Detecting the Collapse of Cooperation in Evolving Networks

Detecting the Collapse of Cooperation in Evolving Networks

Matteo Cavaliere School of Informatics, University of Edinburgh, Scotland, United Kingdom    Guoli Yang School of Informatics, University of Edinburgh, Scotland, United Kingdom    Vincent Danos School of Informatics, University of Edinburgh, Scotland, United Kingdom    Vasilis Dakos Institute of Integrative Biology, Center for Adaptation to a Changing Environment, ETHZ, Zürich, Switzerland, vasilis.dakos@env.ethz.ch
Key Words.:
Evolutionary Games, Cooperation, Critical Transitions, Indicators

The sustainability of structured biological, social, economic and ecological communities are often determined by the outcome of social conflicts between cooperative and selfish individuals (cheaters). Cheaters avoid the cost of contributing to the community and can occasionally spread in the population leading to the complete collapse of cooperation. Although such a collapse often unfolds unexpectedly bearing the traits of a critical transition, it is unclear whether one can detect the rising risk of cheater’s invasions and loss of cooperation in an evolving community. Here, we combine dynamical networks and evolutionary game theory to study the abrupt loss of cooperation as a critical transition. We estimate the risk of collapse of cooperation after the introduction of a single cheater under gradually changing conditions. We observe a systematic increase in the average time it takes for cheaters to be eliminated from the community as the risk of collapse increases. We detect this risk based on changes in community structure and composition. Nonetheless, reliable detection depends on the mechanism that governs how cheaters evolve in the community. Our results suggest possible avenues for detecting the loss of cooperation in evolving communities.

1 Introduction

The sustainability of many biological, social, economic, and ecological communities is determined by the interplay between individual actions and collective dynamics (Levin, 2007). The successful performance of a community is often based on the cooperative attitude of individuals that pay a personal cost to distribute general benefits (Nowak, 2006). Nonetheless, although cooperation favors in general the success of a community, it can also facilitate the appearance of cheaters who take advantage of cooperators, spread in the community, and may even cause its collapse (Levin, 2007). The failure of cooperation in the presence of cheaters has been observed in many systems at different scales. For instance, cooperating cells of Pseudomonas fluorescens build biofilms that help them to grow better while mutant cells (cheaters) - that do not produce similar adhesive factors - take advantage of the existing structure in order to spread and to eventually cause the colony to collapse (Rainey & Rainey, 2003; Popat et al., 2012). Fruiting bodies formed under starvation by cooperative cells of Myxococcus xanthus can also be invaded by cheaters leading to the disruption of the fruiting body structure and forcing the cooperative survivors to reinvest in reconstruction (Travisano & Velicer, 2004). At a different scale and in a different context shifts in cooperation and cheating have been debated to be causes of economic crises, (Haldane et al., 2009) as well as causes for the ”tragedy” of common pool resources, (e.g., fisheries, forests, etc.) (Congleton, 2007; Bowles & Gintis, 2011). In all these systems a long-standing question has been to understand the mechanisms that allow cooperators to resist the reproductive advantage of selfish cheating individuals (Nowak, 2006). Among the many theoretical and experimental studies on the maintenance of cooperation (Nowak, 2006; Ohtsuki et al., 2006; Rand et al., 2011; Sanchez & Gore, 2013), scenarios where strategies co-evolve with population structure (Perc & Szolnoki, 2010) are of particular interest as they show how structural properties in the population can affect the evolution of cooperation (Perc & Szolnoki, 2010; Wardil & Hauert, 2014; Rand et al., 2011; Sanchez & Gore, 2013). For instance, it has been shown that not only the number of cheaters in the community is important, but also how and to whom cheaters are connected (Cavaliere et al., 2012). The interplay between the way individuals are connected and the overall prosperity of a system (Bascompte et al., 2010; Levin, 2007) endogenously determines either the formation or the sudden collapse of cooperative communities (Levin, 2007). Despite our relative good understanding of the conditions that promote the failure of cooperation in a community, it is still difficult to predict whether the appearance of a cheater will cause the eventual loss of cooperation (Congleton, 2007; Bowles & Gintis, 2011). This is because it is hard to identify the underlying conditions that increase the risk of collapse in practice. Thus, it is crucial to develop alternative ways for detecting the risk of collapse of cooperation in a structurally evolving community.

