A Weighted Bipartite Network Framework for Quantifying Molecular Comorbidity: Application to Periodontal Disease

The mathematical formalisms underlying disease-gene overlap metrics in network medicine remain largely implicit. We formalize the disease-gene relation as a weighted bipartite graph represented by a biadjacency matrix W, with edge weights combining DisGeNET confidence scores and pLI values, and derive a pLI-weighted Jaccard coefficient J_w and a hub centrality index H. We establish three formal results: boundedness and reduction to the classical Jaccard index (Proposition 1), strict refinement of the classical index (Proposition 2), and non-metricity of the induced dissimilarity, so comorbidity is not transitive (Proposition 3). We further show that H = W^T k is bounded by the leading singular value of W and derive a perturbation bound for the stability of the hub ranking. The formalism is illustrated on the periodontal diseasome of Martinez-Garcia and Hernandez-Lemus (2024): that study provided the dataset and the biological interpretation; the present note provides the formal structure, the three propositions, and the spectral and perturbation bounds, none of which were considered there. TNF and IL6 achieve the highest centrality (H = 24 and 23), and J_w discriminates comorbidity pairs that classical Jaccard treats as equivalent.