Effective trapping of random walkers in complex networks

Bifractality of fractal scale-free networks

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

The trapping problem of the weighted scale-free treelike networks for two kinds of biased walks

It has been recently reported that trapping problem can characterize various dynamical processes taking place on complex networks. However, most works focused on the case of binary networks, and dynamical processes on weighted networks are poorly understood. In this paper, we study two kinds of biased walks including standard weight-dependent walk and mixed weight-dependent walk on the weighted scale-free treelike networks with a trap at the central node. Mixed weight-dependent walk including non-nearest neighbor jump appears in many real situations, but related studies are much less. By the construction of studied networks in this paper, we determine all the eigenvalues of the fundamental matrix for two kinds of biased walks and show that the largest eigenvalue has an identical dominant scaling as that of the average trapping time (ATT). Thus, we can obtain the leading scaling of ATT by a more convenient method and avoid the tedious calculation. The obtained results show that the weight factor has a significant effect on the ATT, and the smaller the value of the weight factor, the more efficient the trapping process is. Comparing the standard weight-dependent walk with mixed weight-dependent walk, although next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT.

Anomalous behavior of trapping in extended dendrimers with a perfect trap

Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generation-dependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers.

Blind and myopic ants in heterogeneous networks

The diffusion processes on complex networks may be described by different Laplacian matrices due to heterogeneous connectivity. Here we investigate the random walks of blind ants and myopic ants on heterogeneous networks: While a myopic ant hops to a neighbor node every step, a blind ant may stay or hop with probabilities that depend on node connectivity. By analyzing the trajectories of blind ants, we show that the asymptotic behaviors of both random walks are related by rescaling time and probability with node connectivity. Using this result, we show how the small eigenvalues of the Laplacian matrices generating the two random walks are related. As an application, we show how the return-to-origin probability of a myopic ant can be used to compute the scaling behaviors of the Edwards-Wilkinson model, a representative model of load balancing on networks.

Random walks in unweighted and weighted modular scale-free networks with a perfect trap

Designing optimal structure favorable to diffusion and effectively controlling the trapping process are crucial in the study of trapping problem--random walks with a single trap. In this paper, we study the trapping problem occurring on unweighted and weighted networks, respectively. The networks under consideration display the striking scale-free, small-world, and modular properties, as observed in diverse real-world systems. For binary networks, we concentrate on three cases of trapping problems with the trap located at a peripheral node, a neighbor of the root with the least connectivity, and a farthest node, respectively. For weighted networks with edge weights controlled by a parameter, we also study three trapping problems, in which the trap is placed separately at the root, a neighbor of the root with the least degree, and a farthest node. For all the trapping problems, we obtain the analytical formulas for the average trapping time (ATT) measuring the efficiency of the trapping process, as well as the leading scaling of ATT. We show that for all the trapping problems in the binary networks with a trap located at different nodes, the dominating scalings of ATT reach the possible minimum scalings, implying that the networks have optimal structure that is advantageous to efficient trapping. Furthermore, we show that for trapping in the weighted networks, the ATT is controlled by the weight parameter, through modifying which, the ATT can behave superlinearly, linearly, sublinearly, or logarithmically with the system size. This work could help improving the design of systems with efficient trapping process and offers new insight into control of trapping in complex systems.

Random walks in weighted networks with a perfect trap: an application of Laplacian spectra

Trapping processes constitute a primary problem of random walks, which characterize various other dynamical processes taking place on networks. Most previous works focused on the case of binary networks, while there is much less related research about weighted networks. In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and Lévy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For Lévy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing Lévy random walks, which have the minimal average of ATTs taken over all target nodes.

Origin of the hub spectral dimension in scale-free networks

The return-to-origin probability and the first-passage-time distribution are essential quantities for understanding transport phenomena in diverse systems. The behaviors of these quantities typically depend on the spectral dimension d(s). However, it was recently revealed that in scale-free networks these quantities show a crossover between two power-law regimes characterized by d(s) and the so-called hub spectral dimension d(s)((hub)) due to the heterogeneity of connectivities of each node. To understand the origin of d(s)((hub)) from a theoretical perspective, we study a random walk problem on hierarchical scale-free networks by using the renormalization group (RG) approach. Under the RG transformation, not only the system size but also the degree of each node changes due to the scale-free nature of the degree distribution. We show that the anomalous behavior of random walks involving the hub spectral dimension d(s)((hub)) is induced by the conservation of the power-law degree distribution under the RG transformation.

First passage time for random walks in heterogeneous networks

The first passage time (FPT) for random walks is a key indicator of how fast information diffuses in a given system. Despite the role of FPT as a fundamental feature in transport phenomena, its behavior, particularly in heterogeneous networks, is not yet fully understood. Here, we study, both analytically and numerically, the scaling behavior of the FPT distribution to a given target node, averaged over all starting nodes. We find that random walks arrive quickly at a local hub, and therefore, the FPT distribution shows a crossover with respect to time from fast decay behavior (induced from the attractive effect to the hub) to slow decay behavior (caused by the exploring of the entire system). Moreover, the mean FPT is independent of the degree of the target node in the case of compact exploration. These theoretical results justify the necessity of using a random jump protocol (empirically used in search engines) and provide guidelines for designing an effective network to make information quickly accessible.