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A variation of the preferential attachment random graph model of Barabasi and Albert is defined that incorporates planted communities. The graph is built progressively, with new vertices attaching to the existing ones one-by-one. At every step, the incoming vertex is randomly assigned a label, which represents a community it belongs to. This vertex then chooses certain vertices as its neighbors, with the choice of each vertex being proportional to the degree of the vertex multiplied by an affinity depending on the labels of the new vertex and a potential neighbor. It is shown that the fraction of half-edges attached to vertices with a given label converges almost surely for some classes of affinity matrices. In addition, the empirical degree distribution for the set of vertices with a given label converges to a heavy tailed distribution, such that the tail decay parameter can be different for different communities. Our proof method may be of independent interest, both for the classical Barabasi -Albert model and for other possible extensions.
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We propose a random graph model with preferential attachment rule and emph{edge-step functions} that govern the growth rate of the vertex set. We study the effect of these functions on the empirical degree distribution of these random graphs. More sp
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We extend the work of Antunovi{c}, Mossel and R{a}cz on competing types in preferential attachment models to include cases where the types have different fitnesses, which may be either multiplicative or additive. We will show that, depending on the v