# Clique Gossiping

## TL;DR

This paper introduces a framework for clique gossip protocols in networks, analyzing their spectral properties, convergence behavior, and conditions for finite-time averaging, with implications for social, computer, and engineering systems.

## Contribution

It generalizes line graph concepts to clique interactions, proves eigenvalue invariance for certain protocols, and establishes conditions for finite-time convergence based on clique sizes and network divisibility.

## Key findings

- Eigenvalue invariance under permutation of clique sequences when no cycles are present.
- Finite-time convergence achievable with cliques of size dividing the number of nodes.
- Constructed fastest finite-time clique-gossip averaging algorithm.

## Abstract

This paper proposes and investigates a framework for clique gossip protocols. As complete subnetworks, the existence of cliques is ubiquitous in various social, computer, and engineering networks. By clique gossiping, nodes interact with each other along a sequence of cliques. Clique-gossip protocols are defined as arbitrary linear node interactions where node states are vectors evolving as linear dynamical systems. Such protocols become clique-gossip averaging algorithms when node states are scalars under averaging rules. We generalize the classical notion of line graph to capture the essential node interaction structure induced by both the underlying network and the specific clique sequence. We prove a fundamental eigenvalue invariance principle for periodic clique-gossip protocols, which implies that any permutation of the clique sequence leads to the same spectrum for the overall state transition when the generalized line graph contains no cycle. We also prove that for a network with $n$ nodes, cliques with smaller sizes determined by factors of $n$ can always be constructed leading to finite-time convergent clique-gossip averaging algorithms, provided $n$ is not a prime number. Particularly, such finite-time convergence can be achieved with cliques of equal size $m$ if and only if $n$ is divisible by $m$ and they have exactly the same prime factors. A proven fastest finite-time convergent clique-gossip algorithm is constructed for clique-gossiping using size-$m$ cliques. Additionally, the acceleration effects of clique-gossiping are illustrated via numerical examples.