Random walk on the activity-driven model with mutual selection

¹ School of Computer Engineering and Science, Shanghai University - No. 99 Shangda Road, Baoshan District, Shanghai 200-444, PRC
² Shanghai Institute for Advanced Communication and Data Science, Shanghai University No. 99 Shangda Road, Baoshan District, Shanghai 200-444, PRC

^(a) bingbignwang@shu.edu.cn
bingbignmath@gmail.com

Received: 4 August 2018
Accepted: 7 November 2018

Abstract

Random walks are one of the most fundamental types of stochastic processes and have been applied in various domains, such as ranking systems and searching. In this paper, we investigate random walk process unfolding on a generalized version of the activity-driven modelling framework by considering mutual agreement. The model is characterized by a linking function that describes the probability of the existence of an edge, which depends mutually on the fitness of the vertices on both ends of that edge. We investigate two typical forms of linking functions and derive analytically exact expressions for the asymptotic behavior of random walks and the mean first-passage time for the two cases, respectively. We find that, compared with the activity-driven network model, mutual agreement has nontrivial effects on the properties of random walk processes. For the first case, when the fitness of the ends of a link is independent, we find that the capability of vertices to gather walkers is not only related to the vertices' activity, but also determined by their propensity of receiving connections. For the second case, when the creation of the link is determined by whether the sum of the two end points' fitness is larger than a threshold, we find that for vertices with activity larger than a given threshold, the stronger the activity is, the more walkers they will collect in the stationary state. Finally, we confirm our analytical prediction via large-scale numerical simulations performed by controlling flexible parameters. The results presented here contribute to the understanding of the evolution mechanism of the mutual selection network model and give us an insight into their effects on random walks processes.