Emergence of Super Cooperation of Prisoner’s Dilemma Games on Scale-Free Networks

Recently, the authors proposed a quantum prisoner’s dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner’s dilemma (GPD, for short) games based on the weak Prisoner’s dilemma game, the full prisoner’s dilemma game and the normalized Prisoner’s dilemma game, written by GPD_W, GPD_F and GPD_N respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter γ to measure the entangled relationship between the two players. We found that our generalised prisoner’s dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner’s dilemma games.     

Social structure of primate interaction networks facilitates the emergence of cooperation

Animal cooperation has puzzled biologists for a long time as its existence seems to contravene the basic notion of evolutionary biology that natural selection favours ‘selfish’ genes that promote only their own well-being. Evolutionary game theory has shown that cooperators can prosper in populations of selfish individuals if they occur in clusters, interacting more frequently with each other than with the selfish. Here we show that social networks of primates possess the necessary social structure to promote the emergence of cooperation. By simulating evolutionary dynamics of cooperative behaviour on interaction networks of 70 primate groups, we found that for most groups network reciprocity augmented the fixation probability for cooperation. The variation in the strength of this effect can be partly explained by the groups’ community modularity—a network measure for the groups’ heterogeneity. Thus, given selective update and partner choice mechanisms, network reciprocity has the potential to explain socially learned forms of cooperation in primate societies.     

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.     

A dynamic over games drives selfish agents to win–win outcomes

Understanding human institutions, animal cultures and other social systems requires flexible formalisms that describe how their members change them from within. We introduce a framework for modelling how agents change the games they participate in. We contrast this between-game ‘institutional evolution’ with the more familiar within-game ‘behavioural evolution’. We model institutional change by following small numbers of persistent agents as they select and play a changing series of games. Starting from an initial game, a group of agents trace trajectories through game space by navigating to increasingly preferable games until they converge on ‘attractor’ games. Agents use their ‘institutional preferences'' for game features (such as stability, fairness and efficiency) to choose between neighbouring games. We use this framework to pose a pressing question: what kinds of games does institutional evolution select for; what is in the attractors? After computing institutional change trajectories over the two-player space, we find that attractors have disproportionately fair outcomes, even though the agents who produce them are strictly self-interested and indifferent to fairness. This seems to occur because game fairness co-occurs with the self-serving features these agents do actually prefer. We thus present institutional evolution as a mechanism for encouraging the spontaneous emergence of cooperation among small groups of inherently selfish agents, without space, reputation, repetition, or other more familiar mechanisms. Game space trajectories provide a flexible, testable formalism for modelling the interdependencies of behavioural and institutional evolutionary processes, as well as a mechanism for the evolution of cooperation.     

Emotions as infectious diseases in a large social network: the SISa model

Human populations are arranged in social networks that determine interactions and influence the spread of diseases, behaviours and ideas. We evaluate the spread of long-term emotional states across a social network. We introduce a novel form of the classical susceptible–infected–susceptible disease model which includes the possibility for ‘spontaneous’ (or ‘automatic’) infection, in addition to disease transmission (the SISa model). Using this framework and data from the Framingham Heart Study, we provide formal evidence that positive and negative emotional states behave like infectious diseases spreading across social networks over long periods of time. The probability of becoming content is increased by 0.02 per year for each content contact, and the probability of becoming discontent is increased by 0.04 per year per discontent contact. Our mathematical formalism allows us to derive various quantities from the data, such as the average lifetime of a contentment ‘infection’ (10 years) or discontentment ‘infection’ (5 years). Our results give insight into the transmissive nature of positive and negative emotional states. Determining to what extent particular emotions or behaviours are infectious is a promising direction for further research with important implications for social science, epidemiology and health policy. Our model provides a theoretical framework for studying the interpersonal spread of any state that may also arise spontaneously, such as emotions, behaviours, health states, ideas or diseases with reservoirs.     

