Growing networks with two vertex types

Abstract

Growing networks are introduced in which the vertices are allocated one of two possible growth rates; type A with probability p(t), or type B with probability 1−p(t). We investigate the networks using rate equations to obtain their degree distributions. In the first model (I), the network is constructed by connecting an arriving vertex to either a type A vertex of degree k with rate μk, where μ0, or to a type B vertex of degree k with rate k. We study several p(t), starting with p(t) as a constant and then considering networks where p(t) depends on network parameters that change with time. We find the degree distributions to be power laws with exponents mostly in the range 2γ3. In the second model (II), the network is constructed in the same way but with growth rate k for type A vertices and 1 for type B vertices. We analyse the case p(t)=c, where 0c1 is a constant, and again find a power-law degree distribution with an exponent 2γ3.