The Iterative Independent Model

TL;DR

This paper introduces the Iterative Independent Model (IIM), a generalized deterministic network generation process inspired by Structural Balance Theory, which produces graphs with properties similar to real-world social networks.

Contribution

It generalizes previous models by allowing independent cloning or anticloning of vertices, and demonstrates that IIM graphs have key structural properties like spectral gap, clustering, and containment of all fixed subgraphs.

Findings

IIM graphs have spectral gaps bounded away from zero.

IIM graphs exhibit properties like bounded diameter, domination number, and clique number.

All IIM graphs eventually contain any fixed subgraph.

Abstract

Deterministic complex networks that use iterative generation algorithms have been found to more closely mirror properties found in real world networks than the traditional uniform random graph models. In this paper we introduce a new, Iterative Independent Model (IIM), generalizing previously defined models. These models use ideas from Structural Balance Theory to generate edges through a notion of cloning where ``the friend of my friend is my friend'' and anticloning where ``the enemy of my enemy is my friend''. In this paper, we vastly generalize these notions by allowing each vertex added at a given time step to choose independently of the other vertices if it will be cloned or anticloned. While it may seem natural to focus on a randomized model, where we randomly determine whether or not to clone any given vertex, we found the general deterministic model exhibited certain structural properties regardless of the probabilities. This allows applications to then explore the particulars, while having the theoretical model explain the structural phenomenons that occur in all possible scenarios. Throughout the paper we demonstrate that all IIM graphs have spectral gap bounded away from zero, which indicates the clustering properties also found in social networks. Furthermore, we show bounds on the diameter, domination number, and clique number further indicating the well clustered behaviour of IIM graphs. Finally, for any fixed graph $F$ all IIM graphs will eventually contain an induced copy of $F$.