Recent work has suggested that generic patterns in the dynamics of a system can be used to infer proximity to abrupt and unexpected changes termed critical transitions (Scheffer et al., 2009). These dynamical patterns are generic in the sense that they do not depend on the particular system in question, but they are determined by the mathematical phenomenon of critical slowing down (CSD) that occurs prior to local bifurcation points (Strogatz, 2014; Wissel, 1984). A bifurcation point represents a threshold where a qualitative change in the equilibrium of a system takes place: the iconic case is the shift between two alternative equilibria at a crossing of a fold bifurcation. Close to a local bifurcation, CSD implies that the system takes longer to recover back to equilibrium after a disturbance (Scheffer et al., 2009; Van Nes & Scheffer, 2007). In addition, the dynamics of the system become strongly variable (Carpenter & Brock, 2006), and highly autocorrelated (Held & Kleinen, 2004). As such, decreasing recovery rate, rising variance, and rising autocorrelation can all be used as early-warning signals of approaching critical transitions (Scheffer et al., 2009; Dakos et al., 2012), or more general as indicators of loss of resilience (Dakos & Bascompte, 2014). CSD indicators have been identified in a variety of systems: from the collapse of cyanobacteria (Veraart et al., 2012), zooplankton (Drake, 2005), and yeast populations (Dai et al., 2012) in the lab, to changes in trophic structure in lakes (Carpenter & Brock, 2006), as well as prior to abrupt past climatic events (Dakos et al., 2012), and the onset of depression in human patients (van de Leemput et al., 2014). Despite limitations and challenges in detecting resilience indicators (Dakos et al., 2015; Boettiger & Hastings, 2012), growing evidence supports their potential use across different disciplines(Scheffer et al., 2012). The application of such indicators for detecting abrupt transitions in structurally complex communities is still, however, largely unexplored. There are few studies that have highlighted the emergence of tipping points in mutualistic communities of plants and their pollinators (Lever et al., 2014), and the detection of transitions in ecological(Dakos & Bascompte, 2014) and socio-ecological networks (Suweis & D’Odorico, 2014). Nonetheless, these studies assume a static structure that does not allow changes in the interactions among network components.

Here, for the first time to our knowledge, we combine evolutionary game theory and dynamical networks to detect the collapse of cooperation in an evolving (structurally dynamic) community, using resilience indicators for critical transitions. To do this, we adapt a network model that displays cooperation collapses as consequence of cheater’s invasions(Cavaliere et al., 2012). Evolution of cheaters and cooperators is based on social inheritance: newcomers copy strategies and connections of successful role-models. Increased ability of newcomers to link to a high number of individuals already present in the network allows high prosperity but at the risk of facilitating cheater’s invasions that can lead to the collapse of cooperation (Cavaliere et al., 2012). We show that the collapse of cooperation occurs in an abrupt non-linear way that resembles a critical transition. Prior to collapse we estimate a series of indicators (structural and non-structural) that can signal the increasing fragility of cooperation. Our contribution is twofold. First, we demonstrate how the loss of cooperation in evolving communities bears the features of a critical transition. Second, we develop a set of structural-based indicators and state-based indicators for detecting approaching transitions in structurally evolving communities.

