Interactive diversity promotes the evolution of cooperation in structured populations

Evolutionary games on networks traditionally assume that each individual adopts an identical strategy to interact with all its neighbors in each generation. Considering the prevalent diversity of individual interactions in the real society, here we propose the concept of interactive diversity, which allows individuals to adopt different strategies against different neighbors in each generation. We investigate the evolution of cooperation based on the edge dynamics rather than the traditional nodal dynamics in networked systems. The results show that, without invoking any other mechanisms, interactive diversity drives the frequency of cooperation to a high level for a wide range of parameters in both well-mixed and structured populations. Even in highly connected populations, cooperation still thrives. When interactive diversity and large topological heterogeneity are combined together, however, in the relaxed social dilemma, cooperation level is lower than that with just one of them, implying that the combination of many promotive factors may make a worse outcome. By an analytical approximation, we get the condition under which interactive diversity provides more advantages for cooperation than traditional evolutionary dynamics does. Numerical simulations validating the approximation are also presented. Our work provides a new line to explore the latent relation between the ubiquitous cooperation and individuals’ distinct responses in different interactions. The presented results suggest that interactive diversity should receive more attention in pursuing mechanisms fostering cooperation.

Besides exploring structural complexity [31], researchers set out to consider diversities recently [32][33][34][35][36][37][38]. For example, social diversity defines a scenario where the investment of each individual per game relies on the number of its social ties and individuals with more neighbors contribute less to each game. In particular, when social ties follow a scale-free distribution, individuals' investments per game are diverse, which leads to an impressive boost of cooperation [34,35]. Heterogeneous teaching activity models a situation where the rate of strategy imitation depends on both payoff difference and individuals' teaching rates, allowing that individuals with high teaching rates spread their strategies much faster than others do. The remarkable distinction of teaching rates enhances cooperation significantly [36]. Reacting diversity endows individuals with different rates to remove unwanted ties, by which a few individuals sever dissatisfied social ties (that link to defectors) swiftly while others tolerate neighbors' defection. The coexistence of different reacting rates promotes cooperation despite individuals' myopia in their own interests [37]. For a comprehensive review of the diversity, see [38].
Although various diversities receive much attention, so far, most researchers have just focused on the difference of individuals' social features [38], like the number of social ties [35], teaching rates [36] and reacting rates [37]. In these studies, each individual treats all its neighbors as equivalent and adopts an identical strategy against them in each generation [38], which is in stark contrast with many realistic situations [39][40][41][42]. For example, in epidemic dynamics, individuals may take distinct interactive strategies against neighbors to protect themselves from diseases, such as keeping the contact with the healthy while severing links to the infected [39]. In human society, heredity and inheritance endow individuals with distinct properties, like consanguinity, phenotypes and briefs. These individuals may live in harmony with those possessing similar properties while frequently in conflict with the others. A typical example is about parochial altruism that individuals benefit fellow group members at a cost to themselves and adopt a hostile attitude towards those from other groups [40,41]. Besides, in biological systems, human disease missense mutations may cause specific perturbations in particular molecular interactions while leave other interactions undisturbed, which shows the distinct reactions of protein mutants in different interactions [42]. Also, one's emotions, like the sympathy or goodwill to the inferiors and the envy or spite to the superiors, may lead to its distinct acts towards neighbors in different social rank, which has been introduced to explore the evolution of cooperation [43,44]. All the above examples indicate that individuals' actions in an interaction usually depend on both who they are and whom they interact with. Probably, the same individual behaves distinctly in different interactions in one generation.
Taken together, here we interpret the discriminative acts of an individual in different interactions as interactive diversity. From the perspective of networked systems, we investigate the dynamical effects of interactive diversity on the evolution of cooperation based on edge dynamics [45] rather than traditional nodal dynamics explored in other kinds of diversities [35,37]. Put differently, instead of probing how cooperation disseminates over nodes, here we are more interested in how cooperation spreads from one edge to another. The well-mixed and structured populations represented by various complex networks are studied systematically. The obtained results show that interactive diversity fosters cooperation without invoking any other mechanisms.

