Hamming Graph Metrics: A Multi-Scale Framework for Structural Redundancy and Uniqueness in Graphs
R. Scott Johnson
TL;DR
Hamming Graph Metrics (HGM) introduce a multi-scale, tensor-based framework for quantifying structural uniqueness in graphs via exact-$k$ reachability slices. By preserving per-scale dissimilarities through the tensor $\mathcal{B}$ and its Hamming distance, the approach yields permutation-invariant fingerprints, graph-level distributions, and a suite of functionals (e.g., entropy, dispersion) that generalize centrality to full dissimilarity spectra. The paper provides concrete theoretical guarantees, including metric properties, stability under edge perturbations, and extremal results across classical graph families, and demonstrates how HGM connects to classical invariants while offering richer, scale-resolved insights. It also outlines scalable computation (bit-parallel, streaming, and sketching options) and sketches extensions to directed/weighted/dynamic graphs, cross-graph alignment, and tensor-based analysis. Collectively, HGM offers an intrinsic, interpretable geometry of structural patterns that complements traditional centrality and distance measures, with potential impact on resilience analysis, network inference, and multi-graph comparisons.
Abstract
Traditional graph centrality measures effectively quantify node importance but fail to capture the structural uniqueness of multi-scale connectivity patterns -- critical for understanding network resilience and function. This paper introduces Hamming Graph Metrics (HGM), a framework that represents a graph by its exact-$k$ reachability tensor $\mathcal{B}G\in{0,1}^{N\times N\times D}$ with slices $(\mathcal{B}G){:,:,1}=A$ and, for $k\ge 2$, $(\mathcal{B}G){:,:,k}=\mathbf{1}!\left[\sum{t=1}^{k} A^t>0\right]-\mathbf{1}!\left[\sum_{t=1}^{k-1} A^t>0\right]$ (shortest-path distance exactly $k$). Guarantees. (i) Permutation invariance: $d_{\mathrm{HGM}}(π(G),π(H))=d_{\mathrm{HGM}}(G,H)$ for all vertex relabelings $π$; (ii) the tensor Hamming distance $d_{\mathrm{HGM}}(G,H):=|,\mathcal{B}G-\mathcal{B}H,|{1}=\sum{i,j,k}\mathbf{1}!\big[(\mathcal{B}G){ijk}\neq(\mathcal{B}H){ijk}\big]$ is a true metric on labeled graphs; and (iii) Lipschitz stability to edge perturbations with explicit degree-dependent constants (see "Graph-to-Graph Comparison" $\to$ "Tensor Hamming metric"; "Stability to edge perturbations"; Appendix A). We develop: (1) per-scale spectral analysis via classical MDS on double-centered Hamming matrices $D^{(k)}$, yielding spectral coordinates and explained variances; (2) summary statistics for node-wise and graph-level structural dissimilarity; (3) graph-to-graph comparison via the metric above; and (4) analytic properties including extremal characterizations, multi-scale limits, and stability bounds.