Dynamics of multiplayer games on complex networks using territorial interactions

The modeling of evolution in structured populations has been significantly advanced by evolutionary graph theory, which incorporates pairwise relationships between individuals on a network. More recently, a new framework has been developed to allow for multiplayer interactions of variable size in more flexible and potentially changing population structures. While the theory within this framework has been developed and simple structures considered, there has been no systematic consideration of a large range of different population structures, which is the subject of this paper. We consider a large range of underlying graphical structures for the territorial raider model, the most commonly used model in the new structure, and consider a variety of important properties of our structures with the aim of finding factors that determine the fixation probability of mutants. We find that the graphical temperature and the average group size, as previously defined, are strong predictors of fixation probability, while all other properties considered are poor predictors, although the clustering coefficient is a useful secondary predictor when combined with either temperature or group size. The relationship between temperature or average group size and fixation probability is sometimes, however, nonmonotonic, with a directional reverse occurring around the temperature associated with what we term “completely mixed” populations in the case of the hawk-dove game, but not the public goods game.

Figure 1

The fully independent model from [26]. There are $N$ individuals who are distributed over $M$ places such that $I_n$ visits place $P_m$ with probability $p_{nm}$. Individuals interact with one another when they meet; for example, $I_1$ and $I_2$ can interact with one another when they meet in $P_1$.

Figure 2

(a) Population structure represented using a network where nodes represent individuals and places. Individual $I_n$ lives in place $P_n$ and can visit any neighboring places. For example, the home place of $I_1$ is place $P_1$ but it can visit places $P_2,P_3$, and $P_4$. (b) An alternative visualization on a bipartite network where individuals and places are clearly separated.

Figure 3

Fixation probability as a function of (a) clustering coefficient, (b) shortest path length, (c) density, (d) diameter, and (e) average degree for all network simulations for the hawk-dove game. The dark gray dots represent the case of H invading D, the light gray dots represent the case of D invading H, and for both cases, black dots represent simulations with well-mixed networks. The parameter values are $R=10,C=1,V=2$.

Figure 4

Fixation probability as a function of temperature considering the public goods game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. For the selected parameters, the fixation probability for C invading D is effectively zero in all cases, so we have omitted these results. The light gray dots represent the case of D invading C and the black dots represent simulations with well-mixed networks. The parameter values are $R=2,C=1,V=2$.

Figure 5

Fixation probability as a function of temperature considering the hawk-dove game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. The dark gray dots represent the case of H invading D, the light gray dots represent the case of D invading H, and for both cases, black dots represent simulations with well-mixed networks. The parameter values are $R=10,C=1,V=2$.

Figure 6

Fixation probability as a function of temperature considering the fixed fitness game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. The dark gray dots represent the case of A invading B, the light gray dots represent the case of B invading A, and for both cases, black dots represent simulations with well-mixed networks. The parameter values are $R=50,C=1,V=2$.

Figure 7

Fixation probability as a function of average group size considering the public goods game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. The light gray dots represent the case of D invading C and black dots represent simulations with well-mixed networks. The parameter values are $R=2,C=1,V=2$. The C invading D situation is a flat line close to 0.

Figure 8

Fixation probability as a function of average group size considering the hawk-dove game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. The dark gray dots represent the case of H invading D, the light gray dots represent the case of D invading H, and for both cases, black dots represent simulations with well-mixed networks. The parameter values are $R=10,C=1,V=2$.

Figure 9

Fixation probability as a function of average group size considering the fixed fitness game for (a) all networks, (b) Erdös-Rényi, (c) small-world, (d) scale-free, (e) Barabási-Albert, and (f) random regular networks. The dark gray dots represent the case of A invading B, the light gray dots represent the case of B invading A, and for both cases, black dots represent simulations with well-mixed networks. The parameter values are $R=50,C=1,V=2$.

Figure 10

Fixation probability as a function of temperature [(a) public goods game, (b) hawk-dove game] and average group size [(c) public goods game, (d) hawk-dove game] with a closer look at the clustering coefficient (see the vertical bar accompanying each figure), with D invading C and D invading H.

Figure 11

Fixation probability as a function of temperature [(a) public goods game, (b) hawk-dove game] and average group size [(c) public goods game, (d) hawk-dove game] with a closer look at the standard deviation of node temperature (see the vertical bar accompanying each figure), with D invading C and D invading H.

Figure 12

Three-dimensional figure for the fixation probability as a function of clustering coefficient and temperature [(a) public goods game, (b) hawk-dove game] and average group size [(c) public goods game, (d) hawk-dove game] with a closer look at the standard deviation of node temperature (see the vertical bar accompanying each figure), with D invading C and D invading H.