The propagation of failures in networked systems with recovery mechanism can be modeled as failure-recovery propagation processes in complex networks. Considering the adaptive response of nodes against failures, here we study the failure-recovery propagation dynamics in adaptive networks, where an active node intentionally rewires its links to improve its local external environment to reduce failure. We use the effective-degree approach and numerical simulations to analyze the dynamics of adaptive networks. In both synthetic networks and real-world networks, it is found that compared with the static networks where the first-order transition and hysteresis loop exist, the failure-recovery dynamics in adaptive networks exhibits a hysteresis loop in a much smaller parameter region of the failure rates, and with the increase of rewiring rate, the phase transition becomes continuous and the hysteresis loop disappears. This implies an opposite effect of rewiring to that in epidemic dynamics where the first-order transition and hysteresis loops are induced by adaptive rewiring. Counter-intuitively, the adaptive rewiring behaviors do not necessarily result in a reduction in the systems failure size, but even lead to an increase in the failure size of the system. There exists an optimal rewiring rate at which a minimal failure size can be achieved.

The 2025 Bondi Beach mass shooting of Jews was perpetrated by individuals inspired by ISIS (Islamic State) propaganda that increasingly featured anti-Semitic hate content following the October 2023 start of the Israel-Palestine war. There is an urgent need to get ahead of future threats by understanding how and when a newly created piece of hate content will spread systemwide online. We present a two-species coalescence-fragmentation model with susceptible-infected-recovered dynamics that incorporates the following published empirical features: (1) New pieces of hate content tend to be generated and promoted by a subset of in-built communities on less regulated platforms. (2) These ''hate'' communities create links (hyperlinks) with each other and with nonhate communities across all platforms to form dynamically evolving clusters (i.e., coalescence) across which new hate content can then spread. (3) These clusters can get broken up by moderator shutdowns (i.e., fragmentation). We present numerical solutions and derive two levels of approximate mean-field theory: effective medium theory and beyond effective medium theory. Both numerical and analytic solutions reveal that systemwide spreading is governed by reentrant threshold phases: as the fraction of hate communities varies, the system can transition from spreading to no spreading and back to spreading. The derived analytic formulas give explicit insight into how these phase boundaries might be manipulated to prevent systemwide spreading. More broadly, the reentrant phase behavior warns that policies which steadily reduce the number of hate communities can initially succeed but then backfire if pushed further, suggesting that blanket requirements for platforms to simply do ''more'' are oversimplistic.

The 2025 Bondi Beach mass-shooting was perpetrated by individuals inspired by ISIS (Islamic State) propaganda that increasingly featured anti-Semitic hate content following the October 2023 start of the Israel-Palestine war. Similar stories hold for other types of hate attacks, e.g. against Muslims on May 18, 2026. There is an urgent need to get ahead of future threats by understanding how and when a newly created piece of hate content will spread system-wide online. We present a two-species coalescence-fragmentation model with Susceptible-Infected-Recovered dynamics that incorporates the following published empirical features: (1) New pieces of hate content tend to be generated and promoted by a subset of in-built communities on less regulated platforms. (2) These `hate' communities create links (hyperlinks) with each other and with non-hate communities across all platforms to form dynamically evolving clusters (i.e. coalescence) across which new hate content can then spread. (3) These clusters can get broken up by moderator shutdowns (i.e. fragmentation). We present numerical solutions and derive two levels of approximate mean-field theory: Effective Medium Theory (EMT) and Beyond Effective Medium Theory (BEMT). Both numerical and analytic solutions reveal that system-wide spreading is governed by re-entrant threshold phases: as the fraction of hate communities varies, the system can transition from spreading to no-spreading and back to spreading. The derived analytic formulae give explicit insight into how these phase boundaries might be manipulated to prevent system-wide spreading. More broadly, the re-entrant phase behavior warns that policies which steadily reduce the number of hate communities can initially succeed but then backfire if pushed further, suggesting that blanket requirements for platforms to simply do `more' are over-simplistic.

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order alpha where 0 < alpha <= 1. We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as n(k) = p(2p-1)(2-alpha)/p(1-alpha) Gamma(alpha + 2-alpha/1-p) Gamma(k)/Gamma(k-1+alpha+2-alpha/1-p) t for alpha is an element of (0, 1], where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for alpha = 1. The results corresponding to various values of alpha and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both alpha and p, thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.

