Abstract
The unpreventable connections between real networked systems have recently called for an examination of percolation, diffusion or synchronization phenomena in multilayer networks. Here we use network science and game theory to explore interactions in networksofnetworks and model these as a game for gaining importance. We propose a viewpoint where networks choose the connection strategies, in contrast with classical approaches where nodes are the active players. Specifically, we investigate how creating paths between networks leads to different Nash equilibria that determine their structural and dynamical properties. In a wide variety of cases, selecting adequate connections leads to a cooperative solution that allows weak networks to overcome the strongest opponent. Counterintuitively, each weak network can induce a global transition to such cooperative configuration regardless of the actions of the strongest network. This power of the weak reveals a critical dominance of the underdogs in the fate of networksofnetworks.
Introduction
Social, biological, physical and technological systems are composed of a diversity of interacting agents, leading network science, a statistical physics understanding of graph theory, to be a genuine tool for investigating their structure and dynamics^{1,2,3}. Within the framework of social networks^{4}, the topology of the interactions between individuals has been demonstrated to be crucial in, for example, the vanishing of the critical threshold in epidemics^{5,6} or the efficient and fast propagation of innovation^{7}. In a similar manner, the topology of a network itself can be influenced by the dynamical processes occurring in it, giving rise to adaptive mechanisms that rule the evolution of the structure of social networks^{8}.
The emergence of cooperation, defection or altruism can be investigated by linking game theory to network science^{9,10,11,12}. In this way, the intrinsic heterogeneity of social networks, the majority of them showing powerlaw distributions in the number of connections^{1}, has been related in many cases to the emergence of cooperation, contrary to what is observed in homogeneous populations^{13}. Furthermore, highly connected individuals have also been shown to be more prone to collaborate than scarcely connected ones^{14}. Under this framework, the understanding of evolutionary games largely benefited from the methodological tools of network science^{11}. Although attention was initially focused on the interplay between nodes’ strategies and the structure of the underlying (single) network^{15}, more recently, coevolutionary rules have also been related to the emergence of interdependency^{16} and multilayer structures^{17}. However, what if we are concerned about the interests of a network as a whole instead of its nodes? Does it make sense to consider networks competing or collaborating with other networks? The fruitful recent literature about networksofnetworks, or in a more general context about multilayer networks, makes these two questions timely and extremely relevant^{18,19}. A diversity of dynamical processes such as percolation^{20}, diffusion^{21} or synchronization^{22} have been recently reinterpreted by assuming that real networks unavoidably interact with other networks, a contact that may be beneficial or detrimental to each of the networks belonging to the ensemble^{23}.
Here we investigate how m>2 networks compete or cooperate to achieve a relative increase of importance measured as eigenvector centrality, which maximizes their outcome in a variety of dynamical processes. In our competition, networks can vary the way they interact with other networks, evolving in time until they reach a stable situation where all networks refuse to modify their strategy, because any change would lead to a worse result. Importantly, an a priori optimal connection strategy for a given network may not be reachable due to the actions of the competitor networks, which turns the analysis of the final outcome of the networks into a study of Nash equilibria^{24} in a networkofnetworks. With this objective in mind, we define a methodology to analyse the competition among networks of any size or topology, demonstrating that several Nash equilibria can coexist, with some of them benefiting the strongest networks and others benefiting the weaker ones.
In particular, we report the existence of a wide regime of the system parameters in which every weak network can induce the rest to cooperate, to escape from a detrimental Nash equilibrium, taking over the final situation of the whole networkofnetworks. Paradoxically, the strong network cannot reverse this phenomenon. This counterintuitive asymmetry that promotes the cooperation among weak networks is independent of the network structure or the competition rules and it could be applicable to an extensive number of real systems.
Results
Defining the rules of the competition
As a general rule, we consider that nodes belonging to a network will accept a common strategy, which can be justified in terms of a common benefit or the existence of an imposition within a hierarchical organization (see Supplementary Note 1 for details). The following example, based on real networks, illustrates how different strategies can enhance the outcome of a network. The interaction between members of rural communities in southern India was recently investigated through a series of surveys in the framework of a microfinance programme^{25,26}. From those data sets (available at the online version of ref. 25), we constructed the loan networks inside three of those villages (see Fig. 1a), creating a link between individuals i and j if they were willing to lend to or borrow from each other a certain amount of money. Local loan networks constructed as explained above provide much information about the financial resilience of a region. If the local authorities of one village promoted the connections to other regions—for example, through the funding of social events—their loan network would be enhanced and the village would become more prepared to face unexpected natural or financial risks (see refs 27, 28, 29, 30, 31 and Supplementary Note 1). However, what village is the best to connect to if more than one option exists? In addition, more importantly, which village would benefit the most from the creation of new financial channels among them?
