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Understanding the evolution of cooperation in multiplayer games is of vital significance for natural and social systems. An important challenge is that group interactions often leads to nonlinear synergistic effects. However, previous models mainly focus on deterministic nonlinearity where the arise of synergy or discounting effect is determined by certain conditions, ignoring uncertainty and stochasticity in real-world systems. Here, we develop a probabilistic framework to study the cooperative behavior in stochastic nonlinear public goods games. Through both analytical treatment and Monte Carlo simulations, we provide comprehensive understanding of social dilemmas with stochastic nonlinearity in both well-mixed and structured populations. We find that increasing the degree of nonlinearity makes synergy more advantageous when competing with discounting, thereby promoting cooperation. Interestingly, we show that network reciprocity loses effectiveness when the probability of synergy is small. Moreover, group size exhibits nonlinear effects on group cooperation regardless of the underlying structure. Our findings thus provide novel insights into how stochastic nonlinearity influences the emergence of prosocial behavior., Comment: 23 pages, 8 figures

Indirect reciprocity unveils how social cooperation is founded upon moral systems. Under the frame of dyadic games based on individual reputations, the "leading-eight" social norms distinguish themselves in promoting and sustaining cooperation. However, in real societies, people constantly interact within larger groups. They are assessed not only by their own behaviors, but also the reputations of the groups they come from and interact with. Here, we study indirect reciprocity in public goods games with group reputations. We show that varying criteria of group reputation assessment destabilize the consensus within leading-eight populations. We find the presence of an optimal, moderately set social assessment criterion that is most conductive to promoting cooperation. As the payoff of group cooperation increases, the assessment criterion evolves to relaxation following two dynamical patterns, indicating that social prosperity encourages people to be tolerate. Our work identifies the role of group reputation assessment to boost cooperation in social populations.

Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual's activity rate and the possibility of group interaction, we propose a probabilistic activity driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Lastly, we explore the co-evolution of PAD model and higher-order contagion dynamics, and analytically derive the critical conditions for phase transition and bistable phenomenon. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.

Environmental changes play a critical role in determining the evolution of social dilemmas in many natural or social systems. Generally, the environmental changes include two prominent aspects: the global time-dependent fluctuations and the local strategy-dependent feedbacks. However, the impacts of these two types of environmental changes have only been studied separately, a complete picture of the environmental effects exerted by the combination of these two aspects remains unclear. Here we develop a theoretical framework that integrates group strategic behaviors with their general dynamic environments, where the global environmental fluctuations are associated with a nonlinear factor in public goods game and the local environmental feedbacks are described by the `eco-evolutionary game'. We show how the coupled dynamics of local game-environment evolution differs in static and dynamic global environments. In particular, we find the emergence of cyclic evolutions of group cooperation and local environment, which forms an interior irregular loop in the phase plane, depending on the relative changing speed of both global and local environments compared to the strategic change. Our results provide important insights toward how diverse evolutionary outcomes could emerge from the nonlinear interactions between strategies and the changing environments.

The brain's structural connectome supports signal propagation between neuronal elements, shaping diverse coactivation patterns that can be captured as functional connectivity. While the link between structure and function remains an ongoing challenge, the prevailing hypothesis is that the structure-function relationship may itself be gradually decoupled along a macroscale functional gradient spanning unimodal to transmodal regions. However, this hypothesis is strongly constrained by the underlying models which may neglect requisite signaling mechanisms. Here, we transform the structural connectome into a set of orthogonal eigenmodes governing frequency-specific diffusion patterns and show that regional structure-function relationships vary markedly under different signaling mechanisms. Specifically, low-frequency eigenmodes, which are considered sufficient to capture the essence of the functional network, contribute little to functional connectivity reconstruction in transmodal regions, resulting in structure-function decoupling along the unimodal-transmodal gradient. In contrast, high-frequency eigenmodes, which are usually on the periphery of attention due to their association with noisy and random dynamical patterns, contribute significantly to functional connectivity prediction in transmodal regions, inducing gradually convergent structure-function relationships from unimodal to transmodal regions. Although the information in high-frequency eigenmodes is weak and scattered, it effectively enhances the structure-function correspondence by 35% in unimodal regions and 56% in transmodal regions. Altogether, our findings suggest that the structure-function divergence in transmodal areas may not be an intrinsic property of brain organization, but can be narrowed through multiplexed and regionally specialized signaling mechanisms.

