Graph matching algorithms in random graphs with communities

Abstract

Social services such as Facebook, Twitter, and YouTube establish networks by connecting individuals to one another and linking individuals to various services. These services aim to enhance user experiences by uncovering hidden information in the data and providing users with recommendations for new connections. The primary problems in finding hidden information are community detection and graph matching. Community detection involves dividing nodes into suitable groups by using the fact that connections between nodes within the same community are denser than connections between nodes in different communities. Graph matching, on the other hand, focuses on finding similarities by comparing the statistical properties of nodes in each anonymized graph that are correlated. While numerous algorithms have been proposed in both fields, their performance cannot be sufficiently validated through experimentation alone. Results may vary in different experimental datasets, and it can be challenging to explain the success and failure of algorithms. Therefore, theoretical analysis of these algorithms is important. The objective of the thesis is to conduct a theoretical analysis of community detection and graph matching. The structure of the thesis can be divided into two sections. In Section 2, we will propose and analyze a graph matching algorithm that leverages community information, particularly in the Stochastic Block Model (SBM), a probabilistic graph model with a community structure. We will compare the proposed algorithm with other existing ones through both theoretical and experimental assessments. Moving on to Section 3, our analysis will focus on graph matching for community detection. We aim to determine the conditions required for the exact recovery of graph matching between two graphs generated from the correlated Contextual Stochastic Block Model (CSBM). Furthermore, we will investigate the advantages of community detection when correlated data is combined through graph matching under the conditions.