To address these questions in a general framework we consider m networks of N_{i} nodes respectively, where i=A, B, C,... The adjacency matrices G_{i} associated to each network i contain the full information about the connections between their nodes (that is, the specific topology of the networks). The largest eigenvalue λ_{i} of G_{i} is an indicator of the network strength, as explained in ref. 32; thus, we can order them such that λ_{A}>λ_{B}≥ ... ≥λ_{m}. We make use of the eigenvector centrality to determine the importance acquired by each node, which is directly obtained from the eigenvector u_{1} associated with the largest eigenvalue λ_{1} of the adjacency matrix (see ref. 3, Methods and Supplementary Note 2).
In our game, every competitor (that is, network) makes use of up to l undirected links to connect with any of the other networks. The connector nodes are those with the largest centrality (see Methods). The refusal to connect is an accepted strategy. Therefore, there are strategies per competitor and (S_{m,l})^{m} possible combinations of actions. The target of each competitor is to maximize its own eigenvector centrality, calculated as the total importance (or centrality) C_{i} accumulated by all its nodes
where j are the nodes that belong to network i and u_{T} is the eigenvector associated with the largest eigenvalue λ_{T} of the adjacency matrix of the networkofnetworks T, containing all nodes N_{T}=∑_{i} N_{i}. It is noteworthy that we face a zerosum game (∑_{i} C_{i}=1) and the connected system T consists of m networks interconnected through a maximum of m × l connector links. Furthermore, such connector links will influence C_{i} of every network, but not their strength λ_{i}, which is measured when network i is isolated from the rest and is independent of the contest.
As we assume that networks are able to modify their connector links with the aim of acquiring the highest possible centrality C_{i} inside the networkofnetworks T, the final configuration of the system is given by a Nash equilibrium. A Nash equilibrium is the solution of a noncooperative game involving two or more players, in which each competitor is assumed to know the equilibrium strategies of the other players and no competitor has anything to gain by changing only their own strategy^{24,33}. From this general perspective, regardless of the particular rules of the contest, the competition process finishes when a Nash equilibrium is reached, being C_{i} the final payoff of each competitor network.
Figure 1b,c show the competition for centrality in our ‘tale of three villages’. We compute the largest eigenvalue associated with the loan networks within each village and obtain the ranking in strength: λ_{A}>λ_{B}>λ_{C}. The terminology used to indicate how a network (that is, a village) i_{1} decides to connect to a network i_{2} is the following: i_{1}(0) stands for network i_{1} refusing to connect, i_{1}→i_{2} stands for i_{1} connecting to network i_{2} and i_{1}⇆i_{2} stands for i_{1} connecting to network i_{2} and i_{2} connecting to network i_{1}. The arrow → indicates which network decides to create the connector link, but all links are undirected (that is, bidirectional).
Allowing villages to connect through one link (l=1), we obtain two possible Nash equilibria. In one of those, weak villages establish connections with the strong village, {A(0), B→A, C→A}, which plainly benefits the latter (Fig. 1b). Nevertheless, the alternative equilibrium {A→B, B⇆C} allows weak villages to overcome their stronger competitor by connecting with each other. In this scenario, the strong network must connect to B to retain part of the centrality of the whole system (Fig. 1c).
Importantly, the selection of adequate connection strategies goes beyond the competition for centrality. In professional networks, for example, the growth of knowledge of an individual can be modelled to be proportional to the knowledge of his/her acquaintances^{34}, which leads to a final distribution of knowledge that is given by the first eigenvector u_{T} of the adjacency matrix. Translated to a case where independent groups of professionals or researchers can create connections among them, this would indicate that the strategy showed in Fig. 1c would enhance not only the importance of a cluster of professionals, but the relative amount of knowledge acquired by the weakest group compared with the strongest one (see Supplementary Note 1 for several realworld examples dealing with social, technological and biological networks). Furthermore, a wide variety of systems are described by weighted adjacency matrices—transition matrices that include the specificities of the underlying dynamical process—whose vector u_{T} is related to the equilibrium state of the system^{32}. Replicationmutation evolutionary dynamics^{35}, diffusion processes^{21} or disease spreading^{36} are only a few examples where the methodology presented here can be applied with no loss of generality (see Methods).
