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We study the growth of a directed network, in which the growth is constrainedby the cost of adding links to the existing nodes. We propose a newpreferential-attachment scheme, in which a new node attaches to an existingnode i with probability proportional to 1/k_i, where k_i is the number ofoutgoing links at i. We calculate the degree distribution for the outgoinglinks in the asymptotic regime (t->infinity), both analytically and by MonteCarlo simulations. The distribution decays like k c^k/Gamma(k) for large k,where c is a constant. We investigate the effect of thispreferential-attachment scheme, by comparing the results to an equivalentgrowth model with a degree-independent probability of attachment, which givesan exponential outdegree distribution. Also, we relate this mechanism to simplefood-web models by implementing it in the cascade model. We show that thelow-degree preferential-attachment mechanism breaks the symmetry between in-and outdegree distributions in the cascade model. It also causes a faster decayin the tails of the outdegree distributions for both our network growth modeland the cascade model.Comment: 10 pages, 7 figures. A new figure added. Minor modifications made in  the tex

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