Popularity-Driven Networking

TL;DR

This paper models network growth where link formation depends on node popularity, revealing two abrupt phase transitions: giant component emergence and system-wide condensation, with analytical insights into component sizes and degree distribution.

Contribution

It introduces a new model of network growth driven by degree-proportional connection rates, analyzing phase transitions and deriving component and degree distributions.

Findings

Identifies two finite-time phase transitions: percolation and condensation.

Derives exponential degree distribution throughout network evolution.

Provides a criterion for sudden condensation in general homogeneous connection rates.

Abstract

We investigate the growth of connectivity in a network. In our model, starting with a set of disjoint nodes, links are added sequentially. Each link connects two nodes, and the connection rate governing this random process is proportional to the degrees of the two nodes. Interestingly, this network exhibits two abrupt transitions, both occurring at finite times. The first is a percolation transition in which a giant component, containing a finite fraction of all nodes, is born. The second is a condensation transition in which the entire system condenses into a single, fully connected, component. We derive the size distribution of connected components as well as the degree distribution, which is purely exponential throughout the evolution. Furthermore, we present a criterion for the emergence of sudden condensation for general homogeneous connection rates.