More attention has been paid to research of various kinds of fractals due to a great number of applications in different fields over the past years. In this paper, we present stochastic generalized Vicsek fractal networks to model underlying structures on many hyperbranched polymers. Then, we consider random walks, the widely-studied dynamical behavior, on stochastic models built in order to better understand structural properties. Specifically, we analytically derive the closed-form solution to mean first-passage time for random walks, which is a quantity that allows one to better understand the underlying structure of model in question, using a more effective approach compared with the typical computational methods including spectral technique. At the same time, some other fundamental structural parameters including Kirchhoff index and fractal dimension are obtained analytically as well. The results suggest that the fractal characteristic makes the underlying structure of network more loose, and thus leads the efficiency of delivering information in a random-walk-based manner to become lower. Finally, we conduct extensive simulations to demonstrate that theoretical analysis and numerical simulations are in perfect agreement with each other.

Random Walks on Stochastic Generalized Vicsek Fractal Networks: Analytic Solution and Simulations

The mean first-passage time of random walks on a network has been extensively applied in the theory and practice of statistical physics, and its appliaction effects depend on the behavior of first-passage time. Here, we firstly define a graphic operation, namely, rectangle operation, for generating a scale-free network. In this paper, we study the topological structures of our network obtained from the rectangle operation, including degree distribution, clustering coefficient, and diameter. And then, we also consider the characteristic quantities related to the network, including Kirchhoff index and mean first-passage time, where these characteristic quantities can not only be used to evaluate the properties of our network, but also have remarkable applications in science and engineering.

/pdf/mean-first-passage-time-on-scale-free-networks-based-on-29jf2tqoel.pdf

Mean First-Passage Time on Scale-Free Networks Based on Rectangle Operation

Scale-free networks are prevalently observed in a great variety of complex systems, which triggers various researches relevant to networked models of such type. In this work, we propose a family of growth tree networks $\mathcal{T}_{t}$, which turn out to be scale-free, in an iterative manner. As opposed to most of published tree models with scale-free feature, our tree networks have the power-law exponent $\gamma=1+\ln5/\ln2$ that is obviously larger than $3$. At the same time, "small-world" property can not be found particularly because models $\mathcal{T}_{t}$ have an ultra-large diameter $D_{t}$, i.e., $D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$ where $|\mathcal{T}_{t}|$ represents vertex number. In addition, we also study random walks on tree networks $\mathcal{T}_{t}$ and derive exact solution to mean hitting time $\langle\mathcal{H}_{t}\rangle$. The results suggest that analytic formula for quantity $\langle\mathcal{H}_{t}\rangle$ as a function of vertex number $\mathcal{T}_{t}$ has a power-law form, i.e., $\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$ , a characteristic that is rarely encountered in the realm of scale-free tree networks. Lastly, we execute extensive experimental simulations, and find that empirical analysis is in strong agreement with theoretical results.

Scale-free tree network with an ultra-large diameter.

Entropy and enumeration of spanning connected unicyclic subgraphs in self-similar network

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs. 

A general model of hierarchical fractal scale-free networks

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast, in this paper, we report a fact that two arbitrarily chosen networks with identical degree distribution can have completely different other topological structure, such as diameter, spanning trees number, pearson correlation coefficient, and so forth. Besides that, for a given degree distribution (as power-law distribution with exponent $\gamma=3$ discussed here), it is reasonable to ask how many network models with such a constraint we can have. To this end, we generate an ensemble of this kind of random graphs with $P(k)\sim k^{-\gamma}$ ($\gamma=3$), denoted as graph space $\mathcal{N}(p,q,t)$ where probability parameters $p$ and $q$ hold on $p+q=1$, and indirectly show the cardinality of $\mathcal{N}(p,q,t)$ seems to be large enough in the thermodynamics limit, i.e., $N\rightarrow\infty$, by varying values of $p$ and $q$. From the theoretical point of view, given an ultrasmall constant $p_{c}$, perhaps only graph model $N(1,0,t)$ is small-world and other are not in terms of diameter. And then, we study spanning trees number on two deterministic graph models and obtain both upper bound and lower bound for other members. Meanwhile, for arbitrary $p(\neq1)$, we prove that graph model $N(p,q,t)$ does go through two phase transitions over time, i.e., starting by non-assortative pattern and then suddenly going into disassortative region, and gradually converging to initial place (non-assortative point). Among of them, one "null" graph model is built.

An ensemble of random graphs with identical degree distribution

Degree distribution, or equivalently called degree sequence, has been commonly used to study a large number of complex networks in the past few years. This reveals some intriguing results, for instance, the popularity of power-law distribution in most of these networks under consideration. Along such a research line, in this paper, we generate an ensemble of random graphs with an identical degree distribution P(k)∼k-γ (γ=3) as proved shortly, denoted as graph space N(p,q,t), where probability parameters p and q hold on p+q=1. Next, we study some topological structure properties of great interest on each member in the graph space N(p,q,t) using both precisely analytical calculations and extensively numerical simulations, as follows. From the theoretical point of view, given an ultrasmall constant pc, perhaps only the graph model N(1,0,t) is small-world and the others are not in terms of diameter. Then, we obtain exact solutions for a spanning tree number on two deterministic graph models in the graph space N(p,q,t), which gives both upper bound and lower bound for that of other members. Meanwhile, for an arbitrary p(≠1), we prove using the Pearson correlation coefficient that the graph model N(p,q,t) does go through two phase transitions over time, i.e., starting by a nonassortative pattern, then suddenly going into a disassortative region, and gradually converging to an initial position (nonassortative point). Therefore, to some extent, the three topological parameters above can serve as the complementary measures for degree distribution to help us clearly distinguish all members in the graph space N(p,q,t). In addition, one "null" graph model is built.

pdf/an-ensemble-of-random-graphs-with-identical-degree-55xdk6nq49.pdf

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An ensemble of random graphs with identical degree distribution

An ensemble of random graphs with identical degree distribution