Evolutionary dynamics on sequential temporal networks.

Population structure is a well-known catalyst for the evolution of cooperation and has traditionally been considered to be static in the course of evolution. Conversely, real-world populations, such as microbiome communities and online social networks, frequently show a progression from tiny, active groups to huge, stable communities, which is insufficient to be captured by constant structures. Here, we propose sequential temporal networks to characterize growing networked populations, and we extend the theory of evolutionary games to these temporal networks with arbitrary structures and growth rules. We derive analytical rules under which a sequential temporal network has a higher fixation probability for cooperation than its static counterpart. Under neutral drift, the rule is simply a function of the increment of nodes and edges in each time step. But if the selection is weak, the rule is related to coalescence times on networks. In this case, we propose a mean-field approximation to calculate fixation probabilities and critical benefit-to-cost ratios with lower calculation complexity. Numerical simulations in empirical datasets also prove the cooperation-promoting effect of population growth. Our research stresses the significance of population growth in the real world and provides a high-accuracy approximation approach for analyzing the evolution in real-life systems.

Spatial reciprocity in the evolution of cooperation.

Cooperation is key to the survival of all biological systems. The spatial structure of a system constrains who interacts with whom (interaction partner) and who acquires new traits from whom (role model). Understanding when and to what degree a spatial structure affects the evolution of cooperation is an important and challenging topic. Here, we provide an analytical formula to predict when natural selection favours cooperation where the effects of a spatial structure are described by a single parameter. We find that a spatial structure promotes cooperation (spatial reciprocity) when interaction partners overlap role models. When they do not, spatial structure inhibits cooperation even without cooperation dilemmas. Furthermore, a spatial structure in which individuals interact with their role models more often shows stronger reciprocity. Thus, imitating individuals with frequent interactions facilitates cooperation. Our findings are applicable to both pairwise and group interactions and show that strong social ties might hinder, while asymmetric spatial structures for interaction and trait dispersal could promote cooperation.

Coevolution of nonlinear group interactions and strategies in well-mixed and structured populations.

In microbial populations and human societies, the rule of nonlinear group interactions strongly affects the intraspecific evolutionary dynamics, which leads to the variation of the strategy composition eventually. The consequence of such variation may retroact to the rule of the interactions. This correlation indicates that the rule of nonlinear group interactions may coevolve with individuals' strategies. Here, we develop a model to investigate such coevolution in both well-mixed and structured populations. In our model, positive and negative correlations between the rule and the frequency of cooperators are considered, with local and global information. When the correlation refers to the global information, we show that in well-mixed populations, the coevolutionary outcomes cover the scenarios of defector dominance, coexistence, and bi-stability. Whenever the population structure is considered, its impact on the coevolutionary dynamics depends on the type of the correlation: with a negative (positive) correlation, population structure promotes (inhibits) the evolution of cooperation. Furthermore, when the correlation is based on the more accessible local information, we reveal that a negative correlation pushes cooperators into a harsh situation whereas a positive one lowers the barriers for cooperators to occupy the population. All our analytical results are validated by numerical simulations. Our results shed light on the power of the coevolution of nonlinear group interactions and evolutionary dynamics on generating various evolutionary outcomes, implying that the coevolutionary framework may be more appropriate than the traditional cases for understanding the evolution of cooperation in both structureless and structured populations.

Evolution of cooperation with interactive identity and diversity.

Interactive identity and interactive diversity are generally regarded as two typical interaction patterns in living systems. The former describes that in each generation every individual behaves identically to all of its opponents, and the latter allows each individual to behave diversely to its distinct opponents. Most traditional research on the evolution of cooperation, however, has been confined to populations with a uniform interaction pattern. Here we study the cooperation conundrum in a diverse population comprising players with interactive identity and with interactive diversity. We find that in homogeneous networks a small fraction of players taking interactive diversity are enough to stabilize cooperation for a wide range of payoff values even in a noisy environment. When assigned to heterogeneous networks, players in high-degree nodes taking interactive diversity significantly strengthen systems' resilience against the shifty environment and enlarge the survival region of cooperation. However, they fail to establish a homogeneous strategy 'cloud' in the neighborhood and thus can not coordinate players in low-degree nodes to reach a socially optimal cooperation level. The most favorable outcome emerges when players in high-degree nodes take interactive identity and meanwhile others adopt interactive diversity. Our findings reveal the significance of the two typical interaction patterns and could be a good heuristic in coordinating them to achieve the social optimum in cooperation.

