Coexistence in Preferential Attachment Networks

Abstract

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

Condensation in preferential attachment models with location‐based choice

“…However, Theorem 2.4 implies that the point of condensation also has positive probability of occurring at each of the endpoints 1 2 − s and 1 2 + s; since almost surely these values are not locations of any vertex, it follows that there is also a positive probability that there is no persistent hub. Figure 3 shows the results of two simulations for = −0.75 with different behavior: in the first simulation there is rapid convergence of Ψ n to a limit with condensation occurring via a persistent hub, whereas in the second Ψ n shows much slower convergence, apparently towards condensation at 1 2 + s. If ≤ − 7 8 , Theorem 2.3 implies that the location of the jump has full support on (0, 1).…”

“…for some > −1. Equation (1) gives the form of preferential attachment developed by Dorogovtsev, Mendes, and Samukhin in [7] as a generalization of the Barabási and Albert model found in [2], and we shall use this more general form. However, several of the papers referred to in this section, including [2], only consider the case = 0.…”

“…How attractive the vertex is by itself or in comparison to the others also plays a large part. A model incorporating this notion was introduced by Bianconi and Barabási in [3], where they gave each vertex a multiplicative fitness value in their version of (1). They did this in order to add an extra dimension to the competition between vertices that joined the vertex using a generalized preferential attachment mechanism.…”

“…Another variant of preferential attachment is the choice model introduced by Malyshkin and Paquette in [11,12] and Krapivsky and Redner in [10]. Here when a new vertex joins the network it first selects several candidate existing vertices at random using (1), then attaches to one of the candidates according to a deterministic rule, such as always attaching to the candidate of highest degree. In these papers, depending on the parameters, as the number of vertices increases linear or approximately linear growth can be observed in the degree of the largest vertex.…”

Competing first passage percolation on random regular graphs

“…They showed that in contrast to our results when starting from two vertices, a.s., all vertices but a finite number will be occupied by one of the two types. In a different follow up work, the authors of [2] introduced a model that coupled competition with the network growth and demonstrate that in this model often the two competitors will each occupy a linear fraction of the vertices even if one spreads faster than the other.…”