Spreading of cooperative behaviour across interdependent groups

Mixing protocols in the public goods game

If interaction partners in social dilemma games are not selected randomly from the population but are instead determined by a network of contacts, it has far reaching consequences for the evolutionary dynamics. Selecting partners randomly leads to a well-mixed population, where pattern formation is essentially impossible. This rules out important mechanisms that can facilitate cooperation, most notably network reciprocity. In contrast, if interactions are determined by a lattice or a network, then the population is said to be structured, where cooperators can form compact clusters that protect them from invading defectors. Between these two extremes, however, there is ample middle ground that can be brought about by the consideration of temporal networks, mobility, or other coevolutionary processes. The question that we here seek to answer is, when does mixing on a lattice actually lead to well-mixed conditions? To that effect, we use the public goods game on a square lattice, and we consider nearest-neighbor and random mixing with different frequencies, as well as a mix of both mixing protocols. Not surprisingly, we find that nearest-neighbor mixing requires a higher frequency than random mixing to arrive at the well-mixed limit. The differences between the two mixing protocols are most expressed at intermediate mixing frequencies, whilst at very low and very high mixing frequencies the two almost converge. We also find a near universal exponential growth of the average size of cooperator clusters as their fraction increases from zero to one, regardless of whether this increase is due to increasing the multiplication factor of the public goods, decreasing the frequency of mixing, or gradually shifting the mixing from random to nearest neighbors.

Aspiration induced interdependence leads to optimal cooperation level

How to couple different networks is a key issue in interdependent networks, where information sharing and payoff coupling are two frequently used methods. Unlike previous studies, in this paper, we propose a new coupling mode and test its performance in interdependent networks. Specifically, a player tends to seek additional support on another network only if his environment (defined as the proportion of holding different strategies in the neighborhood) is worse enough and exceeds an aspiration level. Conversely, it turns to the traditional version on single network if his environment is pleasing enough (the value of environment is small). Whether to establish additional support will directly influence the range of selecting fittest learning objects. As we can see from numerical results, moderate aspiration introduces diversity into the system and cooperation evolves with the support of network coupling. Besides, we also demonstrate that players with external links on the boundary of cooperative clusters protect internal cooperators and attract more players to cooperate under preferential selection rule.

Individual diversity between interdependent networks promotes the evolution of cooperation by means of mixed coupling

Along with the rapid development of network-based information technology, such as cloud computing, big data, the IoT, and so on, human society has stepped into a new era of complex networks. People's life and production activities depend more and more on various complex networks to ensure security and reliability. The complex interrelationships between human and nature establish a link to explain the cooperation of individual behaviour, especially for individual diversity. However, existing researches mostly ignore the influence of individual diversity on networks involved in individual behaviour to strategy selection. Therefore, it needs further research on how to consider both individual diversity and independent networks in the evolution of cooperative behaviour. To address this issue, we extend a simple game model into the interdependent networks through the mixed coupling (i.e., utility and probability) in this work. Also, we divide the kinds of strategic behaviour of a player in one layer concerning individual diversity. Moreover, there exists an optimal region of mixed coupling between networks such that cooperation can be promoted. Finally, experimental results can open the path to understanding the emergence and maintenance of cooperation within various interconnected and interrelated real-world systems newly.

Evolution of cooperation on large networks with community structure

Cooperation is a major factor in the evolution of human societies. The structure of social networks, which affects the dynamics of cooperation and other interpersonal phenomena, have common structural signatures. One of these signatures is the tendency to organize as groups. This tendency gives rise to networks with community structure, which are composed of distinct modules. In this paper, we study analytically the evolutionary game dynamics on large modular networks in the limit of weak selection. We obtain novel analytical conditions such that natural selection favours cooperation over defection. We calculate the transition point for each community to favour cooperation. We find that a critical inter-community link creation probability exists for given group density, such that the overall network supports cooperation even if individual communities inhibit it. As a byproduct, we present solutions for the critical benefit-to-cost ratio which perform with remarkable accuracy for diverse generative network models, including those with community structure and heavy-tailed degree distributions. We also demonstrate the generalizability of the results to arbitrary two-player games.