Evolutionary dynamics of synergistic and discounted group interactions in structured populations.

The emergence of cooperation between unrelated individuals enables researchers to study how the collective cooperative behavior survives in a world where egoists could get more short-term benefits. The spatial multi-player games, which invoke interactions between individuals who are not directly linked by the interactive networks, are drawing more and more attention in exploring the evolution of cooperation. Here we address the evolutionary dynamics in infinite structured populations with discounted, linear, and synergistic group interactions. The five classical scenarios are recovered from the dynamics: (i) dominating defection, (ii) dominating cooperation, (iii) co-existence, (iv) bi-stability, and (v) neutral variants. For linear interactions, the evolutionary dynamics is equivalent to that in finite as well as the well-mixed counterparts, which can be achieved by a payoff matrix transformation, and it illustrates that the more neighbors there are, the harder the cooperators survive. Yet both cooperation and defection emerge easier in finite populations than in infinite for discounted and synergistic interactions. Counterintuitively, we find that the synergistic group interactions always raise cooperators׳ barriers to occupy the population with the increase of the number of neighbors in infinite structured populations. Our results go against the common belief that synergistic interactions are necessarily beneficial for the cooperative behavior.

Temporal local clustering coefficient uncovers the hidden pattern in temporal networks.

Identifying and extracting topological characteristics are essential for understanding associated structures and organizational principles of complex networks. For temporal networks where the network topology varies with time, beyond the classical patterns such as small-worldness and scale-freeness extracted from the perspective of traditional aggregated static networks, the temporality and simultaneity of time-varying interactions should also be included. Here we extend the traditional analysis on the local clustering coefficient C in static networks and study the dynamical local clustering coefficient of temporal networks. We demonstrate that the temporal local clustering coefficient TC conveys the hidden information of nodes' neighboring connectance when interactions occur at various rhythms. By systematically analyzing various empirical datasets, we find that TC uncovers different interaction patterns in different types of temporal networks. Specifically, we show that TC has a strong positive correlation with C in efficiency-related networks, whereas they are uncorrelated in social activity-related networks. Moreover, TC helps to exclude interference from accidental interactions and reflect the actual clustering properties of network nodes. Our results shed light on the importance of digging into dynamical characteristics to fundamentally understand the underlying temporal structures of real complex systems.

Dynamics of collective cooperation under personalised strategy updates.

Collective cooperation is essential for many social and biological systems, yet understanding how it evolves remains a challenge. Previous investigations report that the ubiquitous heterogeneous individual connections hinder cooperation by assuming individuals update strategies at identical rates. Here we develop a general framework by allowing individuals to update strategies at personalised rates, and provide the precise mathematical condition under which universal cooperation is favoured. Combining analytical and numerical calculations on synthetic and empirical networks, we find that when individuals' update rates vary inversely with their number of connections, heterogeneous connections actually outperform homogeneous ones in promoting cooperation. This surprising property undercuts the conventional wisdom that heterogeneous structure is generally antagonistic to cooperation and, further helps develop an efficient algorithm OptUpRat to optimise collective cooperation by designing individuals' update rates in any population structure. Our findings provide a unifying framework to understand the interplay between structural heterogeneity, behavioural rhythms, and cooperation.

Target control of complex networks: How to save control energy.