2 Results

2.1 The Collapse of Cooperation in a Dynamical Network

Following the work (Cavaliere et al., 2012) we consider a network model with a fixed number of agents (nodes) but with a non-fixed number of links: to whom and to how many neighbours an agent is connected varies during the evolution of the system. Each agent in the network can adopt one of the two strategies of the Prisoner’s Dilemma. A cooperator pays a cost to provide a benefit to all of its neighbours. Cheaters pay no cost and distribute no benefit. For instance, if a cooperator has cooperative neighbours and cheating neighbours, its payoff is . A cheater in the same neighbourhood has payoff . The dynamics are defined by a discrete sequence of update steps (Figure 1). At each update step a newcomer (invader) is added and a randomly chosen existing node is removed so that the number of nodes is constant as . As the newcomer has no specific strategy, a node is selected as a role-model with a probability proportional to its effective payoff = (1+, where specifies the strength of selection and is the sum of pair-wise interactions of each node. For the selection probability is same for all nodes, while increasing makes it more likely that a newcomer chooses a node with a higher payoff. After the newcomer adopts the strategy of the chosen role-model, it also connects with the role-model with a probability and with each of the role-model’s neighbours with a probability . The parameters and are called embedding parameters as they determine the ability of the newcomer to imitate the role-model’s social network.

Figure 1: Evolution in a dynamical network. The dynamics follow three distinct update steps in the considered model: (step i) A role-model is selected based on its effective payoff. (step ii) The invader (newcomer) connects to the role-model with a probability (dashed line), connects to each of its neighbours with a probability (dotted lines) and emulates its strategy. and are called embedding parameters. (step iii) A randomly selected node and all its connections are removed from the network.

More importantly, the embedding parameters and also determine the stability of cooperation (Cavaliere et al., 2012). In particular, high levels of rise the risk of collapse of cooperation (level of is less relevant, Figure in the Supplementary Material). To study how gradually increasing leads to collapse and slowly erodes the stability of cooperation, we perform a series of perturbation experiments. A perturbation experiment is defined as the introduction of an invader (e.g. cheater) in a network where all agents have the opposite strategy (e.g. cooperators).

We start perturbations in a network of all cooperators after we have iterated the model to establish a stable network topology. We then introduce a cheater and we monitor the system until either the cheater fails to invade, recovery, or the cheater invades and cooperation collapses, collapse (Figure 2). Using this approach, we evaluate the fraction of perturbations that lead to the collapse for increasing values of the embedding parameter at different selection strengths .

Figure 2: Recovery and Collapse of Cooperation in Perturbation Experiments. Recovery and collapse of cooperation following the addition of a single invader in a network with agents of the opposite strategy. We show the two possible outcomes of a perturbation obtained by adding a cheater in a network of all cooperators. The first row shows the typical stages (and network topologies) of an unsuccessful perturbation where cooperators resist, while the bottom row shows the typical stages (and network topologies) of a successful perturbation where cheaters invade (in both cases ). Note the distinct network structure in community of all cooperators (highly connected) and in a community of all cheaters (highly fragmented)

This fraction represents the probability of the introduced cheater to spread and trigger the collapse of cooperation. We call the complement of this probability cooperation persistence (1-). We find that cooperation stays uninvadable for a wide range of the control parameter . Only when crosses a certain threshold, persistence decreases rapidly (Figure 3A). Cooperation is doomed to fail and its loss occurs in an abrupt non-linear way. Nonetheless, the exact value of the threshold depends on the selection strength . Higher s cause an earlier but smoother collapse of cooperation, and turn the probability that a single cheater invades successfully almost equal to (Figure 3A).

Once cooperation collapses, its restoration by re-introducing a cooperator in a network of cheaters is difficult. In this reverse scenario, we find that although the probability of a single cooperator to invade a network of cheaters increases for decreasing the embedding parameter , it never reaches (Figure 3B). This means that once cooperation is lost it is difficult to be restored. This indicates that the cheaters dominance equilibrium is much more resilient than the cooperators dominance equilibrium. Restoring cooperation is only possible by inducing a stronger perturbation: a simultaneous invasion of more than a single cooperator (Figure in the Supplementary Material).

