Network growth model with intrinsic vertex fitness

Abstract

We study a class of network growth models with attachment rules governed by intrinsic node fitness. Both the individual node degree distribution and the degree correlation properties of the network are obtained as functions of the network growth rules. We also find analytical solutions to the inverse, design, problems of matching the growth rules to the required (e.g., power-law) node degree distribution and more generally to the required degree correlation function. We find that the design problems do not always have solutions. Among the specific conditions on the existence of solutions to the design problems is the requirement that the node degree distribution has to be broader than a certain threshold and the fact that factorizability of the correlation functions requires singular distributions of the node fitnesses. More generally, the restrictions on the input distributions and correlations that ensure solvability of the design problems are expressed in terms of the analytical properties of their generating functions.

Fitness-based network growth with dynamic feedback

“…[6,7] provide a strong hint that this type of feedback may generate a power-law fixed point. As will be shown below, this conjecture is indeed confirmed by an explicit calculation, with the fixed point characterized by a unique power-law exponent.…”

“…As shown in [7], distributions of node degrees in fitness-based network growth models are generically broader than the "input" distributions of fitnesses. For example, if all nodes have equal fitnesses (a delta-peaked distribution) the resulting degree distribution is exponential, while an exponential distribution of fitnesses leads to stretched exponential distribution of degrees.…”

“…Both the distribution of fitnesses and the connection rules are given by a priori arbitrary functions, thus allowing a considerable amount of tuning in such models. This feature enables fitness-based models to mimic a variety of network topologies, in particular, subject to some constraints, they can be tuned to reproduce a given type of degree distributions and even degree correlation functions [4,5,7]. This tunability makes fitness-based models useful as a modeling tool, but also imparts a degree of arbitrariness which makes them less attractive as a robust explanation for the universality of naturally observed behaviors.…”

“…For example, if all nodes have equal fitnesses (a delta-peaked distribution) the resulting degree distribution is exponential, while an exponential distribution of fitnesses leads to stretched exponential distribution of degrees. Crucially, however, this broadening saturates at power-law distributions, so that a fitness distribution that asymptotically behaves as a power law generates a degree distribution with a matching asymptotic power law tail [6,7].…”

“…Such considerations provided one of the motivations for the study of a different class of growth mechanisms, variously known as hidden variable or fitness-based models [4][5][6][7]. The models of this type are characterized by probabilistic rules for forming connections between nodes based on a static measure of intrinsic node attractiveness, usually termed fitness.…”