In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al. ( Proc. Natl. Acad. Sci. U.S. A., 2007, 104: 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, dependence relationship of fractal dimension and the multifractal parameters on the parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al. ( Nat. Phys, 2006, 2: 275). Then from the shape of the τ (q) and D(q) curves, we find the existence of multifractality in these networks.Last, we find that there exists a linear relationship between the average information dimension < D(1) > and the parameter e.

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): “Sierpinski” WFNs and “Cantor dust” WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks — collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.

No abstract

In this paper, we study the synchronizability of three kinds of dynamical weighted fractal networks (WFNs). These WFNs are weighted Cantor-dust networks, weighted Sierpinski networks and weighted Koch networks. We calculated some features of these WFNs, including average distance ([Formula: see text]), fractal dimension ([Formula: see text]), information dimension ([Formula: see text]), correlation dimension ([Formula: see text]). We analyze two representative types of synchronizable dynamical networks (the type-I and the type-II). There are two indexes ([Formula: see text] and [Formula: see text]) that can be used to characterize the synchronizability of the two types of dynamical network. Here, [Formula: see text] and [Formula: see text] are the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix of the network, respectively. We find that the larger scaling factor [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] implies stronger synchronizability for the type-I dynamical WFNs.

In this paper, we propose a new cross-sample entropy, namely the composite multiscale partial cross-sample entropy (CMPCSE), for quantifying the intrinsic similarity of two time series affected by common external factors. First, in order to test the validity of CMPCSE, we apply it to three sets of artificial data. Experimental results show that CMPCSE can accurately measure the intrinsic cross-sample entropy of two simultaneously recorded time series by removing the effects from the third time series. Then CMPCSE is employed to investigate the partial cross-sample entropy of Shanghai securities composite index (SSEC) and Shenzhen Stock Exchange Component Index (SZSE) by eliminating the effect of Hang Seng Index (HSI). Compared with the composite multiscale cross-sample entropy, the results obtained by CMPCSE show that SSEC and SZSE have stronger similarity. We believe that CMPCSE is an effective tool to study intrinsic similarity of two time series.