Model
In a typical symmetric two-person game, two individuals simultaneously decide whether to cooperate (C) or to defect (D). Depending on their choices, they receive some payoffs, which can be expressed by a general 2×2 payoff matrix [29]. Without loss of generality, we use two parameters T, S to represent the payoff matrix as ( ) 2, three typical social dilemmas, the Prisoner's Dilemma [46], the Snowdrift game [47], and the Stag-hunt game [48], can be captured [49]. In this paper, we study interactive diversity and cooperation in these most used social dilemmas. Figure 1 shows how interactive diversity is introduced in the structured population. Traditionally, without interactive diversity, each individual either cooperates or defects to all its neighbors in each generation [29]. While in the presence of interactive diversity, everyone is allowed to adopt cooperation and defection simultaneously against its different neighbors. As shown in figure 1(b), individual l c cooperates with h 1 and meanwhile defects to h 2 .
In each generation, each individual plays games with all its neighbors with corresponding strategies and accumulates payoffs from all games that it participates in. When it comes to strategy updating, each individual only knows its neighbors' actions towards itself owing to the interactions with these neighbors, while it is not informed how its neighbors interact with the third party. Therefore each individual would update its strategies by imitating how its neighbors behave to it. Here we take l c updating the strategy against h 2 for example ( figure 1(b)). First, a neighbor is randomly chosen from l c ʼs neighborhood, called h 1 . The chosen one could be h 2 , which means that l c possibly treats h 2 in the same way as h 2 does to l c . Next, if l c ʼs accumulated payoff P l c is less than h 1 ʼs accumulated payoff P h 1 , l c imitates h 1 (l c replaces its strategy against h 2 with h 1 ʼs strategy against l c ) with the probability min , 0 and K is the larger one between l c ʼs and h 1 ʼs degree. MK ensures the proper normalization of the probability [29]. Otherwise, l c remains its strategy against h 2 . l c ʼs strategies towards other neighbors are updated in the same way. In fact, all strategies of an individual are independent in some ways. Thus, it is reasonable for each individual to subsequently update its strategies independently by randomly choosing a neighbor to imitate, in accordance with the canonical updating rule [28,29,35]. In this paper, all individuals update their strategies synchronously in the same way as l c (the example mentioned above). And asynchronous update, which usually best describes biological and human processes [31], is also studied for a comparison.
For a comprehensive research, we explore interactive diversity in both well-mixed and structured populations represented by complex networks with different degree of heterogeneity, including regular, random [50], small world [16], and scale-free networks [17]. To further understand the evolutionary dynamics from the microscopic perspective, a double-star graph with common leaves (figure 1), serving as the simplest abstraction of heterogeneous network [34,35,51], is constructed to show the detailed processes of evolution. The doublestar graph has two centers, a left center h 1 with -N 1 leaves and a right center h 2 with -M 1 leaves. Among the leaves of h 1 , --N L 1 leaves l 1 are solely linked to h 1 , and L leaves l c are linked to both h 1 and h 2 . Similarly, --M L 1 leaves l 2 are just linked to h 2 . In this paper, considering interactive diversity, we quantify the frequency of cooperation f C with the average frequency of cooperative strategies [52,53].

Results
By Monte Carlo simulations, we analyze the dynamics of interactive diversity in well-mixed and four structured populations represented by complex networks. Here the small world network is generated in terms of the Watts-Strogatz algorithm [16] with rewiring probability p=0.1, and the scale-free network is generated via the mechanisms of growth and preferential attachment [17]. The population size is 200 for the well-mixed one and 1000 for structured ones with average degree 4. Initially, each individual randomly chooses cooperation or defection in each interaction, and the population has equal amount of cooperative and defective strategies. The mean frequency of cooperation is averaged over5 10 3 generations after a transient time of 10 6 generations, and the equilibrium average is averaged over 10 simulations with 20 different realizations for each kind of network. The obtained results do not change in larger populations or with asynchronous update rules. Later, by theoretical analysis and Monte Carlo simulations on a double-star graph, we clearly elucidate the underlying microscopic mechanisms responsible for the outcomes in complex networks.