Figure 3: Persistence and Restoration of Cooperation. (A) Persistence of cooperation as a function of the embedding parameter for various selection strengths . Note the non-monotonicity of the curves (visible clearly in the inset panel): for a low selection , the persistence of cooperation reaches a maximum before collapsing. As selection becomes stronger, the persistence of cooperation decreases monotonically. (B) Restoration of cooperation as a function of the embedding parameter for various selection strengths . Persistence probabilities are computed as and restoration probabilities are computed as , where is the fraction of perturbations that lead to the successful invasion of the mutant out of perturbation experiments. Each perturbation is done by updating the network of (A) all cooperators or (B) all cheaters for a long time following the addition of a mutant at . (All results are shown at ; see Supplementary Material Figure for different s).

2.2 Structural and Non-structural Indicators for Detecting the Collapse of Cooperation

The abrupt loss of cooperation raises the question whether it is possible to detect it in advance. Measuring the strength of parameter is not only difficult in practice, but also largely uninformative as for a range of low , as there is little sign of change in the persistence of cooperation (Figure 3). Thus, alternative diagnostic tools would be desirable in order to estimate the rising risk of cooperation collapse. Motivated by the possibility to detect upcoming critical transition using generic indicators of resilience (Scheffer et al., 2009), we explore whether we can use similar indicators as precursors of the eroding stability of cooperation.

In particular, we evaluate two broad classes of indicators: those based on the dynamics of the community composition (that we call non-structural) and those based on the community structure (that we call structural). The difference between the two classes is that non-structural indicators reflect changes in the numbers of cheaters and cooperators (demographic changes), whereas structural indicators reflect changes in the interactions between cheaters and cooperators (topological changes). Among a variety of metrics, we estimate two non-structural indicators: a) return rate, the inverse of the time it takes for the system to recover back to its original state of full cooperation after the addition of a single cheater (recall that time is measured in update steps in our model), and b) maximum size, the maximal amount of cheaters possible to invade after the introduction of a single cheater. We also estimate two structural indicators: a) average degree (or connectance), the average number of links of each agent in the network, and b) the relative amount of cooperative interactions , that is the fraction of beneficial interactions between cooperators over the exploitative interactions between cooperators and cheaters (see Methods for details). Other estimated metrics are presented in the Supplementary Material (Section ).

We estimate these four indicators after each perturbation experiment at increasing values of the embedding parameter and for different selection strengths . In Figure 4, we plot the median values from perturbations performed at each (see Supplementary Material Figure for the distributions of all indicators). We find that both non-structural and structural indicators change in distinct ways before cooperation collapses. In particular, we compare trends among indicators up to the value of at which the probability of cooperation persistence drops below . We use this threshold value as the most conservative choice for the onset of cooperation collapse. We observe the strongest changes for all indicators close to the transition except for average degree when selection strength is high (Figure 4D). In general, strong selection () leads to more pronounced and earlier changes in the trends when compared to weak selection (see Figure in the Supplementary Material for different selection strengths). Perhaps most interestingly, return rate decreases before that collapse threshold (Figure 4A,C). As decreasing return rates are signature of critical slowing down prior to local bifurcation points (Van Nes & Scheffer, 2007), this finding indirectly supports that the loss of cooperation in evolving networks might be classified as a critical transition.

Figure 4: Detecting the Loss of Cooperation. Structural and non-structural indicators for detecting the loss of cooperation after the invasion of a single cheater at increasing levels of embedding parameter . Non-structural indicators are 1) return rate (inverse of the number of update steps the system takes to go back to its original state following a perturbation), and 2) max-size (the maximal number of cheaters recorded during a perturbation). Structural indicators are 1) structural coefficient that evaluates the ratio between the cooperative interactions and the exploitative interactions, and 2) average degree (the average number of links per node). Upper row: weak selection, ; lower row: strong selection, . The black curves denote cooperation persistence. The yellow shaded area identifies the values of where cooperation persistence falls below (our defined threshold for cooperation collapse). Each point of the indicators is the median estimated out of the considering perturbation experiments.