Stochastic evolutionary dynamics of direct reciprocity

Evolutionary game theory is the study of frequency-dependent selection. The success of an individual depends on the frequencies of strategies that are used in the population. We propose a new model for studying evolutionary dynamics in games with a continuous strategy space. The population size is finite. All members of the population use the same strategy. A mutant strategy is chosen from some distribution over the strategy space. The fixation probability of the mutant strategy in the resident population is calculated. The new mutant takes over the population with this probability. In this case, the mutant becomes the new resident. Otherwise, the existing resident remains. Then, another mutant is generated. These dynamics lead to a stationary distribution over the entire strategy space. Our new approach generalizes classical adaptive dynamics in three ways: (i) the population size is finite; (ii) mutants can be drawn non-locally and (iii) the dynamics are stochastic. We explore reactive strategies in the repeated Prisoner''s Dilemma. We perform ‘knock-out experiments’ to study how various strategies affect the evolution of cooperation. We find that ‘tit-for-tat’ is a weak catalyst for the emergence of cooperation, while ‘always cooperate’ is a strong catalyst for the emergence of defection. Our analysis leads to a new understanding of the optimal level of forgiveness that is needed for the evolution of cooperation under direct reciprocity.     

Perturbation Centrality and Turbine: A Novel Centrality Measure Obtained Using a Versatile Network Dynamics Tool

Analysis of network dynamics became a focal point to understand and predict changes of complex systems. Here we introduce Turbine, a generic framework enabling fast simulation of any algorithmically definable dynamics on very large networks. Using a perturbation transmission model inspired by communicating vessels, we define a novel centrality measure: perturbation centrality. Hubs and inter-modular nodes proved to be highly efficient in perturbation propagation. High perturbation centrality nodes of the Met-tRNA synthetase protein structure network were identified as amino acids involved in intra-protein communication by earlier studies. Changes in perturbation centralities of yeast interactome nodes upon various stresses well recapitulated the functional changes of stressed yeast cells. The novelty and usefulness of perturbation centrality was validated in several other model, biological and social networks. The Turbine software and the perturbation centrality measure may provide a large variety of novel options to assess signaling, drug action, environmental and social interventions.     

Towards a Methodology for Validation of Centrality Measures in Complex Networks

Background

Living systems are associated with Social networks — networks made up of nodes, some of which may be more important in various aspects as compared to others. While different quantitative measures labeled as “centralities” have previously been used in the network analysis community to find out influential nodes in a network, it is debatable how valid the centrality measures actually are. In other words, the research question that remains unanswered is: how exactly do these measures perform in the real world? So, as an example, if a centrality of a particular node identifies it to be important, is the node actually important?

Purpose

The goal of this paper is not just to perform a traditional social network analysis but rather to evaluate different centrality measures by conducting an empirical study analyzing exactly how do network centralities correlate with data from published multidisciplinary network data sets.

Method

We take standard published network data sets while using a random network to establish a baseline. These data sets included the Zachary''s Karate Club network, dolphin social network and a neural network of nematode Caenorhabditis elegans. Each of the data sets was analyzed in terms of different centrality measures and compared with existing knowledge from associated published articles to review the role of each centrality measure in the determination of influential nodes.

Results

Our empirical analysis demonstrates that in the chosen network data sets, nodes which had a high Closeness Centrality also had a high Eccentricity Centrality. Likewise high Degree Centrality also correlated closely with a high Eigenvector Centrality. Whereas Betweenness Centrality varied according to network topology and did not demonstrate any noticeable pattern. In terms of identification of key nodes, we discovered that as compared with other centrality measures, Eigenvector and Eccentricity Centralities were better able to identify important nodes.     

Reinforcement Learning Explains Conditional Cooperation and Its Moody Cousin

Direct reciprocity, or repeated interaction, is a main mechanism to sustain cooperation under social dilemmas involving two individuals. For larger groups and networks, which are probably more relevant to understanding and engineering our society, experiments employing repeated multiplayer social dilemma games have suggested that humans often show conditional cooperation behavior and its moody variant. Mechanisms underlying these behaviors largely remain unclear. Here we provide a proximate account for this behavior by showing that individuals adopting a type of reinforcement learning, called aspiration learning, phenomenologically behave as conditional cooperator. By definition, individuals are satisfied if and only if the obtained payoff is larger than a fixed aspiration level. They reinforce actions that have resulted in satisfactory outcomes and anti-reinforce those yielding unsatisfactory outcomes. The results obtained in the present study are general in that they explain extant experimental results obtained for both so-called moody and non-moody conditional cooperation, prisoner’s dilemma and public goods games, and well-mixed groups and networks. Different from the previous theory, individuals are assumed to have no access to information about what other individuals are doing such that they cannot explicitly use conditional cooperation rules. In this sense, myopic aspiration learning in which the unconditional propensity of cooperation is modulated in every discrete time step explains conditional behavior of humans. Aspiration learners showing (moody) conditional cooperation obeyed a noisy GRIM-like strategy. This is different from the Pavlov, a reinforcement learning strategy promoting mutual cooperation in two-player situations.     

