Chimera and Anticoordination States in Learning Dynamics

Abstract

In many real-life situations, individuals are dared to simultaneously achieve social objectives of acceptance or approval and strategic objectives of coordination. Since these two objectives may take place in different environments, a two-layer network is the simple and natural framework for the study of such kind of dynamical situations. In this paper we present a model in which the state of the agents corresponds to one of two possible strategies. They change their states by interaction with their neighbors in the network. Inside each layer the agents interact by a social pressure mechanism, while between the layers the agents interact via a coordination game. From an evolutionary approach, we focus on the asymptotic solutions for all-to-all interactions across and inside the layers and for any initial distribution of strategies. We find new asymptotic configurations which do not exist in a single isolated social network analysis. We report the emergence and existence of chimera states in which two different collective states coexist in the network. Namely, one layer reaches a state of full coordination while the other remains in a dynamical state of coexistence of strategies. In addition, the system may also reach a state of global anticoordination where a full coordination is reached inside each layer but with opposite strategies in each of the two network layers. We trace back the emergence of chimera states and global anticoordination states to the agents inertia against social pressure, referred here to as the level of skepticism, along with the degree of risk taken into account in a general coordination game.PACS numbers:

Local connectivity effects in learning and coordination dynamics in a two-layer network

“…The analytical solutions of the mean-field equations in Ref. 6 describe the collective behavior of this evolutionary model. Depending upon the initial conditions and values of the parameters in a fully connected network, the system reaches one four possible different asymptotic states: The asymptotic anticoordination and chimera states are two highlights of this two-layer network.…”

“…The analytical results in Ref. 6 show that the chimera states appear only in herding populations, i.e., T < 0.5, when the coordination game is asymmetric, while the states of anticoordination appear only in skeptical populations, i.e., T > 0.5 for either a symmetric or an asymmetric coordination game. The asymptotic state and the size of the basin of attraction of these states are determined by the particular initial conditions of strategies, the parameter T, and the parameters s and b of the coordination game.…”

“…In such chimera states, there is coexistence of a homogeneous state of coordination in one layer with a heterogeneous state of coexistence of strategies in the other layer. 6 Based on the game-theoretical approach to coordination games, many theoretical, numerical, and experimental studies have been done to elucidate the factors that favor the emergence of coordination, 2,7,8 such as the range of payoffs, the strategy updating rules, and the structures of the network. [9][10][11] There is some experimental evidence of the effect of local interactions as found in Refs.…”

“…Likewise, the literature about chimera states is vast in the field of coupled dynamical systems in networks of interaction, [20][21][22][23][24][25][26][27] but few studies address the existence of chimera states in social networks. 6,28,29 Anticoordination and chimera states are difficult to find in a single component network. However, the framework of a multilayer network seems to be the natural baseline for the emergence of these non-trivial states.…”

“…Indeed, these states are observed in a fully connected two-layer network with particular settings of the initial distribution of strategies, the magnitude of social influence, and the parameters of the coordination game. 6 The aim of this paper is to study the robustness of these states in settings beyond a fully connected network. We will consider random [Erdös-Rényi (ER)] networks inside and between layers.…”