eigenvectors of the adjacencymatrix G:

Muk = Guk − β

α+ βIuk = (κ− β

α+ β)uk, (33)

Note that the eigenvalues of the transition matrix M are related to those of the adjacency matrix asλM = λG − β

α+β . Therefore we obtain an expression of y(t) similar to that obtained for the probability

of being infected in the SI model (Eq. S17):

y(t) =N∑

k=1

ak(0)e[(α+β)λk−β]tuk. (34)

Again Eq. S34 is dominated by the eigenvector associated to the largest eigenvalue λ1, and we cantherefore approximate the probability of being a spreader node as:

y(t) ∼ e[(α+β)λ1−β]tu1. (35)

The exponent of the exponential function determines two different dynamical regimes. For (α+β)λ1−β >0 the rumor spreads over the whole network, while for (α+ β)λ1 − β < 0 the rumor stops spreading. Inthe former case, the probability of being a rumor spreader y(t) at short to moderate times is, once again,proportional to the eigenvector centrality as indicated in Eq. S35.




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2 Analytical approach to the phenomenology of competing net-works and definition of successful strategies

2.1 Analytical derivation of the eigenvector uT,1, the eigenvalue λT,1, and thecompetition time tc,T associated to a network T formed by two connectednetworks A and B

Let us start by defining the terminology to be used during the analytical calculations. Networks A and B,of NA and NB nodes and LA and LB links respectively, form the disconnected network AB of NA +NB

nodes and LA + LB links. We connect them through L connector links to create a new network T ofNT = NA + NB nodes and LT = LA + LB + L links. For simplicity, let us suppose that A and B areweighted but undirected networks (the directed case is studied in section S3.1).

The nodes of network A are numbered from i = 1 to NA and the nodes of network B from i = NA+1

to NT = NA + NB . The adjacency matrix GAB is formed by two diagonal blocks corresponding toGA and GB. The relation between the transition matrix MAB, also formed by two blocks, and GAB,depends on the peculiarities of the process. Let us call λA,i and λB,i the i eigenvalues of the transitionmatrices MA and MB respectively. Let us suppose λA,1 > λB,1 throughout the paper.

Note that the eigenvectors of MA and MB are related to those of MAB as follows: The NA eigenvec-tors of MA are of length NA, the NB eigenvectors of MB are of length NB , and the eigenvectors of MAB

are of length NT . The first i = 1, ..., NA eigenvectors of MAB are equal to those of MA, being their lastNB terms equal to zero. Therefore, λAB,i = λA,i for i = 1, ..., NA. The i = NA + 1, ..., NT eigenvectorsof MAB are equal to those of MB, being their first NA terms equal to zero. Therefore, λAB,i = λB,i−NA

for i = NA + 1, ..., NT . For simplicity in the following calculations, we call uA,i to the first i = 1, ..., NA

eigenvectors uAB,i, and we call uB,i to the last NB eigenvectors uAB,i+NA where i = 1, ..., NB .We call cli=1,...,L the set of pairs kl corresponding to the connector links that attach the connector

nodes k of network A with nodes l of network B to form T . Let us call λT,i the eigenvalues of network T .The adjacency matrix GT corresponding to the union of networks A and B is therefore formed by GAB

with an extra 1 in elements kl ∈ cl and their symmetric positions lk. In particular, let us express thetransition matrix MT as the transition matrix MAB slightly perturbed in the cl positions and theirsymmetric elements. That is, MT = MAB+ εP where Pij = Pji = 0 for ij ∈ cl and Pij = 0 elsewhere.Pij depends on the explicit form of M. The main idea of the analytical calculations is to describe thetotal graph T as a perturbation of graph A by graph B, in a way that the weight of the connector links isε << 1 (and, as it is customary, at the end ε could be replaced by unity [7]). Therefore, as λA,1 > λB,1 byconstruction, the maximum eigenvalue λT,1 will be a perturbation of λA,1 and its associated eigenvectoruT,1 will be a perturbation of uA,1. This methodology is inspired on the perturbation theory of matricesshown in [7], and among other examples was recently applied in the context of Complex Network theoryto characterize the importance of network nodes and links [8], to analyze the impact of a structuralperturbation in the eigenvalues of a network [9], or to the detection of communities [10]. AssumingλA,1 > λB,1, we have

MT · uT,1 = λT,1uT,1 (36)

where

MT = MAB + εP, (37a)

uT,1 = uA,1 + εv1 + ε2v2 + o(ε3), (37b)

λT,1 = λA,1 + εt1 + ε2t2 + o(ε3). (37c)

Taking into account that (i) |uT,1| = 1 ⇒ uA,1 ·v1 = 0, and (ii) uA,1 ·P · uA,1 = 0 because (uA,1)i = 0for i > NA, we include Eqs. S37a, S37b and S37c in Eq. S36 and set equal the different terms of such




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equation according to the orders of ε. We obtain for order ε

uA,1 · (MABv1 +PuA,1) = uA,1 · (λA,1 · v1 + t1 · uA,1) (38a)

⇒ t1 = 0 (38b)

⇒ (MAB − λA,1)v1 = −PuA,1 , (38c)

and for order ε2

uA,1 · (MAB · v2 +Pv1) = uA,1 · (λA,1v2 + t2uA,1) (39a)

⇒ t2 = uA,1 ·P · v1. (39b)

The vector v1 can be obtained numerically solving Eq. S38c. However, it can also be analyticallyexpressed as

v1 =

NT∑k=1

ck · uAB,k =

NA∑k=1

ck · uA,k +

NT∑k=NA+1

ck · uB,k−NA . (40)

