Absorbing phase transition in the coupled dynamics of node and link states in random networks

Abstract

We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states represent positive or negative relationships. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite-size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuations take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a class of finite-size topological transition associated with group splitting in the network of these final frozen configurations.

Empirical social triad statistics can be explained with dyadic homophylic interactions

“…Following Heider’s intuition ( 18 – 41 ), current approaches toward social balance often account for the effect of triangles on social network formation in one way or another. For example, the models in refs.…”

The effect of social balance on social fragmentation

“…The structure of the phase diagram may become richer with long-lived metastable phases. Such a non-equilibrium approach has been considered recently in [ 42 ], where network evolution is not driven by Heider’s balance, but by another aspect of cognitive dissonance. There, fragmentation emerges either as an absorbing steady state of the dynamics, or from an active phase due to fluctuations in systems of finite size.…”

“…Finally, fragmentation may also arise if individuals coevolutionarily rearrange their social ties in response to changes in their 'states' [37,38]. Recent attempts have been made to impose social balance a priori as a constraint on the dynamics of agent states [39], or to drive the network towards balance by a change in the states, but not by the elimination of social tension in unbalanced triads [40][41][42].…”

Topological transition in a coupled dynamic in random networks

“…This is the idea behind the Heider's concept of balanced and unbalanced triadic groups (triangles) of individuals [9,10], which is the base of the structural balance theory [11][12][13]. Outside this triadic structure, the link states have also been studied in a regular network where the iteration with the agent states are responsible to determine the time evolution of the system [14][15][16][17][18].…”