Binary decision-making process is ubiquitous in social life and is of vital significance in many real-world issues, ranging from public health to political campaigns. While continuous opinion evolution independent of discrete choice behavior has been extensively studied, few works unveil how the group binary decision-making result is determined by the coupled dynamics of these two processes. To this end, we propose an agent-based model to study the collective behaviors of individual binary decision-making process through competitive opinion dynamics on social networks. Three key factors are considered: bounded confidence that describes the cognitive scope of the population, stubbornness level that characterizes the opinion updating speed, and the opinion strength that represents the asymmetry power or attractiveness of the two choices. We find that bounded confidence plays an important role in determining competing evolution results. As bounded confidence grows, population opinions experience polarization to consensus, leading to the emergence of phase transition from co-existence to winner-takes-all state under binary decisions. Of particular interest, we show how the combined effects of bounded confidence and asymmetry opinion strength may reverse the initial supportive advantage in competitive dynamics. Notably, our model qualitatively reproduces the important dynamical pattern during a brutal competition, namely, cascading collapse, as observed by real data. Finally and intriguingly, we find that individual cognitive heterogeneity can bring about randomness and unpredictability in binary decision-making process, leading to the emergence of indeterministic oscillation. Our results reveal how the diverse behavioral patterns of binary decision-making can be interpreted by the complicated interactions of the proposed elements, which provides important insights toward competitive dynamics

Confirmation bias and peer pressure both have substantial impacts on the formation of collective decisions. Nevertheless, few attempts have been made to study how the interplay between these two mechanisms affects public opinion evolution. Here we propose an agent-based model of opinion dynamics which incorporates the conjugate effect of confirmation bias (characterized by the population identity scope and individual stubbornness) and peer pressure (described by a susceptibility threshold). We show that the number of opinion fragments first increases and then decreases to one as population identity scope becomes larger in a homogeneous population. Further, in heterogeneous situations, we find that even a small fraction of impressionable individuals who are sensitive to peer pressure could help eliminate public polarization when population identity scope is relatively large. Intriguingly, we highlight the emergence of 'impressionable moderates' who are easily influenced, hold wavering opinions, and are of vital significance in competitive campaigns.

Community detection is a fundamental and important problem in network science, as community structures often reveal both topological and functional relationships between different components of the complex system. In this paper, we first propose a gradient descent framework of modularity optimization called vector-label propagation algorithm (VLPA), where a node is associated with a vector of continuous community labels instead of one label. Retaining weak structural information in vector-label, VLPA outperforms some well-known community detection methods, and particularly improves the performance in networks with weak community structures. Further, we incorporate stochastic gradient strategies into VLPA to avoid stuck in the local optima, leading to the stochastic vector-label propagation algorithm (sVLPA). We show that sVLPA performs better than Louvain Method, a widely used community detection algorithm, on both artificial benchmarks and real-world networks. Our theoretical scheme based on vector-label propagation can be directly applied to high-dimensional networks where each node has multiple features, and can also be used for optimizing other partition measures such as modularity with resolution parameters.

Prisoner's dilemma game is the most commonly used model of spatial evolutionary game which is considered as a paradigm to portray competition among selfish individuals. In recent years, Win-Stay-Lose-Learn, a strategy updating rule base on aspiration, has been proved to be an effective model to promote cooperation in spatial prisoner's dilemma game, which leads aspiration to receive lots of attention. But in many research the assumption that individual's aspiration is fixed is inconsistent with recent results from psychology. In this paper, according to Expected Value Theory and Achievement Motivation Theory, we propose a dynamic aspiration model based on Win-Stay-Lose-Learn rule in which individual's aspiration is inspired by its payoff. It is found that dynamic aspiration has a significant impact on the evolution process, and different initial aspirations lead to different results, which are called Stable Coexistence under Low Aspiration, Dependent Coexistence under Moderate aspiration and Defection Explosion under High Aspiration respectively. Furthermore, a deep analysis is performed on the local structures which cause cooperator's existence or defector's expansion, and the evolution process for different parameters including strategy and aspiration. As a result, the intrinsic structures leading to defectors' expansion and cooperators' survival are achieved for different evolution process, which provides a penetrating understanding of the evolution. Compared to fixed aspiration model, dynamic aspiration introduces a more satisfactory explanation on population evolution laws and can promote deeper comprehension for the principle of prisoner's dilemma., Comment: 17 pages, 13 figures

Community detection is of fundamental significance for understanding the topology characters and the spreading dynamics on complex networks. While random walk is widely used and is proven effective in many community detection algorithms, there still exists two major defects: (i) the maximal length of random walk is too large to distinguish the clustering information if using the average step of all possible random walks; (ii) the useful community information at all other step lengths are missed if using a pre-assigned maximal length. In this paper, we propose a novel community detection method based on the first passage probabilities (FPPM), equipped with a new similarity measure that incorporates the complete structural information within the maximal step length. Here the diameter of the network is chosen as an appropriate boundary of random walks which is adaptive to different networks. Then we use the hierarchical clustering to group the vertices into communities and further select the best division through the corresponding modularity values. Finally, a post-processing strategy is designed to integrate the unreasonable small communities, which significantly improves the accuracy of community division. Surprisingly, the numerical simulations show that FPPM performs best compared to several classic algorithms on both synthetic benchmarks and real-world networks, which reveals the universality and effectiveness of our method.