Including Eq. S40 in S38c, and multiplying both sides by uAB,k from the left, we obtain ck = 0 for

1 < k ≤ NA (because uA,kPuA,1 = 0 ∀k) and ck =uB,k−NA

PuA,1

λA,1−λB,k−NAfor k > NA. We know c1 = 0 because

uA,1 · v1 = 0. All this yields

v1 =

NB∑k=1

uA,1PuB,k

λA,1 − λB,k· uB,k , (41)

and including Eq. S41 in Eq. S37b, and Eq. S39b in Eq. S37c, we finally obtain an analytical expressionfor the eigenvector uT,1 and its associated eigenvalue λT,1:

uT,1 = uA,1 + ε

NB∑k=1

uA,1PuB,k

λA,1 − λB,k· uB,k + o(ε2) , (42a)

λT,1 = λA,1 + ε2NB∑k=1

(uA,1PuB,k)2

λA,1 − λB,k+ o(ε3) . (42b)

The terms k = 1 of both summations are the most relevant ones because λA,1 − λB,k > λA,1 − λB,1 fork > 1. Furthermore, note that uA,1PuB,1 =

∑cl (uA,1)i · Pij · (uB,1)j .

The competition time is another important dynamical property of a population evolving on a network,which verifies [2]

tc,T ∝(ln

λT,1

λT,2

)−1

. (43)

In order to obtain an analytical expression for λT,2, the second eigenvalue of MT, we develop a similarcalculation to the one dedicated to λT,1, assuming λA,1 > λB,1 > λA,2. This yields

λT,2 = λB,1 + ε2NA∑k=1

( uA,kPuB,1)2

λB,1 − λA,k+ o(ε3) , (44)

where the two first terms of the summation are the most relevant ones because λB,1−λA,k > λB,1−λA,2

for k > 2. Introducing Eqs. S42b and S44 in Eq. S43, and developing in powers of ε, we obtain for the




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competition time

tc,T ∝(ln

λT,1

λT,2

)−1

∼(ln

λA,1

λB,1

)−1 [1 + ε2

( NA∑k=1

(uA,kPuB,1)2

λB,1(λB,1 − λA,k)

−NB∑k=1

(uA,1PuB,k)2

λA,1(λA,1 − λB,k)

)]+ o(ε4) . (45)

Finally, let us particularize the present results for the transition matrix shown in Eq. S8, M = (R−µ)I+(µ/S)G. γi is the set of eigenvalues of matrix G, γi ≥ γi+1, and wi is the corresponding set ofeigenvectors. The eigenvalues of both matrices are related as λi = (R−µ)+ µ

S γi, and it is easy to check thatthe eigenvectors of the adjacency matrix are also eigenvectors of the transition matrix. Furthermore, Pij =Pji = µ/S for ij ∈ cl and Pij = 0 elsewhere, from where we obtain that uA,1PuB,1 = µ

S

∑cl (uA,1)i ·

(uB,1)j .

2.2 Study of a numerical example: application of the analytical results

In the former section we presented the analytical work to obtain the main dynamical properties of anetwork T formed by two connecting undirected networks A and B, which can be summarized as

uT,1 = uA,1 + ε

NB∑k=1

akuB,k + o(ε2) , (46a)

λT,1 = λA,1 + ε2NB∑k=1

(uA,1PuB,k)ak + o(ε3) , (46b)

where ak = (uA,1PuB,k)/(λA,1 − λB,k), and

tc,T ∝(ln

λA,1

λB,1

)−1 [1 + ε2

(− (λA,1 + λB,1)(uA,1PuB,1)

2

λA,1λB,1(λA,1 − λB,1)

+

NA∑k=2

(uA,kPuB,1)2

λB,1(λB,1 − λA,k)−

NB∑k=2

(uA,1PuB,k)2


)]+ o(ε4) .

(47)

The eigenvector uT,1 determines the outcome of the competition between networks A and B. Cen-trality A (CA) and centrality B (CB) are the fractions of the total centrality that remain in the nodes ofnetwork A and B after the connection, and are obtained as

CA =

∑NA

i=1 (uT,1)i∑NT

i=1 (uT,1)i, (48a)

CB = 1− CA . (48b)

This way, the goal of the competition between networks is to increase their C as much as possible.Regarding Eqs. S46 and taking into account that ak > ak+1, the final outcome of the competition dependsmainly on a1: CA → 1 when a1 → 0, and CB will grow when a1 grows. Therefore, the main propertiesinvolved in the competition are (i) the highest eigenvalues associated to both networks, λA,1 and λB,1,




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Figure S1. Competition between two Barabasi-Albert networks [11] A and B ofNA = NB = 1000 nodes and LA = LB = 2000 links, connected by a unique link in all possibleconfigurations (103 × 103). For each network, nodes are numbered according to its ranking incentrality. In this example, the transition matrix is Eq. S8, where parameters are R = 2, µ = 0.01, andS = 20, and the population starts uniformly spread over all nodes. The axes represent the connectorlinks in network A (X-axis) and network B (Y-axis). (a) Centrality A: Fraction of population that endsthe competition in network A. (b) Highest eigenvalue associated to the total network T . (c)Competition time tc,T .

and (ii) uA,1PuB,1, a quantity that is proportional to the centralities of the connector nodes when thenetworks are still disconnected, and to the number of connector links.

