Preferential attachment random graphs with general weight function

Abstract

Start with graph G 0 ≡ (V 1, V 2) with one edge connecting the two vertices V 1, V 2. Now create a new vertex V 3 and attach it (i.e., add an edge) to V 1 or V 2 with equal probability. Set G 1 ≡ (V 1, V 2, V 3). Let G n ≡ (V 1, V 2,… V n+2) be the graph after n steps, n ≥ 0. For each i, 1 ≤ i ≤ n+2, let d n(i) be the number of vertices in G n to which V i is connected. Now create a new vertex V n+3 and attach it to V i in G n with probability proportional to w(d n(i)), 1 ≤ i ≤ n+2, where w(·) is a function from N ≡ (1, 2, 3,…) to (0,∞). In this paper, some results on behavior of the degree sequence (d n(i))n≥1, i≥1 and the empirical distribution (Figure presented.) are derived. Our results indicate that the much discussed power-law growth of d n(i) and power law decay of (Figure presented.) hold essentially only when the weight function w(·) is asymptotically linear. For example, if w(x) = cx 2 then for i ≥ 1, limn d n(i) exists and is finite with probability (w.p.) 1 and (Figure presented.), and if w(x) = cx p, 1/2 < p < 1 then for i ≥ 1, d n(i) grows like (log n)q where q = (1 − p)−1. The main tool used in this paper is an embedding in continuous time of pure birth Markov chains. © A K Peters, Ltd.