Evaluating the Adequacy of Gravity Models as a Description of Human Mobility for Epidemic Modelling

Gravity models have a long history of use in describing and forecasting the movements of people as well as goods and services, making them a natural basis for disease transmission rates over distance. In agent-based micro-simulations, gravity models can be directly used to represent movement of individuals and hence disease. In this paper, we consider a range of gravity models as fits to movement data from the UK and the US. We examine the ability of synthetic networks generated from fitted models to match those from the data in terms of epidemic behaviour; in particular, times to first infection. For both datasets, best fits are obtained with a two-piece ‘matched’ power law distance distribution. Epidemics on synthetic UK networks match well those on data networks across all but the smallest nodes for a range of aggregation levels. We derive an expression for time to infection between nodes in terms of epidemiological and network parameters which illuminates the influence of network clustering in spread across networks and suggests an approximate relationship between the log-likelihood deviance of model fit and the match times to infection between synthetic and data networks. On synthetic US networks, the match in epidemic behaviour is initially poor and sensitive to the initially infected node. Analysis of times to infection indicates a failure of models to capture infrequent long-range contact between large nodes. An assortative model based on node population size captures this heterogeneity, considerably improving the epidemiological match between synthetic and data networks.     

Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks

In online social media networks, individuals often have hundreds or even thousands of connections, which link these users not only to friends, associates, and colleagues, but also to news outlets, celebrities, and organizations. In these complex social networks, a ‘community’ as studied in the social network literature, can have very different meaning depending on the property of the network under study. Taking into account the multifaceted nature of these networks, we claim that community detection in online social networks should also be multifaceted in order to capture all of the different and valuable viewpoints of ‘community.’ In this paper we focus on three types of communities beyond follower-based structural communities: activity-based, topic-based, and interaction-based. We analyze a Twitter dataset using three different weightings of the structural network meant to highlight these three community types, and then infer the communities associated with these weightings. We show that interesting insights can be obtained about the complex community structure present in social networks by studying when and how these four community types give rise to similar as well as completely distinct community structure.     

Partner Choice Drives the Evolution of Cooperation via Indirect Reciprocity

Indirect reciprocity potentially provides an important means for generating cooperation based on helping those who help others. However, the use of ‘image scores’ to summarize individuals’ past behaviour presents a dilemma: individuals withholding help from those of low image score harm their own reputation, yet giving to defectors erodes cooperation. Explaining how indirect reciprocity could evolve has therefore remained problematic. In all previous treatments of indirect reciprocity, individuals are assigned potential recipients and decide whether to cooperate or defect based on their reputation. A second way of achieving discrimination is through partner choice, which should enable individuals to avoid defectors. Here, I develop a model in which individuals choose to donate to anyone within their group, or to none. Whereas image scoring with random pairing produces cycles of cooperation and defection, with partner choice there is almost maximal cooperation. In contrast to image scoring with random pairing, partner choice results in almost perfect contingency, producing the correlation between giving and receiving required for cooperation. In this way, partner choice facilitates much higher and more stable levels of cooperation through image scoring than previously reported and provides a simple mechanism through which systems of helping those who help others can work.     

Relational Diversity Promotes Cooperation in Prisoner’s Dilemma Games

Relational diversity can be characterized by heterogeneous distributions of tie strengths in social networks and this diversity is present not only among humans, but throughout the animal world. We account for this observation by analyzing two network datasets from Facebook. We measure the strength of a tie by calculating the extent of overlap of friends between the two individuals. Based on the previous findings in human experiments, we argue that it is very unlikely that players will allocate their investments equally to their neighbors. There is a tendency that players prefer to donate more to their intimate friends. We find that if players preferentially allocate their investments to their good friends, cooperation will be promoted in PDG. We proved that the facilitation of the cooperative strategy relies mostly on the cooperative allies between best friends, resulting in the formation of cooperative clusters which are able to prevail against the defectors even when there is a large cost to cooperate. Moreover, we discover that the effect of relational diversity cannot be analyzed by adopting classical complex networks models, because neither of the artificial networks is able to produce networks with diverse distributions of tie strengths. It is of vital importance to introduce real social networks to study the influence of diverse relations especially when it comes to humans. This research proposes a brand new perspective to understand the influence of social relations on the emergence of cooperation in evolutionary prisoner’s dilemma games.     

