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

TL;DR

This study investigates the percolation of strongly connected clusters in directed systems across various dimensions, revealing fractal properties below the upper critical dimension and mean-field behavior above it, with unique size distribution features on complete graphs.

Contribution

It provides a comprehensive analysis of SCC percolation in finite dimensions and complete graphs, highlighting the transition from fractal to mean-field behavior and discovering a double-scaling structure in size distributions.

Findings

Critical SCCs are fractal with dimension $d_{SCC}$ below $d_u=6$.

Size distribution follows a hyperscaling relation $ au_{SCC}=1+d/d_{SCC}$ below $d_u$.

Complete graphs show a double-scaling structure with different exponents for large and small SCCs.

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 $\sim s^{-\tau_{\rm SCC}}$ with the hyperscaling relation $\tau_{\rm SCC}=1+d/d_{\rm SCC}$. 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 $\tau_{\rm SCC}=4$, while small SCCs follow a distinct power law with exponent $\tau'=1$. 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.