2.3 Assessing the Consistency and Early Detection of the Indicators

Trends in the indicators are useful only if they are consistently correlated to changes of the embedding parameter independently from the strength of selection . We test for such consistency by computing trends in the indicators up to the threshold where the cooperation persistence drops below (Figure 4). We first generate sequences of indicator values by randomly selecting a value from any of the perturbations at each . Second, we computed the correlation (quantified by Kendall rank coefficient) between the constructed indicator sequences and the values of (see Methods). By comparing mean values of the estimated distributions of Kendall s, we found that the trends in indicators are generally consistent across different selection strengths (Figure 5): the mean values keep the same signs. However, when comparing the means and standard deviations of the estimated distributions of Kendall s, we found that return rate, max-size and are stronger when selection strength increases. Average degree is the strongest indicator regardless of whether the selection is weak or strong (Figure 5).

Figure 5: Consistency of the indicators at different selection strengths. Mean values and standard deviation values of the distributions for Kendall rank coefficients for increasing selection strengths . Non-structural indicators are 1) return rate and 2) max-size; and the structural indicators are 1) structural coefficient and 2) average degree. The average degree appears to be the most consistent indicator as its distribution is almost always positive across all selection strengths. In general, the consistence strength is enhanced as the selection increases.

We also analyze how “early” an indicator accurately signals the increasing risk of collapse. Or, in other words, how far from the collapse changes in the indicators significantly signal the rising risk of cooperation loss. We do this by using receiver operator characteristic (ROC) curves (Boettiger & Hastings, 2012) that estimate true positive and false positive rates for all possible cut-off levels of the indicators (see Methods). The larger the area under the ROC curve (AUC), the more accurately an indicator identifies the risk of cheater’s invasions. Areas below mean that the indicator trend carries no accurate information about the risk of invasion. We compare the estimated area of the ROC curves for each indicator across a range of observation window sizes (Figure 6). The size of the observation window is inversely related to the distance from the collapse threshold. Under weak selection, all indicators are poorly signaling the increasing risk of collapse. Only at big window sizes that include estimates close to the collapse, the accuracy of the indicators increases (above window size). Notably, average degree is inaccurately detecting proximity to collapse for most window sizes (Figure 6A). When selection is strong (Figure 6B), all indicators more accurately signal the rising risk of cooperation loss compared to the case at weak selection (Figure 6A).

Figure 6: Accuracy in the early detection of the indicators. We plot the area under the receiver operator characteristic (ROC) curves (AUC) for structural and non-structural indicators at weak and strong selections ( and ) with the change of observational window. Non-structural indicators are 1) return rate and 2) max-size, and the structural indicators are 1) structural coefficient and 2) average degree. An observational window defines the distance from the transition: a large window corresponds to comparing changes in the indicators including values close to the collapse of cooperation. An AUC of represents a perfect test; an area below represents a worthless test. When the selection strength is weak, the average degree displays misleading prediction information as the window size increases. In this scenario, the structural coefficient is generally the most accurate indicator. At strong selection, all indicators are more accurate than those at weak selection.

3 Methods

3.1 Model

We consider a network model with a fixed number of agents (nodes) but with a non-fixed number of links: to whom and to how many neighbours an agent is connected varies during the evolution of the system. Each agent in the network adopts one of the two strategies of the Prisoner’s Dilemma. A cooperator pays a cost to provide a benefit to all of its neighbours; but the cheaters pay no cost and distribute no benefit. For instance, if a cooperator has cooperative neighbours and cheating neighbours, its payoff is . A cheater in the same neighbourhood has payoff . We use in our numerical experiments. The dynamics of the system are defined by a discrete sequence of update steps (Figure 1). At each update step a new node (a newcomer) is added and a randomly chosen existing node is removed from the system so that the number of nodes is constant (in the main text we assume ). As the newcomer has no specific strategy, a node is selected as a role-model with a probability proportional to its effective payoff = (1+, where specifies the intensity of selection and is the sum of pair-wise interactions of each node. For the selection probability is same for all nodes, while increasing makes it more likely that a newcomer chooses a node with a higher payoff. A newcomer has the same strategy of the role-model and connects with it with a probability as well as with each of its neighbours with a probability (that means is the probability that a newcomer connects to all neighbours of the role-model). The parameters and are called embedding parameters as they explicitly determine the ability of the newcomer to imitate the role-model’s social network. In particular, the embedding parameter determines the ability of newcomers to link to a high number of agents in the network. In this paper we focus on a version of the model described in the work (Cavaliere et al., 2012), where the newcomers always copy (no mutation is allowed) the strategy of the selected role-model. The degree of competition, selection strength, is controlled by the parameter , which determines the effective payoff. The standard model (Cavaliere et al., 2012) where newcomers can select a different strategy than the one used by the role-model has been analyzed in the Supplement (Section ).