Clustering in large networks does not promote upstream reciprocity

Upstream reciprocity (also called generalized reciprocity) is a putative mechanism for cooperation in social dilemma situations with which players help others when they are helped by somebody else. It is a type of indirect reciprocity. Although upstream reciprocity is often observed in experiments, most theories suggest that it is operative only when players form short cycles such as triangles, implying a small population size, or when it is combined with other mechanisms that promote cooperation on their own. An expectation is that real social networks, which are known to be full of triangles and other short cycles, may accommodate upstream reciprocity. In this study, I extend the upstream reciprocity game proposed for a directed cycle by Boyd and Richerson to the case of general networks. The model is not evolutionary and concerns the conditions under which the unanimity of cooperative players is a Nash equilibrium. I show that an abundance of triangles or other short cycles in a network does little to promote upstream reciprocity. Cooperation is less likely for a larger population size even if triangles are abundant in the network. In addition, in contrast to the results for evolutionary social dilemma games on networks, scale-free networks lead to less cooperation than networks with a homogeneous degree distribution.     

Social Networks,the ‘Work’ and Work Force of Chronic Illness Self-Management: A Survey Analysis of Personal Communities

Self-management support forms a central aspect of chronic Illness management nationally and globally. Evidence for the success of self-management support has mainly focussed on individually-centred outcomes of behavioural change. While it is recognised that social network members play an important role there is currently a gap in knowledge regarding who provides what type of support and under what circumstances. This is relevant for understanding the division of labour and the meeting of needs for those living with a long-term condition. We therefore took a network approach to explore self-management support conceptualising it as types of illness ‘work’ undertaken within peoples’ social networks. 300 people from deprived areas and with chronic illnesses took part in a survey conducted in 2010 in the North West of England. A concentric circles diagram was used as a research tool with which participants identified 2,544 network members who contributed to illness management. The results provide an articulation of how social network members are substantially involved in illness management. Whilst partners and close family make the highest contributions there is evidence of inputs from a wide range of relationships. Network member characteristics (type of relationship, proximity, frequency of contact) impact on the amount of illness work undertaken in peoples’ networks. In networks with ‘no partner’ other people tend to contribute more in the way of illness related work than in networks with a partner. This indicates a degree of substitutability between differently constituted networks, and that the level and type of input by different members of a network might change according to circumstances. A network perspective offers an opportunity to redress the balance of an exclusively individual focus on self-management because it addresses the broader set of contributions and resources available to people in need of chronic illness management and support.     

Network 'small-world-ness': a quantitative method for determining canonical network equivalence

Background

Many technological, biological, social, and information networks fall into the broad class of ‘small-world’ networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction (‘small/not-small’) rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network''s small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model – the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.

Methodology/Principal Findings

We defined a precise measure of ‘small-world-ness’ S based on the trade off between high local clustering and short path length. A network is now deemed a ‘small-world’ if S>1 - an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.

Conclusions/Significance

We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.     

Efficient and robust coding in heterogeneous recurrent networks

Cortical networks show a large heterogeneity of neuronal properties. However, traditional coding models have focused on homogeneous populations of excitatory and inhibitory neurons. Here, we analytically derive a class of recurrent networks of spiking neurons that close to optimally track a continuously varying input online, based on two assumptions: 1) every spike is decoded linearly and 2) the network aims to reduce the mean-squared error between the input and the estimate. From this we derive a class of predictive coding networks, that unifies encoding and decoding and in which we can investigate the difference between homogeneous networks and heterogeneous networks, in which each neurons represents different features and has different spike-generating properties. We find that in this framework, ‘type 1’ and ‘type 2’ neurons arise naturally and networks consisting of a heterogeneous population of different neuron types are both more efficient and more robust against correlated noise. We make two experimental predictions: 1) we predict that integrators show strong correlations with other integrators and resonators are correlated with resonators, whereas the correlations are much weaker between neurons with different coding properties and 2) that ‘type 2’ neurons are more coherent with the overall network activity than ‘type 1’ neurons.     