Controlling complex networks has received much attention in the past two decades. In order to control complex networks in practice, recent progress is mainly focused on the control energy required to drive the associated system from an initial state to any final state within finite time. However, one of the major challenges when controlling complex networks is that the amount of control energy is usually prohibitively expensive. Previous explorations on reducing the control energy often rely on adding more driver nodes to be controlled directly by external control inputs, or reducing the number of target nodes required to be controlled. Here we show that the required control energy can be reduced exponentially by appropriately setting the initial states of uncontrollable nodes for achieving the target control of complex networks. We further present the energy-optimal initial states and theoretically prove their existence for any structure of network. Moreover, we demonstrate that the control energy could be saved by reducing the distance between the energy-optimal states set and the initial states of uncontrollable nodes. Finally, we propose a strategy to determine the optimal time to inject the control inputs, which may reduce the control energy exponentially. Our conclusions are all verified numerically, and shed light on saving control energy in practical control.

Impact of misinformation in the evolution of collective cooperation on networks.

Human societies are organized and developed through collective cooperative behaviours. Based on the information in their environment, individuals can form collective cooperation by strategically changing unfavourable surroundings and imitating superior behaviours. However, facing the rampant proliferation and spreading of misinformation, we still lack systematic investigations into the impact of misinformation on the evolution of collective cooperation. Here, we study this problem by classical evolutionary game theory. We find that the existence of misinformation generally impedes the emergence of collective cooperation on networks, although the level of cooperation is slightly higher for weak social cooperative dilemma below a proven threshold. We further show that this possible advantage diminishes as social connections become denser, suggesting that the detrimental effect of misinformation further increases when 'social viscosity' is low. Our results uncover the quantitative effect of misinformation on suppressing collective cooperation, and pave the way for designing possible mechanisms to improve collective cooperation.

Imitation dynamics on networks with incomplete information.

Imitation is an important learning heuristic in animal and human societies. Previous explorations report that the fate of individuals with cooperative strategies is sensitive to the protocol of imitation, leading to a conundrum about how different styles of imitation quantitatively impact the evolution of cooperation. Here, we take a different perspective on the personal and external social information required by imitation. We develop a general model of imitation dynamics with incomplete information in networked systems, which unifies classical update rules including the death-birth and pairwise-comparison rule on complex networks. Under pairwise interactions, we find that collective cooperation is most promoted if individuals neglect personal information. If personal information is considered, cooperators evolve more readily with more external information. Intriguingly, when interactions take place in groups on networks with low degrees of clustering, using more personal and less external information better facilitates cooperation. Our unifying perspective uncovers intuition by examining the rate and range of competition induced by different information situations.

Time delays modulate the stability of complex ecosystems.

What drives the stability, or instability, of complex ecosystems? This question sits at the heart of community ecology and has motivated a large body of theoretical work exploring how community properties shape ecosystem dynamics. However, the overwhelming majority of current theory assumes that species interactions are instantaneous, meaning that changes in the abundance of one species will lead to immediate changes in the abundances of its partners. In practice, time delays in how species respond to one another are widespread across ecological contexts, yet the impact of these delays on ecosystems remains unclear. Here we derive a new body of theory to comprehensively study the impact of time delays on ecological stability. We find that time delays are important for ecosystem stability. Large delays are typically destabilizing but, surprisingly, short delays can substantially increase community stability. Moreover, in stark contrast to delay-free systems, delays dictate that communities with more abundant species can be less stable than ones with less abundant species. Finally, we show that delays fundamentally shift how species interactions impact ecosystem stability, with communities of mixed interaction types becoming the most stable class of ecosystem. Our work demonstrates that time delays can be critical for the stability of complex ecosystems.

Constructing temporal networks with bursty activity patterns.

Human social interactions tend to vary in intensity over time, whether they are in person or online. Variable rates of interaction in structured populations can be described by networks with the time-varying activity of links and nodes. One of the key statistics to summarize temporal patterns is the inter-event time, namely the duration between successive pairwise interactions. Empirical studies have found inter-event time distributions that are heavy-tailed, for both physical and digital interactions. But it is difficult to construct theoretical models of time-varying activity on a network that reproduce the burstiness seen in empirical data. Here we develop a spanning-tree method to construct temporal networks and activity patterns with bursty behavior. Our method ensures any desired target inter-event time distributions for individual nodes and links, provided the distributions fulfill a consistency condition, regardless of whether the underlying topology is static or time-varying. We show that this model can reproduce burstiness found in empirical datasets, and so it may serve as a basis for studying dynamic processes in real-world bursty interactions.