Interactive diversity in complex networks
Let us first focus on how interactive diversity affects the evolution of cooperation in the well-mixed population. In the SH domain (-< < S 1 0 , < < T 0 1), previous work has demonstrated that in the infinite well-mixed population without the interactive diversity, the critical value of the initial density of cooperators is + -( ) S T S 1 above which cooperators dominate while below which defectors take over the whole population [54]. Here, we set the initial cooperator density to be 0.5 (apply to all the simulations in figure 2). When 0 . 5 , meaning that under our initial setting cooperators finally die out (dominate). However, when interactive diversity is introduced, our results show that it can lead to the emergence of cooperation even when -> T S 1 (see the first column of figure 2), thus indicating that interactive diversity enlarges the survival ranges of cooperation. Similar results are also found in PD where the population inevitably evolves to full defection in the traditional model. When it comes to the SG domain, interactive diversity elevates the final level of cooperation. An intuitive explanation for these results is that, interacting diversely makes it possible for individuals to react to opponents' cooperative acts with cooperation and defective acts with defection. It helps to maintain individuals' incomes and reduce the exploitation by defectors, which weakens the definite advantage of defectors in the well-mixed population. We find a similar case in previous studies [55]: individuals with low strategy values and high image scores benefit from their mutual donation while refuse to contribute to defectors with the highest strategy values and low image scores, which leads to the extinction of defectors.
Then we employ the population structures represented by regular, random, small world, and scale-free networks. Due to the similar results obtained in the first three networks (middle three contours in figure 2(c)), here we analyze them together. We find that, in both the Prisoner's Dilemma and the Stag-hunt game, cooperation is elevated significantly beyond that supported network reciprocity alone. More to the point, the moderate topological heterogeneity embedded in random and small world networks seems to enhance the facilitative effects of interactive diversity on the evolution of cooperation. Here we do not stress this point too much. For these outcomes in the structured populations, the stable cooperative clusters are responsible. Inside the clusters, individuals preserve high payoffs by their mutual cooperation. Meanwhile, they defect to defectors outside to get out of exploitation. Consequently, individuals in the boundary of the clusters resist the invasion of defectors even in the harsh situation where the temptation to defect is sizable. Moreover, we observe the slight decrease of the cooperation level in SG, and a brief explanation is given below. The payoff structure of SG determines that it is better for individuals to adopt the opposite strategies of their opponents, which inhabits the formation of compact clusters of cooperators [53]. Meanwhile, since individuals are able to react differently to different neighbors in our model, the preferred opposite strategy pattern along a link is far more pervasive than that in the traditional case. These two together lead to the shrinkage in the level of cooperation.
Further we incorporate large topological heterogeneity in our study. By virtue of the greatest heterogeneity possessed by scale-free networks, interactive diversity guarantees the largest survival domain of cooperation in both PD and SH. However, in the relaxed social dilemma, such as the upper-left domain of PD, cooperation and defection coexist. Such coexistence is in contrast with the domination of cooperators in the traditional setting [29] as well as that in structured populations without large topological heterogeneity. In other words, when the interactive diversity, the frequency of cooperation f C in the well-mixed population, regular, random, small world, and scale-free networks are plotted as contours of T and S. Detailed difference of f C shown in (c) is generated by subtracting those in (a) from those in (b) correspondingly. In each panel, the upper-right quadrant corresponds to the Snowdrift game (SG, , the lower-left one to the Stag-Hunt game (SH, -< < < < S T 1 0 1 ), and the lower-right one to the Prisoner's Dilemma (PD, temptation to defect is small, the combination of interactive diversity and large topological heterogeneity reversely decreases cooperation level, implying that combining many promotive factors may make a worse outcome [25]. The evolution on the double-star graph later helps to make this mechanism clear, as figure 5 shows.
Next, we explore interactive diversity in the highly connected populations. Taking the Prisoner's Dilemma as a typical example, we consider this problem with rescaled payoff matrix where r denotes the ratio of the cost over the net benefit of cooperation [56]. Figure 3 shows that interactive diversity maintains cooperation in the highly connected populations irrespective of population structures. Traditionally, the evolution of cooperation is extremely susceptible to the average degree. Increasing the number of social ties makes cooperators possibly be exploited by more defectors, which weakens the resistance of cooperator clusters to defectors' invasion. In particular, the approximation to the well-mixed limit consistently causes the sharp decline of cooperation [57]. However, in our model, increasing average degrees towards the fully connected networks does not offer defectors such an advantage to exploit more cooperators. As we have mentioned, individuals inside the clusters reduce the exploitation by defectors outside, which actually consolidates the clusters [9]. Therefore, considerable cooperative acts remain in the population.
