Abstract
We study the voter model, under node and link update, and the related invasion process ona single strongly connected component of a directed network. We implement an analyticaltreatment in the thermodynamic limit using the heterogeneous mean-field assumption.From the dynamical rules at the microscopic level, we find the equations for the evolutionof the relative densities of nodes in a given state on heterogeneous networks with arbitrarydegree distribution and degree–degree correlations. We prove that conserved quantities asweighted linear superpositions of spin states exist for all three processes and,for uncorrelated directed networks, we derive their specific expressions. We alsodiscuss the time evolution of the relative densities that decay exponentially to ahomogeneous stationary value given by the conserved quantity. The conservation lawsobtained in the thermodynamic limit for a system that does not order in that limitdetermine the probabilities of reaching the absorbing state for a finite system. Thecontribution of each degree class to the conserved quantity is determined by alocal property. Depending on the dynamics, the highest contribution is associatedwith influential nodes reaching a large number of outgoing neighbors, not tooinfluenceable ones with a low number of incoming connections, or both at the same time.
Highlights
The voter model on strongly connected componentsIn the voter model under node update (VM), each node of a network can exist in one of two possible states, 1 or 04
We study the voter model, under node and link update, and the related invasion process on a single strongly connected component of a directed network
In an isolated and strongly connected component, the voter dynamics keeps an active dynamical state in the thermodynamic limit, but it leads to a consensus in a finite network as it happens on undirected networks
Summary
In the voter model under node update (VM), each node of a network can exist in one of two possible states, 1 or 04. The interactions in the voter dynamics are instantaneously asymmetric since the updates always go in the same direction once the original node is chosen independently of the undirectionality of the substrate. The straightforward generalization of the voter model on directed networks under node update consists of selecting a node at random, and assigning to it the state of one of its incoming neighbors, chosen at random.
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