Abstract
Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness representing initial inhomogeneities among vertices and the fitness influences the attractiveness of a vertex in an additive way. We consider a heavy-tailed fitness distribution and show that the model exhibits a phase transition depending on the tail exponent of the fitness distribution. In the weak disorder regime, one of the old vertices has maximal degree irrespective of fitness, while for strong disorder the vertex with maximal degree has to satisfy the right balance between fitness and age. Our methods use martingale methods to show concentration of degree evolutions as well as extreme value theory to control the fitness landscape.
Highlights
A distinctive feature of real-world networks is their inhomogeneity, characterized in particular through the presence of hubs
The existence of hubs in a network is closely linked to the scale-free property, that is, the proportion of nodes in the network with degree k scales as a power law k−τ for some τ > 1
Somewhat related is a model of preferential attachment with random initial degree, for which [8] show convergence of empirical fitness distributions, but the structure of these networks is very different from the additive fitness case due to large out-degrees
Summary
A distinctive feature of real-world networks is their inhomogeneity, characterized in particular through the presence of hubs. We consider the model with additive fitness, where a vertex is chosen with probability proportional to the sum of its degree and its intrinsic fitness Both models of multiplicative and additive fitness can be seen as a way to understand how a random perturbation of the attractiveness of a vertex (due to natural inhomogeneities in the system) changes the behavior of a standard preferential attachment model. Somewhat related is a model of preferential attachment with random (possibly heavy-tailed) initial degree, for which [8] show convergence of empirical fitness distributions, but the structure of these networks is very different from the additive fitness case due to large out-degrees. PF (·) := P(· | (Fi)i∈N) and expectation EF [·] := E[· | (Fi)i∈N]
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