Social reinforcement mechanism, which characterizes the promoting effects when exposing to multiple sources in social contagion process, is ubiquitous in information-technology ecosystem and has aroused great attention in recent years. While the impacts of social reinforcement on single-layer networks are well-documented, extension to multilayer networks is needed to study how reinforcement from different social circles influences the spreading dynamics. To this end, we incorporate multilayer social reinforcement into ignorant-spreader-ignorant (SIS) model on multiplex networks. Our theoretical analysis combines pairwise method and mean-field theory and agrees well with large-scale simulations. Surprisingly, we find this complex social contagion mechanism triggers the emergence of bistability phenomena, where extinction and outbreak states coexist. In particular, the hysteresis loop of stationary prevalence occurs in this bistable region, explaining why the fight towards the spread of rumors is protracted and difficult in modern society. Further, we show that the final state of bistable regions depends on the initial density of adopters, the critical value of which decreases as the contagion transmissibility or the multilayer reinforcement increases. In particular, we highlight two possible conditions for the outbreak of social contagion: to possess large contagion transmissibility, or to possess large initial density of adopters with strong multilayer reinforcement. Our results unveil the non-negligible power of social reinforcement on multiplex networks, which sheds lights on designing efficient strategies in spreading behaviors such as marketing and promoting innovations.

In recent years, social media has increasingly become an important platform for political campaigns, especially elections. It remains elusive how exactly public discourse is driven by the intricate interplay between individual socio-cognitive biases, dueling campaign efforts, and social media platforms. We examine this complex socio-political process by integrating observed retweet networks from the 2016 political networks with an agent-based model of political opinion formation and network structure. Here we show that the range of political viewpoints individuals are willing to consider is a key determinant in the formation of polarized networks and the emergence of echo chambers. We also find that winning majority support in public discourse is determined by both the effort exerted by campaigns and the relative ideological positioning of opposing campaigns. Our results demonstrate how public discourse and political polarization can be modeled as an interactive process of shifting individual opinions, evolving social networks, and political campaigns.

Competitive information diffusion on large-scale social networks reveals fundamental characteristics of rumor contagions and has profound influence on public opinion formation. There has been growing interest in exploring dynamical mechanisms of the competing evolutions recently. Nevertheless, the impacts of population homophily, which determines powerful collective human behaviors, remains unclear. In this paper, we incorporate homophily effects into a modified competitive ignorant-spreader-ignorant (SIS) rumor diffusion model with generalized population preference. Using microscopic Markov chain approach, we first derive the phase diagram of competing diffusion results and examine how competitive information spreads and evolves on social networks. We then explore the detailed effects of homophily, which is modeled by a rewiring mechanism. Results show that homophily promotes the formation of divided "echo chambers" and protects the disadvantaged information from extinction, which further changes or even reverses the evolutionary advantage, i.e., the difference of final proportions of the competitive information. We highlight the conclusion that the reversals may happen only when the initially disadvantaged information has stronger transmission ability, owning diffusion advantage over the other one. Our framework provides profound insight into competing dynamics with population homophily, which may pave ways for further controlling misinformation and guiding public belief systems. Moreover, the reversing condition sheds light on designing effective competing strategies in many real scenarios.

Feedback loops between population dynamics of individuals and their ecological environment are ubiquitously found in nature, and have shown profound effects on the resulting eco-evolutionary dynamics. Incorporating linear environmental feedback law into replicator dynamics of two-player games, recent theoretical studies shed light on understanding the oscillating dynamics of social dilemma. However, detailed effects of more general nonlinear feedback loops in multi-player games, which is more common especially in microbial systems, remain unclear. Here, we focus on ecological public goods games with environmental feedbacks driven by nonlinear selection gradient. Unlike previous models, multiple segments of stable and unstable equilibrium manifolds can emerge from the population dynamical systems. We find that a larger relative asymmetrical feedback speed for group interactions centered on cooperators not only accelerates the convergence of stable manifolds, but also increases the attraction basin of these stable manifolds. Furthermore, our work offers an innovative manifold control approach: by designing appropriate switching control laws, we are able to steer the eco-evolutionary dynamics to any desired population states. Our mathematical framework is an important generalization and complement to coevolutionary game dynamics, and also fills the theoretical gap in guiding the widespread problem of population state control in microbial experiments.

