Toward the Eigenvalue Power Law

Abstract

Many graphs arising in various real world networks exhibit the so called “power law” behavior, i.e., the number of vertices of degree i is proportional to i−β, where β> 2 is a constant (for most real world networks β≤ 3). Recently, Faloutsos et al. [18] conjectured a power law distribution for the eigenvalues of power law graphs. In this paper, we show that the eigenvalues of the Laplacian of certain random power law graphs are close to a power law distribution. First we consider the generalized random graph model G(d) =(V,E), where d=(d1, ..., dn) is a given sequence of expected degrees, and two nodes vi, vj ∈V share an edge in G(d) with probability pi, j=didj /$\sum^{n}_{k=1}$dk, independently [9]. We show that if the degree sequence d follows a power law distribution, then some largest Θ(n1/β) eigenvalues of L(d) are distributed according to the same power law, where L(d) represents the Laplacian of G(d). Furthermore, we determine for the case β ∈(2,3) the number of Laplacian eigenvalues being larger than i, for any i = ω(1), and compute how many of them are in some range (i,(1+e) i), where i=ω(1) and e>0 is a constant. Please note that the previously described results are guaranteed with probability 1–o(1/n). We also analyze the eigenvalues of the Laplacian of certain dynamically constructed power law graphs defined in [2,3], and discuss the applicability of our methods in these graphs.