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1 Successful strategies for competing networks Contents 1 Interplay between the eigenvector centrality and the dynamical properties of complex networks Definition of eigenvector centrality Basic properties of dynamical processes on complex networks Dynamics towards equilibrium: the role of the transition matrix, its associated eigenvalues and eigenvectors Competition time Applications to population dynamics: RNA neutral networks Applications to epidemics: The susceptible-infected model Applications to rumor spreading: The Maki-Thompson model Analytical approach to the phenomenology of competing networks and definition of successful strategies Analytical derivation of the eigenvector u T,1, the eigenvalue λ T,1, and the competition time t c,t associated to a network T formed by two connected networks A and B Study of a numerical example: application of the analytical results Summary of successful strategies in competing networks Application of the results to generic competing networks Competition between two directed networks Competition between N networks Study of competition on large networks through exactly solvable examples 17 5 Competition strategies based on increasing the maximum eigenvalue associated to the competing networks Successful strategies based on increasing the number of nodes or links Successful strategies based on local network rewiring Competition in real networks 23 NATURE PHYSICS 1

2 1 Interplay between the eigenvector centrality and the dynamical 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 the node s neighbours by giving a score to the node that is proportional to the scores of all its connected nodes. In this way, the mathematical definition of the eigenvector centrality x k of a node k is related to an iterative process were the centrality is calculated as the sum of the centralities of its neighbours: x k = γ 1 j G kj x j, (1) where γ is a constant, x k is the eigenvector centrality of node k and G kj are the components of the adjacency matrix. In matrix notation Eq. S1 reads γ x = G x so that x can be expressed as a linear combination of the eigenvectors v k of the adjacency matrix G, being γ k the set of the corresponding eigenvalues. Note that Eq. S1 can be regarded as an iterative process that begins at t = 0 with a set of initial conditions x 0. No matter what the values of x 0 are, the value of x k (t) at t will be proportional to the eigenvector v 1 associated to the dominant eigenvalue γ 1. Therefore, the eigenvector centrality is directly obtained from the eigenvector v 1 of the adjacency matrix G. The concept of eigenvector centrality can be easily extended to weighted connectivity matrices, where the weight of the connections w kj is proportional to the amount of interaction between two nodes. In the case of a weighted connectivity matrix W, we have that λ x = W x, and the eigenvector centrality will be, in this case, obtained from eigenvector u 1 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 generic dynamical process that takes place on a complex network (adapted from [2]) Dynamics towards equilibrium: the role of the transition matrix, its associated eigenvalues and eigenvectors A variety of dynamical processes occurring in a network can be mathematically described as n (t + 1) = M n (t), (2) where n (t) is a vector whose components are the state of each node at time t (for example, the population of individuals at each node), and M, with M ij 0, is a transition matrix that contains the peculiarities of the dynamical process. M is a primitive matrix. For this reason, its largest eigenvalue is positive, it verifies that λ 1 > λ i, i>1, and its associated eigenvector is also positive (i.e., all its elements are positive). Therefore, the dynamics becomes m n (t) =M t n (0) = ( n (0) u i )λ t i u i, (3) where n (0) is the initial condition and u i 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 u 1 i=1 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION ( ) n (t) lim t ( n (0) u 1 )λ t = u 1, (4) 1 while its associated eigenvalue λ 1 yields the growth rate at the asymptotic equilibrium. If n (t) is normalized such that n (t) = 1 after each iteration, n (t) u 1 when t. This way, there is a correspondence between the eigenvector centrality and the asymptotic state of the system at equilibrium: both quantities are proportional to the eigenvector associated to the largest eigenvalue of the transition matrix M Competition time The distance (t) to the equilibrium state at time t starting with an initial condition n (0) can be written as (t) = M t n (0) m ( ) t n (0) u i λi ( n (0) u 1 )λ t u 1 = u 1 i n (0) u 1 λ 1. (5) In order to estimate the time to reach the equilibrium state, we fix a tolerance ξ, and define the competition time t c as the number of iterations required for (t c ) <ξ. For generic cases, t c can be extracted from Eq. S5 and approximated by t c i=2 ln n (0) u 2 n (0) u 1 ln ξ ln λ 1 λ 2 (6) because of the exponentially fast suppression of the contributions due to higher-order terms (λ i λ i+1, i). Therefore, we finally obtain t c (ln λ 1 /λ 2 ) 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 weighted networks. However, for the sake of clarity, and without any loss of generality, we suppose the following transition matrix throughout the simulations of the paper: M =(R µ)i + µ S G, (8) where I is the identity matrix and G is the adjacency matrix of a connected graph, whose elements are G ij = 1 if nodes i and j are connected and G ij = 0 otherwise. M describes a process modelling a population that, each time step: a) replicates at each node with a growing rate R>1, b) its daughter individuals leave the node with probability µ, being 0 <µ 1, and c) the probability of remaining alive after leaving a node of degree k i is k i /S [2]. In particular, the competition times shown in Fig. 4 of the main 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 RNA molecules evolving through point mutations on a neutral network. An RNA sequence is composed of the combination of four nucleotides (guanine, adenine, uracil and cytosine) and a neutral network represents all nodes with different primary structures (genotypes) that fold into the same secondary structure (as a proxy for the phenotype). Two nodes, or genotypes, are connected when their sequences differ in only one nucleotide, which allows a molecule to evolve by one point mutation over the space of genotypes. If the mutation pushes the molecule out of the neutral network, it is discarded. In particular, the parameters of Eq. S8 are translated to RNA evolutionary dynamics as follows: R stands for the growing rate of each molecule per generation, µ for the mutation rate and µ/s = µ/(3l) for the probability of mutating to a viable neighbour. Finally, L is the number of nucleotides per sequence [2, 3]. NATURE PHYSICS 3

