Network reconstruction by the stationary distribution of random walk process

It is known that the stationary distribution of the random walk process is dependent on the structure of the network. This could provide us a solution of the network reconstruction. However, the stationary distribution of the random walk process can only reflect the relative size of node degrees directly, how to infer the real connection is still a problem. In this paper, we will propose a method to reconstruct network by the random walk process, which can reconstruct the total number of links, degree sequence and links sequentially. In our method, only the stationary distribution is used, and no data of the evolution process is needed, such as the first passage time. We perform our method on some network models and real-world network, the results indicate our method can reconstruct networks accurately, even when we can not get the exact stationary distribution. Introduction. – Many real-world complex systems can be considered as complex networks, such as biology [1,2], psychology [3], social and economic systems [4]. The dynamics in these networks are generally determined by their structures, so revealing the structure of network systems is one of the important ways to study the dynamics of complex systems [5]. However, not all the structures can be detected directly in practice, such as some biology network systems. To obtain the structure of such systems, the data-driven reconstruction is usually used [6,7]. Generally speaking, that is inferring the connection of an unknown network by analyzing the feedback information of some dynamics in that system. Previous studies have already obtain many significant results with various reconstruction techniques. These reconstruction techniques are often based on data analysis of the time series of network dynamics, and the widely used dynamics are the ones that can be expressed as some first-order non-linear differential equations. For example, Timme et al reveal network connectivity by response dynamics [8], Wang et al infer the network structure with noise-bridge dynamics [9], Levnajic et al reconstruct network from random phase-resetting [10], and Yu et al es(a)E-mail: [email protected] (b)E-mail: [email protected] timate topology of networks with phase oscillators [11]. Besides, some special dynamics are also used in biology studies [12–16], such as DBNs [17] and gene regulatory [18]. Recent works extend reconstruction techniques to game theory [19], mean field theory [20], compressed sensing [21], epidemic spreading [22,23] and many other techniques [24–26]. Even in noisy network observations, some method can also be used to estimate the true network properties [27]. In addition, the link prediction methods are also one kind of the network reconstruction, which are usually used in recommender systems [28,29]. A difference is that the link prediction methods often use the information of the known links to infer the missing ones, however, the dynamic data is usually used for the traditional network reconstruction As a basic conclusion of the Markov process on networks, the stationary distribution of the random walk process depends on the network structure [30]. This correlation of the network structure and dynamic could also provide us a method of the network reconstruction. However, this correlation can only reflect the relative size of node degrees, how to infer the real connection is still a problem. In this paper, we will propose a method based on random walk process to infer the network connection. Only the stationary distribution is used in our method, the information of the evolution of random walk process