Metric symmetry and distance distribution functions on graphs

After reviewing various notions of symmetry in graph theory, which are typically defined by the connections between vertices, we demonstrate that traditional concepts of symmetry, such as vertex transitivity, can be too restrictive for certain applications. For instance, in some areas of graph analysis, symmetry based on metric properties (such as average distances between vertices) may be more appropriate, particularly in social network analysis or economic fraud detection. This paper focuses on developing metric-based symmetry concepts by introducing mathematical analysis tools, all related to the central idea of the distance distribution function, to group vertices according to their distance-related properties within the graph. In particular, we prove several results that show, under certain compactness properties for the set of distribution functions of all the vertices in an infinite graph, that it is always possible to group these vertices into a finite number of classes with the desired accuracy based on distances.