Next, we use numerical simulations in order to analyze the applicability of Eqs. S46 and S47, obtainedfor the perturbed case ε << 1, to generic cases where the weight of the connector links ε is the same asthat of the rest of the links. Figure 1b of the main text and Fig. S1 show a competition between twoBarabasi-Albert scale-free networks [11] A and B of 1000 nodes and 2000 links each. The centrality ofnetwork A follows Eq. S48a, where uT,1 was calculated as the first eigenvector of the transition matrixMT. X and Y-axes represent the connector nodes of A and B, with the node number ordered from higherto lower centrality. The phenomenology shown in Figs. 1b and S1 shows excellent agreement with Eqs.S46 and S47. When the connector nodes are the most central (i.e., uA,1PuB,1 is maximum and thereforeso is a1), network B shows its best results in centrality, λT,1 has its maximum value, and the competitiontime tc,T reaches a minimum because the first term of the perturbative part of Eq. S47 prevails over therest. Second, high centrality in the connector nodes of A combined with low centrality in the connectornodes of B maintains an intermediate value of the eigenvalue λC,1 due to the secondary terms k > 1 inEqs. S46a and S46b (which depend on uA,1 but not on uB,1). In addition, the time tc,T shows relativelylow values due to the fact that the third term of the perturbative part of Eq. S47 (which also depends on




11

uA,1 but not on uB,1) prevails over the rest. Third, when the connector links join peripheral nodes of bothnetworks, centrality distributes over network A and CA → 1, uT,1 → uA,1, λT,1 → λA,1. In this case, thetime tc,T reaches the intermediate value of tc,T ∝ (lnλA,1/λB,1)

−1. Interestingly, the competition timereaches its maximum when the connector links join central nodes of network B with peripheral nodesof network A, as the second term of the perturbative part of Eq. S47 prevails over the rest (becauseit depends on uB,1 but not on uA,1). Finally, note that, despite CA > CB in all possible cases, CB

varies from 10−5 to 0.43 depending on the connector nodes, which reflects the importance of choosingthe adequate connections. Similar results (not shown here) were obtained for other network models.

It is remarkable that the expression of uT,1 can be approximated, up to first order, to a linearcombination of uA,1 and uB,1 (because the terms k > 1 in Eq. S46a are less relevant and mainly affectthe connector nodes). These facts yield that the distribution of centrality over the different nodes of eachnetwork after the connection is, to some extent, proportional to that obtained before the connection. Thatis, the competition influences drastically the total centrality that at the end remains in each network,but as the internal topology of each network is not strongly disturbed by the new links, as far as theyare not too many and the networks are sufficiently large, the capacity of each network to distribute itscentrality between its own nodes is maintained.

2.3 Summary of successful strategies in competing networks

In summary, the results obtained allow us to develop a general framework of strategies to follow by thecompetitors A and B (with λA,1 > λB,1) in order to obtain as much centrality as possible. If we definethe strong (weak) network as the one with higher (lower) λ1:

1. Connecting the most central nodes of two networks optimizes centrality of the weak network andminimizes the competition time.

2. Connecting the most peripheral nodes of two networks optimizes centrality of the strong networkand shows large competition times.

3. Increasing the number of connector links reinforces centrality of the weak network.

4. Any variation that increases the maximum eigenvalue associated to a network (e.g. suitable internalrewiring, addition of nodes or links, etc.) increases its centrality.

Finally, we have seen that, unless λA,1 and λB,1 are almost equal, the network with a higher eigenvalueassociated always obtains a higher centrality. This enables us to call strong network to the one with thelarger first eigenvalue, and weak network to the other one.

From all above, we stress that the goal of each competitor is not really to overcome the adversary,but to obtain the optimum outcome (eigenvector centrality in this case) according to the characteristicsof the competition.




12

3 Application of the results to generic competing networks

Although for clarity we have focused on the competition between two undirected networks, the method-ology presented here can be applied to competing dynamics in generic networks. In particular, we sketchthe generalization for directed networks and the case of N > 2 competing networks.

3.1 Competition between two directed networks

The calculation presented in section 2.1 of the Supplementary Information to obtain the eigenvector uT,1,the eigenvalue λT,1, and the competition time tc,T for two interconnected undirected networks, can begeneralized to two directed networks A and B as follows. We must take into account that in this casethe transition matrices MA, MB, MAB and MT are not symmetric. As a consequence, the left uL

k andright ul eigenvectors of these matrices do not coincide but are biorthonormal, that is, uL

k ·ul = 1, ∀ k = l,and uL

k ·ul = 0, ∀ k = l. By following the same steps as in the undirected case, and multiplying by uLAB,k

from the left when we multiplied by uAB,k from the left in section 2.1, we obtain

uT,1 = uA,1 + ε

NB∑k=1

dkuB,k + o(ε2) , (49a)

λT,1 = λA,1 + ε2NB∑k=1

(uLA,1PuB,k)dk + o(ε3) , (49b)

where dk = (uLB,kPuA,1)/(λA,1 − λB,k), and

tc,T ∝(ln

λT,1

λT,2

)−1

∼(ln

λA,1

λB,1

)−1 [1 + ε2

( NA∑k=1

(uLB,1PuA,k)(u

LA,kPuB,1)

λB,1(λB,1 − λA,k)

−NB∑k=1

(uLA,1PuB,k)(u

LB,kPuA,1)


)]+ o(ε4) . (50)

As we can easily check, when we impose the transition matrices to be symmetric, that is, whenwe make the left and right eigenvectors become equal, Eqs. S49a, S49b and S50 yield the results forundirected networks presented in Eqs. S42a, S42b and S45. Please note that the final outcome of thecompetition is given by the size of dk, and its most relevant term d1 depends exclusively on the righteigenvector associated to the strong network uA,1 and the left eigenvector associated to the weak networkuLB,1. Furthermore, to first order only the connector links that go from the strong network to the weak

network are important. If we call central (peripheral) nodes the nodes with high (low) centrality measuredas the right eigenvectors of the transition matrix, and left-central (left-peripheral) nodes the nodes withhigh (low) centrality measured as the left eigenvectors of the transition matrix, the successful strategiesbecome

1. Connecting the most central nodes of the strong network to the most left-central nodes of the weaknetwork optimizes centrality of the weak network and minimizes the competition time.