Thresholds in the resilience of modular social networks to invasion by defectors

Cooperative interactions constitute the backbone of many biological and social systems. Since cooperation is prone to exploitation, these systems must incorporate mechanisms that prevent the spreading of defective behaviors. One such mechanism is modularity, i.e., the tendency of a social network to be organized in modules, where individuals within a module tend to interact strongly among themselves while avoiding interacting with individuals from other modules. This structure allows cooperation to prevail by having modules of cooperative individuals with a limited exposure to defectors. To address the rate and shape of the effect of modularity on the resilience of cooperation, here we study a variant of the Prisoner's Dilemma on modular networks. Our simulations reveal a sharp transition between a resilient and a vulnerable regime as modularity exceeds a critical threshold. By using a simplified mathematical model, we show that the observed threshold is equivalent to the epidemic threshold found in a certain class of SIR models. This allows us to derive an explicit condition under which a cooperative society is expected to be resilient to invasive defectors.

Bribery games on interdependent complex networks

Bribe demands present a social conflict scenario where decisions have wide-ranging economic and ethical consequences. Nevertheless, such incidents occur daily in many countries across the globe. Harassment bribery constitute a significant sub-set of such bribery incidents where a government official demands a bribe for providing a service to a citizen legally entitled to it. We employ an evolutionary game-theoretic framework to analyse the evolution of corrupt and honest strategies in structured populations characterized by an interdependent complex network. The effects of changing network topology, average number of links and asymmetry in size of the citizen and officer population on the proliferation of incidents of bribery are explored. A complex network topology is found to be beneficial for the dominance of corrupt strategies over a larger region of phase space when compared with the outcome for a regular network, for equal citizen and officer population sizes. However, the extent of the advantage depends critically on the network degree and topology. A different trend is observed when there is a difference between the citizen and officer population sizes. Under those circumstances, increasing randomness of the underlying citizen network can be beneficial to the fixation of honest officers up to a certain value of the network degree. Our analysis reveals how the interplay between network topology, connectivity and strategy update rules can affect population level outcomes in such asymmetric games.

Social contagions on interdependent lattice networks

Although an increasing amount of research is being done on the dynamical processes on interdependent spatial networks, knowledge of how interdependent spatial networks influence the dynamics of social contagion in them is sparse. Here we present a novel non-Markovian social contagion model on interdependent spatial networks composed of two identical two-dimensional lattices. We compare the dynamics of social contagion on networks with different fractions of dependency links and find that the density of final recovered nodes increases as the number of dependency links is increased. We use a finite-size analysis method to identify the type of phase transition in the giant connected components (GCC) of the final adopted nodes and find that as we increase the fraction of dependency links, the phase transition switches from second-order to first-order. In strong interdependent spatial networks with abundant dependency links, increasing the fraction of initial adopted nodes can induce the switch from a first-order to second-order phase transition associated with social contagion dynamics. In networks with a small number of dependency links, the phase transition remains second-order. In addition, both the second-order and first-order phase transition points can be decreased by increasing the fraction of dependency links or the number of initially-adopted nodes.

Bribery games on inter-dependent regular networks

We examine a scenario of social conflict that is manifest during an interaction between government servants providing a service and citizens who are legally entitled to the service, using evolutionary game-theory in structured populations characterized by an inter-dependent network. Bribe-demands by government servants during such transactions, called harassment bribes, constitute a widespread form of corruption in many countries. We investigate the effect of varying bribe demand made by corrupt officials and the cost of complaining incurred by harassed citizens, on the proliferation of corrupt strategies in the population. We also examine how the connectivity of the various constituent networks affects the spread of corrupt officials in the population. We find that incidents of bribery can be considerably reduced in a network-structured populations compared to mixed populations. Interestingly, we also find that an optimal range for the connectivity of nodes in the citizen's network (signifying the degree of influence a citizen has in affecting the strategy of other citizens in the network) as well as the interaction network aids in the fixation of honest officers. Our results reveal the important role of network structure and connectivity in asymmetric games.