4 1.4 Applications to epidemics: The susceptible-infected model In the framework of epidemic dynamics, one example of the dynamical implications of the eigenvector centrality is represented by the susceptible-infected (SI) model [4]. In this model, a disease propagates through a network of individuals whose dynamical state can be either susceptible or infected by the disease. 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 I k (t), where S k (t) = 1 I k (t) is the probability of being susceptible (i.e., not infected). Considering β as the infection rate between susceptible and infected individuals, the probability that a node k becomes infected between times t and t + dt is proportional to the number of neighbours that are already infected β j G kji j, being G the adjacency matrix. Since only susceptible individuals can get infected, the dynamics of S k (t) and I k (t) can be described by a set of N differential equations, N being the total number of individuals (nodes): = βs k G kj I j = βs k G kj (1 S j ), (9) ds k dt di k dt = βs k j G kj I j = β(1 I k ) j j j G kj I j, (10) with S k + I k = 1. If the disease starts at a small number of nodes, in the limit of large system size N and ignoring quadratic terms, Eq. S10 becomes: di k dt = β j G kj I j, (11) which in matrix form reads di dt = βg I, (12) I being a vector of components I k. The temporal evolution of I can be expressed as a linear combination of the eigenvectors u k of the transition matrix, which, in this case, coincides with the adjacency matrix G: N I(t) = a k (t) u k, (13) where u k are the eigenvectors of the λ k eigenvalues of G. Then d I(t) dt = N da k (t) dt Comparing terms multiplying u k we obtain: which has the solution N N u k = βg a k (t) u k = β λ k a k (t) u k, (14) da k dt = βλ k a k, (15) a k (t) =a k (0)e βλ kt. (16) If we substitute Eq. S16 into Eq. S13 we arrive to the final expression on I(t): I(t) = N a k (0)e βλkt u k. (17) 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION Since the largest eigenvalue λ 1 dominates over the others, we can approximate the infected population as: I(t) e βλ kt u 1. (18) Thus, for t the exponential term leads to I 1, i.e. the whole population gets infected at the final state. Nevertheless, for low to intermediate time scales (t << ), the term u 1, i.e. precisely the eigenvector 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 of rumor/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 of connections 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 to the eigenvector centrality. This point is illustrated using the Maki-Thompson (MK) model [6], a variation of the initial model introduced by Daley and Kendal (where similar results are obtained). The MK model considers three different kinds of actors in a population of N individuals: a) ignorants I, who never heard about the rumor, b) spreaders S, who spread the rumor to their contacts and c) stiflers R, who have heard the rumor but decide to stop its diffusion. Three kinds of interactions can be defined when two individuals are in contact: S + I α 2S, (19) indicating that when a spreader meets an ignorant, the ignorant becomes a spreader of the rumor with a probability α; S + S β S + R, (20) saying that when two spreaders meet, one of them realizes that the rumor does not have novelty and becomes a stifler with a probability β; and S + R β 2R, (21) indicating that when a spreader meets a stifler, the former becomes a stifler with a probability β. Any other 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 increment of each population for short time intervals will be: I tαsi/n, (22) S t( αsi N βs2 N βsr ), (23) N R t( βs2 N + βsr ). (24) N 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) NATURE PHYSICS 5

