Testing properties of signed graphs

TL;DR

This paper introduces sublinear algorithms for property testing in signed graphs, focusing on balance, clusterability, and signed triangle freeness, in both dense and bounded-degree models, with a key contribution on testing clusterability.

Contribution

It initiates the study of property testing in signed graphs and provides the first sublinear algorithm for testing clusterability in the bounded-degree model.

Findings

Existence of sublinear algorithms for testing key properties of signed graphs.

Development of a sublinear algorithm for testing clusterability in the bounded-degree model.

Contrast with non-testability of k-clusterability with sublinear queries.

Abstract

In graph property testing the task is to distinguish whether a graph satisfies a given property or is "far" from having that property, preferably with a sublinear query and time complexity. In this work we initiate the study of property testing in signed graphs, where every edge has either a positive or a negative sign. We show that there exist sublinear algorithms for testing three key properties of signed graphs: balance (or 2-clusterability), clusterability and signed triangle freeness. We consider both the dense graph model, where we can query the (signed) adjacency matrix of a signed graph, and the bounded-degree model, where we can query for the neighbors of a node and the sign of the connecting edge. Our algorithms use a variety of tools from graph property testing, as well as reductions from one setting to the other. Our main technical contribution is a sublinear algorithm for testing clusterability in the bounded-degree model. This contrasts with the property of k-clusterability which is not testable with a sublinear number of queries. The tester builds on the seminal work of Goldreich and Ron for testing bipartiteness.