Theory and algorithms for clusters of cycles in graphs for material networks

Abstract

Analysis of complex networks, particularly material networks such as the carbon skeleton of hydrocarbons generated in hydrocarbon pyrolysis in carbon-rich systems, is essential for effectively describing, modeling, and predicting their features. An important and the most challenging part of this analysis is the extraction and effective description of cycles, when many of them coalesce into complex clusters. A deterministic minimum cycle basis (MCB) is generally non-unique and biased to the vertex enumeration. The union of all MCBs, called the set of relevant cycles, is unique, but may grow exponentially with the graph size. To resolve these issues, we propose a method to sample an MCB uniformly at random. The output MCB is statistically well-defined, and its size is proportional to the number of edges. We review and advance the theory of graph cycles from previous works of Vismara, Gleiss et al., and Kolodzik et al. In particular, we utilize the polyhedron-interchangeability (pi) and short loop-interchangeability (sli) classes to partition the relevant cycles. We introduce a postprocessing step forcing pairwise intersections of relevant cycles to consist of a single path. This permits the definition of a dual graph whose nodes are cycles and edges connect pairs of intersecting cycles. The pi classes identify building blocks for crystalline structures. The sli classes group together sets of large redundant cycles. We present the application to an amorphous hydrocarbon network, where we (i) theorize how the number of relevant cycles may explode with system size and (ii) observe small polyhedral structures related to diamond.