The dissemination of information and opinion to the public has never been easier due to the easy access to platforms, making it necessary to study the responses to public information by a competing population. We study the impact of an expert’s opinion on the performance of a population including some die-hard fans of the expert and cautious agents who will adapt to trust or to ignore the expert’s opinion, within the framework of the minority game (MG). The expert collects information by probing a subpopulation for its minority opinion without the knowledge of its composition. When the information is not disseminated, the expert’s opinion has a higher accuracy than the intrinsic success rate of the population. When the information is disseminated, the population could enhance its success rate by making use of the wasted winning quotas through the actions of the fans and the adaptive ability of the cautious agents, as long as the fraction of the fans is not too big to turn the expert’s opinion into the losing option. There exists an optimal fraction of followers, the fans plus those cautious agents adapted to trust the expert, for the population to be the most harmonious with the highest success rate allowed by MG. The phenomena observed by numerical simulations motivated us to develop a theory, using the intrinsic success rate SrI, a quantity related to how the expert collects his information, and the average adapted level of trust f¯ among the cautious agents as inputs. The theory gives excellent agreement with simulation results, even when an approximated form of f¯ is invoked. The work paves the way for studying and formulating theories for the dissemination of other forms of information and opinion.

The cooperative behavior in a population engaging in Public Goods Games (PGG) with players in dynamic groups of various sizes and memberships, and a learning mechanism for switching strategy by comparing payoffs between two randomly chosen players each engaging in their own group is studied. We demonstrate that the model has the merit of allowing for an analytic treatment. Upon averaging over all possible group sizes and group constituents in PGG, we derive an analytic expression of the time evolution of the frequency of cooperation fc(t). Starting from an initial cooperative level fc(0), the population will evolve either to a 100% cooperative (AllC) or a 100% non-cooperative (AllD) state depending on the sign of a parameter K∝(r〈1/g〉−1), where r is the multiplicative factor in PGG and 〈1/g〉 is the inverse first moment of the group size distribution of players engaging in PGG. For K>0 (K<0), the system approaches an AllC (AllD) state with an exponential temporal behavior of exp(−2|K|t/M), where M is the population size. The parameter K suggests that cooperation can be promoted by suitably adjusting the multiplicative factor r to be above a critical value rc≡1/〈1/g〉. All the features predicted by the analytic results, including fc(t), the effects of the multiplicative factor r and group size distribution, and how the time constant depends on the parameter K, are confirmed by detailed numerical simulations of the model. The work sheds light on how the varying group sizes, especially the smaller groups, help promote cooperation in a PGG setting.

We present a minimal yet empirically-grounded theory for the spread of online harms (e.g. misinformation, hate) across current multi-platform social media and future Metaverses. New physics emerges from the interplay between the intrinsic heterogeneity among online communities and platforms, their clustering dynamics generated through user-created links and sudden moderator shutdowns, and the contagion process. The theory provides an online `R-nought' criterion to prevent system-wide spreading; it predicts re-entrant spreading phases; it establishes the level of digital vaccination required for online herd immunity; and it can be applied at multiple scales.

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equation that determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

The evolution of cooperation is studied within the context of an evolutionary snowdrift game on three special networks chosen to have the same uniform degree but with different extents of spatial correlations. The cooperative behaviors on the networks differ in where the phase transitions take place and the frequencies of cooperation in the mixed phase. With only the spatial correlations being different, the study allows us to gauge the accuracy of different theoretical approaches and shed light on the choice of theoretical approaches in handling spatial correlations. It is found that analyzing the last surviving patterns often provides an understanding in the transitions in the cooperative behavior. We also construct a theoretical framework to describe the dynamical process. For different approaches of incorporating the agents’ spatial correlations, the local configuration approximation captures all the features observed in numerical simulations, while the commonly used pair approximation is too crude for quantitative purposes in studying cooperation, especially on networks with more complicated spatial correlations.

Non-Markovian spontaneous recovery processes with a time delay (memory) are ubiquitous in the real world. How does the non-Markovian characteristic affect failure propagation in complex networks? We consider failures due to internal causes at the nodal level and external failures due to an adverse environment, and develop a pair approximation analysis taking into account the two-node correlation. In general, a high failure stationary state can arise, corresponding to large-scale failures that can significantly compromise the functioning of the network. We uncover a striking phenomenon: memory associated with nodal recovery can counter-intuitively make the network more resilient against large-scale failures. In natural systems, the intrinsic non-Markovian characteristic of nodal recovery may thus be one reason for their resilience. In engineering design, incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist catastrophic failures.

