Effects of preference for attachment to low-degree nodes on the degree distributions of a growing directed network and a simple food-web model

We study the growth of a directed network, in which the growth is constrained by the cost of adding links to the existing nodes. We propose a preferential-attachment scheme, in which a new node attaches to an existing node i with probability Π(k_i)∝k_i⁻¹, where k_i is the number of outgoing links at i. We calculate the degree distribution for the outgoing links in the asymptotic regime (t→∞), n_k^*, both analytically and by Monte Carlo simulations. The distribution decays like kμ^k∕Γ(k) for large k, where μ is a constant. We investigate the effect of this preferential-attachment scheme, by comparing the results to an equivalent growth model with a degree-independent probability of attachment, which gives an exponential outdegree distribution. Also, we relate this mechanism to simple food-web models by implementing it in the cascade model. We show that the low-degree preferential-attachment mechanism breaks the symmetry between in- and outdegree distributions in the cascade model. It also causes a faster decay in the tails of the outdegree distributions for both our network growth model and the cascade model.