Trees = Networks ?!?

TL;DR

This paper establishes a reversible mapping between trees and weighted graphs, revealing their duality and enabling enhanced analysis of complex networks, including community detection and structural characterization.

Contribution

It introduces a complete, invertible mapping between trees and weighted graphs, extending to unweighted networks, and demonstrates its applications in network analysis and community detection.

Findings

Reversible mapping between trees and graphs is possible.

Network measurements can reveal tree structures.

The method aids in community detection.

Abstract

This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without loss of information. The extension of this methodology to more general networks, including unweighted structures, is also discussed and illustrated. It is shown that the identified duality between trees and graphs underlies several key concepts and issues of current interest in complex networks, including comprehensive characterization of trees and community detection. For instance, additional information about tree structures (e.g. phylogenetic trees) can be immediately obtained by taking into account several off-the-shelf network measurements -- such as the clustering coefficient, degree correlations and betweenness centrality. At the same time, the hierarchical structure of networks, including the respective communities, becomes clear when the network is represented in terms of the respective tree. Indeed, the network-tree mapping described in this work provides a simple and yet effective means of community detection.