The results above are obtained in the setting that individuals randomly choose strategies against different neighbors initially (with random initial strategies) and everyone updates all its strategies synchronously. Considering that individuals know little about their neighbors in the beginning, instead of treating neighbors differently, they may adopt an identical strategy against neighbors initially (with same initial strategies). Furthermore, we probe how strategy updating proportions p affect the evolution of cooperation. For = p 25%, each individual is set to update one fourth of its strategies in each generation. The presented results in figure 4 demonstrate that the evolution is robust for strategy updating proportions and individuals' initial states.

Interactive diversity on the double-star graph
By scrutinizing the detailed evolutionary process on the double-star graph, we make two questions understood: how cooperative clusters form and expand; why cooperation shrinks in the relaxed situations when interactive diversity and topological heterogeneity are combined together. The answer for the first question well explains the emergence of cooperation in the harsh situations. Here the payoff structure is the same as that in figure 3. Two situations, the relaxed one for a small r and the harsh one for a large r, are quantitatively analyzed respectively in the traditional setting (without interactive diversity, see appendix A) and in our model (with interactive diversity, see appendix B). In figure 5, in each situation, three most common evolutionary processes, including one in the traditional setting and two in our model, are elaborated. The theoretical cooperation level corresponding to each process is shown in table 1.
We start with the evolution in the relaxed situation. In the traditional setting, defectors on hubs easily become the victims of their own success due to the dissemination of defection in their neighborhoods, and then lose the occupation of hubs [35,58,59]. Here on the double-star graph, we get a critical cost-to-benefit ratio below which cooperators easily wipe out all defectors and dominate in the population (A 1  A 5 in figure 5(a), see appendix A).
In the presence of interactive diversity, each individual is allowed to adopt cooperation and defection simultaneously. Especially in the games played by individuals sitting on hubs, cooperation and defection are observed simultaneously. Indeed, the dissemination of defection causes inevitably losses to them. Nevertheless, In the harsh situation, the sizeable temptation to defect and the great loss from exploitation weaken the viability of cooperators. In the traditional setting, defectors on hubs directly invade their nearby sites, resulting in the extinction of cooperators (A 1  A 3 in figure 5(b)).
With interactive diversity, cooperation however emerges. The formation and expand of cooperative clusters are responsible for this. The former one can be explained through the process  B 22 B 32 in figure 5(b). The latter one is shown in  B 21 B 51 . On the one hand, relying on the mutual cooperation inside the clusters and the resistance to exploitation by defectors outside, individuals in the boundary of the clusters hold their advantages over individuals outside. These advantages make the dissemination of cooperation feasible. On the other hand, for individuals outside the clusters, in the traditional setting, imitating cooperative behaviors means to be exploited by all neighboring defectors. However, interactive diversity enables them to imitate cooperative acts just in some interactions, which reduces the risk to adopt cooperation and facilitates the further spread of cooperation.
Moreover, the analytical dynamics above (the graph sets in the dashed box and the stable values in the right side of solid box) is validated by our simulation results (the lines in the solid box of figure 5) in both relaxed and harsh situations. Here we have to note that the steady trapped states shown in figure 5 actually result from the update rule by which individuals only imitate those performing better. When an irrational choice for one to learn the less successful individuals is allowed, as induced by the application of the Fermi rule, population escapes from such trapped states. Despite the different equilibriums, both the average frequency of cooperation and the microscopic mechanisms responsible for these results still remain consistent.
In figure 6, we show the cooperation level throughout the entire range of r on double-star graphes with two different graph sizes. When individuals interact identically, as r increases, f C is predicted to decrease sharply near r * since the cooperator suffers from its occupation to a hub and this suffering inhibits the further spread of cooperation to its neighbors. With interactive diversity, individuals adjust their strategies partly and cooperation spreads easily even for large r, resulting in the smooth curve of f C . Analytically, r * well predicts when interactive diversity provides more advantages for cooperation than the traditional dynamics does (see appendix B). As figure 6 shows, no matter who occupies the larger center initially, interactive diversity always benefits cooperation for r above r * . Moreover, the promotion is enhanced further in both relaxed and harsh situations when a cooperator is located on the larger center in the beginning. Other graph sizes lead to similar results. Although we study this problem on the double-star graph, we have checked that the microscopic mechanisms explained above are also applicable in both well-mixed and general structured populations.
