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 ( ln ϵ n ) / ϵ for homogeneous networks with a large finite n . By contrast, it scales as ( ln ( ϵ μ 1 2 n / μ 2 ) ) / ϵ 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 n c o ∼ ϵ − 1 / α above which the state of all nodes becomes the target state in logarithmic time. Here, α = 1 for homogeneous networks and α = ( γ − 1 ) / 2 for scale-free networks with a degree exponent 2 < γ < 3 .