Transportation dynamics on coupled networks with limited bandwidth

The communication networks in real world often couple with each other to save costs, which results in any network does not have a stand-alone function and efficiency. To investigate this, in this paper we propose a transportation model on two coupled networks with bandwidth sharing. We find that the free-flow state and the congestion state can coexist in the two coupled networks, and the free-flow path and congestion path can coexist in each network. Considering three bandwidth-sharing mechanisms, random, assortative and disassortative couplings, we also find that the transportation capacity of the network only depends on the coupling mechanism, and the fraction of coupled links only affects the performance of the system in the congestion state, such as the traveling time. In addition, with assortative coupling, the transportation capacity of the system will decrease significantly. However, the disassortative coupling has little influence on the transportation capacity of the system, which provides a good strategy to save bandwidth. Furthermore, a theoretical method is developed to obtain the bandwidth usage of each link, based on which we can obtain the congestion transition point exactly.

Utility Evaluation Based on One-To-N Mapping in the Prisoner's Dilemma Game for Interdependent Networks

In the field of evolutionary game theory, network reciprocity has become an important means to promote the level of promotion within the population system. Recently, the interdependency provides a novel perspective to understand the widespread cooperation behavior in many real-world systems. In previous works, interdependency is often built from the direct or indirect connections between two networks through the one-to-one mapping mode. However, under many realistic scenarios, players may need much more information from many neighboring agents so as to make a more rational decision. Thus, beyond the one-to-one mapping mode, we investigate the cooperation behavior on two interdependent lattices, in which the utility evaluation of a focal player on one lattice may not only concern himself, but also integrate the payoff information of several corresponding players on the other lattice. Large quantities of simulations indicate that the cooperation can be substantially promoted when compared to the traditionally spatial lattices. The cluster formation and phase transition are also analyzed in order to explore the role of interdependent utility coupling in the collective cooperation. Current results are beneficial to deeply understand various mechanisms to foster the cooperation exhibited inside natural, social and engineering systems.

Reconstructing direct and indirect interactions in networked public goods game

Network reconstruction is a fundamental problem for understanding many complex systems with unknown interaction structures. In many complex systems, there are indirect interactions between two individuals without immediate connection but with common neighbors. Despite recent advances in network reconstruction, we continue to lack an approach for reconstructing complex networks with indirect interactions. Here we introduce a two-step strategy to resolve the reconstruction problem, where in the first step, we recover both direct and indirect interactions by employing the Lasso to solve a sparse signal reconstruction problem, and in the second step, we use matrix transformation and optimization to distinguish between direct and indirect interactions. The network structure corresponding to direct interactions can be fully uncovered. We exploit the public goods game occurring on complex networks as a paradigm for characterizing indirect interactions and test our reconstruction approach. We find that high reconstruction accuracy can be achieved for both homogeneous and heterogeneous networks, and a number of empirical networks in spite of insufficient data measurement contaminated by noise. Although a general framework for reconstructing complex networks with arbitrary types of indirect interactions is yet lacking, our approach opens new routes to separate direct and indirect interactions in a representative complex system.

A novel framework of classical and quantum prisoner's dilemma games on coupled networks

Evolutionary games on multilayer networks are attracting growing interest. While among previous studies, the role of quantum games in such a infrastructure is still virgin and may become a fascinating issue across a myriad of research realms. To mimick two kinds of different interactive environments and mechanisms, in this paper a new framework of classical and quantum prisoner's dilemma games on two-layer coupled networks is considered. Within the proposed model, the impact of coupling factor of networks and entanglement degree in quantum games on the evolutionary process has been studied. Simulation results show that the entanglement has no impact on the evolution of the classical prisoner's dilemma, while the rise of the coupling factor obviously impedes cooperation in this game, and the evolution of quantum prisoner's dilemma is greatly impacted by the combined effect of entanglement and coupling.

Extreme events in multilayer, interdependent complex networks and control

We investigate the emergence of extreme events in interdependent networks. We introduce an inter-layer traffic resource competing mechanism to account for the limited capacity associated with distinct network layers. A striking finding is that, when the number of network layers and/or the overlap among the layers are increased, extreme events can emerge in a cascading manner on a global scale. Asymptotically, there are two stable absorption states: a state free of extreme events and a state of full of extreme events, and the transition between them is abrupt. Our results indicate that internal interactions in the multiplex system can yield qualitatively distinct phenomena associated with extreme events that do not occur for independent network layers. An implication is that, e.g., public resource competitions among different service providers can lead to a higher resource requirement than naively expected. We derive an analytical theory to understand the emergence of global-scale extreme events based on the concept of effective betweenness. We also articulate a cost-effective control scheme through increasing the capacity of very few hubs to suppress the cascading process of extreme events so as to protect the entire multi-layer infrastructure against global-scale breakdown.