To this end, we have explained the non-monotonic results shown in figure 2. Here we want to stress that the impact of interactive diversity on the evolution of cooperation is less straightforward than that of other diversities like topological diversity [29]. In case of topological diversity, individuals' difference helps to establish a sort of strategy homogeneity around hubs, where successful hubs spread strategies to their neighborhoods. This strategy homogeneity is harmful to defector hubs while it is beneficial to cooperator hubs, leading to a higher cooperation level around the highly-connected nodes and then the boom of cooperation in the whole population [35,60]. However, interactive diversity enables successful individuals to enforce cooperation and defection simultaneously to their neighbors, breaking the strategy homogeneity. Hence the largest cooperation level may not accompany individuals with many social ties, and the diverse system behavior is observed.  Situations

Discussion and conclusions
We have investigated interactive diversity in both well-mixed and structured populations represented by various networks, focusing specifically on the effect of individuals' distinct responses against different neighbors on the evolution of cooperation. Without invoking other mechanisms like recognizing each other from past interactions [61] or being informed how neighbors interact with the third party [55], individuals independently update each of their strategies by imitating what successful neighbors apply to them. We find that, in the well-mixed population, with interactive diversity, cooperation emerges in a wide parameter range under three most adopted game metaphors. In the structured populations, interactive diversity drives the cooperation to a high level in PD and SH. The facilitative effects are also identified on networks associated with high average degrees, which provides an instrumental clue on explaining how cooperation thrives on highly connected social populations [16,62,63]. All of these have demonstrated that interactive diversity benefits cooperation. When interactive diversity is introduced in the scale-free network, cooperation is elevated in the harsh situation while reduced in the relaxed case. Note that both interactive diversity and large topological heterogeneity facilitate cooperation solely. The reduction implies that combining many promotive factors together may make the outcome worse than that with each single one, verifying the conclusion in previous studies [25].
We elaborate on the evolutionary process on a double-star graph which serves as the simplest abstraction of heterogeneous networks, to explore the underlying mechanisms leading to outcomes above. On the one hand, interactive diversity helps to form stable cooperative clusters in the boundary of which individuals cooperate with cooperators inside and meanwhile defect to defectors outside. Compared with traditional cooperator clusters [9], here individuals in the boundary can reduce the exploitation and resist the invasion of defectors in harsh social dilemmas. On the other hand, interactive diversity enables individuals to imitate cooperative behaviors in a fraction of interactions, which lowers the loss caused by the strategy turning from defection to cooperation. Therefore, in the harsh social dilemma, dissemination of cooperation is still feasible. These two together well explain the promotive effects of interactive diversity on the emergence of cooperation.
Actually, these mechanisms can be easily understood from the perspective of edge dynamics. Different from the traditional dynamics where edges connecting to a specific node are closely related, here interactive diversity considers the independence of edges linking to a focal node to some extent. In the network, cooperation disseminates from one edge to another rather than from one node to another, which effectively reduces the fluctuation of relevant properties of the associated node. Put differently, interactive diversity facilitates the permeation of edges of mutual cooperation, indicating that how two individuals treat each other may have a more profound impact on the evolution of cooperation than whether they are cooperators or defectors.
As for the combination of topological heterogeneity and interactive diversity, the former one endows individuals on hubs with chance to participate in more games, and the latter one enable them to adopt cooperation and defection simultaneously. Although these individuals behave defectively in some interactions and incur losses, they still keep advantages over their neighbors. Consequently, defection survives under the shelter of individuals on hubs. A previous work [52] has probed the effects of simultaneous adoption of cooperation and defection in a square lattice and ring lattice in the Prisoner's Dilemma. However, previous conclusions are incomplete due to the lack of further investigations in other structured populations. Here we systematically study these problems in structured populations represented by various complex networks with three typical two-player games. We find the close relation between interactive diversity and edges, such as the connectivity stressing edges' number and heterogeneity stressing degree distribution. And we understand the evolutionary dynamics on networks from the point of edges. We have to point out that, when individuals are modeled to behave differently, they need to remember their strategies to each neighbor, which might be costly. The consideration of a cognition cost assigned to memorize the information may enrich the work and bring some interesting results [64][65][66]. Moreover, the combination with other factors, like individuals' reputation [67] and the evaluation to the payoffs [68,69] may provide an efficient response to different opponents' strategies. Especially, with intention recognition, individuals recognize opponents's intentions on the basis of the previous interactions or in some other ways, and then make decisions according to such recognized intentions [66,70,71], which well achieves interactive diversity and is worth a further investigation. Interactive diversity, consisting of diverse interaction types [72], constitutes realistic representations of the real world and should receive more attention.
