A Statistical Performance Analysis of Graph Clustering Algorithms

Abstract

Measuring graph clustering quality remains an open problem. To address it, we introduce quality measures based on comparisons of intra-and inter-cluster densities, an accompanying statistical test of the significance of their differences and a step-by-step routine for clustering quality assessment. Our null hypothesis does not rely on any generative model for the graph, unlike modularity which uses the configuration model as a null model. Our measures are shown to meet the axioms of a good clustering quality function, unlike the very commonly used modularity measure. They also have an intuitive graph-theoretic interpretation, a formal statistical interpretation and can be easily tested for significance. Our work is centered on the idea that well clustered graphs will display a significantly larger intra-cluster density than inter-cluster density. We develop tests to validate the existence of such a cluster structure. We empirically explore the behavior of our measures under a number of stress test scenarios and compare their behavior to the commonly used modularity and conductance measures. Empirical stress test results confirm that our measures compare very favorably to the established ones. In particular, they are shown to be more responsive to graph structure and less sensitive to sample size and breakdowns during numerical implementation and less sensitive to uncertainty in connectivity. These features are especially important in the context of larger data sets or when the data may contain errors in the connectivity patterns.

A density-based statistical analysis of graph clustering algorithm performance

“…Here, we slightly modify the procedure to generate inter-cluster edges. In our previous article [35], we varied the proportion of vertices inside and outside each cluster that shared an edge. Here, we vary edge probabilities.…”

“…Here, we streamline our statistical test. In our previous article [35], we conducted two separate tests. We formulated two null hypotheses,K intra = K andK inter = K, to avoid the effects of a possible correlation between K intra andK inter (K is a graph constant, not the result of a clustering).…”

A Statistical Test of Heterogeneous Subgraph Densities to Assess Clusterability

“…The links between density and clustered patterns of connectivity were shown in Miasnikof et al [26]. Under such a pattern of connectivity, it is expected that the densities of induced subgraphs obtained by sampling vertices within a neighborhood will exhibit, on average, higher densities than the graph's global density.…”

Graph Distances and Clustering

“…We then examine the relationship between mean Jaccard [10], Otsuka-Ochiai [17] and Burt's distances [2], on one hand, and intra-cluster density [14,13,15,16] within each cluster, on the other. Because these distances are pairwise measures, we compare their mean value for a given cluster to the cluster's internal density.…”

“…At the cluster level, this distance takes the form of subsets of densely connected vertices. The link between clustering and density has been discussed in depth, recently [14,15,13,16]. In this article, our ultimate goal is to transform a graph's adjacency matrix into a |V | × |V | similarity or distance matrix D = [d ij ], where the distance between each pair of vertices is given by the element d ij .…”