Multiple effect of social influence on cooperation in interdependent network games

The social influence exists widely in the human society, where individual decision-making process (from congressional election to electronic commerce) may be affected by the attitude and behavior of others belonging to different social networks. Here, we couple the snowdrift (SD) game and the prisoner's dilemma (PD) game on two interdependent networks, where strategies in both games are associated by social influence to mimick the majority rule. More accurately, individuals' strategies updating refers to social learning (based on payoff difference) and above-mentioned social influence (related with environment of interdependent group), which is controlled by social influence strength s. Setting s = 0 decouples the networks and returns the traditional network game; while its increase involves the interactions between networks. By means of numerous Monte Carlo simulations, we find that such a mechanism brings multiple influence to the evolution of cooperation. Small s leads to unequal cooperation level in both games, because social learning is still the main updating rule for most players. Though intermediate and large s guarantees the synchronized evolution of strategy pairs, cooperation finally dies out and reaches a completely dominance in both cases. Interestingly, these observations are attributed to the expansion of cooperation clusters. Our work may provide a new understanding to the emergence of cooperation in intercorrelated social systems.

Inferring the mesoscale structure of layered, edge-valued, and time-varying networks

Many network systems are composed of interdependent but distinct types of interactions, which cannot be fully understood in isolation. These different types of interactions are often represented as layers, attributes on the edges, or as a time dependence of the network structure. Although they are crucial for a more comprehensive scientific understanding, these representations offer substantial challenges. Namely, it is an open problem how to precisely characterize the large or mesoscale structure of network systems in relation to these additional aspects. Furthermore, the direct incorporation of these features invariably increases the effective dimension of the network description, and hence aggravates the problem of overfitting, i.e., the use of overly complex characterizations that mistake purely random fluctuations for actual structure. In this work, we propose a robust and principled method to tackle these problems, by constructing generative models of modular network structure, incorporating layered, attributed and time-varying properties, as well as a nonparametric Bayesian methodology to infer the parameters from data and select the most appropriate model according to statistical evidence. We show that the method is capable of revealing hidden structure in layered, edge-valued, and time-varying networks, and that the most appropriate level of granularity with respect to the additional dimensions can be reliably identified. We illustrate our approach on a variety of empirical systems, including a social network of physicians, the voting correlations of deputies in the Brazilian national congress, the global airport network, and a proximity network of high-school students.

Measuring and modeling correlations in multiplex networks

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

Synchronization Analysis of Master-Slave Probabilistic Boolean Networks

In this paper, we analyze the synchronization problem of master-slave probabilistic Boolean networks (PBNs). The master Boolean network (BN) is a deterministic BN, while the slave BN is determined by a series of possible logical functions with certain probability at each discrete time point. In this paper, we firstly define the synchronization of master-slave PBNs with probability one, and then we investigate synchronization with probability one. By resorting to new approach called semi-tensor product (STP), the master-slave PBNs are expressed in equivalent algebraic forms. Based on the algebraic form, some necessary and sufficient criteria are derived to guarantee synchronization with probability one. Further, we study the synchronization of master-slave PBNs in probability. Synchronization in probability implies that for any initial states, the master BN can be synchronized by the slave BN with certain probability, while synchronization with probability one implies that master BN can be synchronized by the slave BN with probability one. Based on the equivalent algebraic form, some efficient conditions are derived to guarantee synchronization in probability. Finally, several numerical examples are presented to show the effectiveness of the main results.

Group Size Effect on Cooperation in One-Shot Social Dilemmas II: Curvilinear Effect

In a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.

On topological indices of poly oxide, poly silicate, DOX, and DSL networks

Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study different interconnection networks and derive analytical closed results of the general Randić index (Rα(G)) for α = 1, [Formula: see text], –1, [Formula: see text] only, for dominating oxide network (DOX), dominating silicate network (DSL), and regular triangulene oxide network (RTOX). All of the studied interconnection networks in this paper are motivated by the molecular structure of a chemical compound, SiO4. We also compute the general first Zagreb, ABC, GA, ABC4, and GA5 indices and give closed formulae of these indices for these interconnection networks.