In this part, we provide a theoretical analysis about the evolutionary dynamics on the double-star graph in the traditional setting. The double-star graph is depicted in the main text, where the number of common leaves l c (L) is much less than the connectivity of two centers (N M , ). We begin with the assumption of > N M . Initially, a defector occupies the left center h 1 and a cooperator occupies the right center h 2 . Over leaves l 1 , l c , and l 2 , defectors and cooperators are distributed equally. The evolutionary processes in relaxed and harsh situations are studied and the details are provided below. For the convenience of description, we use h 1 (h l l l , , , c ). In the beginning (A 1 of figure 5(a)), the payoffs (P) of h 1 and h 2 are given by The payoff of a cooperator (defector) l 1 is = -P r and the update rule with which each individual imitates successful neighbors with the probability proportional to their payoff difference, Ds are more likely to spread to l 1 than to h 2 . A similar case happens to l c . This means that h 1 ʼs neighbors on leaves will turn into defectors before h 2 , leading to the deterioration of h 1 ʼs neighborhood. Meanwhile, h 2 spreads Cs to its neighbors. At this stage (A 2 of figure 5(a)), both l c and h 1 have payoffs + r 1 . By both h 1 and l c reversely imitate h 2 . Compared with h 1 , l c has less neighbors to imitate. In other words, l c will imitate h 2 more frequently and turn into a cooperator before h 1 (A 3 of figure 5(a)). After that, h 1 is also inevitably invaded. Once it happens, the invader on h 1 obtains Given that defectors l 1 get = + P r 1 l 1 , the invader can stabilize on h 1 (A 4 of figure 5(a)) whenever where r * is a critical cost-to-benefit ratio. Since both N and M are much larger than L, we get > * r r 1 and > * r r 2 , which demonstrates the existence of r * . For < * r r , > P P h l 1 1 whereby Cs of h 1 spread to l 1 further (A 5 of figure 5(a)). If r is slightly above r * , due to < P P h l 1 1 , Cs can never spread to l 1 , resulting in = figure 5(a)). Here we stress that the common leaves l c play an important role in the evolutionary dynamics. The traditional double-star graph exhibiting no loops makes the simple analytic treatments possible [35]. However, when it comes to PD rather than PGGs (Public Goods Games), without the mutual cooperation with common neighboring cooperators, the cooperator can never stabilize on the left center, which actually can not explain the boost of cooperation in heterogeneous networks [28,29].
As r increases, the initial difference between h 1 ʼs and h 2 ʼs payoff h h 1 2 rises. Meanwhile, the value decreases, which indicates that Ds of h 1 may not spread to l 1 before to h 2 . In particular, in the harsh situation, h 2 is invaded initially and defectors dominate in the population (A 1 →A 3 of figure 5(b)).

Appendix B. Evolutionary dynamics on the double-star graph with interactive diversity
All settings are similar to those in appendix A except that individuals are allowed to behave differently against different neighbors. Because various types of individuals is involved in the graph and their states are diverse, the evolutionary dynamics is quite intricate. Here we simplify the evolution scenarios and give two most typical processes separately in both relaxed and harsh situations. In each situation, the average cooperation level lies between the theoretical values of these two processes. Before our analysis, we give an intuitive understanding about how interactive diversity affects individuals' evolution. p ( < < p 0 1) denotes the probability that a cooperator h 2 with M neighbors imitates a defector h 1 . h 2 is assumed to have a larger payoff than all its neighbors except h 1 , suggesting that h 2 can only change its strategy by imitateing h 1 . In the traditional setting, h 2 turns to a defector with probability p and remains unchanged with probabilityp 1 . With interactive diversity, in one generation, the probability for h 2 turning into a defector is p M (updating M strategies successfully) and the probability for h 2 to remain unchanged is -( ) p 1 M (failing to update any strategy). The relations > p p M and -> -( ) ( ) p p 1 1 M indicate that such cases are less likely to happen. Instead, h 2 is likely to update its strategies in Mp interactions in average.
