A Fast Algorithm to Find All High Degree Vertices in Graphs with a Power Law Degree Sequence

Abstract

We develop a fast method for finding all high degree vertices of a connected graph with a power law degree sequence. The method uses a biassed random walk, where the bias is a function of the power law c of the degree sequence.

Let G(t) be a t-vertex graph, with degree sequence power law c ≥ 3 generated by a generalized preferential attachment process which adds m edges at each step. Let S _a be the set of all vertices of degree at least t ^a in G(t). We analyze a biassed random walk which makes transitions along undirected edges {x,y} proportional to (d(x)d(y))^b, where d(x) is the degree of vertex x and b > 0 is a constant parameter. Choosing the parameter b = (c − 1)(c − 2)/(2c − 3), the random walk discovers the set S _a completely in $\widetilde{O}(t^{1-2ab(1-\epsilon)})$ steps with high probability. The error parameter ε depends on c,a and m. We use the notation $\tilde O(x)$ to mean O(x log^k x) for some constant k > 0.

The cover time of the entire graph G(t) by the biassed walk is $\widetilde{O}(t)$. Thus the expected time to discover all vertices by the biassed walk is not much higher than in the case of a simple random walk Θ(t logt).

The standard preferential attachment process generates graphs with power law c = 3. Choosing search parameter b = 2/3 is appropriate for such graphs. We conduct experimental tests on a preferential attachment graph, and on a sample of the underlying graph of the www with power law c ~3 which support the claimed property.