Directed networks with at least three connections per node completely defy the mathematical rules that experts thought governed their internal cycles.
The number of loops inside a random network was long suspected to follow a consistent pattern regardless of how many connections each point had. A recent conjecture predicted these cycle-factors would behave predictably as long as the graph was regular. A sharp breaking point actually exists where the rule works perfectly for two connections but fails for every single number higher than that. The discovery forces a total rethink of how we model complex systems like electrical grids or computer networks. Predicting the stability of these systems now requires entirely new formulas that account for this sudden jump in complexity.
Counterexamples to an Extremal Conjecture for Random Cycle-Factors
arXiv · 2604.26101
Christoph, Draganić, Girão, Hurley, Michel, and Müyesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph on $n$ vertices is uniquely maximised by the disjoint union of $n/d$ copies of the complete looped digraph $K_d^\circ$, with value $(n/d)H_d$ [FOCS 2025]. We disprove this conjecture in the strongest possible range. For every $d\ge 3$ and every multiple $n=kd$ with $k\ge 2$, we construct a directed $d$-regular