Universal Properties of Growing Networks

TL;DR

This paper explores the universal behaviors of growing networks, revealing a phase diagram with power-law cluster size distributions, a continuous but infinite-order percolation transition, and unique properties at criticality.

Contribution

It introduces a universal framework for understanding phase transitions and cluster distributions in growing networks with attachment probabilities p_k.

Findings

Power law decay of cluster size distribution in the non-percolating phase

Continuous but infinite-order percolation transition

Discontinuous average cluster size at the transition

Abstract

Networks growing according to the rule that every new node has a probability p_k of being attached to k preexisting nodes, have a universal phase diagram and exhibit power law decays of the distribution of cluster sizes in the non-percolating phase. The percolation transition is continuous but of infinite order and the size of the giant component is infinitely differentiable at the transition (though of course non-analytic). At the transition the average cluster size (of the finite components) is discontinuous.