Heterogeneous Coupling between Interdependent Lattices Promotes the Cooperation in the Prisoner's Dilemma Game

In the research realm of game theory, interdependent networks have extended the content of spatial reciprocity, which needs the suitable coupling between networks. However, thus far, the vast majority of existing works just assume that the coupling strength between networks is symmetric. This hypothesis, to some extent, seems inconsistent with the ubiquitous observation of heterogeneity. Here, we study how the heterogeneous coupling strength, which characterizes the interdependency of utility between corresponding players of both networks, affects the evolution of cooperation in the prisoner's dilemma game with two types of coupling schemes (symmetric and asymmetric ones). Compared with the traditional case, we show that heterogeneous coupling greatly promotes the collective cooperation. The symmetric scheme seems much better than the asymmetric case. Moreover, the role of varying amplitude of coupling strength is also studied on these two interdependent ways. Current findings are helpful for us to understand the evolution of cooperation within many real-world systems, in particular for the interconnected and interrelated systems.

Dependency links can hinder the evolution of cooperation in the prisoner's dilemma game on lattices and networks

Networks with dependency links are more vulnerable when facing the attacks. Recent research also has demonstrated that the interdependent groups support the spreading of cooperation. We study the prisoner's dilemma games on spatial networks with dependency links, in which a fraction of individual pairs is selected to depend on each other. The dependency individuals can gain an extra payoff whose value is between the payoff of mutual cooperation and the value of temptation to defect. Thus, this mechanism reflects that the dependency relation is stronger than the relation of ordinary mutual cooperation, but it is not large enough to cause the defection of the dependency pair. We show that the dependence of individuals hinders, promotes and never affects the cooperation on regular ring networks, square lattice, random and scale-free networks, respectively. The results for the square lattice and regular ring networks are demonstrated by the pair approximation.

Enhancing robustness of coupled networks under targeted recoveries

Coupled networks are extremely fragile because a node failure of a network would trigger a cascade of failures on the entire system. Existing studies mainly focused on the cascading failures and the robustness of coupled networks when the networks suffer from attacks. In reality, it is necessary to recover the damaged networks, and there are cascading failures in recovery processes. In this study, firstly, we analyze the cascading failures of coupled networks during recoveries. Then, a recovery robustness index is presented for evaluating the resilience of coupled networks to cascading failures in the recovery processes. Finally, we propose a technique aiming at protecting several influential nodes for enhancing robustness of coupled networks under the recoveries, and adopt six strategies based on the potential knowledge of network centrality to find the influential nodes. Experiments on three coupling networks demonstrate that with a small number of influential nodes protected, the robustness of coupled networks under the recoveries can be greatly enhanced.

Conformity enhances network reciprocity in evolutionary social dilemmas

The pursuit of highest payoffs in evolutionary social dilemmas is risky and sometimes inferior to conformity. Choosing the most common strategy within the interaction range is safer because it ensures that the payoff of an individual will not be much lower than average. Herding instincts and crowd behaviour in humans and social animals also compel to conformity in their own right. Motivated by these facts, we here study the impact of conformity on the evolution of cooperation in social dilemmas. We show that an appropriate fraction of conformists within the population introduces an effective surface tension around cooperative clusters and ensures smooth interfaces between different strategy domains. Payoff-driven players brake the symmetry in favour of cooperation and enable an expansion of clusters past the boundaries imposed by traditional network reciprocity. This mechanism works even under the most testing conditions, and it is robust against variations of the interaction network as long as degree-normalized payoffs are applied. Conformity may thus be beneficial for the resolution of social dilemmas.

Learning and coordinating in a multilayer network

We introduce a two layer network model for social coordination incorporating two relevant ingredients: a) different networks of interaction to learn and to obtain a pay-off, and b) decision making processes based both on social and strategic motivations. Two populations of agents are distributed in two layers with intralayer learning processes and playing interlayer a coordination game. We find that the skepticism about the wisdom of crowd and the local connectivity are the driving forces to accomplish full coordination of the two populations, while polarized coordinated layers are only possible for all-to-all interactions. Local interactions also allow for full coordination in the socially efficient Pareto-dominant strategy in spite of being the riskier one.