6 6 dy dt = yαx βy2 βy(1 y x) =(α + β)xy βy, (26) dz = βy(1 x). (27) dt We can extend Eqs. S25-S27 into a network of contacts just by considering that interactions between individuals are defined by an adjacency matrix G: dx k dt dy k dt =(α + β)x k = αx k G kj y j, (28) j G kj y j βy k, (29) j dz k dt = β(1 x k) j G kj y j, (30) where now, x k, y k and z k are the probabilities that node k is, respectively, ignorant, spreader or stifler. Assuming that at t = 0 the number of individuals that are prone to transmit the rumor is low, in the limit of large number of nodes N the probability of a node to be a spreader is: dy k dt =(α + β) j G kj y j βy k = j [(α + β)g kj βδ kj ]y j, (31) δ kj being the Kronecker delta. In matrix form, Eq. S31 reads d y dt =(α + β)m y, (32) where M is a transition matrix given by M = G α+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 express y as a linear combination of the eigenvectors u k of M, which are also the eigenvectors of the adjacency matrix G: M u k = G u k β α + β I u k =(κ β α + β ) u k, (33) Note that the eigenvalues of the transition matrix M are related to those of the adjacency matrix as λ M = λ G 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) = β N a k (0)e [(α+β)λk β]t u k. (34) Again Eq. S34 is dominated by the eigenvector associated to the largest eigenvalue λ 1, and we can therefore approximate the probability of being a spreader node as: y (t) e [(α+β)λ 1 β]t u 1. (35) The exponent of the exponential function determines two different dynamical regimes. For (α+β)λ 1 β> 0 the rumor spreads over the whole network, while for (α + β)λ 1 β<0 the rumor stops spreading. In the 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. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION 7 2 Analytical approach to the phenomenology of competing networks and definition of successful strategies 2.1 Analytical derivation of the eigenvector u T,1, the eigenvalue λ T,1, and the competition time t c,t associated to a network T formed by two connected networks A and B Let us start by defining the terminology to be used during the analytical calculations. Networks A and B, of N A and N B nodes and L A and L B links respectively, form the disconnected network AB of N A + N B nodes and L A + L B links. We connect them through L connector links to create a new network T of N T = N A + N B nodes and L T = L A + L B + L links. For simplicity, let us suppose that A and B are weighted but undirected networks (the directed case is studied in section S3.1). The nodes of network A are numbered from i = 1 to N A and the nodes of network B from i = N A+1 to N T = N A + N B. The adjacency matrix G AB is formed by two diagonal blocks corresponding to G A and G B. The relation between the transition matrix M AB, also formed by two blocks, and G AB, depends on the peculiarities of the process. Let us call λ A,i and λ B,i the i eigenvalues of the transition matrices M A and M B respectively. Let us suppose λ A,1 >λ B,1 throughout the paper. Note that the eigenvectors of M A and M B are related to those of M AB as follows: The N A eigenvectors of M A are of length N A, the N B eigenvectors of M B are of length N B, and the eigenvectors of M AB are of length N T. The first i =1,..., N A eigenvectors of M AB are equal to those of M A, being their last N B terms equal to zero. Therefore, λ AB,i = λ A,i for i =1,..., N A. The i = N A +1,..., N T eigenvectors of M AB are equal to those of M B, being their first N A terms equal to zero. Therefore, λ AB,i = λ B,i NA for i = N A +1,..., N T. For simplicity in the following calculations, we call u A,i to the first i =1,..., N A eigenvectors u AB,i, and we call u B,i to the last N B eigenvectors u AB,i+NA where i =1,..., N B. We call {cl} i=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 λ T,i the eigenvalues of network T. The adjacency matrix G T corresponding to the union of networks A and B is therefore formed by G AB with an extra 1 in elements kl {cl} and their symmetric positions lk. In particular, let us express the transition matrix M T as the transition matrix M AB slightly perturbed in the {cl} positions and their symmetric elements. That is, M T = M AB + ɛp where P ij = P ji 0 for ij {cl} and P ij = 0 elsewhere. P ij depends on the explicit form of M. The main idea of the analytical calculations is to describe the total graph T as a perturbation of graph A by graph B, in a way that the weight of the connector links is ɛ << 1 (and, as it is customary, at the end ɛ could be replaced by unity [7]). Therefore, as λ A,1 >λ B,1 by construction, the maximum eigenvalue λ T,1 will be a perturbation of λ A,1 and its associated eigenvector u T,1 will be a perturbation of u A,1. This methodology is inspired on the perturbation theory of matrices shown in [7], and among other examples was recently applied in the context of Complex Network theory to characterize the importance of network nodes and links [8], to analyze the impact of a structural perturbation in the eigenvalues of a network [9], or to the detection of communities [10]. Assuming λ A,1 >λ B,1, we have where M T u T,1 = λ T,1 u T,1 (36) M T = M AB + ɛp, u T,1 = u A,1 + ɛ v 1 + ɛ 2 v 2 + o(ɛ 3 ), λ T,1 = λ A,1 + ɛt 1 + ɛ 2 t 2 + o(ɛ 3 ). (37a) (37b) (37c) Taking into account that (i) u T,1 =1 u A,1 v 1 = 0, and (ii) u A,1 P u A,1 = 0 because ( u A,1 ) i =0 for i>n A, we include Eqs. S37a, S37b and S37c in Eq. S36 and set equal the different terms of such NATURE PHYSICS 7