3.2 Evaluation of the Indicators

We estimate two classes of indicators: structural and non-structural. Non-structural indicators consider only the population composition, while structural indicators consider the structure of the network. We compute structural and non-structural indicators for each perturbation experiment. A perturbation experiment consists of the introduction of a mutant in a network where all agents have the opposite strategy. We only compute the indicators in the scenario of the invasion of a single cheater in a network of all cooperators. This is done with the following procedure. A network of all cooperators is updated for steps to remove transients. We then introduce a cheater newcomer. The system is updated until one of the two outcomes is reached: either recovery, the cheater fails to invade, or collapse, the cheater invades and cooperation collapses. We perform of these perturbation experiments for each value of the embedding parameter increasing from to at an increment of . We use different selection strengths (), but in the main text we report the indicator results for weak () and strong selection strength (). We identify by the beginning of a perturbation (the addition of the cheater) and with the end of a perturbation (either recovery or collapse). For each perturbation experiment, we compute the indicators as follows regardless of whether a perturbation will lead to recovery or collapse.

  • Non-structural indicators

    - Return rate is , where the Return time is the number of update steps the system takes to go back to its original state following a perturbation. Hence, if is the step at which a perturbation starts and is the step at which the population comes back to its original state, then the return time is defined as . If the perturbation is successful, then the return rate is defined to be .

    - Max-size (of cheaters) is the maximal number of cheaters recorded during a perturbation to a network of cooperators. It is the size of the population if the perturbation leads to collapse.

  • Structural indicators.

    - Structural coefficient :

    (1)

    where is the total number of links (counted twice), is the total number of links in the network at step . Intuitively evaluates the ratio between the beneficial payoff (generated by purely cooperative interactions) and the detrimental payoff (generated by cheaters connected to cooperators). This coefficient is a simplified version of the structural parameter studied in the work (Tarnita et al., 2009b; Nowak et al., 2010).

    - Average degree is the average number of links per node recorded during a perturbation to a network of cooperators.

3.3 Consistency of Indicators - Kendall’s coefficient

Given two pairs of data, e.g., and , we say they are concordant if and or if and ; otherwise if and or and then we say they are discordant. Suppose we have two sequences (corresponding to two variables), each with data points, the Kendall’s coefficient between the two variables is computed using the number of concordant (cp) and discordant pairs (dp):

(2)

If the agreement between the two variable is perfect the coefficient has a value of ; if the disagreement is perfect then it has a value of ; while if the two variables are independent then it is . The coefficient can be interpreted as the probability of observing concordant pairs minus the probability of observing discordant pairs (Conover & Conover, 1980).

Let’s denote as the value of for which the persistence of cooperation is . For each indicator, we compute the Kendall’s coefficient by constructing indicator sequence whose length is the number of s considered between and and each corresponding value is randomly chosen from perturbations. In this way, a large number of sequences can be constructed until all data are exhausted following a bootstrap approach without replacement. The distribution is fitted as a Gaussian one (Figure in the Supplementary Material) through those obtained the Kendall coefficients between each of these constructed indicator sequences and the corresponding embedding parameter s.

3.4 Accuracy of Indicators - ROC curves

The observational window is from to , and . For each in Figure 6, we plot the area under the ROC curve (AUC) that is computed using all (and only) the data between and . Specifically, for each indicator we compute two distinct ROC curves in the following manner —that correspond to two possible discrimination conditions.