Intergroup conflict and rational decision making

The literature has been relatively silent about post-conflict processes. However, understanding the way humans deal with post-conflict situations is a challenge in our societies. With this in mind, we focus the present study on the rationality of cooperative decision making after an intergroup conflict, i.e., the extent to which groups take advantage of post-conflict situations to obtain benefits from collaborating with the other group involved in the conflict. Based on dual-process theories of thinking and affect heuristic, we propose that intergroup conflict hinders the rationality of cooperative decision making. We also hypothesize that this rationality improves when groups are involved in an in-group deliberative discussion. Results of a laboratory experiment support the idea that intergroup conflict –associated with indicators of the activation of negative feelings (negative affect state and heart rate)– has a negative effect on the aforementioned rationality over time and on both group and individual decision making. Although intergroup conflict leads to sub-optimal decision making, rationality improves when groups and individuals subjected to intergroup conflict make decisions after an in-group deliberative discussion. Additionally, the increased rationality of the group decision making after the deliberative discussion is transferred to subsequent individual decision making.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Synchronization on complex networks of networks

In this paper, pinning synchronization on complex networks of networks is investigated, where there are many subnetworks with the interactions among them. The subnetworks and their connections can be regarded as the nodes and interactions of the networks, respectively, which form the networks of networks. In this new setting, the aim is to design pinning controllers on the chosen nodes of each subnetwork so as to reach synchronization behavior. Some synchronization criteria are established for reaching pinning control on networks of networks. Furthermore, the pinning scheme is designed, which shows that the nodes with very low degrees and large degrees are good candidates for applying pinning controllers. Then, the attack and robustness of the pinning scheme are discussed. Finally, a simulation example is presented to verify the theoretical analysis in this paper.

Nonlinear growth and condensation in multiplex networks

Different types of interactions coexist and coevolve to shape the structure and function of a multiplex network. We propose here a general class of growth models in which the various layers of a multiplex network coevolve through a set of nonlinear preferential attachment rules. We show, both numerically and analytically, that by tuning the level of nonlinearity these models allow us to reproduce either homogeneous or heterogeneous degree distributions, together with positive or negative degree correlations across layers. In particular, we derive the condition for the appearance of a condensed state in which one node in each layer attracts an extensive fraction of all the edges.

Synchronization in output-coupled temporal Boolean networks

This paper presents an analytical study of synchronization in an array of output-coupled temporal Boolean networks. A temporal Boolean network (TBN) is a logical dynamic system developed to model Boolean networks with regulatory delays. Both state delay and output delay are considered, and these two delays are assumed to be different. By referring to the algebraic representations of logical dynamics and using the semi-tensor product of matrices, the output-coupled TBNs are firstly converted into a discrete-time algebraic evolution system, and then the relationship between the states of coupled TBNs and the initial state sequence is obtained. Then, some necessary and sufficient conditions are derived for the synchronization of an array of TBNs with an arbitrary given initial state sequence. Two numerical examples including one epigenetic model are finally given to illustrate the obtained results.

From a single network to a network of networks

Network science has attracted much attention in recent years due to its interdisciplinary applications. We witnessed the revolution of network science in 1998 and 1999 started with small-world and scale-free networks having now thousands of high-profile publications, and it seems that since 2010 studies of ‘network of networks’ (NON), sometimes called multilayer networks or multiplex, have attracted more and more attention. The analytic framework for NON yields a novel percolation law for n interdependent networks that shows that percolation theory of single networks studied extensively in physics and mathematics in the last 50 years is a specific limit of the rich and very different general case of n coupled networks. Since then, properties and dynamics of interdependent and interconnected networks have been studied extensively, and scientists are finding many interesting results and discovering many surprising phenomena. Because most natural and engineered systems are composed of multiple subsystems and layers of connectivity, it is important to consider these features in order to improve our understanding of such complex systems. Now the study of NON has become one of the important directions in network science. In this paper, we review recent studies on the new emerging area—NON. Due to the fast growth of this field, there are many definitions of different types of NON, such as interdependent networks, interconnected networks, multilayered networks, multiplex networks and many others. There exist many datasets that can be represented as NON, such as network of different transportation networks including flight networks, railway networks and road networks, network of ecological networks including species interacting networks and food webs, network of biological networks including gene regulation network, metabolic network and protein–protein interacting network, network of social networks and so on. Among them, many interdependent networks including critical infrastructures are embedded in space, introducing spatial constraints. Thus, we also review the progress on study of spatially embedded networks. As a result of spatial constraints, such interdependent networks exhibit extreme vulnerabilities compared with their non-embedded counterparts. Such studies help us to understand, realize and hopefully mitigate the increasing risk in NON.