Competition and cooperation to overcome the strongest
Figure 2 shows a complete numerical description of the competition among m=3 generic scalefree networks A, B and C of largest eigenvalues λ_{A}>λ_{B}>λ_{C}. For the sake of clarity, only one connector link is allowed per network (l=1) and, therefore, each competitor has m different connection strategies (that is, connecting to any of the other m−1 networks or refusing to connect). In the case of three networks, 27 combinations of strategies are possible for each realization—choice of networks—among which only those that verify the conditions to be Nash equilibria are taken as solutions to the contest. Figure 1b,c show that more than one final solution can coexist, which poses two relevant questions: (i) is the coexistence of solutions a general phenomenon? And if this is the case, (ii) does the final outcome of each player vary substantially depending on the solution reached?
Figure 2a,b address question (i) and show the solution profiles as the strength of network A (that is, λ_{A}) is increased. Figure 2a shows a scenario that we found to be general: the coexistence of Nash equilibria, among which two of them, called X_{0} and X_{∞}, are especially relevant (see Fig. 2b):
On the other hand, Fig. 2c shows the centrality for the two existing Nash equilibria as the strength of A is increased for a particular choice of B and C. For a wide range of values of λ_{A}, the centralities attained by each player strongly depend on the specific solution of the contest, which answers question (ii) and highlights the importance of choosing an adequate connection strategy.
To check the relevance of this result, in Fig. 2d we show the relative centrality variability (ΔC), a measure of how much a player outcome improves by getting to its optimal solution:
where the maxima and minima are computed among all coexisting Nash equilibria. The ΔC_{i} of each player i takes values ranging from zero (if all solutions lead to the same centrality) to infinity (if the worstcase solution leads to zero centrality for that player). All three networks show values of the order of ΔC∼0.8, which means that the final centrality of each competing network can vary up to almost twofold depending on the solution reached.
In view of all, we can identify different competition regimes depending on the relative strength between the strong network A and the rest of competitors. For very large strengths of A, that is, when λ_{A}>λ_{B⇆C}, the only existing Nash equilibrium corresponds to X_{∞}: small networks avoid mutual cooperation and tend to connect to the largest one, which dominates the contest. A critical transition occurs at λ_{A}=λ_{B⇆C} below which X_{∞} and X_{0} coexist in a bistable manner. Interestingly, within this region, cooperation between the weak networks always leads to their best outcome. Finally, a smoother transition occurs around λ_{A}∼λ_{B↔C}, being B↔C the network resulting from connecting B and C through a single link: when the strength of A approaches that of B, mutual cooperation between B and C becomes dominant and equilibrium X_{∞} is no longer possible. At the same time, other Nash equilibria may appear (grey region of Fig. 2b). See Supplementary Notes 2 and 3 for a full analytical treatment of this phenomenon.
Weak networks can induce migration between equilibria
As explained, in a Nash equilibrium every competitor that may change strategy would decrease its centrality. However, as there are coexisting Nash equilibria, it may be worth assuming a temporal loss of centrality if the final situation leads to an improvement in the outcome. In this way, we study the potential migration between equilibria X_{0} and X_{∞} in the regime in which they coexist (λ_{A}<λ_{B⇆C}). In Fig. 3a we show that the migration from X_{∞} to X_{0} can be provoked by the weakest network C individually, whereas Fig. 3b shows that the strong network cannot trigger the system to migrate from X_{0} to X_{∞}. As a consequence, when multiple networks compete for centrality, a weak network by itself might escape from a detrimental equilibrium and push the whole system towards a much more beneficial one with the cost of a onestep transient during which its centrality is decreased. On the contrary, this migrationstrategy is not accessible to the strong network, resulting in a natural asymmetry in the context of networksofnetworks that benefits the final outcome of weak networks and provides them with a flexibility not permitted to the strongest competitors.
