fivevertexscale

Fivevertexscale is a graph-theoretic descriptor that quantifies the occurrence and arrangement of five-vertex induced subgraphs within a finite simple graph. It serves as a local-structure metric that captures motif composition and diversity in the network, providing a compact summary of five-vertex patterns beyond basic degree-based measures.

The standard definition uses the set of all induced subgraphs on five vertices. For a graph G with n vertices, there are N = binomial(n, 5) such subgraphs. Each induced subgraph is classified according to its isomorphism class among the 34 unlabeled graphs on five vertices. Let c_i be the count of subgraphs isomorphic to the i-th class, and p_i = c_i / N the corresponding probability. The fivevertexscale is commonly represented by the distribution p = (p_1, ..., p_34). Scalar variants derived from this distribution include the entropy H(G) = - sum_i p_i log p_i, and the concentration measure s2(G) = sum_i p_i^2.

Properties and interpretation: The vector p is invariant under graph isomorphism and reflects motif composition independent of labeling. A high entropy indicates a rich diversity of five-vertex motifs, while a low entropy (or high s2) signals domination by a few motif types. These scalar summaries give a single-number or low-dimensional view of local structure.

Computation and applications: Exact counting requires enumerating all five-vertex induced subgraphs and performing isomorphism tests, which is computationally intensive for large graphs. Approaches include optimized counting algorithms and randomized sampling. Fivevertexscale is used in network analysis, chemistry, and bioinformatics to compare real networks with random models and to detect anomalous motif patterns. See also graph motifs, subgraph counting, and motif entropy.