Random graphs with clustering

TL;DR

This paper introduces a generalized random graph model that accurately incorporates clustering, providing exact solutions for network properties like component sizes and phase transitions, addressing a key challenge in network physics.

Contribution

It presents a novel, solvable model of clustered networks, extending standard random graphs to include transitivity and enabling precise analysis of their structural properties.

Findings

Exact solutions for component sizes and phase transitions.

Model captures clustering and transitivity in networks.

Identifies critical points for giant component formation.

Abstract

We offer a solution to a long-standing problem in the physics of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity -- the propensity for two neighbors of a network node also to be neighbors of one another. We show how standard random graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant component forms, and position of the phase transition for percolation on the network.