2. Connecting the most peripheral nodes of the strong network to the most left-peripheral nodes ofthe weak network optimizes centrality of the strong network and shows large competition times.




13

3. Increasing the number of connector links from the strong to the weak network reinforces centralityof the weak network.

4. Any variation that increases the maximum eigenvalue associated to a network (e.g. suitable internalrewiring, addition of nodes or links, etc.) increases its centrality.

Figure S2 shows an example of a real competition between two directed networks. It represents thelinks between US political blogs in the year 2004 [12]. Nodes (blogs) are classified as liberal (blue) andconservative (red), the former belonging to network A and the latter to network B. Since links in a webpage only have outgoing directions, the whole network is directed, showing a strong interactions (highnumber links) inside each network, but also between networks. The fact that both networks are connectedhas crucial implications in, for example, Internet navigation between blogs. The liberal community hasa larger first eigenvalue associated of λA

1 = 33.87, and the conservative community has λB1 = 26.75. As

a consequence, the liberal community benefits from λA1 > λB

1 and acquires a centrality of CA = 0.743(CB = 0.257). Note that an adequate linking strategy between blogs, based of the basic rules explainedabove, may crucially affect the centrality retained by each network.

Figure S2. Complex network of the US political blogosphere in the year 2004 (see Ref. [12]for details). A giant connected component of N = 1490 political blogs is divided into two communities:blue nodes (NA = 758) represent those blogs classified as liberal, while red nodes (NB = 732) areconservative blogs. The liberal community has a larger first eigenvalue of λA

1 = 33.87, being λB1 = 26.75

in the case of the conservative community. We can see how the liberal community benefits fromλA1 > λB

1 and takes the 74.3% of the whole network centrality (CA = 0.743 and CB = 0.257).




14

3.2 Competition between N networks

The competition between N > 2 networks is a very rich phenomenon that deserves a detailed study farbeyond the scope of this work. However, we can present here a simplified example to show that the mainhighlights of the methodology presented in this paper are also applicable to the more complex systemof N networks. Let us suppose that a network C is connected with a network AB that consists of twonetworks A and B joined by several connector links, and λA,1 > λB,1 > λC,1. The expression for theeigenvector centrality and the first eigenvalue associated to network AB follow Eqs. S49a and S49b,where we suppose ε = 1:

uAB,1 = uA,1 +

NB∑k=1

uLB,kPuA,1

λA,1 − λB,k· uB,k + o(ε) , (51a)

λAB,1 = λA,1 +

NB∑k=1

(uLA,1PuB,k)(u

LB,kPuA,1)

λA,1 − λB,k+ o(ε) . (51b)

If we study the connection of network C to network AB as a perturbation, in the way already shownin former calculations, we obtain

uABC,1 = uAB,1 + ε

NC∑l=1

uLC,lPuAB,1

λAB,1 − λC,l· uC,l + o(ε2) . (52)

Substituting Eqs. S51a and S51b in Eq. S52, yields

uABC,1 = uA,1 +

NB∑k=1

bkuB,k + ε

NC∑l=1

cluC,l + o(ε2)

= uA,1 +

NB∑k=1

uLB,kPuA,1

λA,1 − λB,k· uB,k

+ ε

NC∑l=1

[uLC,lPuA,1

λAB,1 − λC,l+

NB∑k=1

(uLB,kPuA,1)(u

LC,lPuB,k)

(λAB,1 − λC,l)(λAB,1 − λB,k)

]· uC,l + o(ε2) . (53)

The terms b1 and c1 of the summations are the most relevant ones because λA,1 −λB,k > λA,1 −λB,1 fork > 1, λAB,1 − λB,1 > λAB,1 − λB,k for k > 1, and λAB,1 − λC,1 > λAB,1 − λC,l for l > 1. Maintainingonly these terms, and approximating λAB,1 ∼ λA,1, we finally obtain

uABC,1 ∼ uA,1 + b1uB,1 + εc1uC,1

= uA,1 +uLB,1PuA,1

λA,1 − λB,1· uB,1

+ ε

[uLC,1PuA,1

λA,1 − λC,1+

(uLB,1PuA,1)(u

LC,1PuB,1)

(λA,1 − λC,1)(λA,1 − λB,1)

]· uC,1 . (54)

Regarding Eq. S54, we see that, in order to optimize its final centrality, network C must maximize c1.The first term of c1 is the most relevant one, and reflects the connection of network C with the strongestnetwork A. The second term of c1 is due to the connection between C and B, and is negligible unlessthe connections between A and B were favourable for the latter (remember that λA,1 > λB,1 > λC,1). Insummary, analysing c1 as we already did in the case of the competition between two networks, it is clear




15

Figure S3. RNA neutral network corresponding to the sequences that fold into thestructure .(.(....).). -where the symbol ‘.’ represents unpaired nucleotides and ‘(’ or ‘)’ representpaired nucleotides-. The size of each node is proportional to the number of molecules sharing such agenotype when the mutation-selection equilibrium state has been reached. Sub-network A (green nodes)contains sequences whose first nucleotide of the primary structure is adenine, in sub-network B (rednodes) genotypes begin with guanine and in sub-network C with uracil. The number of differentgenotypes (nodes) per sub-network are NA = 61, NB = 62 and NC = 16, while λ1,A = 7.90,λ1,B = 7.79, and λ1,C = 6.00. In this particular case, while two very similar sub-networks accumulatemost of the population (CA = 0.45, and CB = 0.44), a much smaller sub-network C withλ1,C << λ1,B ∼ λ1,A is also well represented (CB = 0.11) because of the high number of connectionsfrom C to sub-networks A and B, and also between sub-networks A and B (see Eq. S54).

that network C must follow the strategies already explained in section 2.3 for undirected networks andsection 3.1 for directed networks: connecting the central nodes of A and B with those of C will increasethe centrality of C, while peripheral connections will dramatically diminish the centrality of C.

