About the Discriminant Power of the Subgraph Centrality and Other Centrality Measures About the Discriminant Power of the Subgraph Centrality and Other Centrality Measures(Working paper)

The discriminant power of centrality indices for the degree, eigenvector, closeness, betweenness and subgraph centrality is analyzed. It is defined by the number of graphs for which the standard deviation of the centrality of its nodes is zero. On the basis of empirical analysis it is concluded that the subgraph centrality displays better discriminant power than the rest of centralities. We also propose some new conjectures about the types of graphs for which the subgraph centrality does not discriminate among nonequivalent nodes.

💡 Research Summary

The paper investigates how well five widely used node‑centrality measures—degree, eigenvector, closeness, betweenness, and the subgraph centrality introduced by Estrada and Rodríguez‑Velázquez (2005)—are able to distinguish non‑equivalent vertices in simple connected graphs. The authors formalize “discriminant power” as the proportion of graphs for which the standard deviation of a given centrality across all vertices equals zero; in other words, a centrality has low discriminant power if it assigns the same value to every node in many graphs.

To evaluate this, the authors exhaustively enumerate all connected graphs with 5, 6, 7, and 8 vertices, amounting to 12 103 graphs in total. For each graph they compute the five centrality vectors and record whether the standard deviation is zero. The results are summarized in Table 1. Subgraph centrality distinguishes nodes in almost all graphs: only 2 of the 5‑node graphs, 6 of the 6‑node graphs, 7 of the 7‑node graphs, and 10 of the 8‑node graphs have zero standard deviation. By contrast, degree and eigenvector centralities fail to discriminate in 17, 15, and 12 graphs respectively, closeness in 15 graphs, and betweenness in 12 graphs.

A detailed inspection of the “undistinguishable” cases reveals that they are essentially walk‑regular graphs—graphs in which, for every walk length ℓ, the number of closed walks starting at any vertex is the same. Walk‑regularity includes vertex‑transitive and distance‑regular graphs, but not all walk‑regular graphs are distance‑regular. The six‑node examples (cycle, complete graph, octahedral graph, utility graph, and the 3‑prism) are all walk‑regular; the seven‑node and eight‑node examples are likewise walk‑regular, with the eight‑node set consisting of the ten walk‑regular graphs on eight vertices.

The paper revisits an earlier conjecture (Conjecture 1) that “if subgraph centrality fails to distinguish nodes, then none of the other four centralities will either.” This conjecture was recently disproved by Puck Rombach and Porter, who exhibited walk‑regular but not distance‑regular graphs where subgraph centrality is uniform while closeness or betweenness are not. In response, the authors propose two stronger conjectures. Conjecture 2 states that, after excluding walk‑regular graphs that are not distance‑regular, a zero standard deviation for subgraph centrality implies zero standard deviation for degree, closeness, betweenness, and eigenvector centralities. Conjecture 3 asserts an equivalence: a graph has uniform subgraph centrality if and only if it is walk‑regular. The forward direction is immediate from the definition of subgraph centrality; the reverse direction is non‑trivial and forms the core of the new hypothesis.

The contributions of the work are threefold. First, it provides a systematic empirical comparison of discriminant power, demonstrating that subgraph centrality is the most sensitive measure for detecting structural differences among vertices in small graphs. Second, it identifies walk‑regular graphs as the principal class of structures where any of the five centralities lose discriminative ability, highlighting a structural limitation that practitioners should be aware of when interpreting centrality scores. Third, it refines the theoretical landscape by disproving an earlier conjecture and formulating new conjectures that link walk‑regularity to the behavior of subgraph centrality.

Future research directions suggested include extending the analysis to larger graphs and random graph ensembles, exploring algorithmic applications of subgraph centrality for node ranking and community detection, and deepening the theoretical understanding of walk‑regular graphs in the context of centrality measures.