Let us first discuss the evolution in the relaxed situation. The initial stages are similar to those in appendix A: Ds of h 1 spread to l l , c 1 , and Cs of h 2 spread to l 2 (B 2 in figure 5(a)). Next, both l c and h 1 imitate h 2 . Because of the connectivity of l c , l c becomes a cooperator before h 1 .
Here we consider the first process: h 1 imitates h 2 successfully and adopts C against h 2 (B 31 in figure 5(a)). Besides, h 1 is expected to adopt Cs to additional L neighbors l c (the exact number may not be L while it does not affect our analysis) and n 1 neighbors l 1 . The expected value of n 1 is The value of n 1 suggests that n 1 is small. The payoff of l 1 associated with those n 1 interactions is = + P r 1 l 1 . We easily get the payoff difference -- Sooner, h 1 places Cs on n 1 corresponding leaves, which increases h 1 ʼs payoff in return and meanwhile enhances h 1 ʼs advantage over leaves l 1 . Since the payoff of h 1 is still lower than that of h 2 , h 1 continues imitating h 2 and adopting Cs in some interactions. The cycle of imitation and spread of Cs gradually increases h 1 ʼs payoff until it rises to or slightly exceeds h 2 ʼs payoff (B 41 in figure 5(a)). At this stage, the payoffs of h h , 1 2 can be approximated as = = P P M h h 1 2 . That is, h 1 adopts Cs to L leaves l c , --M L 1 leaves l 1 and h 1 . The stable frequency of cooperation is The second process is similar except that h 1 insists on defecting to h 2 (B 32 in figure 5(a)). Once h 1 gets the highest payoff in its neighborhood (B 42 in figure 5(a)), it reversely turns h 2 into a defector (B 52 in figure 5(a)).
Actually, P h 1 may exceed --M r 1 (the maximum payoff of a cooperator h 2 ) slightly. That is, h 1 cooperates in -M 2 interactions and defects to h 2 . The stable frequency of cooperation is When r is a little above r * , in the traditional setting, = is near f C R 2 (both N and M are much larger than L). However, in our model, the average f C lies between f C R 1 and f C R 2 . Equivalently, the average f C is larger than f C in the traditional setting. Therefore, r * is also a critical value above which interactive diversity provides more advantages for cooperation.
In the harsh situation, although the cooperator h 2 is more vulnerable to nearby defector h 1 , h 2 just adopts Ds in a few interactions. Assume that h 2 adopts Ds in n 2 interactions. Due to the rapid deterioration of h 1 ʼs neighborhood, h 1 will not be influential in h 2 . Two typical processes are considered. In the first one, h 2 insists on cooperating with h 1 (B 21 in figure 5(b)). At this stage, = = + P P r 1 In such a case, Cs spread to h 1 from h 2 and then to l c through h 1 . The subsequent process (B 31 →B 41 in figure 5(b)) is similar to the first process in the relaxed situation (B 31 →B 41 in figure 5(a)). The stable frequency of cooperation is The second process considers that h 2 adopts D to h 1 (B 22 in figure 5(b)). In such a case, h 2 and its -M n 2 neighbors form a stable cooperative cluster while Cs can never spread to h 1 . Since h 2 adopts Ds against h 1 , n 2 is 1 at least. Taking = n 1 2 , the stable frequency of cooperation is Actually, n 2 relies on the cost-to-benefit ratio and the time length of the evolution. Both these values increase n 2 and the actual cooperation level deviates much from the theoretical value as r increases. Furthermore, if a cooperator occupies the larger center initially ( > M N ), = -* r L N L still exists and the evolutionary dynamics remains nearly unchanged in the traditional case. With interactive diversity, in the relaxed situation, since h 1 ʼs payoff never exceeds h 2 ( > M N ), h 1 imitates h 2 until it precludes all Ds. Finally, defection becomes extinct. In the harsh situation, cooperation is still more favorable compared with the case without interactive diversity. Taken together, interactive diversity always supports the evolution of cooperation.