Overall consequences on the networkofnetworks
Extensive numerical work yields that the strength λ_{T} of the networkofnetworks T following the equilibrium X_{0} is always larger than that of the solution X_{∞} (that is, λ_{T}(X_{0})>λ_{T}(X_{∞}), see Supplementary Note 4). Importantly, an increase of λ_{T} is related to enhanced growth at the equilibrium for a wide range of dynamical processes^{35}, a reduction of the critical coupling strength for the emergence of synchronization (as ∼1/λ_{T})^{37} or the emergence of a giant component in percolation phenomena^{38}. Therefore, the natural tendency towards cooperation among weak networks presented in this work also improves efficiency and growth of the whole system. Going back to the three villages example, the analysis of the Nash equilibria reveals that the X_{0} equilibrium (Fig. 1c) leads to a higher global λ_{T} than X_{∞} (Fig. 1b; that is, λ_{T}(X_{0})=4.78>λ_{T}(X_{∞})=4.57, for l=1). At the end, this is good news for all villages, as a higher λ_{T} enhances the strength of the whole ensemble^{39}. It is noteworthy that these results can be used not only for description but, more importantly, for prescription of how networks can maximize their outcome when interacting with other networks and how the appearance of new interactions among isolated networks influences the structural and dynamical properties of real systems.
Finally, the migration between equilibria described above could have a suggestive counterpart in a wide variety of situations where a relationship based on subjugation to a powerful leader naturally migrated towards a new and more productive model based on cooperation (see refs 40, 41 and Supplementary Note 4 for an illustrative historical example).
Discussion
In summary, we propose combining network science and game theory to analyse the election of interconnecting strategies in a zerosum game where players are not single agents but networks. The creation of paths between interacting networks leads to different Nash equilibria, some of them benefiting the strong competitor and some of them reinforcing the underdogs. Counterintuitively, we showed that transitions between coexisting Nash equilibria are restricted to the weakest competitors, which in practice rule the contest, whereas the strongest network is unable to change the status quo in its own interest.
Importantly, most assumptions of our model can be modified to describe more realistic scenarios without causing qualitative changes (see Supplementary Note 5 for details). When each player is permitted to connect to the rest of networks through more than one link (that is, l>1), the number of strategy combinations grows as l^{m(m−1)} for fixed m. However, the number of coexisting solutions and the phenomenology observed are totally equivalent to those obtained for l=1. The same applies for random (Erdös–Rényi) network topologies, networks of any size and networks with different abilities—that is, when certain networks can connect through a larger number of connector links (or even more weighted links) than the other competitors.
Furthermore, extending the analysis to mixed Nash equilibria, where any noninteger distribution of the l connector links is allowed, does not alter the results and provides a probabilistic nature to the game that broadens its applicability. When more than 3 networks are considered in the contest, new kinds of Nash equilibria arise, but again, we observe the existence of wide regions of the parameter space where weak networks rule the whole networkofnetworks. In addition, different definitions of the centralitybased payoffs have been analysed—such as betweenness or closeness centrality—and only those closely related to the eigenvector centrality lead to a rich phenomenology in the number of Nash equilibria and the effect of such equilibria on the payoffs of the competitors.
Finally, in some social and economical networks the connection strategies may be influenced by the individual motivations of the connector nodes, resulting in a potential conflict with the collective interests. As a first step in understanding such complex scenarios, we introduced a payoff following ref. 42, which includes both individual and collective contributions (see Supplementary Note 6): we conclude that cooperation between the weak networks and their control of the game is a frequent outcome that can appear at relatively small (or even zero) levels of collective incentive, although the quantitative details significantly depend on the topology of the networks.
The robustness of the phenomenology presented here, added to its potential applicability to real cases, makes this ‘power of the weak’ a valuable fact to consider in future modelling of technological, biological or sociological processes on networks.
Methods
Measuring the importance of nodes and networks
We use the eigenvector centrality x_{k} for quantifying the importance of a node k in a network, which can be obtained as an iterative process that sums the centralities of all neighbours of k:
where λ is a constant, x_{k} is the eigenvector centrality of node k and G_{kj} are the components of the adjacency matrix, which could be both binary or weighted^{43}. In matrix notation equation (5) reads λ x=Gx so that x can be expressed as a linear combination of the eigenvectors u_{k} of the adjacency matrix G, being λ_{k} the set of the corresponding eigenvalues. Equation (5) can be regarded as an iterative process that begins at t=0 with a set of initial conditions x_{0}. Regardless of the values of x_{0}, the value of x(t) at t→∞ will be proportional to the eigenvector u_{1} associated with the largest eigenvalue λ_{1}. Therefore, the eigenvector centrality is directly obtained from the eigenvector u_{1} of the adjacency matrix G, which also holds for weighted adjacency matrices. As explained in the main text, the centrality accumulated by each network is obtained as the fraction of centrality accumulated by their nodes. Finally, we use λ_{1} as the measure of the network strength, as it is related to a series of dynamical properties of networks and, in turn, it increases with the number of nodes, links and the average degree of the network^{44}.
