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

Abstract

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 II(k(i)) approximately k(-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 --> infinity, n(k) both analytically and by Monte Carlo simulations. The distribution decays like kmu(k)/Tau(k) for large k, where 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.

The Role of Diverse Strategies in Sustainable Knowledge Production

“…Our work generalizes the models of Barabasi et al [ 9 ] and Sevim et al [ 10 ] to study large-scale cooperation in online communities. The present analysis on attention networks can also be applied to model a variety of other online collective behaviors such thread browsing [ 24 ], photo tagging [ 25 – 27 ], and news sharing [ 28 ].…”

“…As type A users prefer easy questions (low-degree nodes) and type B users favor difficult questions (high-degree nodes), when these two types of users move to a new question from old questions, they bring connections to the new node from old nodes of very different degrees. Therefore, we can naively assume that strategy A corresponds to “preferential attachment” [ 9 ], in which the rich get richer, and strategy B corresponds to the reversed process of “preferential attachment” [ 10 ], in which the attractiveness of a node decreases with its degree. The reversed “preferential attachment” process has been observed in systems featured by the strong competition between nodes for limited resources, such as food webs [ 18 ], power grids[ 19 ], and airport networks [ 20 ].…”

“…Substituting this condition into Eq 3 we derive the scaling relationship between the number of new and old nodes Note that if an old node generates many new nodes, then there will be a stronger competition between these new nodes for edges. As a result, the cost of linking to an existing node is proportional to its degree [ 10 ]. Meanwhile, it is reasonable to assume that new questions cannot obtain an infinite number of answers but have a limited “quota” that approximates constant C .…”

“…In attention networks nodes are questions and edges are the successive answering activities of users that connect two questions. We developed a theoretical network model in which the co-existence of two answering strategies B and A are represented by the mixture of two linking dynamics, i.e., preferential attachment [ 9 ] and its “reversed” process [ 10 ]. The mathematical analysis of this model not only supports the existence of a trade-off between the two strategies, but also describes the consequences when the mixing ratio in a community deviates from the optimal value.…”

Linear de-preferential urn models

“…Linear de-preferential urn models 1177 Recently, there has been some interest in random graphs [9], [50], [51], where attachment probabilities of a new vertex are decreasing functions of the degree of the existing vertices. In most cases, such models also lead to negative reinforcement.…”