Deterministic and stochastic cooperation transitions in evolutionary games on networks

Nagi Khalil, I. Leyva, J. A. Almendral, and I. Sendiña-Nadal

Phys. Rev. E 107, 054302 – Published 5 May 2023

Abstract

Although the cooperative dynamics emerging from a network of interacting players has been exhaustively investigated, it is not yet fully understood when and how network reciprocity drives cooperation transitions. In this work, we investigate the critical behavior of evolutionary social dilemmas on structured populations by using the framework of master equations and Monte Carlo simulations. The developed theory describes the existence of absorbing, quasiabsorbing, and mixed strategy states and the transition nature, continuous or discontinuous, between the states as the parameters of the system change. In particular, when the decision-making process is deterministic, in the limit of zero effective temperature of the Fermi function, we find that the copying probabilities are discontinuous functions of the system's parameters and of the network degrees sequence. This may induce abrupt changes in the final state for any system size, in excellent agreement with the Monte Carlo simulation results. Our analysis also reveals the existence of continuous and discontinuous phase transitions for large systems as the temperature increases, which is explained in the mean-field approximation. Interestingly, for some game parameters, we find optimal “social temperatures” maximizing or minimizing the cooperation frequency or density.

Received 5 October 2022
Revised 13 February 2023
Accepted 17 April 2023

DOI:https://doi.org/10.1103/PhysRevE.107.054302

Physics Subject Headings (PhySH)

Competition Continuous phase transition Discontinuous phase transition Nonequilibrium statistical mechanics Stochastic processes

Monte Carlo methods Prisoner's Dilemma

Interdisciplinary PhysicsNetworksStatistical Physics & Thermodynamics

Authors & Affiliations

Nagi Khalil ^1,*, I. Leyva^1,2, J. A. Almendral ^1,2, and I. Sendiña-Nadal ^1,2

¹Complex Systems Group & GISC, Universidad Rey Juan Carlos, Móstoles, 28933 Madrid, Spain
²Center for Biomedical Technology, Universidad Politécnica de Madrid, Pozuelo de Alarcón, 28223 Madrid, Spain

^*nagi.khalil@urjc.es

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Vol. 107, Iss. 5 — May 2023

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Images

Figure 1
Raster plots for the evolutionary dynamics of a population on a random regular graph for two different game settings displaying quasiabsorbing states (top panel, $S = - 0.4, T = 0.55$ ) and mixed strategy states (bottom panel, $S = 0.4, T = 1.55$ ). Blue (yellow) [light gray (dark gray)] colors encode defection (cooperation) strategies. The population size is $N = 600$ and each player is randomly connected to $k = 3$ neighbors. The rest of the parameters are $R = 1$ and $P = θ = 0$ .
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Figure 2
Deterministic cooperation transitions: (a) Average cooperation observed in Monte Carlo simulations of random regular graphs with $N = 1000$ nodes and $k = 3$ as a function of $T$ for three different values of $S$ (see the legend). Vertical lines correspond to $T$ values solving Eq. (14) for each one of the values of $S$ used ( $S = - 0.4$ , red continuous; $S = 0$ , dotted blue; and $S = 0.4$ , dashed black). The left red (right black) circle corresponds to the game dynamical settings used in Fig. 1 [(b)]. (b) Average cooperation as a function of $T$ when $S = 0.4$ for $N = 1000$ networks with different degree distributions and increasingly longer degree sequences. Vertical gray dashed lines correspond to the $T$ fulfilling Eq. (14) for the case of the ER $〈 k 〉 = 4$ networks ( $k = {2 : 12}$ in our ensembles), which encloses the previous $k = {3}, {4}, {4, 7}$ cases. See the main text for the rest of the parameters.
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Figure 3
(a) Average cooperation observed in Monte Carlo simulations of graphs with $N = 600$ nodes and ER and SF topologies of different $〈 k 〉$ values. Vertical dashed gray lines correspond to the possible transitions for ER, $〈 k 〉 = 8$ , which includes the $〈 k 〉 = 4, 6$ cases. Game parameters: $S = 0$ , $θ = 0$ . (b) Raster plot realization for the game and graph conditions represented by the blue dot in (a) after discarding a transient of $t = 10^{4}$ time steps.
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Figure 4
Stochastic cooperation transitions in a random regular network with $N = 1000$ nodes and $k = 3$ . (a), (b) Average cooperation as a function of $T$ for several values of the pseudotemperature $θ$ and (a) $S = - 0.4$ and (b) $S = 0.4$ . In both panels, vertical dashed lines correspond to the $T$ values predicted by Eq. (14) using the constant $S$ value and the rest of parameters. (c), (d) Average or probability of full cooperation as a function of $θ$ for several values of $T$ resulting from different Monte Carlo simulations for (c) $S = - 0.4$ and (d) $S = 0.4$ . (e), (f) Scatter plots showing the average cooperation or probability of complete cooperation as a function of $θ$ for (e) $T = 0.65$ and $S = - 0.4$ [black central curve in panel (c)] and (f) $T = 1.2$ and $S = 0.4$ [blue curve, second from above in panel (d)]. Insets show the corresponding standard deviation. The rest of the parameters in all panels are $R = 1$ and $P = 0$ .
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Figure 5
Mean field and the effective potential in the limit $θ ≪ N$ . (a)–(c) Cooperation density in the $(T, S)$ plane for a complete graph and effective temperatures (a) $θ = 0$ , (b) $θ = 0.05$ , and (c) $θ = 0.5$ . (d)–(f) Effective potential functions $V (ρ)$ for the parameter settings marked with letters from A to F in panel (a). Each panel corresponds to the same effective temperature as in the upper panel, and the color code is the same as for the symbols labeled in panel (a). Notice the different vertical scales. (g)–(i) Cooperation density in the $(T, S)$ plane for a random regular graph with $k = 3$ and the same temperatures as in the top and middle panels. The white dashed lines in panel (g) are solutions to Eq. (14) for the $(n, m) \leq k$ pairs (1,1),(1,2),(2,2),(3,2). The initial cooperation level is set to $50 %$ and networks have $N = 100$ nodes.
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Figure 6
Average cooperation as a function of the effective temperature $θ$ for a complete graph with $N = 100$ nodes. Each dot corresponds to different Monte Carlo simulations (50 in total), and the red curve is the sample average. Inset shows the standard deviation in the average cooperation. The game parameters are $T = 1.2$ and $S = 0.4$ [point E in Fig. 5]. The blue dashed line for $θ \leq 0.45$ is the analytical value of the cooperation given by Eq. (34) when the effective potential (32) has a local minimum $ρ \in (0, 1)$ , while for $θ > 0.45$ , the effective potential has only one minimum located at $ρ_{m} = 1$ .
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