Degree mixing in multilayer networks impedes the evolution of cooperation

Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play different roles in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider two-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.

Emergence of blind areas in information spreading

Recently, contagion-based (disease, information, etc.) spreading on social networks has been extensively studied. In this paper, other than traditional full interaction, we propose a partial interaction based spreading model, considering that the informed individuals would transmit information to only a certain fraction of their neighbors due to the transmission ability in real-world social networks. Simulation results on three representative networks (BA, ER, WS) indicate that the spreading efficiency is highly correlated with the network heterogeneity. In addition, a special phenomenon, namely Information Blind Areas where the network is separated by several information-unreachable clusters, will emerge from the spreading process. Furthermore, we also find that the size distribution of such information blind areas obeys power-law-like distribution, which has very similar exponent with that of site percolation. Detailed analyses show that the critical value is decreasing along with the network heterogeneity for the spreading process, which is complete the contrary to that of random selection. Moreover, the critical value in the latter process is also larger than that of the former for the same network. Those findings might shed some lights in in-depth understanding the effect of network properties on information spreading.

Evolutionary origin of asymptotically stable consensus

Consensus is widely observed in nature as well as in society. Up to now, many works have focused on what kind of (and how) isolated single structures lead to consensus, while the dynamics of consensus in interdependent populations remains unclear, although interactive structures are everywhere. For such consensus in interdependent populations, we refer that the fraction of population adopting a specified strategy is the same across different interactive structures. A two-strategy game as a conflict is adopted to explore how natural selection affects the consensus in such interdependent populations. It is shown that when selection is absent, all the consensus states are stable, but none are evolutionarily stable. In other words, the final consensus state can go back and forth from one to another. When selection is present, there is only a small number of stable consensus state which are evolutionarily stable. Our study highlights the importance of evolution on stabilizing consensus in interdependent populations.

Biased imitation in coupled evolutionary games in interdependent networks

We explore the evolutionary dynamics of two games—the Prisoner's Dilemma and the Snowdrift Game—played within distinct networks (layers) of interdependent networks. In these networks imitation and interaction between individuals of opposite layers is established through interlinks. We explore an update rule in which revision of strategies is a biased imitation process: individuals imitate neighbors from the same layer with probability p, and neighbors from the second layer with complementary probability 1 − p. We demonstrate that a small decrease of p from p = 1 (which corresponds to forbidding strategy transfer between layers) is sufficient to promote cooperation in the Prisoner's Dilemma subpopulation. This, on the other hand, is detrimental for cooperation in the Snowdrift Game subpopulation. We provide results of extensive computer simulations for the case in which layers are modelled as regular random networks, and support this study with analytical results for coupled well-mixed populations.

Moderate intra-group bias maximizes cooperation on interdependent populations

Evolutionary game theory on spatial structures has received increasing attention during the past decades. However, the majority of these achievements focuses on single and static population structures, which is not fully consistent with the fact that real structures are composed of many interactive groups. These groups are interdependent on each other and present dynamical features, in which individuals mimic the strategy of neighbors and switch their partnerships continually. It is however unclear how the dynamical and interdependent interactions among groups affect the evolution of collective behaviors. In this work, we employ the prisoner's dilemma game to investigate how the dynamics of structure influences cooperation on interdependent populations, where populations are represented by group structures. It is found that the more robust the links between cooperators (or the more fragile the links between cooperators and defectors), the more prevalent of cooperation. Furthermore, theoretical analysis shows that the intra-group bias can favor cooperation, which is only possible when individuals are likely to attach neighbors within the same group. Yet, interestingly, cooperation can be even inhibited for large intra-group bias, allowing the moderate intra-group bias maximizes the cooperation level.