8 equation according to the orders of ɛ. We obtain for order ɛ u A,1 (M AB v 1 + P u A,1 )= u A,1 (λ A,1 v 1 + t 1 u A,1 ) t 1 =0 (M AB λ A,1 ) v 1 = P u A,1, (38a) (38b) (38c) and for order ɛ 2 u A,1 (M AB v 2 + P v 1 )= u A,1 (λ A,1 v 2 + t 2 u A,1 ) t 2 = u A,1 P v 1. (39a) (39b) The vector v 1 can be obtained numerically solving Eq. S38c. However, it can also be analytically expressed as N T N A N T v 1 = c k u AB,k = c k u A,k + c k u B,k NA. (40) k=n A +1 Including Eq. S40 in S38c, and multiplying both sides by u AB,k from the left, we obtain c k = 0 for 1 <k N A (because u A,k P u A,1 =0 k) and c k = u B,k N A P u A,1 λ A,1 λ B,k NA u A,1 v 1 = 0. All this yields v 1 = for k>n A. We know c 1 = 0 because u A,1 P u B,k λ A,1 λ B,k u B,k, (41) and including Eq. S41 in Eq. S37b, and Eq. S39b in Eq. S37c, we finally obtain an analytical expression for the eigenvector u T,1 and its associated eigenvalue λ T,1 : u A,1 P u B,k u T,1 = u A,1 + ɛ u B,k + o(ɛ 2 ), λ A,1 λ B,k λ T,1 = λ A,1 + ɛ 2 (42a) ( u A,1 P u B,k ) 2 λ A,1 λ B,k + o(ɛ 3 ). (42b) The terms k = 1 of both summations are the most relevant ones because λ A,1 λ B,k >λ A,1 λ B,1 for k>1. Furthermore, note that u A,1 P u B,1 = cl ( u A,1) i P ij ( u B,1 ) j. The competition time is another important dynamical property of a population evolving on a network, which verifies [2] ( t c,t ln λ ) 1 T,1. (43) λ T,2 In order to obtain an analytical expression for λ T,2, the second eigenvalue of M T, we develop a similar calculation to the one dedicated to λ T,1, assuming λ A,1 >λ B,1 >λ A,2. This yields N A λ T,2 = λ B,1 + ɛ 2 ( u A,k P u B,1 ) 2 + o(ɛ 3 ), (44) λ B,1 λ A,k where the two first terms of the summation are the most relevant ones because λ B,1 λ A,k >λ B,1 λ A,2 for k>2. Introducing Eqs. S42b and S44 in Eq. S43, and developing in powers of ɛ, we obtain for the 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION competition time t c,t ( ln λ T,1 λ T,2 ( ln λ A,1 λ B,1 ) 1 ) 1 [ 1+ɛ 2( N A ( u A,1 P u B,k ) 2 λ A,1 (λ A,1 λ B,k ) ( u A,k P u B,1 ) 2 λ B,1 (λ B,1 λ A,k ) )] + o(ɛ 4 ). (45) Finally, let us particularize the present results for the transition matrix shown in Eq. S8, M =(R µ)i +(µ/s)g. {γ i } is the set of eigenvalues of matrix G, γ i γ i+1, and { w i } is the corresponding set of eigenvectors. The eigenvalues of both matrices are related as λ i =(R µ)+ µ S γ i, and it is easy to check that the eigenvectors of the adjacency matrix are also eigenvectors of the transition matrix. Furthermore, P ij = P ji = µ/s for ij {cl} and P ij = 0 elsewhere, from where we obtain that u A,1 P u B,1 = µ S cl ( u A,1) i ( u B,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 a network T formed by two connecting undirected networks A and B, which can be summarized as k=2 u T,1 = u A,1 + ɛ a k u B,k + o(ɛ 2 ), λ T,1 = λ A,1 + ɛ 2 ( u A,1 P u B,k )a k + o(ɛ 3 ), where a k =( u A,1 P u B,k )/(λ A,1 λ B,k ), and t c,t ( ln λ ) 1 [ A,1 1+ɛ 2( (λ A,1 + λ B,1 )( u A,1 P u B,1 ) 2 λ B,1 λ A,1 λ B,1 (λ A,1 λ B,1 ) + N A ( u A,k P u B,1 ) 2 NB λ B,1 (λ B,1 λ A,k ) ( u A,1 P u B,k ) 2 )] + o(ɛ 4 ). λ A,1 (λ A,1 λ B,k ) k=2 (46a) (46b) (47) The eigenvector u T,1 determines the outcome of the competition between networks A and B. Centrality A (C A ) and centrality B (C B ) are the fractions of the total centrality that remain in the nodes of network A and B after the connection, and are obtained as NA i=1 C A = ( u T,1) i NT i=1 ( u, (48a) T,1) i C B =1 C A. (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 a k >a k+1, the final outcome of the competition depends mainly on a 1 : C A 1 when a 1 0, and C B will grow when a 1 grows. Therefore, the main properties involved in the competition are (i) the highest eigenvalues associated to both networks, λ A,1 and λ B,1, NATURE PHYSICS 9

