Explosive Percolation and Entropy of k-mers on Networks using a parallel Monte Carlo algorithm

Explosive percolation on networks is a phenomenon where edge addition causes a sudden shift from a non-percolating to a percolating state, unlike classical percolation [1]. Entropy sharply peaks near the phase transition with differences in sharpness and magnitude depending on the lattice structure. Using finitesize scaling theory [2], numerical and analytical methods are used to find the phase-transition thresholds. Since the explosive percolation algorithm requires information from the previous step (memory) and computes more comparisons, considerable computing resources are needed than the classical percolation algorithm. Moreover, as k-mer size increases, the required resources also increase . In this work, we numerically studied the explosive percolation and entropy of k-mers on square networks using the Monte Carlo (MC) method implemented in C++ [3, 4]. Using OpenMP, we improve the sequential code using different techniques, significantly improving computational task performance by taking advantage of multicore processors. Simulations were performed on square networks of size L = 128, 256, 512, 1024 and 2048 with M = 106 MC simulations. We computed the entropy and percolation functions, percolation thresholds, and critical exponents characterizing this phenomenon and the explosive phase transition. The finite-size scaling theory helps us find the percolation thresholds as L and k Grow.