Neural network approach to phase transitions in clique percolation

Clique percolation focuses on the connectivity of complete subgraphs rather than the local connectivity of edges or nodes, leading to distinct phase transition behaviors compared to traditional percolation models. Traditional simulation methods often require large system sizes and high computational costs, making them impractical for reliable results. This paper uses neural network methods to study the (k,l)-clique percolation phase transition on Moore lattices and Erdős–Rényi (ER) random networks. For Moore lattices, five methods were compared: (1) the 4N×N matrix-based convolutional neural network (CNN), (2) the 4N2×1 vector-based fully connected neural network (FCNN), (3) the largest cluster-based CNN, (4) the adjacency matrix-based CNN, and (5) the graph convolutional network (GCN). For ER random networks, two methods were used: (1) GCN based on edge connection and (2) the CNN with fixed nodes on a square lattice and random edge connections. The results show that (1) As k and l increase, identifying phase transition behavior becomes more difficult. (2) In Moore lattices, the 4N×N matrix-based CNN and the largest cluster-based CNN are effective methods. (3) For ER networks, GCN effectively identifies the (k,l)-clique percolation phase transition, while the fixed lattice CNN successfully detects the (2,1)-clique and (3,1)-clique transitions.