Scaling of cluster heterogeneity in percolation transitions

Emergence of biconnected clusters in explosive percolation

By introducing a simple competition mechanism for bond insertion in random graphs, explosive percolation exhibits a sharp phase transition with rich critical phenomena. We investigate high-order connectivity in explosive percolation using an event-based ensemble, focusing on biconnected clusters, where any two sites are connected by at least two independent paths. Our numerical analysis confirms that explosive percolation with different intracluster bond competition rules shares the same percolation threshold and universality, with biconnected clusters percolating simultaneously with simply connected clusters. However, the volume fractal dimension d_{f}^{'} of biconnected clusters varies depending on the competition rules of intracluster bonds. The size distribution of biconnected clusters exhibits double-scaling behavior: large clusters follow the standard Fisher exponent derived from the hyperscaling relation τ^{'}=1+1/d_{f}^{'}, while small clusters display a modified Fisher exponent τ_{0}<τ^{'}. These findings provide insights into the intricate nature of connectivity in explosive percolation.

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model

After a sudden quench from the disordered high-temperature T_{0}→∞ phase to a final temperature well below the critical point T_{F}≪T_{c}, the nonconserved order parameter dynamics of the two-dimensional ferromagnetic Ising model on a square lattice initially approaches the critical percolation state before entering the coarsening regime. This approach involves two timescales associated with the first appearance (at time t_{p_{1}}>0) and stabilization (at time t_{p}>t_{p_{1}}) of a giant percolation cluster, as previously reported. However, the microscopic mechanisms that control such timescales are not yet fully understood. In this paper, to study their role on each time regime after the quench (T_{F}=0), we distinguish between spin flips that decrease the total energy of the system from those that keep it constant, the latter being parametrized by the probability p. We show that observables such as the cluster size heterogeneity H(t,p) and the typical domain size ℓ(t,p) have no dependence on p in the first time regime up to t_{p_{1}}. Furthermore, when energy-decreasing flips are forbidden while allowing constant-energy flips, the kinetics is essentially frozen after the quench and there is no percolation event whatsoever. Taken together, these results indicate that the emergence of the first percolating cluster at t_{p_{1}} is completely driven by energy decreasing flips. However, the time for stabilizing a percolating cluster is controlled by the acceptance probability of constant-energy flips: t_{p}(p)∼p^{-1} for p≪1 (at p=0, the dynamics gets stuck in a metastable state). These flips are also the relevant ones in the later coarsening regime where dynamical scaling takes place. Because the phenomenology on the approach to the percolation point seems to be shared by many 2D systems with a nonconserved order parameter dynamics (and certain cases of conserved ones as well), our results may suggest a simple and effective way to set, through the dynamics itself, t_{p_{1}} and t_{p} in such systems.

Scaling behavior of information entropy in explosive percolation transitions

An explosive percolation transition is the abrupt emergence of a giant cluster at a threshold caused by a suppression of the growth of large clusters. In this paper, we consider the information entropy of the cluster-size distribution, which is the probability distribution for the size of a randomly chosen cluster. It has been reported that information entropy does not reach its maximum at the threshold in explosive percolation models, a result seemingly contrary to other previous results that the cluster-size distribution shows power-law behavior and the cluster-size diversity (number of distinct cluster sizes) is maximum at the threshold. Here, we show that this phenomenon is due to the fact that the scaling form of the cluster-size distribution is given differently below and above the threshold. We also establish the scaling behaviors of the first and second derivatives of the information entropy near the threshold to explain why the first derivative has a negative minimum at the threshold and the second derivative diverges negatively (positively) at the left (right) limit of the threshold, as predicted through previous simulation.

Statistics of geometric clusters in Potts model: statistical mechanics approach

The percolation of Potts spins with equal values in Potts model on graphs (networks) is considered. The general method for finding the Potts clusters' size distributions is developed. It allows full description of percolation transition when a giant cluster of equal-valued Potts spins appears. The method is applied to the short-ranged q-state ferromagnetic Potts model on the Bethe lattices with the arbitrary coordination number z . The analytical results for the field-temperature percolation phase diagram of geometric spin clusters and their size distribution are obtained. The last appears to be proportional to that of the classical non-correlated bond percolation with the bond probability, which depends on temperature and Potts model parameters.

Dynamical cluster size heterogeneity

Only recently has the essential role of the percolation critical point been considered on the dynamical properties of connected regions of aligned spins (domains) after a sudden temperature quench. In equilibrium, it is possible to resolve the contribution to criticality by the thermal and percolative effects (on finite lattices, while in the thermodynamic limit they merge at a single critical temperature) by studying the cluster size heterogeneity, H_{eq}(T), a measure of how different the domains are in size. We extend this equilibrium measure here and study its temporal evolution, H(t), after driving the system out of equilibrium by a sudden quench in temperature. We show that this single parameter is able to detect and well-separate the different time regimes, related to the two timescales in the problem, namely the short percolative and the long coarsening one.

Domain-size heterogeneity in the Ising model: Geometrical and thermal transitions

A measure of cluster size heterogeneity (H), introduced by Lee et al. [Phys. Rev. E 84, 020101 (2011)] in the context of explosive percolation, was recently applied to random percolation and to domains of parallel spins in the Ising and Potts models. It is defined as the average number of different domain sizes in a given configuration and a new exponent was introduced to explain its scaling with the size of the system. In thermal spin models, however, physical clusters take into account the temperature-dependent correlation between neighboring spins and encode the critical properties of the phase transition. We here extend the measure of H to these clusters and, moreover, present new results for the geometric domains for both d=2 and 3. We show that the heterogeneity associated with geometric domains has a previously unnoticed double peak, thus being able to detect both the thermal and percolative transitions. An alternative interpretation for the scaling of H that does not introduce a new exponent is also proposed.

Epidemic threshold of the susceptible-infected-susceptible model on complex networks

We demonstrate that the susceptible-infected-susceptible (SIS) model on complex networks can have an inactive Griffiths phase characterized by a slow relaxation dynamics. It contrasts with the mean-field theoretical prediction that the SIS model on complex networks is active at any nonzero infection rate. The dynamic fluctuation of infected nodes, ignored in the mean field approach, is responsible for the inactive phase. It is proposed that the question whether the epidemic threshold of the SIS model on complex networks is zero or not can be resolved by the percolation threshold in a model where nodes are occupied in degree-descending order. Our arguments are supported by the numerical studies on scale-free network models.

Cluster-size heterogeneity in the two-dimensional Ising model

We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.

Scaling of cluster heterogeneity in the two-dimensional Potts model

Cluster heterogeneity, the number of clusters of mutually distinct sizes, has been recently studied for explosive percolation and standard percolation [H. K. Lee et al., Phys. Rev. E 84, 020101(R) (2011); J. D. Noh et al., Phys. Rev. E 84, 010101(R) (2011)]. In this work we study the scaling of various quantities related with cluster heterogeneity in a broader context of two-dimensional q-state Potts model. We predict, via an analytic approach, the critical exponents for most of the measured quantities, and confirm these predications for various q values using extensive Monte Carlo simulations.