We use a cut-off to define true/false positive/negatives as follows. Given two arbitrary points between and , with , we evaluate the values of the indicator at and at and denote them by and , respectively. We denote by and the values of the cooperation persistence at and , respectively. Consequently, we can define the true/false positive/negative by considering two possible (symmetric) discriminatory conditions.

Condition (I):

  • If and , then it is a true positive (TP).

  • If and , then it is a false positive (FP).

  • If and , then it is a true negative (TN).

  • If and , then it is a false negative (FN).

Condition (II):

  • If and , then it is a true positive (TP).

  • If and , then it is a false positive (FP).

  • If and , then it is a true negative (TN).

  • If and , then it is a false negative (FN).

For a given cut-off , we can produce a large number of random indicator pairs and corresponding cooperation pairs , which are picked between and . And then we can evaluate, for each indicator, the total number TP, FP, TN and FN, using either condition (I) or condition (II). Once these values have been obtained, we compute the sensitivity and specificity for each indicator as:

(3)
(4)

The calculated pair (sensitivity, 1-specificity) constitutes a single point of the ROC curve. Repeating the described process for all possible cut-offs (from very small to very large), we can obtain a set of pairs (sensitivity, 1-specificity) that constitute a full ROC curve.

Each indicator has two possible symmetric ROC curves, depending whether the condition used to determine FP, FN, TP, TN is either (I), or (II). For convention, we choose the one that gives the largest AUC when computed within the largest observational window, and use that condition to compute the ROC for all other observational windows. For completeness, we report the ROC curves in the Supplementary Material (Figure ).

4 Discussion

Recognizing the conditions that favor the spreading of cheating and the collapse of cooperation in a community has been a major objective in the study of complex adaptive systems(Levin, 2007, 2010; Bowles & Gintis, 2011). Here we approach this issue from a different perspective and look at the dynamics of cheaters, cooperators, and their interactions to infer the risk of cooperation collapse. Specifically, we argue that the collapse of cooperation resembles a critical transition in our structurally evolving communities - an abrupt change in equilibrium state at the crossing of a threshold. Although, it is difficult to analytically show the nature of this transition, we find that return rate (i.e., the inverse of the time necessary for a cheater to get expelled from the community) decreases prior to the collapse in our numerical experiments (Figure 4). Slow return rates resemble the critical slowing down that is hallmark of proximity to bifurcations points (Scheffer et al., 2009; Wissel, 1984), but it clearly appears in our evolving complex communities.

Apart from decreasing return rates, we found that the abrupt loss of cooperation can be detected in advance by a handful of structural as well as non-structural indicators that are not linked to critical slowing down (Figure 4). We show that the maximum size of invaders is increasing relatively to the risk of collapse. Intuitively, this indicator reflects the disturbance (in terms of invader numbers) that the community can tolerate without collapsing. On the other hand, we use, for the first time, structural indicators for detecting instabilities in network dynamics. The few studies on critical transitions in networks assume fixed topologies and use non-structural indicators to detect species extinctions (Dakos et al., 2015; Suweis & D’Odorico, 2014). Network metrics, like connectance, assortativity, or clustering, have been used as structural indicators applied on spatial dynamical models where network structure was derived from cross-correlation matrices (Tirabassi et al., 2014; Viebahn & Dijkstra, 2014). However, these matrices were static. In our communities, network structure is not static but it co-evolves with the spreading of strategies depending on the expected differences in payoff between cheaters and cooperators. Clearly, the dynamic nature of the structure of our communities permits to test indirect alternative metrics for estimating the risk of abrupt transitions. Among the structural indicators we tested (Figure 4 and Figure in the Supplementary Material), we show that the decreasing ratio of beneficial interactions (between cooperators) to detrimental ones (between cheaters and cooperators) can be successfully used to detect the increasing risk of cheater’s invasions (Figure 4). Moreover such indicator is the most accurate when the strength of selection is weak (Figure 6)