Reactivity of complex communities can be more important than stability.

Understanding stability-whether a community will eventually return to its original state after a perturbation-is a major focus in the study of various complex systems, particularly complex ecosystems. Here, we challenge this focus, showing that short-term dynamics can be a better predictor of outcomes for complex ecosystems. Using random matrix theory, we study how complex ecosystems behave immediately after small perturbations. Our analyses show that many communities are expected to be 'reactive', whereby some perturbations will be amplified initially and generate a response that is directly opposite to that predicted by typical stability measures. In particular, we find reactivity is prevalent for complex communities of mixed interactions and for structured communities, which are both expected to be common in nature. Finally, we show that reactivity can be a better predictor of extinction risk than stability, particularly when communities face frequent perturbations, as is increasingly common. Our results suggest that, alongside stability, reactivity is a fundamental measure for assessing ecosystem health.

The evolution of strategy in bacterial warfare via the regulation of bacteriocins and antibiotics.

Bacteria inhibit and kill one another with a diverse array of compounds, including bacteriocins and antibiotics. These attacks are highly regulated, but we lack a clear understanding of the evolutionary logic underlying this regulation. Here, we combine a detailed dynamic model of bacterial competition with evolutionary game theory to study the rules of bacterial warfare. We model a large range of possible combat strategies based upon the molecular biology of bacterial regulatory networks. Our model predicts that regulated strategies, which use quorum sensing or stress responses to regulate toxin production, will readily evolve as they outcompete constitutive toxin production. Amongst regulated strategies, we show that a particularly successful strategy is to upregulate toxin production in response to an incoming competitor's toxin, which can be achieved via stress responses that detect cell damage (competition sensing). Mirroring classical game theory, our work suggests a fundamental advantage to reciprocation. However, in contrast to classical results, we argue that reciprocation in bacteria serves not to promote peaceful outcomes but to enable efficient and effective attacks.

Generative models of simultaneously heavy-tailed distributions of interevent times on nodes and edges.

Intervals between discrete events representing human activities, as well as other types of events, often obey heavy-tailed distributions, and their impacts on collective dynamics on networks such as contagion processes have been intensively studied. The literature supports that such heavy-tailed distributions are present for interevent times associated with both individual nodes and individual edges in networks. However, the simultaneous presence of heavy-tailed distributions of interevent times for nodes and edges is a nontrivial phenomenon, and its origin has been elusive. In the present study, we propose a generative model and its variants to explain this phenomenon. We assume that each node independently transits between a high-activity and low-activity state according to a continuous-time two-state Markov process and that, for the main model, events on an edge occur at a high rate if and only if both end nodes of the edge are in the high-activity state. In other words, two nodes interact frequently only when both nodes prefer to interact with others. The model produces distributions of interevent times for both individual nodes and edges that resemble heavy-tailed distributions across some scales. It also produces positive correlation in consecutive interevent times, which is another stylized observation for empirical data of human activity. We expect that our modeling framework provides a useful benchmark for investigating dynamics on temporal networks driven by non-Poissonian event sequences.

Autonomy promotes the evolution of cooperation in prisoner's dilemma.

Population structure has been widely reported to foster cooperation in spatially structured populations, where individuals interact with all of their network neighbors defined by the spatial structure in each generation. However, most results rely on the assumption that individuals strictly interact with all of their neighbors during evolution. In reality, human beings, with sophisticated psychology, are willing to interact with some of their neighbors from time to time. Thus, individuals may not play games with all neighbors due to their psychological factors. Here we investigate how the autonomy, one of the basic psychological needs, affects the fate of cooperators in various social networks. By constructing a dynamical effective network, we find that the introduction of autonomy favors cooperative behavior. Further systematical studies by eliminating heterogeneity and the dynamic characteristics of the network reveal that autonomy plays a pivotal role in the evolution of cooperation. Moreover, we find that a moderate effective network degree, defined by the product of the original network degree and the level of autonomy, maximizes the cooperation on networks connecting individuals with fixed neighbors. Our results offer a possible way for organizations to improve individuals' cooperation and shed light on the importance of individuals' psychology on the evolution of cooperation.

