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 new preferential-attachment scheme, in which a new node attaches to an existing node i with probability proportional to 1/k_i, 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->infinity), both analytically and by Monte Carlo simulations. The distribution decays like k c^k/Gamma(k) for large k, where c 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.