Models of contagion dynamics, originally developed for infectious diseases, have proven relevant to the study of information, news, and political opinions in online social systems. Modelling diffusion processes and predicting viral information cascades are important problems in network science. Yet, many studies of information cascades neglect the variation in infectivity across different pieces of information. Here, we employ early-time observations of online cascades to estimate the infectivity of distinct pieces of information. Using simulations and data from real-world Twitter retweets, we demonstrate that these estimated infectivities can be used to improve predictions about the virality of an information cascade. Developing our simulations to mimic the real-world data, we consider the effect of the limited effective time for transmission of a cascade and demonstrate that a simple model for slow but non-negligible decay of the infectivity captures the essential properties of retweet distributions. These results demonstrate the interplay between the intrinsic infectivity of a tweet and the complex network environment within which it diffuses, strongly influencing the likelihood of becoming a viral cascade., Comment: 16 pages, 10 figures

The stability of Boolean networks has attracted much attention due to its wide applications in describing the dynamics of biological systems. During the past decades, much effort has been invested in unveiling how network structure and update rules will affect the stability of Boolean networks. In this paper, we aim to identify and control a minimal set of influential nodes that is capable of stabilizing an unstable Boolean network. By minimizing the largest eigenvalue of a modified non-backtracking matrix, we propose a method using the collective influence theory to identify the influential nodes in Boolean networks with high computational efficiency. We test the performance of collective influence on four different networks. Results show that the collective influence algorithm can stabilize each network with a smaller set of nodes than other heuristic algorithms. Our work provides a new insight into the mechanism that determines the stability of Boolean networks, which may find applications in identifying the virulence genes that lead to serious disease.

When analyzing the statistical and topological characteristics of complex networks, an effective and convenient way is to compute the centralities for recognizing influential and significant nodes or structures, yet most of them are restricted to local environment or some specific configurations. In this paper we propose a new centrality for nodes based on the von Neumann entropy, which allows us to investigate the importance of nodes in the view of spectrum eigenvalues distribution. By presenting the performances of this centrality with network examples in reality, it is shown that the von Neumann entropy node centrality is an excellent index for selecting crucial nodes as well as classical ones. Then to lower down the computational complexity, an approximation calculation to this centrality is given which only depends on its first and second neighbors. Furthermore, in the optimal spreader problem and reducing average clustering coefficients, this entropy centrality presents excellent efficiency and unveil topological structure features of networks accurately. The entropy centrality could reduce the scales of giant connected components fastly in Erdos-Renyi and scale-free networks, and break down the cluster structures efficiently in random geometric graphs. This new methodology reveals the node importance in the perspective of spectrum, which provides a new insight into networks research and performs great potentials to discover essential structural features in networks., Comment: 11 pages, 7 figures, 3 tables

Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Levy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Levy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a Levy random walk is the most efficient strategy. Our results give us a deeper understanding of Levy random walks and show the importance of considering the topological structure of multiplex networks when trying to find efficient navigation strategies., Comment: 11 pages and 6 figures

The evolution of network structure and the spreading of epidemic are common coexistent dynamical processes. In most cases, network structure is treated either static or time-varying, supposing the whole network is observed in a same time window. In this paper, we consider the epidemic spreading on a network consisting of both static and time-varying structures. At meanwhile, the time-varying part and the epidemic spreading are supposed to be of the same time scale. We introduce a static and activity driven coupling (SADC) network model to characterize the coupling between static (strong) structure and dynamic (weak) structure. Epidemic thresholds of SIS and SIR model are studied on SADC both analytically and numerically with various coupling strategies, where the strong structure is of homogeneous or heterogeneous degree distribution. Theoretical thresholds obtained from SADC model can both recover and generalize the classical results in static and time-varying networks. It is demonstrated that weak structures can make the epidemics break out much more easily in homogeneous coupling but harder in heterogeneous coupling when keeping same average degree in SADC networks. Furthermore, we show there exists a threshold ratio of the weak structure to have substantive effects on the breakout of the epidemics. This promotes our understanding of why epidemics can still break out in some social networks even we restrict the flow of the population.