Evolution of cooperation on temporal networks.

Population structure is a key determinant in fostering cooperation among naturally self-interested individuals in microbial populations, social insect groups, and human societies. Traditional research has focused on static structures, and yet most real interactions are finite in duration and changing in time, forming a temporal network. This raises the question of whether cooperation can emerge and persist despite an intrinsically fragmented population structure. Here we develop a framework to study the evolution of cooperation on temporal networks. Surprisingly, we find that network temporality actually enhances the evolution of cooperation relative to comparable static networks, despite the fact that bursty interaction patterns generally impede cooperation. We resolve this tension by proposing a measure to quantify the amount of temporality in a network, revealing an intermediate level that maximally boosts cooperation. Our results open a new avenue for investigating the evolution of cooperation and other emergent behaviours in more realistic structured populations.

Energy cost for controlling complex networks with linear dynamics.

Examining the controllability of complex networks has received much attention recently. The focus of many studies is commonly trained on whether we can steer a system from an arbitrary initial state to any final state within finite time with admissible external inputs. In order to accomplish the control at the minimum cost, we must study how much control energy is needed to reach the desired state. At a given control distance between the initial and final states, existing results have offered the scaling behavior of lower bounds of the minimum energy in terms of the control time. However, to reach an arbitrary final state at a given control distance, the minimum energy is actually dominated by the upper bound, whose analytic expression still remains elusive. Here we theoretically show the scaling behavior of a precise upper bound of the minimum energy in terms of the time required to achieve control. Apart from validating the analytical results with numerical simulations, our findings are applicable to any number of nodes that receive inputs directly and any types of networks with linear dynamics. Moreover, more precise analytical results for the lower bound of the minimum energy are derived with the proposed method. Our results pave the way for implementing realistic control over various complex networks with the minimum control cost.

Asymmetric migration decreases stability but increases resilience in a heterogeneous metapopulation.

Many natural populations are spatially distributed, forming a network of subpopulations linked by migration. Migration patterns are often asymmetric and heterogeneous, with important consequences on the ecology and evolution of the species. Here we investigate experimentally how asymmetric migration and heterogeneous structure affect a simple metapopulation of budding yeast, formed by one strain that produces a public good and a non-producer strain that benefits from it. We study metapopulations with star topology and asymmetric migration, finding that all their subpopulations have a higher fraction of producers than isolated populations. Furthermore, the metapopulations have lower tolerance to challenging environments but higher resilience to transient perturbations. This apparent paradox occurs because tolerance to a constant challenge depends on the weakest subpopulations of the network, while resilience to a transient perturbation depends on the strongest ones.

Evolutionary dynamics of general group interactions in structured populations.

The evolution of populations is influenced by many factors, and the simple classical models have been developed in a number of important ways. Both population structure and multiplayer interactions have been shown to significantly affect the evolution of important properties, such as the level of cooperation or of aggressive behavior. Here we combine these two key factors and develop the evolutionary dynamics of general group interactions in structured populations represented by regular graphs. The traditional linear and threshold public goods games are adopted as models to address the dynamics. We show that for linear group interactions, population structure can favor the evolution of cooperation compared to the well-mixed case, and we see that the more neighbors there are, the harder it is for cooperators to persist in structured populations. We further show that threshold group interactions could lead to the emergence of cooperation even in well-mixed populations. Here population structure sometimes inhibits cooperation for the threshold public goods game, where depending on the benefit to cost ratio, the outcomes are bistability or a monomorphic population of defectors or cooperators. Our results suggest, counterintuitively, that structured populations are not always beneficial for the evolution of cooperation for nonlinear group interactions.