Accurately and quickly calculating the weighted spectral distribution

Abstract

The weighted spectral distribution (WSD) is a measure defined on the normalized Laplacian spectrum. It can be used for comparing complex networks with different sizes (number of nodes) and provides a sensitive discrimination of the structural robustness of complex networks. In this paper, we design an algorithm for the accurate calculation of the WSD in large-scale complex networks by utilizing characteristics of the graph structure. As an extension to Sylvester's Law of Inertia for the calculation of the WSD based on spectral characteristics, our algorithm exhibits a higher time efficiency when applied to large-scale complex networks.

Structural Decomposition Model for the Evolution of AS-Level Internet Topologies

“…This paper chooses spectral property = ∑ (1 − ) ‖ ‖ to describe the distance, where ( = 1,2, ⋯ , ‖ ‖) are all the eigenvalues of the normalized Laplacian matrix of graph [34] because our recent work has demonstrated that the spectral property is a good indicator of the average path length [26]. However, the former can be calculated faster using the circle enumeration method than the latter [28].…”

“…Recently, we have partitioned the Internet transit and stub AS nodes into seven categories by analyzing the physical mean of the normalized Laplacian spectral properties [24][25][26][27][28][29]. However, these works only provided a static node classification and have not yet structurally distinguished the core from the periphery.…”

Stability of weighted spectral distribution in a pseudo tree-like network model

“…[22,23] In general, the sum over the 4cycles represents the quasi-randomness of the graph in a nonregular case. [22,23] Therefore, four is considered to be the suitable value of the parameter N. [4][5][6]…”

“…[2] Moreover, Wu et al [3] rigorously demonstrated that the feature is asymptotically independent of the order of regular graphs in which each node has an identical expected degree. However, our recent work [4,5] found numerically that the stability of the natural connectivity does not work in scale-free networks. The WSD is another spectral graph feature defined on the normalized Laplacian spectrum that strongly corresponds to the distribution of random N-cycles in a network.…”

“…[6] Fay et al [7] designed a calculation algorithm for the WSD using Sylvester's law of inertia which is restricted to networks with less than 30000 nodes. Using 4-cycle structures, Jiao et al [4] proposed a fast algorithm by which the WSD can be accurately calculated in scale-free networks with more than millions of nodes. The WSD captures the local connectivity of diverse networks.…”