The conventional notion of community that favors a high ratio of internal edges to outbound edges becomes invalid when each vertex participates in multiple communities. Such a behavior is commonplace in social networks. The significant overlaps among communities make most existing community detection algorithms ineffective. The lack of effective and efficient tools resulted in very few empirical studies on large-scale detection and analyses of overlapping community structure in real social networks. We developed recently a scalable and accurate method called the Partial Community Merger Algorithm (PCMA) with linear complexity and demonstrated its effectiveness by analyzing two online social networks, Sina Weibo and Friendster, with 79.4 and 65.6 million vertices, respectively. Here, we report in-depth analyses of the 2.9 million communities detected by PCMA to uncover their complex overlapping structure. Each community usually overlaps with a significant number of other communities and has far more outbound edges than internal edges. Yet, the communities remain well separated from each other. Most vertices in a community are multi-membership vertices, and they can be at the core or the peripheral. Almost half of the entire network can be accounted for by an extremely dense network of communities, with the communities being the vertices and the overlaps being the edges. The empirical findings ask for rethinking the notion of community, especially the boundary of a community. Realizing that it is how the edges are organized that matters, the f-core is suggested as a suitable concept for overlapping community in social networks. The results shed new light on the understanding of overlapping community.

The options for agents susceptible to an infection of taking preventive measures with a cost or to expose themselves to an acceptable level of risk of being infected are incorporated into an epidemic model within a game theoretical framework. A susceptible agent decides on his action by assessing the risk posed by his neighborhood, the cost-to-benefit ratio c∕b and the infection probability. In a well-mixed population, the evolution and long time limit of the densities of infected and susceptible agents are studied for different values of the parameters, and the results can be described well by a set of dynamical equations in all cases. The epidemic model is also studied within a dynamical 1-group situation in a population where only agents who are simultaneously present in the common place can make contact. While the 1-group grows and fragments in time repeatedly, the infection profiles show different behavior: from a resurgent behavior following the 1-group dynamics to one that decouples from the grouping dynamics and everything in between. Formulating the co-evolving dynamics mathematically is a challenging task. We demonstrate the necessity of setting up separate sets of dynamical equations for the growing and fragmented stages. The work sheds light on the debate around whether vaccinations should be imposed and whether unvaccinated students should be allowed to go to school amid the recent measles outbreak, as well as on the proper mathematical formulation of co-evolving problems involving contacts among agents only in a popular place.

Understanding how systems with many semi-autonomous parts reach a desired target is a key question in biology (e.g., Drosophila larvae seeking food), engineering (e.g., driverless navigation), medicine (e.g., reliable movement for brain-damaged individuals), and socioeconomics (e.g., bottom-up goal-driven human organizations). Centralized systems perform better with better components. Here, we show, by contrast, that a decentralized entity is more efficient at reaching a target when its components are less capable. Our findings reproduce experimental results for a living organism, predict that autonomous vehicles may perform better with simpler components, offer a fresh explanation for why biological evolution jumped from decentralized to centralized design, suggest how efficient movement might be achieved despite damaged centralized function, and provide a formula predicting the optimum capability of a system’s components so that it comes as close as possible to its target or goal.

We study the effects of biased selection on evolutionary games in finite populations. Biased selection is induced by including a number of invariant agents, who do not evolve, into an otherwise competing and evolving population. The invariant agents react differently when they encounter different types of variant agents, helping the cooperators to attain a higher payoff than the defectors by a difference Δ. The probability of a single cooperator invading and taking over a population of all defectors is evaluated based on the Moran process. When M invariant agents are introduced into the evolving population of size N, their presence helps promote the replacement of defectors by cooperators, in comparison with the case of unbiased selection with M=0. While the prisoner’s dilemma cannot sustain a cooperative population without the invariant agents, it is possible for cooperation to emerge with biased selection. We show that for a given N (M), there is a threshold of M (N) above (below) which an initial population of all defectors evolve into one of all cooperators. The dependence of the threshold on the game’s payoffs, intensity of selection, and the values of M and Δ is studied. Beyond the prisoner’s dilemma, biased selection also promotes cooperation in a bigger part of payoff parameter space corresponding to other types of games.

One of the biggest challenges in unravelling the complexity of living systems, is to fully understand the neural logic that translates sensory input into the highly nonlinear motor outputs that are observed when simple organisms crawl. Recent work has shown that organisms such as larvae that exhibit klinotaxis (i.e., orientation through lateral movements of portions of the body) can perform normal exploratory practices even in the absence of a brain. Abdominal and thoracic networks control the alternation between crawls and turns. This motivates the search for decentralized models of movement that can produce nonlinear outputs that resemble the experiments. Here, we present such a complex system model, in the form of a population of decentralized decision-making components (agents) whose aggregate activity resembles that observed in klinotaxis organisms. Despite the simplicity of each component, the complexity created by their collective feedback of information and actions akin to proportional navigation, drives the model organism towards a specific target. Our model organism’s nonlinear behaviors are consistent with empirically observed reorientation rate measures for Drosophila larvae as well as nematode C. elegans.