This calculation can be repeated recursively to include more networks to the system, but the resultsare similar: the idea behind this preliminary study is the case of a competition between N networkswhere each network must face the rest as a unique opponent, and follow the already explained successfulstrategies for competition between two networks. However, note that when several networks are ableto choose their connections with other networks at the same time, the problem becomes extraordinarilycomplex, as social behaviors such as cooperation or defection between the different competitors can leadto strikingly different results. This promising subject deserves a detailed study under the focus of GameTheory.

Finally, it is worth noticing that when N networks are present in the contest, each network i has adifferent competition parameter Ωi ∈ [−1, 1]. Following the same methodology just presented, each Ωi

can be calculated following the definition obtained for N=2 networks (Eq. S64) but assuming that the




16

rest of the N − 1 networks are a unique opponent.A clarifying application of a competition between N > 2 undirected networks is that of the dynamics

of a population of RNA molecules evolving through point mutations on a neutral network (see Fig. S3).The definition of an RNA neutral network and the equations describing the evolution of the populationof RNA sequences are given in Section S1.3. Due to a mutational process, a given genotype -or node-can accumulate a high number of RNA molecules in the mutation-selection equilibrium state, whileothers may be underpopulated. RNA neutral networks are very modular, and it is usual to identify sub-networks which differ either in the folding energy, in the nature of the base-pairs, or in key nucleotides incertain positions of the sequence [3]. Figure S3 shows an example of a small RNA neutral network madeof sequences of 12 nucleotides, with the particularity that three sub-networks are clearly recognizable.With the methodology proposed in this paper, it is possible to understand how the sub-networks havecompeted for acquiring the highest number of molecules when the equilibrium state has been reached, andto identify which sub-network is taking advantage from the distribution of connector links. This is justan example, but a general study of this phenomenon could yield key information about the still widelyunknown genotype-phenotype relationship, not only in RNA networks, but also in metabolic networksand regulatory circuits [13].




17

4 Study of competition on large networks through exactly solv-able examples

The analytical results obtained for the eigenvector uT,1, the eigenvalue λT,1, and the competition timetc,T associated to a network T formed by two connected networks were sketched in the main text of thispaper and fully developed in Eqs. S46 and S47. While they are of general application, they do not yieldqualitative information about how the system behaves for large network sizes. Since many of the realnetworks in nature and society have a very high number of nodes, this collateral subject deserves a carefulanalysis. For this purpose, we make use of the simplest system that shows sufficient degree heterogeneityto present all connecting strategies shown in Fig. 1a of the main text: two star networks connected byone link. Furthermore, we take advantage from the fact that this system is analytically solvable in twoparticular cases: the competition between one star network of m nodes versus one of (a) 3 nodes and (b)m− 1 nodes.

Let us consider a star network A formed by m nodes, 1 central node with connectivity m − 1 andm − 1 peripheral nodes with connectivity 1, and a star network B formed by mB nodes. To study theinfluence of the network size, we fix mB and let m → ∞. CA(mA,mB) and CB(mA,mB) are the fractionof centralities when two networks of mA and mB nodes are connected. Before connection, the adjacencymatrix of an isolated star of m nodes has eigenvalues

γi = √m− 1, 0, 0, 0, . . . , 0, 0,−

√m− 1 . (55)

If the transition matrix is Eq. S8, its eigenvalues λi are obtained from

λi = (R− µ) +µ

Sγi , (56)

and the eigenvector associated to the maximum eigenvalue is

uA,1 = (2m− 2)−1/2(√m− 1, 1, 1, 1, . . . , 1, 1) . (57)

The outcome of the competition of a star A of m nodes versus (i) a star B of mB = 100 nodes is shownin Fig. 2a of the main text, and versus (ii) a star B of mB = 3 nodes is shown in Fig. S4. The lattercase was chosen because it is analytically solvable: Table S1 presents the first and second eigenvalues ofthe adjacency matrix associated to the total network T depending on the four possible strategies CC,CP , PC and PP , while Table S2 presents the corresponding network centralities CA(m, 3) and CB(m, 3).We can see both in the analytical expressions and in Fig. S4a that CA → 1 when m → ∞ for all fourstrategies, as expected. Furthermore, it verifies that

CPPA > CPC

A > CCPA > CCC

A for m > 5 ,

λCCT,1 > λCP

T,1 > λPCT,1 > λPP

T,1 for m > 3 ,

tPCc,T > tPP

c,T > tCPc,T > tCC

c,T for m > 3 . (58a)

for the centrality, the maximum eigenvalue, and the competition time respectively. All these relationsare in full agreement with the analytical results summarized in Eqs. S46 and S47.