Selection of the specific connector nodes
As explained in ref. 32, the centrality of the connector nodes linking independent networks may be crucial in the final distribution of centrality. The central nodes (C) of a network are those nodes with the largest eigenvector centrality, whereas the peripheral nodes (P) are those nodes with very low centrality (see Supplementary Note 2 for more details). When connecting two networks, the connector nodes allow to distinguish between central–central (CC) connections or peripheral–peripheral (PP) connections. Importantly, when a networkofnetworks is broken into disconnected components, the cluster with the largest eigenvalue acquires all centrality, whereas the rest of the (weak) components accumulate zero centrality. PP connections lead to a scenario close to the disconnected case, thus pushing almost the totality of the centrality towards the strong network. Accordingly, any connection strategy based on PP links is practically equivalent to refusing to connect with any other network. For this reason, we have restricted the networks’ strategies to CC connections or no connections (refusal to connect).
Finally, it is worth noting that the rules of the competition allow the networks to connect through more than one link (for example, in A⇆B). For simplicity, throughout the examples studied in this work, we represent w connector links between networks as one link of weight w. However, in certain systems, links with weight larger than one could have no real meaning. In those cases, the second (third, etc.) link between two networks should be built between their second (third, etc.) most central nodes, maintaining the phenomenology qualitatively unchanged.
The adjacency matrix and the dynamical processes
A variety of dynamical processes occurring in a network can be mathematically described as n(t+1)=Mn(t), 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 matrix that contains the peculiarities of the dynamical process. As M is a primitive matrix, its largest eigenvalue is positive, it verifies that λ_{1}>λ_{i}, ∀ i>1 and its associated eigenvector is also positive. Therefore, the dynamics of the whole system is given by
where n(0) is the initial condition and u_{i} the ith eigenvectors of M. From equation (6) we obtain that the system evolves towards an asymptotic state independent of the initial condition and proportional to the first eigenvector u_{1}:
whereas 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→∞ and there is a correspondence between the eigenvector centrality and the asymptotic state of the system at equilibrium: both the eigenvector centrality and the asymptotic state of the system are proportional to the eigenvector associated with the largest eigenvalue of the transition matrix M.
Regarding the phenomenology presented in this work, in case we were concerned about a specific dynamical process, we would replace the adjacency matrix G by the transition matrix M, obtaining the eigenvector centrality retained by each network with no loss of generality.
Data availability
All relevant data are available from the authors upon request.
Additional information
How to cite this article: Iranzo, J. et al. Competition among networks highlights the power of the weak. Nat. Commun. 7, 13273 doi: 10.1038/ncomms13273 (2016).
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Acknowledgements
We thank S. Ahnert, A. Arenas, S. Boccaletti, D. Crespo, J. Cuesta, L. Dudbridge, J. GarciaOjalvo, S. Manrubia, Y. Moreno, F. PabloMartí, D. Papo, M.A. Porter, F. PuenteSánchez, A. Sánchez and A. Sanz for fruitful conversations, and J.F. Nash for being a source of inspiration. J.A. acknowledges financial support from Spanish MINECO (projects FIS201127569 and FIS201457686), J.I. is supported by intramural funds of the US Department of Health and Human Services (to the National Library of Medicine) and J.M.B. is funded by Spanish MINECO (project FIS201341057).
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J.A., J.I. and J.M.B. conceived and designed the project. J.I. performed the numerical simulations, J.M.B. analysed the real data sets and J.A. developed the analytical work. J.A., J.I. and J.M.B. wrote the manuscript.
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Iranzo, J., Buldú, J. & Aguirre, J. Competition among networks highlights the power of the weak. Nat Commun 7, 13273 (2016). https://doi.org/10.1038/ncomms13273
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