Evolutionary games with a self-questioning action updating mechanism have recently been mapped onto the Ising model. We discuss here an alternative viewpoint in making a connection between two fields of research that each has much to offer to the other.

Multiple stable states, hysteresis, sensitivity to initial distributions, and a control algorithm for promoting cooperation are studied in an evolutionary prisoner’s dilemma with agents connected into a regular random network. A system could evolve into states of different cooperative frequencies xc in different runs, even starting with the same initial cooperative frequency xc(in) and payoff parameters. For a large reward R, some values of xc(in) either take the system to a group of low cooperative frequency (LCF) states or to a few high cooperative frequency (HCF) states. These states differ by their network structures, with cooperative players connected into ring-like structure in LCF states and compact clusters in HCF states. Hysteresis in xc is observed when R is swept down and up, when the final state of the previous R is used as the initial state of the next R. The analysis led us to propose a closed pack cluster algorithm that gives HCF states effectively. The algorithm intervenes the system at some point in time by selectively switching some non-cooperative D-agents into cooperative C-agents at the peripheral of an existing cluster of C-agents. It ensures protection of a small C-cluster from which more cooperation can be induced. Practically, a governing body may first allow a society to evolve freely and then derive suitable policy to promote selected pockets of good practices for attaining a higher level of common good.

Resources are limited in epidemic containment; how to optimally allocate the limited resources in suppressing the epidemic spreading has been a challenging problem. To find an effective resource allocation strategy, we take the infectiousness of each infected node into consideration. By studying the interplay between the resource allocation and epidemic spreading, we find that the spreading dynamics of epidemic is affected by the preferential resource allocation. There are double phase transitions of the fraction of infected nodes, which are different from the classical epidemic model. More importantly, we find that the preferential resource allocation has double-edged sword effects on the disease spreading. When there is a small transmission rate, the infected fraction at the steady state decreases with the increment of degree of resource allocation preference, which indicates that resources of the healthy nodes should be allocated preferentially to the high infectious nodes to constrain the disease spreading. Moreover, when there is a large transmission rate, the fraction of infected nodes at the steady state increases with the increment of the degree of the preference, but the resource allocation is determined by the stage of epidemic spreading. Namely, in the early stage of the disease spreading, resources should be allocated preferentially to the high infectious nodes similar to the case of a small transmission rate. While after the early stage, resources should be allocated to the low infectious nodes. Based on the findings, we propose a simple resource allocation strategy that can adaptively change with the current fraction of infected nodes and the disease can be suppressed to the most extent under the proposed strategy.

Anisotropic strategies are introduced and studied numerically and analytically in the context of prisoner’s dilemma (PD) played in a system of connected agents for their effects on the evolution of cooperation. For an agent competing with k neighbors, an anisotropic strategy has k entries representing the different actions against each of the opponents. The commonly used cooperative and non-cooperative strategies against all opponents are isotropic and special cases of anisotropic strategies. For evolutionary PD based on a death–birth process in dynamically formed competing groups, selection results in a totally non-cooperative state but the dynamics is altered by the anisotropic strategies. For agents connected by static networks having a uniform degree, the cooperative level exhibits plateaux as the temptation payoff increases. The plateaux represent the dominance of different classes of anisotropic strategies with decreasing cooperative entries. Dynamical equations with results in good agreement with simulation results on the evolution of strategies and the long time behavior are constructed. For static networks with a spread in degrees, the plateaux disappear except for the one corresponding to a totally cooperative population. For all cases, anisotropic strategies are found to maintain a cooperative level no less than that in a corresponding system in which only isotropic strategies are in play for high temptation payoffs. Anisotropic strategies suggest an alternative mechanism for enhancing cooperation in PD and other games that is worthy of further investigations.

What happens when you slow down the delivery of information in large-scale complex systems that operate faster than the blink of an eye? This question just adopted immediate commercial, legal and political importance following U.S. regulators’ decision to allow an intentional 350 microsecond delay to be added in the ultrafast network of financial exchanges. However there is still no scientific understanding available to policymakers of the potential system-wide impact of such delays. Here we take a first step in addressing this question using a minimal model of a population of competing, heterogeneous, adaptive agents which has previously been shown to produce similar statistical features to real markets. We find that while certain extreme system-level behaviors can be prevented by such delays, the duration of others is increased. This leads to a highly non-trivial relationship between delays and system-wide instabilities which warrants deeper empirical investigation. The generic nature of our model suggests there should be a fairly wide class of complex systems where such delay-driven extreme behaviors can arise, e.g. sub-second delays in brain function possibly impacting individuals’ behavior, and sub-second delays in navigational systems potentially impacting the safety of driverless vehicles.