Crossover phenomenon in adversarial attacks on voter model

Abstract

A recent study (Chiyomaru and Takemoto 2022 Phys. Rev. E 106 014301) considered adversarial attacks conducted to distort voter model dynamics in networks. This method intervenes in the interaction patterns of individuals and induces them to be in a target opinion state through a small perturbation ε. In this study, we investigate adversarial attacks on voter dynamics in random networks of finite size n. The exit probability P +1 to reach the target absorbing state and the mean time τ n to reach consensus are analyzed in the mean-field approximation. Given ε > 0, the exit probability P +1 converges asymptotically to unity as n increases. The mean time τ n to reach consensus scales as for homogeneous networks with a large finite n. By contrast, it scales as for heterogeneous networks with a large finite n, where µ 1 and µ 2 represent the first and second moments of the degree distribution, respectively. Moreover, we observe the crossover phenomenon of τ n from a linear scale to a logarithmic scale and find above which the state of all nodes becomes the target state in logarithmic time. Here, α = 1 for homogeneous networks and for scale-free networks with a degree exponent .