10 10 Figure S1. Competition between two Barabási-Albert networks [11] A and B of N A = N B = 1000 nodes and L A = L B = 2000 links, connected by a unique link in all possible configurations ( ). For each network, nodes are numbered according to its ranking in centrality. In this example, the transition matrix is Eq. S8, where parameters are R = 2, µ =0.01, and S = 20, and the population starts uniformly spread over all nodes. The axes represent the connector links in network A (X-axis) and network B (Y-axis). (a) Centrality A: Fraction of population that ends the competition in network A. (b) Highest eigenvalue associated to the total network T. (c) Competition time t c,t. and (ii) u A,1 P u B,1, a quantity that is proportional to the centralities of the connector nodes when the networks 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, obtained for the perturbed case ɛ << 1, to generic cases where the weight of the connector links ɛ is the same as that of the rest of the links. Figure 1b of the main text and Fig. S1 show a competition between two Barabási-Albert scale-free networks [11] A and B of 1000 nodes and 2000 links each. The centrality of network A follows Eq. S48a, where u T,1 was calculated as the first eigenvector of the transition matrix M T. X and Y-axes represent the connector nodes of A and B, with the node number ordered from higher to 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., u A,1 P u B,1 is maximum and therefore so is a 1 ), network B shows its best results in centrality, λ T,1 has its maximum value, and the competition time t c,t reaches a minimum because the first term of the perturbative part of Eq. S47 prevails over the rest. Second, high centrality in the connector nodes of A combined with low centrality in the connector nodes of B maintains an intermediate value of the eigenvalue λ C,1 due to the secondary terms k>1 in Eqs. S46a and S46b (which depend on u A,1 but not on u B,1 ). In addition, the time t c,t shows relatively low values due to the fact that the third term of the perturbative part of Eq. S47 (which also depends on 10 NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION 11 u A,1 but not on u B,1 ) prevails over the rest. Third, when the connector links join peripheral nodes of both networks, centrality distributes over network A and C A 1, u T,1 u A,1, λ T,1 λ A,1. In this case, the time t c,t reaches the intermediate value of t c,t (ln λ A,1 /λ B,1 ) 1. Interestingly, the competition time reaches its maximum when the connector links join central nodes of network B with peripheral nodes of network A, as the second term of the perturbative part of Eq. S47 prevails over the rest (because it depends on u B,1 but not on u A,1 ). Finally, note that, despite C A >C B in all possible cases, C B varies from 10 5 to 0.43 depending on the connector nodes, which reflects the importance of choosing the adequate connections. Similar results (not shown here) were obtained for other network models. It is remarkable that the expression of u T,1 can be approximated, up to first order, to a linear combination of u A,1 and u B,1 (because the terms k>1 in Eq. S46a are less relevant and mainly affect the connector nodes). These facts yield that the distribution of centrality over the different nodes of each network after the connection is, to some extent, proportional to that obtained before the connection. That is, 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 they are not too many and the networks are sufficiently large, the capacity of each network to distribute its centrality 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 the competitors A and B (with λ A,1 >λ B,1 ) in order to obtain as much centrality as possible. If we define the strong (weak) network as the one with higher (lower) λ 1 : 1. Connecting the most central nodes of two networks optimizes centrality of the weak network and minimizes the competition time. 2. Connecting the most peripheral nodes of two networks optimizes centrality of the strong network and 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 internal rewiring, addition of nodes or links, etc.) increases its centrality. Finally, we have seen that, unless λ A,1 and λ B,1 are almost equal, the network with a higher eigenvalue associated always obtains a higher centrality. This enables us to call strong network to the one with the larger 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 characteristics of the competition. NATURE PHYSICS 11

12 3 Application of the results to generic competing networks Although for clarity we have focused on the competition between two undirected networks, the methodology presented here can be applied to competing dynamics in generic networks. In particular, we sketch the 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 u T,1, the eigenvalue λ T,1, and the competition time t c,t for two interconnected undirected networks, can be generalized to two directed networks A and B as follows. We must take into account that in this case the transition matrices M A, M B, M AB and M T are not symmetric. As a consequence, the left ū L k and right ū l eigenvectors of these matrices do not coincide but are biorthonormal, that is, u L k u l = 1, k = l, and u L k u l = 0, k l. By following the same steps as in the undirected case, and multiplying by u L AB,k from the left when we multiplied by u AB,k from the left in section 2.1, we obtain u T,1 = u A,1 + ɛ d k u B,k + o(ɛ 2 ), λ T,1 = λ A,1 + ɛ 2 ( u L A,1P u B,k )d k + o(ɛ 3 ), (49a) (49b) where d k =( u L B,k P u A,1)/(λ A,1 λ B,k ), and t c,t ( ln λ T,1 λ T,2 ) 1 ( ln λ ) 1 [ A,1 1+ɛ 2( N A ( u L B,1 P u A,k)( u L A,k P u B,1) λ B,1 λ B,1 (λ B,1 λ A,k ) ( u L A,1 P u B,k)( u L B,k P u A,1) λ A,1 (λ A,1 λ B,k ) )] + o(ɛ 4 ). (50) As we can easily check, when we impose the transition matrices to be symmetric, that is, when we make the left and right eigenvectors become equal, Eqs. S49a, S49b and S50 yield the results for undirected networks presented in Eqs. S42a, S42b and S45. Please note that the final outcome of the competition is given by the size of d k, and its most relevant term d 1 depends exclusively on the right eigenvector associated to the strong network u A,1 and the left eigenvector associated to the weak network u L B,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 measured as the right eigenvectors of the transition matrix, and left-central (left-peripheral) nodes the nodes with high (low) centrality measured as the left eigenvectors of the transition matrix, the successful strategies become 1. Connecting the most central nodes of the strong network to the most left-central nodes of the weak network 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 of the weak network optimizes centrality of the strong network and shows large competition times. 12 NATURE PHYSICS

13 SUPPLEMENTARY INFORMATION 3. Increasing the number of connector links from the strong to the weak network reinforces centrality of the weak network. 4. Any variation that increases the maximum eigenvalue associated to a network (e.g. suitable internal rewiring, addition of nodes or links, etc.) increases its centrality. Figure S2 shows an example of a real competition between two directed networks. It represents the links between US political blogs in the year 2004 [12]. Nodes (blogs) are classified as liberal (blue) and conservative (red), the former belonging to network A and the latter to network B. Since links in a web page only have outgoing directions, the whole network is directed, showing a strong interactions (high number links) inside each network, but also between networks. The fact that both networks are connected has crucial implications in, for example, Internet navigation between blogs. The liberal community has a larger first eigenvalue associated of λ A 1 = 33.87, and the conservative community has λ B 1 = As a consequence, the liberal community benefits from λ A 1 >λ B 1 and acquires a centrality of C A =0.743 (C B =0.257). Note that an adequate linking strategy between blogs, based of the basic rules explained above, 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 (N A = 758) represent those blogs classified as liberal, while red nodes (N B = 732) are conservative blogs. The liberal community has a larger first eigenvalue of λ A 1 = 33.87, being λ B 1 = in the case of the conservative community. We can see how the liberal community benefits from λ A 1 >λ B 1 and takes the 74.3% of the whole network centrality (C A =0.743 and C B =0.257). NATURE PHYSICS 13

