Quantum walks as a probe of structural anomalies in graphs

TL;DR

This paper demonstrates how quantum walks can efficiently detect structural anomalies in various graph configurations, offering potential for network analysis and anomaly detection.

Contribution

It introduces methods using quantum walks to identify specific structural anomalies in graphs, such as complete subgraphs, shared vertices, and extra edges, with demonstrated quantum speedup.

Findings

Quantum walks can locate complete subgraphs with speedup.

Quantum walks can identify shared vertices between graphs.

Quantum walks can detect edges that alter bipartite structure.

Abstract

We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external vertices, are connected by edges. In the basic star graph, these are the only edges. If we now connect a subset of the external vertices to form a complete subgraph, a quantum walk can be used to find these vertices with a quantum speedup. Thus, under some circumstances, a quantum walk can be used to locate where the connectivity of a network changes. We also look at the case of two stars connected at one of their external vertices. A quantum walk can find the vertex shared by both graphs, again with a quantum speedup. This provides an example of using a quantum walk in order to find where two networks are connected. Finally, we use a quantum walk on a complete bipartite graph to find an extra edge that destroys the bipartite nature of the graph.