There are two facts that worth a deeper study for larger sizes of the networks. The first one is thecharacterization, depending on the connecting strategy, of the migration of centrality from network Bto network A when the latter increases its size and, as a consequence, its first eigenvalue. Particularly,we focus on how the network size is reinforcing the role played by the connecting strategies. In Fig. 2a(inset) of the main text we measure the gain of centrality ∆CA in network A when its size grows fromm = mB − 1 to m = mB + 1,

∆CA = CA(mB + 1,mB)− CA(mB − 1,mB) , (59)




18

10 100m

0,4

0,6

0,8

1

CA

10 100m

1,99

1,99005

10 100m

1e+05

2e+05t c,

T

λ T,1,λ

T,2

(a) (b)

(c)

Figure S4. Outcome of the competition between a star A of m nodes and a star B ofmB = 3 connected by one link. The four possible strategies are plotted: central-central CC (blackcurve), central-peripheral CP (red dashed curve), peripheral-central PC (green dashed-dotted curve),and peripheral-peripheral PP (blue curve). The transition matrix used is Eq. S8 and the parametervalues are R = 2, µ = 0.01, and S = 500. Note that all curves plotted in this figure were obtainedanalytically. (a) Centrality CA of network A. (b) First and second eigenvalues λT,1 and λT,2 for thetotal network T . (c) Competition time tc,T obtained from Eq. S7.

for increasing values of mB . This quantity can be obtained explicitly, because the value ∆CA can beexpressed in terms of solutions of the system formed by one star A of m nodes versus one star B of m−1nodes, which is analytically solvable ∀m -see Tables S3 and S4-, as follows:

∆CA = CA(mB + 1,mB)− CA(mB − 1,mB)

= CA(mB + 1,mB)− CB(mB ,mB − 1)

= CA(mB + 1,mB) + CA(mB ,mB − 1)− 1 . (60a)

Including the values shown in Tables S3 and S4 in the former expression, we obtain explicitly the valuesof ∆CA plotted in Fig. 2a (inset) of the main text. Finally, developing the expressions obtained for ∆CA

in powers of m, we obtain

∆CCCA ∼ (1/4)m

−1/2B + (3/8)m−1

B − o(m−3/2B ) → 0 when mB → ∞ , (61a)

∆CNNA ∼ 1− 2m

−1/2B + 2m−1

B − o(m−3/2B ) → 1 when mB → ∞ , (61b)

as can be checked in Fig. 2a (inset) of the main text. As a result, the nature of the migration of centralityfor the large network limit is totally different depending on how the networks are connected: connectingthe central nodes of both networks leads to a smooth transition of centrality from network B to networkA, and therefore the weak network (that of lower λ1) can maintain a reasonable fraction of the totalcentrality even when it is considerably smaller than the strong network. On the contrary, connecting twoperipheral nodes converts the competition into a winner-takes-all process when networks reach a certainsize. Interestingly, it would become a phase transition in the limit mB → ∞.




19

The second relevant quantity to analyze is how the CC and PP strategies affect the competitiontime tc,T when networks increase their size. The competition time is maximum when the size of bothnetworks is equal, that is for m = mB , independently of the strategy, coinciding with a minimum in thedistance between the two highest eigenvalues of the system (see Eq. S7). For m = mB , we can see thatγC,1−γC,2 ∼ 1/mB → 0 when m → ∞ for PP , while γC,1−γC,2 → 1 for CC. From this fact, and takinginto account that γC,2 ∼

√m for m ≤ mB , we develop Eq. S7 in powers of m making use of Eq. S8 and

obtain for m = mB that

tCCc,T ∝

√m+ S(

R

µ− 1)− 1/2 + o(m−1/2) , (62a)

tPPc,T ∝ m3/2 + S(

R

µ− 1)m+ 1/2 + o(m−3/2) . (62b)

Therefore, for two competing star networks of the same size m,

tPPc,T

tCCc,T

∼ m → ∞ when m → ∞ , (63)

that is, the competition time for the PP strategy is much larger than that of the CC strategy, as it canbe observed in Fig. S4c, as well as in Figs. 4a-b-c of the main text. Finally, note that neither tPP

c,T nor tCCc,T

are strongly dependent to first order of the explicit form of MT . Furthermore, although the functionaldependence of the competition time with the size of a network varies with its topology [2], extensivenumerical verifications with different types of networks yield that the two main results obtained in thissection for our simple system are qualitatively general. A good example of the ubiquitous applicability ofsuch results is the striking similarity between the curves of centrality and competition time respectivelyplotted in Fig. 2a and Fig. 4a of the main text for the competition between two star networks, and theones plotted in Figs. 2b-c and Figs. 4b-c for networks of different structures and sizes.

Connecting strategy γ1, γ2

CC√m/2 + 1± 1/2

√m2 − 4m+ 12

CP√m/2 + 1± 1/2

√m2 − 4m+ 8

PC√m/2 + 1± 1/2

√m2 − 8m+ 20

PP√MaxRoot(x3 − (2 +m)x2 + (3m− 3)x+ (2−m) = 0)√2ndRoot(x3 − (2 +m)x2 + (3m− 3)x+ (2−m) = 0)

Table S1. Maximum eigenvalue γ1 and second eigenvalue γ2 of the adjacency matrix for thesystem formed by one star of m nodes and one star of 3 nodes connected by one link. Fourpossible connecting strategies: CC (central node of network A with central node of network B), CP(central node of network A with peripheral node of network B), PC (peripheral node of network A withcentral node of network B), and PP (peripheral node of network A with peripheral node of network B).