14 3.2 Competition between N networks The competition between N>2networks is a very rich phenomenon that deserves a detailed study far beyond the scope of this work. However, we can present here a simplified example to show that the main highlights of the methodology presented in this paper are also applicable to the more complex system of N networks. Let us suppose that a network C is connected with a network AB that consists of two networks A and B joined by several connector links, and λ A,1 >λ B,1 >λ C,1. The expression for the eigenvector centrality and the first eigenvalue associated to network AB follow Eqs. S49a and S49b, where we suppose ɛ = 1: u L B,k u AB,1 = u A,1 + P u A,1 u B,k + o(ɛ), λ A,1 λ B,k λ AB,1 = λ A,1 + (51a) ( u L A,1 P u B,k)( u L B,k P u A,1) λ A,1 λ B,k + o(ɛ). (51b) If we study the connection of network C to network AB as a perturbation, in the way already shown in former calculations, we obtain N C u L C,l u ABC,1 = u AB,1 + ɛ P u AB,1 u C,l + o(ɛ 2 ). (52) λ AB,1 λ C,l l=1 Substituting Eqs. S51a and S51b in Eq. S52, yields N C u ABC,1 = u A,1 + b k u B,k + ɛ c l u C,l + o(ɛ 2 ) l=1 u L B,k = u A,1 + P u A,1 u B,k λ A,1 λ B,k [ N C u L C,l P u A,1 ( u L B,k + ɛ + P u A,1)( u L C,l P u ] B,k) u C,l + o(ɛ 2 ). (53) λ AB,1 λ C,l (λ AB,1 λ C,l )(λ AB,1 λ B,k ) l=1 The terms b 1 and c 1 of the summations are the most relevant ones because λ A,1 λ B,k >λ A,1 λ B,1 for k>1, λ AB,1 λ B,1 >λ AB,1 λ B,k for k>1, and λ AB,1 λ C,1 >λ AB,1 λ C,l for l>1. Maintaining only these terms, and approximating λ AB,1 λ A,1, we finally obtain u ABC,1 u A,1 + b 1 u B,1 + ɛc 1 u C,1 = u A,1 + u L B,1 P u A,1 u B,1 λ A,1 λ B,1 [ u L C,1 P u A,1 + ɛ + ( u L B,1 P u A,1)( u L C,1 P u ] B,1) u C,1. (54) λ A,1 λ C,1 (λ A,1 λ C,1 )(λ A,1 λ B,1 ) Regarding Eq. S54, we see that, in order to optimize its final centrality, network C must maximize c 1. The first term of c 1 is the most relevant one, and reflects the connection of network C with the strongest network A. The second term of c 1 is due to the connection between C and B, and is negligible unless the connections between A and B were favourable for the latter (remember that λ A,1 >λ B,1 >λ C,1 ). In summary, analysing c 1 as we already did in the case of the competition between two networks, it is clear 14 NATURE PHYSICS

15 SUPPLEMENTARY INFORMATION 15 Figure S3. RNA neutral network corresponding to the sequences that fold into the structure.(.(...).). -where the symbol. represents unpaired nucleotides and ( or ) represent paired nucleotides-. The size of each node is proportional to the number of molecules sharing such a genotype 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 (red nodes) genotypes begin with guanine and in sub-network C with uracil. The number of different genotypes (nodes) per sub-network are N A = 61, N B = 62 and N C = 16, while λ 1,A =7.90, λ 1,B =7.79, and λ 1,C =6.00. In this particular case, while two very similar sub-networks accumulate most of the population (C A =0.45, and C B =0.44), a much smaller sub-network C with λ 1,C << λ 1,B λ 1,A is also well represented (C B =0.11) because of the high number of connections from 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 and section 3.1 for directed networks: connecting the central nodes of A and B with those of C will increase the 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 results are similar: the idea behind this preliminary study is the case of a competition between N networks where each network must face the rest as a unique opponent, and follow the already explained successful strategies for competition between two networks. However, note that when several networks are able to choose their connections with other networks at the same time, the problem becomes extraordinarily complex, as social behaviors such as cooperation or defection between the different competitors can lead to strikingly different results. This promising subject deserves a detailed study under the focus of Game Theory. Finally, it is worth noticing that when N networks are present in the contest, each network i has a different competition parameter Ω i [ 1, 1]. Following the same methodology just presented, each Ω i can be calculated following the definition obtained for N=2 networks (Eq. S64) but assuming that the NATURE PHYSICS 15

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 population of RNA sequences are given in Section S1.3. Due to a mutational process, a given genotype -or nodecan accumulate a high number of RNA molecules in the mutation-selection equilibrium state, while others may be underpopulated. RNA neutral networks are very modular, and it is usual to identify subnetworks which differ either in the folding energy, in the nature of the base-pairs, or in key nucleotides in certain positions of the sequence [3]. Figure S3 shows an example of a small RNA neutral network made of 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 have competed for acquiring the highest number of molecules when the equilibrium state has been reached, and to identify which sub-network is taking advantage from the distribution of connector links. This is just an example, but a general study of this phenomenon could yield key information about the still widely unknown genotype-phenotype relationship, not only in RNA networks, but also in metabolic networks and regulatory circuits [13]. 16 NATURE PHYSICS

