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 Sa 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 Sa completely in Õ(t1−2ab(1−ɛ) ) steps with high probability. The error parameter ɛ depends on c, a and m. We use the notation Õ(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 Õ(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 log t). 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.