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

Abstract

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 nonzero 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.

Maximum matchings in scale-free networks with identical degree distribution

“…As shown in [31], for all g ≥ 1, H g has the same degree sequence as that of F g . Thus, H g is scale-free with the power exponent γ = 3, identical to that of F g .…”

“…7. The non-fractal scale-free network is also self-similar, which can also be generated in an alternative approach [31]. Similar to its fractal counterpart F g , in H g , g ≥ 1, the initial four vertices created at g = 1 have the largest degree, which are call hub vertices.…”

“…In this section, we study maximum matchings in the non-fractal counterpart H g of F g , which has an infinite fractal dimension [31]. Both F g and H g have the same degree sequence for all g ≥ 1, but H g has perfect matchings.…”

“…After introducing related notations, in what follows we will study maximum matchings of two scale-free networks with the same degree sequence [31]: one is fractal and large-world, while the other is non-fractal and small-world. We will show that for both networks, the properties of their maximum matchings differ greatly.…”

“…In order to answer the above-raised problem, in this paper, we present an analytical study of maximum matchings in two scale-free networks with identical degree distribution [31], and show that the first network has no perfect matchings, while the second network has many. For the first network, we derive an exact expression for the size of maximum matchings, and provide a recursive solution to the number of maximum matchings.…”