Clustering Compresses Attractors in Watts-Strogatz Threshold Boolean Networks

Abstract

AbstractDoes higher clustering shorten attractor periods? We examine whether the global clustering coefficient C, a direct measure of triangle density, predicts attractor lengths in synchronous, signed-threshold Boolean networks on Watts-Strogatz (WS) graphs. We generate 330 directed, signed WS networks spanning sizes N = 10-100 and mean degrees [Formula], simulate dynamics from M = 100 random initial states per graph with exact cycle detection, and summarise each graph by the average log attractor period (equivalently, the geometric mean period). Our primary analysis relates this log-period summary to C while adjusting for N, [Formula], the mean directed shortest path (MSP), and including nonlinear size-degree and clustering-degree interactions. Main findingHigher clustering robustly shortens attractor periods: a 0.10 increase in C (C [isin] [0, 1]) corresponds to an {approx} 13-15% lower expected geometric mean period, and moving from C = 0.000 to C = 0.460 yields an {approx} 50% reduction, holding other properties fixed. The effect persists when the linear C term is replaced by a nonlinear function of C, and it replicates in held-out graph instances (graphs not used to fit the model). Shorter periods are not explained by an increase in fixed points under the strict comparator (>); rather, higher triangle density shifts mass from long cycles to medium-length cycles. SignificanceIn threshold-like logic, settling speed and oscillatory stability are central to computation and control. Our results provide a direct, quantitative link between triangle density and these long-run behaviours, showing that C acts as a structural lever on temporal complexity. All figures and tables are reproducible from the accompanying code, data, and analysis scripts.