17 SUPPLEMENTARY INFORMATION 4 Study of competition on large networks through exactly solvable examples The analytical results obtained for the eigenvector u T,1, the eigenvalue λ T,1, and the competition time t c,t associated to a network T formed by two connected networks were sketched in the main text of this paper and fully developed in Eqs. S46 and S47. While they are of general application, they do not yield qualitative information about how the system behaves for large network sizes. Since many of the real networks in nature and society have a very high number of nodes, this collateral subject deserves a careful analysis. For this purpose, we make use of the simplest system that shows sufficient degree heterogeneity to present all connecting strategies shown in Fig. 1a of the main text: two star networks connected by one link. Furthermore, we take advantage from the fact that this system is analytically solvable in two particular 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 and m 1 peripheral nodes with connectivity 1, and a star network B formed by m B nodes. To study the influence of the network size, we fix m B and let m. C A (m A,m B ) and C B (m A,m B ) are the fraction of centralities when two networks of m A and m B nodes are connected. Before connection, the adjacency matrix of an isolated star of m nodes has eigenvalues {γ i } = { m 1, 0, 0, 0,...,0, 0, m 1}. (55) If the transition matrix is Eq. S8, its eigenvalues {λ i } are obtained from and the eigenvector associated to the maximum eigenvalue is λ i =(R µ)+ µ S γ i, (56) u A,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 m B = 100 nodes is shown in Fig. 2a of the main text, and versus (ii) a star B of m B = 3 nodes is shown in Fig. S4. The latter case was chosen because it is analytically solvable: Table S1 presents the first and second eigenvalues of the 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 C A (m, 3) and C B (m, 3). We can see both in the analytical expressions and in Fig. S4a that C A 1 when m for all four strategies, as expected. Furthermore, it verifies that C PP A >CA PC >CA CP >CA CC for m>5, λ CC T,1 >λ CP T,1 >λ PC T,1 >λ PP T,1 for m>3, t PC c,t >t PP c,t >t CP c,t >t CC c,t for m>3. (58a) for the centrality, the maximum eigenvalue, and the competition time respectively. All these relations are 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 the characterization, depending on the connecting strategy, of the migration of centrality from network B to 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 C A in network A when its size grows from m = m B 1 to m = m B + 1, C A = C A (m B +1,m B ) C A (m B 1,m B ), (59) NATURE PHYSICS 17

18 1 (a) (b) C A 0,8 0,6 0,4 1,99005 λ T,1, λ T, m 2e+05 1,99 (c) m t c,t 1e m Figure S4. Outcome of the competition between a star A of m nodes and a star B of m B =3connected by one link. The four possible strategies are plotted: central-central CC (black curve), 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 parameter values are R = 2, µ =0.01, and S = 500. Note that all curves plotted in this figure were obtained analytically. (a) Centrality C A of network A. (b) First and second eigenvalues λ T,1 and λ T,2 for the total network T. (c) Competition time t c,t obtained from Eq. S7. for increasing values of m B. This quantity can be obtained explicitly, because the value C A can be expressed in terms of solutions of the system formed by one star A of m nodes versus one star B of m 1 nodes, which is analytically solvable m -see Tables S3 and S4-, as follows: C A = C A (m B +1,m B ) C A (m B 1,m B ) = C A (m B +1,m B ) C B (m B,m B 1) = C A (m B +1,m B )+C A (m B,m B 1) 1. (60a) Including the values shown in Tables S3 and S4 in the former expression, we obtain explicitly the values of C A plotted in Fig. 2a (inset) of the main text. Finally, developing the expressions obtained for C A in powers of m, we obtain CA CC (1/4)m 1/2 B + (3/8)m 1 B o(m 3/2 B ) 0 when m B, (61a) CA NN 1 2m 1/2 B +2m 1 B o(m 3/2 B ) 1 when m B, (61b) as can be checked in Fig. 2a (inset) of the main text. As a result, the nature of the migration of centrality for the large network limit is totally different depending on how the networks are connected: connecting the central nodes of both networks leads to a smooth transition of centrality from network B to network A, and therefore the weak network (that of lower λ 1 ) can maintain a reasonable fraction of the total centrality even when it is considerably smaller than the strong network. On the contrary, connecting two peripheral nodes converts the competition into a winner-takes-all process when networks reach a certain size. Interestingly, it would become a phase transition in the limit m B. 18 NATURE PHYSICS