Information spreading in online social communities has attracted tremendous attention due to its utmost practical values in applications. Despite that several individual-level diffusion data have been investigated, we still lack the detailed understanding of the spreading pattern of information. Here, by comparing information flows and social links in a blog community, we find that the diffusion processes are induced by three different spreading mechanisms: social spreading, self-promotion and broadcast. Although numerous previous studies have employed epidemic spreading models to simulate information diffusion, we observe that such models fail to reproduce the realistic diffusion pattern. In respect to users behaviors, strikingly, we find that most users would stick to one specific diffusion mechanism. Moreover, our observations indicate that the social spreading is not only crucial for the structure of diffusion trees, but also capable of inducing more subsequent individuals to acquire the information. Our findings suggest new directions for modeling of information diffusion in social systems and could inform design of efficient propagation strategies based on users behaviors.

Detecting spreading outbreaks in social networks with sensors is of great significance in applications. Inspired by the formation mechanism of human's physical sensations to external stimuli, we propose a new method to detect the influence of spreading by constructing excitable sensor networks. Exploiting the amplifying effect of excitable sensor networks, our method can better detect small-scale spreading processes. At the same time, it can also distinguish large-scale diffusion instances due to the self-inhibition effect of excitable elements. Through simulations of diverse spreading dynamics on typical real-world social networks (facebook, coauthor and email social networks), we find that the excitable senor networks are capable of detecting and ranking spreading processes in a much wider range of influence than other commonly used sensor placement methods, such as random, targeted, acquaintance and distance strategies. In addition, we validate the efficacy of our method with diffusion data from a real-world online social system, Twitter. We find that our method can detect more spreading topics in practice. Our approach provides a new direction in spreading detection and should be useful for designing effective detection methods.

Identifying highly susceptible individuals in spreading processes is of great significance in controlling outbreaks. In this paper, we explore the susceptibility of people in susceptible-infectious-recovered (SIR) and rumor spreading dynamics. We first study the impact of community structure on people's susceptibility. Despite that the community structure can reduce the infected population given same infection rates, it will not deterministically affect nodes' susceptibility. We find the susceptibility of individuals is sensitive to the choice of spreading dynamics. For SIR spreading, since the susceptibility is highly correlated to nodes' influence, the topological indicator k-shell can better identify highly susceptible individuals, outperforming degree, betweenness centrality and PageRank. In contrast, in rumor spreading model, where nodes' susceptibility and influence have no clear correlation, degree performs the best among considered topological measures. Our finding highlights the significance of both topological features and spreading mechanisms in identifying highly susceptible population.

The topic of finding effective strategy to halt virus in complex network is of current interest. We propose an immunization strategy for seasonal epidemics that occur periodically. Based on the local information of the infection status from the previous epidemic season, the selection of vaccinated nodes is optimized gradually. The evolution of vaccinated nodes during iterations demonstrates that the immunization tends to locate in both global hubs and local hubs. We analyze the epidemic prevalence by a heterogeneous mean-field method and present numerical simulations of our model. This immunization performs superiorly to some other previously known strategies. Our work points out a new direction in immunization of seasonal epidemics., Comment: 8 pages, 9 figures

A number of predictors have been suggested to detect the most influential spreaders of information in online social media across various domains such as Twitter or Facebook. In particular, degree, PageRank, k-core and other centralities have been adopted to rank the spreading capability of users in information dissemination media. So far, validation of the proposed predictors has been done by simulating the spreading dynamics rather than following real information flow in social networks. Consequently, only model-dependent contradictory results have been achieved so far for the best predictor. Here, we address this issue directly. We search for influential spreaders by following the real spreading dynamics in a wide range of networks. We find that the widely-used degree and PageRank fail in ranking users' influence. We find that the best spreaders are consistently located in the k-core across dissimilar social platforms such as Twitter, Facebook, Livejournal and scientific publishing in the American Physical Society. Furthermore, when the complete global network structure is unavailable, we find that the sum of the nearest neighbors' degree is a reliable local proxy for user's influence. Our analysis provides practical instructions for optimal design of strategies for "viral" information dissemination in relevant applications., Comment: 12 pages, 7 figures

We introduce a non-interacting boson model to investigate topological structure of complex networks in the present paper. By exactly solving this model, we show that it provides a powerful analytical tool in uncovering the important properties of real-world networks. We find that the ground state degeneracy of this model is equal to the number of connected components in the network and the square of coefficients in the expansion of ground state gives the averaged time for a random walker spending at each node in the infinite time limit. Furthermore, the first excited state appears always on its largest connected component. To show usefulness of this approach in practice, we carry on also numerical simulations on some concrete complex networks. Our results are completely consistent with the previous conclusions derived by graph theory methods., Comment: 4 pages, 3 figures