The distribution of first hitting times of randomwalks on Erdős–Rényi networks

Abstract

Analytical results for the distribution of first hitting times of random walks on Erdős–Rényi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has already visited before. At this point, the path terminates. The path length, namely the number of steps, d, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, . The results are found to be in excellent agreement with numerical simulations. It is found that the distribution follows a product of an exponential distribution and a Rayleigh distribution. The mean, median and standard deviation of this distribution are also calculated, in terms of the network size and its mean degree. The termination of an RW path may take place either by backtracking to the previous node or by retracing of its path, namely stepping into a node which has been visited two or more time steps earlier. We obtain analytical results for the probabilities, pb and pr, that the cause of termination will be backtracking or retracing, respectively. It is shown that in dilute networks the dominant termination scenario is backtracking while in dense networks most paths terminate by retracing. We also obtain expressions for the conditional distributions and , for those paths which are terminated by backtracking or by retracing, respectively. These results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

The distribution of first hitting times of random walks on directed Erdős–Rényi networks

“…The distribution of last hitting times of SAWs on a directed ER network, obtained from Eq. (13), is also shown (dashed-dotted line). As expected, the last hitting times are significantly longer than the first hitting times on the same network.…”

“…For comparison we also present the corresponding distribution of RWs on an undirected ER network (dashed line) with the same value of c (based on Ref. [13]). It is found that the first hitting times of RWs on directed ER networks are much longer than those of 28).…”

“…For comparison we also present the tail distribution of first hitting times of RWs on an undirected ER network (dashed line), obtained using the analytical expression derived in Ref. [13], as well as of NBWs on undirected ER networks (dotted line), obtained using the analytical expression derived in Ref. [39].…”

Revealing the microstructure of the giant component in random graph ensembles

“…For networks with an exponential component in the degree distribution as described in Eq. (22), the degree distribution conditioned on the finite components takes the form…”

“…It would help to obtain results pertaining only to the giant component of such systems, without the contributions from finite components which often amount to trivial contaminations or (unwanted) distortions of results. Examples that come to mind are localization phenomena in sparse matrix spectra [17,18] (where finite components of a random graph support eigenvectors that are trivially localized), properties of random walks [19][20][21][22] (where a random walker chosen to start a walk on one of the finite components will never be able to explore an appreciable fraction of the entire network), or the spread of diseases or cascading failures [23][24][25][26][27][28] (where an initial failure or initial infection occurring on a finite component will never lead to a global system failure or the outbreak of an epidemic). Component-size distributions in the percolation problem on complex networks [27,29] will likewise contain a component originating from clusters that were finite, before nodes or edges were randomly removed from the network.…”