20

Connecting strategy α β

CC γ21,CC + (m− 1)γ1,CC − 2 + (2− 2m)/γ1,CC γ1,CC + 2

CP γ31,CP + (m− 1)γ2

1,CP − 2γ1,CP + (2− 2m) γ21,CP + γ1,CP

PC γ31,PC + (m− 1)γ2

1,PC − 3γ1,PC + 4− 3m γ1,PC + 2

PP γ41,PP + (m− 1)γ3

1,PP − 3γ21,PP + (4− 3m)γ1,PP + 1 + (m− 2)/γ1,PP γ2

1,PP + γ1,PP

Table S2. Centrality of network A and centrality of network B for the system formed byone star of m nodes and one star of 3 nodes connected by one link. The expressions that yieldthe centralities are CA = α/(α+ β) and CB = 1− CA = β/(α+ β). For simplicity, some are written asfunctions of the eigenvalues printed in Table S1. Connecting strategies are defined in the caption ofTable S1.

Connecting strategy γ1 γ2

CC√m− 1 +

√m− 1

√m− 1−

√m− 1

CP√m− 1 +

√2

√m− 1−

√2

PC√m

√m− 2

PP√m/2 + 1/2

√m2 − 4m+ 12

√m− 2

Table S3. Maximum eigenvalue γ1 and second eigenvalue γ2 of the adjacency matrix forthe system formed by one star of m nodes and one star of m− 1 nodes connected by onelink. Connecting strategies are defined in the caption of Table S1.

Connecting strategy α β

CC√m− 1γ1,CC +

√m− 1 + 1 γ1,CC +m− 2

CP γ21,CP +

√3 + 2

√2γ1,CP +

√2(m− 2) γ2

1,CP + γ1,CP

PC m+√m m+

√m− 2

PP γ31,PP + γ2

1,PP γ21,PP + γ1,PP

Table S4. Centrality of network A and centrality of network B for the system formed byone star of m nodes and one star of m− 1 nodes connected by one link. The expressions thatyield the centralities are CA = α/(α+ β) and CB = 1− CA = β/(α+ β). For simplicity, some arewritten as functions of the eigenvalues printed in Table S3. Connecting strategies are defined in thecaption of Table S1.




21

5 Competition strategies based on increasing the maximum eigen-value associated to the competing networks

As we have already seen, the successful strategies for a competing network are based on (i) choosingappropriate connector links with the opponent, or (ii) increasing its associated first eigenvalue λ1 as muchas possible. In a number of situations the competing networks are not allowed to choose or modify thespecific connections between them. This could be the case of, for example, spatial networks where nodesclose to the boundaries are the connectors, technological networks with nodes specialized in connectingdifferent communities inside the network, or simply networks that are reluctant to modify its connectornodes. In this section, we will focus on the strategy of increasing the first eigenvalue via two efficientmethods: the addition of nodes or links to the networks, and the local rewiring of the existing links.

5.1 Successful strategies based on increasing the number of nodes or links

An important fact when competing networks are able to gain nodes from external sources (belonging ornot to other competitors), or increasing the number of internal links, is how to place them in order tooptimize the final outcome in the fight for centrality. From Eq. S42b (undirected networks) and Eq. S49b(directed networks), it is straightforward to see that the way to maximize the first eigenvalue associatedto network A when one node is added is to connect it to the node with the highest centrality (uA,1)i inthe undirected case and (uA,1)i · (uL

A,1)i in the directed case. Furthermore, adding internal links will beoptimum when we connect two nodes i and j of A such that the product of centralities (uA,1)i · (uA,1)j ismaximum in the undirected case and (uA,1)i · (uL

A,1)i · (uA,1)j · (uLA,1)j is maximum in the directed case.

These results coincide with the influence of nodes and links in the eigenvalue associated to a networkalready studied in [8,14], but we present them in this context for completeness. In summary, the strategiesto optimize the final outcome of the competition when the networks allow gaining nodes become:

1. Connecting external nodes to the most central/left-central nodes of one network optimizes themaximum eigenvalue associated to it and therefore its final centrality in the competition.

2. Erasing very central/left-central nodes of one network maximally diminishes the maximum eigen-value associated to it and therefore its final centrality in the competition.

3. As a consequence of the former two strategies, when networks are allowed to steal nodes from theircompetitors, the optimum strategy is to obtain the most central/left-central nodes of the competitorand connect them to the own central/left-central nodes.

Finally, the strategies to optimize the final outcome when the networks are allowed to gain internallinks become:

1. Connecting the most central/left-central nodes of one network with internal links optimizes themaximum eigenvalue associated to it and therefore its final centrality in the competition.

2. Erasing the links that connect the most central/left-central nodes of one network maximally dimin-ishes the maximum eigenvalue associated to it and therefore its final centrality in the competition.

5.2 Successful strategies based on local network rewiring

Under many real conditions, increasing the number of nodes or links is not possible, and therefore theonly option for a network to overcome the other is to reorganize its internal structure.

Figures 2b-c and 4b-c of the main text show the results of a competition for centrality between twonetworks A and B, where the weak network (B) is allowed to reorganize by rewiring its internal links.




22

(a) (b)

(c) (d)

Figure S5. Final state of network B after following a rewiring process that promotes theenhancement of its associated first eigenvalue λB,1. Network A (not shown here) has beengenerated by a Barabasi-Albert algorithm [11], leading to a network with λA,1 = 6.764. Network B hasfour different initial structures: regular (a), small-world (b), random (c) and scale-free (d). In all casesnetwork B has N = 200 nodes and L = 400 links, the same as network A. Both networks are initiallyisolated and, then, connected with a unique link that joins those nodes with the highest centrality (CCstrategy). Next, network B follows an internal rewiring process that increases its associated firsteigenvalue λB,1 by randomly reshuffling its internal connections. Only permutations that increase λB,1

are accepted. The only restriction is to maintain the network connected.

