Characteristic times of biased random walks on complex networks

Abstract

We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k α , where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to visit a given node for the first time, and iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times is different from the value αmin = −1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of αmin in the range [−1, −0.5], while disassortative networks have αmin in the range [−0.5, 0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value αmin, which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks, by means of an appropriate tuning of the motion bias.PACS numbers: 89.75. Hc, 05.40.Fb, 89.75.Kd In the last decade or so the quantitative analysis of networks having different origin and function, including social networks, the human brain, the Internet, the World Wide Web, has revealed that all these systems exhibit comparable structural properties at different scales, and are more similar to each other than expected [1,2]. It has been found that the structural complexity of networks from the real world usually has a significant impact on the dynamical processes occurring over them, including opinion dynamics [3], epidemics [4] and synchronization [5].Random walks are the simplest way to explore a network, and are one of the most widely studied class of processes on complex networks [6,7]. Different kinds of random walks have been used to implement efficient local search strategies [8,9], and also to reveal the presence of hierarchies and network communities [10,11]. Particular attention has been devoted to the study of the characteristic times associated to random walks, such as the mean return times, or the mean first passage times, respectively the average time the walker takes to come back to the starting node or to hit a given node [12]. Such characteristic times can be determined analytically for random walks on regular lattices [13], but their calculation for graphs with heterogeneous structures is still the object of active research [14,15]. Recent results include the derivation of analytic expressions for the characteristic times of unbiased random walks on Erdös-Rényi random graphs [16], on fractal networks [17][18][19][20] and on particular classes of scale-free graphs [21]. To date, only approximate solutions are available for random walks on real networks [22][23][24][25].A class of random walks which is particularly int...

Spectra of random stochastic matrices and relaxation in complex systems

“…Our approach allows to disentangle contributions to the spectral density related to extended and localized states, respectively, allowing to differentiate between time-scales associated with transport processes and those associated with the dynamics of local rearrangements. There are numerous processes, both natural and artificial, which can be understood in terms of random walks on complex networks [1][2][3], including the spread of diseases in social networks [4,5], the transmission of information in communication networks (e.g.…”

“…Our approach allows to disentangle contributions to the spectral density related to extended and localized states, respectively, allowing to differentiate between time-scales associated with transport processes and those associated with the dynamics of local rearrangements. There are numerous processes, both natural and artificial, which can be understood in terms of random walks on complex networks [1][2][3], including the spread of diseases in social networks [4,5], the transmission of information in communication networks (e.g.[6]), search algorithms [7,8], the out-of-equilibrium dynamics of glassy systems at low temperatures as described in terms of hopping between long-lived states in state space [9][10][11], the dynamics of major conformational changes in macro-molecules [12], or cell-signalling through protein-protein interaction networks [13], to name but a few. For reviews that cover several of these topics, see e.g.…”

Persistent Random Search on Complex Networks

“…As the GMFPT is feature of a state, or node taken as target, obtained by averaging over different starting points, one can average across GMFPTs for all states and get a property of the whole chain or network which was introduced as Graph MFPT (GrMFPT) [25]. Thus, the GrMFPT for well connected networks (with a good convergence rate) would be…”

Cross-Frequency Interactions During Information Flow in Complex Brain Networks Are Facilitated by Scale-Free Properties

“…Random walks 5 , the processes by which randomly-moving objects wander away from their starting location are commonly used to describe diffusion processes. In the past decades, there has been considerable progress in characterizing first passage times, or the amount of time it takes a random walker to reach a target [6][7][8][9] . However, previous works have neglected the study of the temporal dynamics of the information flow in the network, which depends on how the walkers move and not just on their arrival time.…”