Dynamics of link states in complex networks: The case of a majority rule

Abstract

Motivated by the idea that some characteristics are specific to the relations between individuals and not to the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model in fully connected networks, square lattices, and Erdös-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link dynamics have no counterpart under traditional node dynamics in the same topologies.

Fragmentation transition in a coevolving network with link-state dynamics

“…However, both reference [17] and this paper are limited to states defined on the links. A natural step further would be to consider mixed dynamics, with states defined both on the nodes and on the links and a certain coupling between them.…”

“…It is, therefore, a measure of the plasticity of the topology. When p is zero the network is static and only the majority rule dynamics takes place (as studied in [17]), while in the opposite situation, p = 1, there is only rewiring.…”

“…In order to characterize the system at different times it is useful to consider the density of nodal interfaces ρ as an order parameter [17], defined as the fraction of pairs of connected links that are in different states. If k i is the degree of node i, and k…”

“…In Fig. 8 we plot the survival probability P s (t), i.e, the probability that a realization did not reach the ordered state (ρ = 0) up to time t. When p = 0 we have P s = 1 for all times, meaning that all realizations (except for a few runs with the smallest size N = 500, as reported in [17]) fall into a disordered configuration characterized by a constant value of ρ > 0, as we shall discussed in detail in the next section. For p = 0.01, p = 0.05 and p = 0.10 [ Figs.…”

“…A large number of dynamics and rewiring rules have been studied [25][26][27][28][29][30][31]. As in [17], we consider a link-state dynamics where each link can be in one of two equivalent states and they are updated according to the majority rule or zero-temperature Glauber dynamics [32][33][34][35], in which links adopt the state of the majority of their neighboring links in the network. Additionally, we define a rewiring mechanism inspired by the case of competing languages.…”