Effect of Resource Spatial Correlation and Hunter-Fisher-Gatherer Mobility on Social Cooperation in Tierra del Fuego

This article presents an agent-based model designed to explore the development of cooperation in hunter-fisher-gatherer societies that face a dilemma of sharing an unpredictable resource that is randomly distributed in space. The model is a stylised abstraction of the Yamana society, which inhabited the channels and islands of the southernmost part of Tierra del Fuego (Argentina-Chile). According to ethnographic sources, the Yamana developed cooperative behaviour supported by an indirect reciprocity mechanism: whenever someone found an extraordinary confluence of resources, such as a beached whale, they would use smoke signals to announce their find, bringing people together to share food and exchange different types of social capital. The model provides insight on how the spatial concentration of beachings and agents’ movements in the space can influence cooperation. We conclude that the emergence of informal and dynamic communities that operate as a vigilance network preserves cooperation and makes defection very costly.     

Contagion of Cooperation in Static and Fluid Social Networks

Cooperation is essential for successful human societies. Thus, understanding how cooperative and selfish behaviors spread from person to person is a topic of theoretical and practical importance. Previous laboratory experiments provide clear evidence of social contagion in the domain of cooperation, both in fixed networks and in randomly shuffled networks, but leave open the possibility of asymmetries in the spread of cooperative and selfish behaviors. Additionally, many real human interaction structures are dynamic: we often have control over whom we interact with. Dynamic networks may differ importantly in the goals and strategic considerations they promote, and thus the question of how cooperative and selfish behaviors spread in dynamic networks remains open. Here, we address these questions with data from a social dilemma laboratory experiment. We measure the contagion of both cooperative and selfish behavior over time across three different network structures that vary in the extent to which they afford individuals control over their network ties. We find that in relatively fixed networks, both cooperative and selfish behaviors are contagious. In contrast, in more dynamic networks, selfish behavior is contagious, but cooperative behavior is not: subjects are fairly likely to switch to cooperation regardless of the behavior of their neighbors. We hypothesize that this insensitivity to the behavior of neighbors in dynamic networks is the result of subjects’ desire to attract new cooperative partners: even if many of one’s current neighbors are defectors, it may still make sense to switch to cooperation. We further hypothesize that selfishness remains contagious in dynamic networks because of the well-documented willingness of cooperators to retaliate against selfishness, even when doing so is costly. These results shed light on the contagion of cooperative behavior in fixed and fluid networks, and have implications for influence-based interventions aiming at increasing cooperative behavior.     

Species,Habitats, Society: An Evaluation of Research Supporting EU's Natura 2000 Network

The Natura 2000 network is regarded as one of the conservation success stories in the global effort to protect biodiversity. However, significant challenges remain in Natura 2000 implementation, owing to its rapid expansion, and lack of a coherent vision for its future. Scientific research is critical for identifying conservation priorities, setting management goals, and reconciling biodiversity protection and society in the complex political European landscape. Thus, there is an urgent need for a comprehensive evaluation of published Natura 2000 research to highlight prevalent research themes, disciplinary approaches, and spatial entities. We conducted a systematic review of 572 scientific articles and conference proceedings focused on Natura 2000 research, published between 1996 and 2014. We grouped these articles into ‘ecological’ and ‘social and policy’ categories. Using a novel application of network analysis of article keywords, we found that Natura 2000 research forms a cohesive small-world network, owing to the emphasis on ecological research (79% of studies, with a strong focus on spatial conservation planning), and the underrepresentation of studies addressing ‘social and policy’ issues (typically focused on environmental impact assessment, multi-level governance, agri-environment policy, and ecosystem services valuation). ‘Ecological’ and ‘social and policy’ research shared only general concepts (e.g., Natura 2000, Habitats Directive) suggesting a disconnection between these disciplines. The UK and the Mediterranean basin countries dominated Natura 2000 research, and there was a weak correlation between number of studies and proportion of national territory protected. Approximately 40% of ‘social and policy’ research and 26% of ‘ecological’ studies highlighted negative implications of Natura 2000, while 21% of studies found positive social and biodiversity effects. We emphasize the need for designing inter- and transdisciplinary research in order to promote a social-ecological understanding of Natura 2000, and advance EU conservation policies.