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The authors construct a unified network formulation of pairwise thermal conductance—spanning contact, near-contact, and bond interactions—and apply persistent homology filtered by conductance relative to the sink boundary. They prove that an edge's dissipation share is mathematically identical to the logarithmic sensitivity of the network's effective conductance, yielding exact closed-form solutions for finite single-edge perturbations. This enables simultaneous, diagnostic attribution of energy dissipation and mechanistic idealisation ratios across complex composite microstructures within a single graph.
Perturbing a single edge in a complex transport network can now be predicted in exact closed form because local dissipation share is mathematically identical to global conductance sensitivity.
We formulate the thermal conductance $g_e$ of a particle pair as three branch types -- contact, near-contact and bond -- within a single construction, and we introduce the filtration that takes this $g_e$ as its axis (the conductance filtration) together with persistence taken relative to the sink boundary. We then show that the dissipation share is exactly the logarithmic sensitivity of the effective conductance, and that a finite change of a single edge is predicted exactly in closed form. The allocation of the dissipation among branch types, and the idealisation ratio of each mechanism (interface, matrix and particle), are thereby obtained as diagnostic quantities on one and the same graph representation.