Effect of Local Topological Changes on Resistance in Tunably-Disordered Networks

Disordered materials occur naturally and also provide a broader design space than ordered or crystalline structures. We investigate a two-dimensional disordered network metamaterial constructed from a Delaunay triangulation of an underlying point cloud. Small perturbations in the point cloud induce discrete topological changes. One such change we identify is a Delaunay flip, in which two neighboring Delaunay triangles that form a convex quadrilateral structure with their common edge being one of the two quadrilateral diagonals exchange this diagonal for the other diagonal. These topological changes can cause substantial jumps in the effective resistance measured diagonally across the network, when the change is located near the source or the sink node. The jumps are explained analytically by showing that the change in effective resistance from edge removal or addition depends on the voltage drop across that edge. However, Delaunay flips have less impact on global resistance measurements and in larger networks. These local topological changes are relevant for finite-sized samples and experimentally-measurable properties such as electrical transport. Global characterizations of the network disorder or topology lack the location-specificity of our observed effects on network transport, and thus may be inadequate for predicting certain experimentally measurable transport properties in disordered network metamaterials, highlighting the importance of localized regions in material design.

💡 Research Summary

The paper investigates how minute perturbations in the underlying point cloud of a two‑dimensional disordered network metamaterial lead to discrete topological changes that dramatically affect electrical transport. The networks are constructed by first generating a set of N random points inside a bounded box, then iteratively applying Lloyd’s algorithm to move each point to the centroid of its Voronoi cell. After L iterations, the points are connected via a Delaunay triangulation, producing a planar graph whose edges are treated as ohmic resistors with resistance proportional to edge length (Rij = ρ ℓij/A).

The authors focus on two specific local topological events: (1) the addition of an edge along the network boundary, and (2) a Delaunay flip, where two neighboring triangles sharing a diagonal are re‑triangulated by swapping that diagonal for the other one in the convex quadrilateral formed by the four vertices. Both events occur naturally as the point cloud evolves toward an ordered configuration, but they have markedly different impacts on the measured effective resistance R_eff, defined as the voltage drop between a designated source node (northeast corner) and sink node (southwest corner) divided by a fixed injected current (10 mA).

Numerical simulations reproduce the “jumps” in R_eff reported in earlier work (Obrero et al., Ref.