Related papers: Effective dimensions and percolation in hierarchic…
We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size),…
Scale-free percolation is a percolation model on $\mathbb{Z}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs.…
Hierarchical networks actually have many applications in the real world. Firstly, we propose a new class of hierarchical networks with scale-free and fractal structure, which are the networks with triangles compared to traditional…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal…
Many complex networks in nature have directed links, a property that affects the network's navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in- and…
The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal…
Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced…
The fractal and self-similarity properties are revealed in many real complex networks. However, the classical information dimension of complex networks is not practical for real complex networks. In this paper, a new information dimension…
We consider the effects of spatial correlations in a two-dimensional site percolation model. By generalizing the Newman-Ziff Monte Carlo algorithm to include spatial correlations, percolation thresholds and fractal dimensions of percolation…
The concept of scale-free networks has been widely applied across natural and physical sciences. Many claims are made about the properties of these networks, even though the concept of scale-free is often vaguely defined. We present tools…
We demonstrate analytically and numerically the possibility that the fractal property of a scale-free network cannot be characterized by a unique fractal dimension and the network takes a multifractal structure. It is found that the mass…
We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the…
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…
We constructs a new network by superposition of hexahedron , which are scale-free, highly sparse,disassortative ,and maximal planar graphs. The network degree distribution, agglomeration coefficient and degree of correlation are computed…
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the…
Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal $\Lambda $. With rigorous mathematical results we…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
Topological Data Analysis (TDA) uses insights from topology to create representations of data able to capture global and local geometric and topological properties. Its methods have successfully been used to develop estimations of fractal…
In many real complex networks, the fractal and self-similarity properties have been found. The fractal dimension is a useful method to describe fractal property of complex networks. Fractal analysis is inadequate if only taking one fractal…