How Symmetric Are Real-World Graphs? A Large-Scale Study

Abstract

Abstract:The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the "network redundancy" measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

Analysis of Open Source Operating System Evolution: A Perspective from Package Dependency Network Motif

“…However, at present, in the field of software engineering, there is little research on the software network motif, especially the open source operating system software package network motif, and it has not yet been applied in the process of version evolution analysis. Therefore, this paper investigates open source operating system evolution from the perspective of the package dependency network motif for the sake of discovering potential cooperation and reusing schema in software development [4,9,[14][15][16][17][18][19][20]. Specifically, this manuscript makes three main contributions:…”

IntraClusTSP—An Incremental Intra-Cluster Refinement Heuristic Algorithm for Symmetric Travelling Salesman Problem

“…Symmetric graphs have many real-life applications as vehicle routing, warehouse logistics, planning circuit boards, virtual networking [1][2][3][4]. The goal of the base Symmetric Traveling Salesman Problem (sTSP) is to find the shortest Hamiltonian cycle in a graph.…”

Group User Profile Modeling Based on Neural Word Embeddings in Social Networks

“…Recommender systems provide users with personalized service based on people's perceived experience and socialized behaviors. Therefore, how to acquire personalized rich interests is still a challenging problem in most recommender systems [2,3].…”