A Finite Pattern Method for Edge-Based Distance Invariants on Self-Similar Fractal Networks

The exact analytical computation of edge-based distance invariants on self-similar fractal networks presents a fundamental challenge in fractal network theory: while vertex-based invariants have been extensively studied, their edge counterparts require handling boundary contributions and distance interactions across self-similar copies, for which no systematic method has been established. This paper proposes the finite pattern method, a general technique that leverages strict self-similarity to reduce the infinite global sum over edge pairs to a finite system of recursive equations. The method consists of three steps: classification of all relative position patterns between pairs of basic self-similar copies, derivation of edge-distance formulas for each pattern, and construction of linear recurrences governing the pattern counts. We demonstrate the method on the Sierpinski hexagon skeleton network, a canonical self-similar fractal, and obtain the exact closed-form expression for its edge-Wiener index—the core edge-based distance invariant. The twelve fundamental patterns arising from the six-fold rotational symmetry are fully enumerated, and the resulting recurrence system is solved explicitly. The finite pattern method is independent of the specific geometry of the Sierpinski hexagon and applies to a broad class of deterministic fractal networks generated by iterated function systems, providing a unified framework for the exact solution of edge-based distance invariants on self-similar structures.