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

Abstract

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 biased 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 that adds m edges at each step. Let Sa be the set of all vertices of degree at least ta in G(t). We analyze a biased random walk that makes transitions along undirected edges {x, y} with probabilities proportional to (d(x)d(y))b, where d(x) is the degree of vertex x and b > 0 is a constant parameter. With parameter b = (c − 1)(c − 2)/(2c − 3), the random walk discovers the set Sa completely in steps with high probability. The error parameter ε depends on c, a, and m. The cover time of the entire graph G(t) by the biased walk is . Thus the expected time to discover all vertices by the biased walk is not much higher than the Θ(tlog t) cover time of a simple random walk. The standard preferential attachment process generates graphs with power law c = 3. The 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 World Wide Web with power law c ≈ 3 that support the claimed property.