In all cases, a gain in λB,1 is reflected as an increase of the centrality of the weak network B, whichcan eventually overcome network A when λB,1 > λA,1. To obtain these results, network B undergoes arewiring process with the following steps:

1. We depart from two isolated networks A and B where λA,1 > λB,1.

2. We connect them with a unique link that is placed following any of the four proposed strategies(CC,PP,CP, PC).

3. We let the weak network B to randomly reshuffle one of its internal links.

4. In the case that λB,1 increases by the rewiring, we accept the new configuration.

5. If λB,1 decreases, we discard the rewiring and return to the previous configuration.

6. We repeat the process from step 3.

In all cases, i.e., different connecting strategies or different (initial) network topologies, the competitionfor centrality is ruled by Eq. S42a while the competition time tc,T is given by Eq. S45.

It is worth noting the final structure of network B. We observe that after sufficiently high numberof rewirings, the process leads to the emergence of a “rich-club” formed by few central nodes stronglyconnected between them (see Fig. S5), no matter what the initial structure is. This is, in fact, theconsequence of optimizing λ1 [8, 14] and shows that, in the aim of gaining centrality, a primary strategywould be to link the most influential nodes of each network.




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6 Competition in real networks

Two paradigmatic examples have been analyzed in detail: the Zachary karate club [15] and the bottlenosedolphin community of Doubtful Sound [16]. They both represent interactions inside a social group withtwo well-defined networks A and B inside. Alternative interpretations of centrality in these two real casescould be, apart from the importance of the node/network in the social interactions, the probability ofspreading an idea or rumor in the case of the karate network or the probability of early infection by adisease in the dolphin network.

The competition parameter Ω is

Ω =2(CA − Cmin

A )

CmaxA − Cmin

A

− 1 , (64)

and reflects whether the real structure of connector links is favoring any of the networks (see maintext). This parameter depends on Cmax

A (CmaxB ) and Cmin

A (CminB ), which are the highest and lowest

centrality that could achieve network A (B) by rewiring the L connector links that exist in the realcase. Note that the calculation of Cmax

A and CminA is a problem with an extremely high time complexity,

since all possible connections between two networks of size NA and NB via L links must be checked. ForL << NA, NB , the number of connector links to analyze goes with (NA×NB)

L. Taking into account thatthe estimation of centrality depends on the calculation of the first eigenvector, which has a complexityof O(NA · NB) ∼ O(N2), the search for the optimal configuration for any of the networks goes withO(N2L+2), which makes it extremely dependent on the number of connector links L. Nevertheless, wecan reduce the complexity of the algorithm by designing a suboptimal approach to the search of Cmax

A

and CminA . This can be achieved by computing, sequentially, the optimal configuration for an increasing

number of connector links L′ = 1, ..., L. Once the best configuration for L′ = 1 is found, we fix theconnector link and calculate the best configuration for L′ = 2, and so on. This way, we have to checka number of possible configurations equal to L · N1 · N2 which leads to ∼ O(N2) and reduces the timecomplexity of the competition parameter Ω to O(N4). The results obtained for Ω in the two real casesare summarized in Table S5. Interestingly, we obtain a competition parameter close to zero Ω = −0.015in the case of the Zachary karate club. This fact indicates that none of the networks has taken advantageof an adequate strategy in the distribution of connector links. As we can observe in Fig. S6a, bothnetworks remain equal in centrality in the real case. This indicates that, apart from network importance,the propagation of dynamical processes such as the spreading of an idea/rumor or a disease would havesimilar consequences in both networks. Interestingly, we can also calculate which connector links wouldincrease the importance of any of the two networks up to its highest level (Fig. S6b-c).

Such a balance is not reported in the dolphin network (see Fig. 3 of the main text), where thecompetition parameter reaches a value of Ω = +0.70 and indicates that network A, in addition to itshigher λ1, is much more benefited from the structure of the connector links. This is due to the nonexistenceof social connections between dolphins that are leaders (highest centrality) at their networks, which resultsin the peripheral nature of the connector nodes (see Fig. 3a). As explained above, connecting peripheralnodes is the strategy that benefits the strong network (network A in this case).




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Network NA NB λT,1 λA,1 λB,1 CA CB CmaxA Cmin

B CminA Cmax

B Ω

Karate club 16 18 6.73 5.66 5.66 0.47 0.53 0.76 0.24 0.19 0.81 -0.015

Dolphins 42 20 7.20 7.17 5.87 0.95 0.05 0.99 0.01 0.71 0.29 +0.70

Table S5. Summary of the main parameters of two real networks: the Zachary karate clubnetwork [15] and the dolphin network [16]. Both real networks are, in turn, formed by two welldefined networks A and B. Specifically, the parameters shown in the table are: size of the networks NA

and NB , first eigenvalue associated to the total network (λT,1), network A (λA,1) and network B (λB,1),and centrality of A (CA) and B (CB) in the real distribution of links. Cmax

A and CmaxB are the

centralities obtained by a rewiring process of the connector links that maximizes, respectively, thecentrality of networks A and B (note that Cmax

A,B = 1− CminB,A ). Finally, we calculate the competition

parameter Ω, which indicates which network benefits from the structure of the connector links.

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(c)

Figure S6. Competition in real networks: the case of the Zachary karate club. (a) Realdistribution of centrality in the real network known as the Zachary karate club [15] (see Table S5 fordetails). The shape of the nodes shows the network they belong to: circles for network A and squares fornetwork B. Connections between networks are indicated by sine-shaped links. The size of the nodes isproportional to their centrality. (b) Distribution of centrality where connector links L (the same numberas in the real situation) are redistributed in a way that maximizes centrality of network A. Coloursindicate if nodes are increasing (green) or decreasing (red) their centrality with regard to the real case.(c) Same procedure as in (b) but, in this case, the connector links maximize the centrality of network B.




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