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=(d ₁, ..., d _n ) is a given sequence of expected degrees, and two nodes v _i , v _j ∈V share an edge in G(d) with probability p _{i, j} =d _i d _j /$\sum^{n}_{k=1}$ d _k , independently [9]. We show that if the degree sequence d follows a power law distribution, then some largest Θ(n ^1/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+ε) i), where i=ω(1) and ε>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.

The research was performed while the author visited the Department of Mathematics at University of California, San Diego. Partly supported by the German Research Foundation under contract EL-399/1-1.