Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect

A vast variety of real-life networks display the ubiquitous presence of scale-free phenomenon and small-world effect, both of which play a significant role in the dynamical processes running on networks. Although various dynamical processes have been investigated in scale-free small-world networks, analytical research about random walks on such networks is much less. In this paper, we will study analytically the scaling of the mean first-passage time (MFPT) for random walks on scale-free small-world networks. To this end, we first map the classical Koch fractal to a network, called Koch network. According to this proposed mapping, we present an iterative algorithm for generating the Koch network, based on which we derive closed-form expressions for the relevant topological features, such as degree distribution, clustering coefficient, average path length, and degree correlations. The obtained solutions show that the Koch network exhibits scale-free behavior and small-world effect. Then, we investigate the standard random walks and trapping issue on the Koch network. Through the recurrence relations derived from the structure of the Koch network, we obtain the exact scaling for the MFPT. We show that in the infinite network order limit, the MFPT grows linearly with the number of all nodes in the network. The obtained analytical results are corroborated by direct extensive numerical calculations. In addition, we also determine the scaling efficiency exponents characterizing random walks on the Koch network.

Anomalous behavior of trapping on a fractal scale-free network

It is known that the heterogeneity of scale-free networks helps enhancing the efficiency of trapping processes performed on them. In this paper, we show that transport efficiency is much lower in a fractal scale-free network than in non-fractal networks. To this end, we examine a simple random walk with a fixed trap at a given position on a fractal scale-free network. We calculate analytically the mean first-passage time (MFPT) as a measure of the efficiency for the trapping process, and obtain a closed-form expression for MFPT, which agrees with direct numerical calculations. We find that, in the limit of a large network order $V$, the MFPT $<T>$ behaves superlinearly as $<T > sim V^{{3/2}}$ with an exponent 3/2 much larger than 1, which is in sharp contrast to the scaling $<T > sim V^{theta}$ with $theta leq 1$, previously obtained for non-fractal scale-free networks. Our results indicate that the degree distribution of scale-free networks is not sufficient to characterize trapping processes taking place on them. Since various real-world networks are simultaneously scale-free and fractal, our results may shed light on the understanding of trapping processes running on real-life systems.

Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)sim k^{-gamma}$, where the degree exponent $gamma$ describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various $gamma in (2,1+frac{ln 3}{ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $gamma$ has no effect on APL $d$ of RSFTs: In the full range of $gamma$, $d$ behaves as a logarithmic scaling with the number of network nodes $N$ (i.e. $d sim ln N$), which is in sharp contrast to the well-known double logarithmic scaling ($d sim ln ln N$) previously obtained for uncorrelated scale-free networks with $2 leq gamma <3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $gamma$.

Different thresholds of bond percolation in scale-free networks with identical degree sequence

Generally, the threshold of percolation in complex networks depends on the underlying structural characterization. However, what topological property plays a predominant role is still unknown, despite the speculation of some authors that degree distribution is a key ingredient. The purpose of this paper is to show that power-law degree distribution itself is not sufficient to characterize the threshold of bond percolation in scale-free networks. To achieve this goal, we first propose a family of scale-free networks with the same degree sequence and obtain by analytical or numerical means several topological features of the networks. Then, by making use of the renormalization group technique we determine the threshold of bond percolation in our networks. We find an existence of non-zero thresholds and demonstrate that these thresholds can be quite different, which implies that power-law degree distribution does not suffice to characterize the percolation threshold in scale-free networks.

Exact solution for mean first-passage time on a pseudofractal scale-free web

The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially complex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.

Effects of accelerating growth on the evolution of weighted complex networks

Many real systems possess accelerating statistics where the total number of edges grows faster than the network size. In this paper, we propose a simple weighted network model with accelerating growth. We derive analytical expressions for the evolutions and distributions for strength, degree, and weight, which are relevant to accelerating growth. We also find that accelerating growth determines the clustering coefficient of the networks. Interestingly, the distributions for strength, degree, and weight display a transition from scale-free to exponential form when the parameter with respect to accelerating growth increases from a small to large value. All the theoretical predictions are successfully contrasted with extensive numerical simulations.

Random Sierpinski network with scale-free small-world and modular structure

In this paper, we define a stochastic Sierpinski gasket, on the basis of which we construct a network called random Sierpinski network (RSN). We investigate analytically or numerically the statistical characteristics of RSN. The obtained results reveal that the properties of RSN is particularly rich, it is simultaneously scale-free, small-world, uncorrelated, modular, and maximal planar. All obtained analytical predictions are successfully contrasted with extensive numerical simulations. Our network representation method could be applied to study the complexity of some real systems in biological and information fields.

Topologies and Laplacian spectra of a deterministic uniform recursive tree

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.

Transition from fractal to non-fractal scalings in growing scale-free networks

Real networks can be classified into two categories: fractal networks and non-fractal networks. Here we introduce a unifying model for the two types of networks. Our model network is governed by a parameter $q$. We obtain the topological properties of the network including the degree distribution, average path length, diameter, fractal dimensions, and betweenness centrality distribution, which are controlled by parameter $q$. Interestingly, we show that by adjusting $q$, the networks undergo a transition from fractal to non-fractal scalings, and exhibit a crossover from `large to small worlds at the same time. Our research may shed some light on understanding the evolution and relationships of fractal and non-fractal networks.

Exact solution of mean geodesic distance for Vicsek fractals

The Vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important. In this paper, the exact formula for the mean geodesic distance of Vicsek fractals is found. The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractals considered. The obtained exact solution exhibits that the mean geodesic distance approximately increases as an exponential function of the number of nodes, with the exponent equal to the reciprocal of the fractal dimension. The closed-form solution is confirmed by extensive numerical calculations.