Tests about the consistency and accuracy of the estimated trends (Figure 5 and 6) strongly imply that there is high uncertainty in detecting robustly the loss of cooperation in an evolving community. Such uncertainty in the detection of critical transitions of resilience has been reported to be caused by a variety of factors (e.g. heterogeneity, non-linear dynamics, measurement error, strong environmental stochasticity (Dakos et al., 2015; Boettiger et al., 2013)). In our communities, we found that detection is strongly dependent at least on the underlying level of selection strength. This does not come as a surprise. For weaker selection strength, cooperators are less disfavored allowing them to persist even at high levels of the embedding parameter . This implies that all indicators, but average degree, can “pick up” the increase of fragility only at very high values of close to the collapse (Figure 5).

Clearly, our approach is limited as we considered the dynamics of the communities after a single invasion. A more realistic scenario would allow for constant invasions of cheaters, or mutations of cheaters to cooperators as well. In that case, cooperation would collapse and recover in an infinite sequence of such events (Cavaliere et al., 2012). When we tested such more realistic scenario of long-term evolution with endogenous mutations, we found that most of the indicators could still detect the increasing risk of cooperation loss (Section in the Supplementary Material). Similarly, although we only tested a handful of indicators, it appears that alternative metrics could also be used as potential indicators to detect the cooperation loss (Figure in the Supplementary Material). We also found that some of the considered indicators (e.g. average degree) can detect the collapse of cooperation even if in the absence of cheater invasions (Figure in the Supplementary Material).

Despite our work is based on a specific dynamical network model, the approach and the indicators we developed can be easily extended to other evolutionary models of dynamical structured populations (Lieberman et al., 2005; Tarnita et al., 2009b, a). In these models, it is straightforward to measure the same or similar indicators and to evaluate their consistency. For example, average degree appears to be a good indicator when the selection is either weak or strong, but it is the relative number of interactions between cooperators and cheaters that accurately detects whether cheaters have better (or worse) chances to invade (Figure 6). Similar features of the indicators may be presented in other models of evolving structured populations, where the structural coefficients have been shown to be key for studying the long-term level of cooperation (Tarnita et al., 2009b; Nowak et al., 2010).

The presented results suggest that it is possible to evaluate the fragility of cooperation by monitoring interactions and following the behaviour of individuals in a community. In practice, however, such information might be difficult to obtain. Identifying cheaters from cooperators, or even following their interactions in time seems a daunting task. Still, the difficulty for evaluating the proposed signals may depend on the specific application, the type, and the scale of the community in question. For instance, bacterial communities are emerging as a promising experimental tool to validate hypotheses in ecology and evolution (Velenich & Gore, 2012). Testing the proposed indicators may be done with cooperative cells that produce public goods. In such bacterial colonies, in principle one could perform a series of perturbation experiments by introducing cells of cheaters (or invaders) and measuring return rate. Recent papers have showed how structured cellular communities can be used to measure critical slowing down in deteriorating populations (Dai et al., 2013), even in the presence of cheaters exploiting public good producing yeast (Chen et al., 2014).

Perhaps the most novel conclusion of this work is that we can identify the progressive loss of cooperation by combining non-structural and structural indicators. Although the patterns we report might be idiosyncratic to our model assumptions, they still confer a promising pattern. As such our study paves the way for testing and developing similar indicators in a variety of evolving dynamical networks, ranging from biological systems to ecological communities and even socio-economic networks.

5 Acknowledgements

We acknowledge helpful discussions with C.E. Tarnita on the structural indicators. V. Dakos was supported by a MarieCurie IntraEuropean postdoctoral fellowship and M. Cavaliere was supported by Engineering and Physical Sciences Research Council grant number EP/J02175X/1.

6 Author contributions statement

M. Cavaliere. and V.Dakos conceived the study and wrote the paper. G.Yang conducted the computational experiments. All authors analysed the results and reviewed the paper.

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