19 SUPPLEMENTARY INFORMATION The second relevant quantity to analyze is how the CC and PP strategies affect the competition time t c,t when networks increase their size. The competition time is maximum when the size of both networks is equal, that is for m = m B, independently of the strategy, coinciding with a minimum in the distance between the two highest eigenvalues of the system (see Eq. S7). For m = m B, we can see that γ C,1 γ C,2 1/m B 0 when m for PP, while γ C,1 γ C,2 1 for CC. From this fact, and taking into account that γ C,2 m for m m B, we develop Eq. S7 in powers of m making use of Eq. S8 and obtain for m = m B that t CC c,t m + S( R µ 1) 1/2+o(m 1/2 ), t PP c,t m 3/2 + S( R µ 1)m +1/2+o(m 3/2 ). (62a) (62b) Therefore, for two competing star networks of the same size m, t PP c,t t CC c,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 can be observed in Fig. S4c, as well as in Figs. 4a-b-c of the main text. Finally, note that neither t PP c,t nor tcc c,t are strongly dependent to first order of the explicit form of M T. Furthermore, although the functional dependence of the competition time with the size of a network varies with its topology [2], extensive numerical verifications with different types of networks yield that the two main results obtained in this section for our simple system are qualitatively general. A good example of the ubiquitous applicability of such results is the striking similarity between the curves of centrality and competition time respectively plotted in Fig. 2a and Fig. 4a of the main text for the competition between two star networks, and the ones plotted in Figs. 2b-c and Figs. 4b-c for networks of different structures and sizes. Connecting strategy γ 1,γ 2 CC m/2+1± 1/2 m 2 4m + 12 CP m/2+1± 1/2 m 2 4m +8 PC m/2+1± 1/2 m 2 8m + 20 PP MaxRoot(x3 (2 + m)x 2 + (3m 3)x + (2 m) = 0) 2ndRoot(x3 (2 + m)x 2 + (3m 3)x + (2 m) = 0) Table S1. Maximum eigenvalue γ 1 and second eigenvalue γ 2 of the adjacency matrix for the system formed by one star of m nodes and one star of 3 nodes connected by one link. Four possible 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 with central node of network B), and PP (peripheral node of network A with peripheral node of network B). NATURE PHYSICS 19

20 Connecting strategy α β CC γ1,cc 2 +(m 1)γ 1,CC 2 + (2 2m)/γ 1,CC γ 1,CC +2 CP γ1,cp 3 +(m 1)γ2 1,CP 2γ 1,CP + (2 2m) γ1,cp 2 + γ 1,CP PC γ1,p 3 C +(m 1)γ2 1,P C 3γ 1,P C +4 3m γ 1,P C +2 PP γ1,p 4 P +(m 1)γ3 1,P P 3γ2 1,P P + (4 3m)γ 1,P P +1+(m 2)/γ 1,P P γ1,p 2 P + γ 1,P P Table S2. Centrality of network A and centrality of network B for the system formed by one star of m nodes and one star of 3 nodes connected by one link. The expressions that yield the centralities are C A = α/(α + β) and C B =1 C A = β/(α + β). For simplicity, some are written as functions of the eigenvalues printed in Table S1. Connecting strategies are defined in the caption of Table S1. Connecting strategy γ 1 γ 2 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 m 2 4m + 12 m 2 Table S3. Maximum eigenvalue γ 1 and second eigenvalue γ 2 of the adjacency matrix for the system formed by one star of m nodes and one star of m 1 nodes connected by one link. Connecting strategies are defined in the caption of Table S1. Connecting strategy α β CC m 1γ1,CC + m 1+1 γ 1,CC + m 2 CP γ1,cp γ 1,CP + 2(m 2) γ1,cp 2 + γ 1,CP PC m + m m + m 2 PP γ1,p 3 P + γ2 1,P P γ1,p 2 P + γ 1,P P Table S4. Centrality of network A and centrality of network B for the system formed by one star of m nodes and one star of m 1 nodes connected by one link. The expressions that yield the centralities are C A = α/(α + β) and C B =1 C A = β/(α + β). For simplicity, some are written as functions of the eigenvalues printed in Table S3. Connecting strategies are defined in the caption of Table S1. 20 NATURE PHYSICS

21 SUPPLEMENTARY INFORMATION 5 Competition strategies based on increasing the maximum eigenvalue associated to the competing networks As we have already seen, the successful strategies for a competing network are based on (i) choosing appropriate connector links with the opponent, or (ii) increasing its associated first eigenvalue λ 1 as much as possible. In a number of situations the competing networks are not allowed to choose or modify the specific connections between them. This could be the case of, for example, spatial networks where nodes close to the boundaries are the connectors, technological networks with nodes specialized in connecting different communities inside the network, or simply networks that are reluctant to modify its connector nodes. In this section, we will focus on the strategy of increasing the first eigenvalue via two efficient methods: 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 or not to other competitors), or increasing the number of internal links, is how to place them in order to optimize 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 associated to network A when one node is added is to connect it to the node with the highest centrality ( u A,1 ) i in the undirected case and ( u A,1 ) i ( u L A,1 ) i in the directed case. Furthermore, adding internal links will be optimum when we connect two nodes i and j of A such that the product of centralities ( u A,1 ) i ( u A,1 ) j is maximum in the undirected case and ( u A,1 ) i ( u L A,1 ) i ( u A,1 ) j ( u L A,1 ) j is maximum in the directed case. These results coincide with the influence of nodes and links in the eigenvalue associated to a network already studied in [8,14], but we present them in this context for completeness. In summary, the strategies to 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 the maximum 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 eigenvalue 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 their competitors, the optimum strategy is to obtain the most central/left-central nodes of the competitor and connect them to the own central/left-central nodes. Finally, the strategies to optimize the final outcome when the networks are allowed to gain internal links become: 1. Connecting the most central/left-central nodes of one network with internal links optimizes the maximum 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 diminishes 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 the only 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 two networks A and B, where the weak network (B) is allowed to reorganize by rewiring its internal links. NATURE PHYSICS 21