Percolation transition of strongly connected clusters in finite dimensions and on complete graphs

Abstract

We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimension $d=2$ to $7$ and complete graphs. Below the upper critical dimension $d_u=6$, the critical SCCs exhibit nontrivial fractal dimension $d_{\rm SCC}$, and the size distribution scales as $\sim s^{-τ_{\rm SCC}}$ with the hyperscaling relation $τ_{\rm SCC}=1+d/d_{\rm SCC}$. For $d \ge d_u$, mean-field behavior is recovered with $d_{\rm SCC}/d=1/3$, consistent with complete-graph results. However, in contrast to hypercubic lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by mean-field value $τ_{\rm SCC}=4$, while small SCCs follow a distinct power law with exponent $τ'=1$. At criticality, the giant in- and out-clusters are also fractal, sharing the same dimension as standard percolation clusters. These results show that critical SCCs remain well-defined fractal objects across dimensions, while their approach to the mean-field limit involves nontrivial changes in cluster statistics.