We report the critical point for site percolation for the "explosive" type for 2D square lattices using Monte Carlo simulations and compare it to the classical well known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the "explosive" site percolation threshold as pc = 0.695 and we find evidence that "explosive" site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. We do not attempt to determine the order of the explosive transition. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large L. I.INTRODUCTIONRandom percolation (RP) [1][2][3][4][5][6][7] is a well studied model for phase transitions in statistical physics and condensed matter physics. It consists in randomly occupying sites (site percolation) or bonds (bond percolation) on a given lattice with probability p. Neighboring occupied sites merge to form clusters. As p increases, new clusters are formed or coalesce, until at a critical value, p c , a giant component emerges, transversing the system, the wellknown infinite percolating cluster. Classical percolation and all of its variants (continuum percolation, site-bond percolation, bootstrap percolation, invasion percolation, etc [1]), are known to be continuous transitions, exhibiting a divergent correlation length ξ at p c . Thus, there was a stupendous surprise that recently, Achlioptas et al [8], proposed a new method for the occupation of sites which produces "explosive" transitions: when filling sequentially an empty lattice with occupied sites, instead of randomly occupying a site or bond, we choose two candidates and investigate which one of them leads to the smaller clustering. The one that does this is kept as a new occupied site on the lattice while the second one is discarded. This procedure considerably slows down the emergence of the giant component, which is now formed abruptly, thus the term "explosive". Although it is not clear if "explosive" percolation is a first-order transition [9][10][11] or continuous transition with a very small β exponent [12], it is certainly a very sharp transition with a lot of interesting and unusual properties and has stimulated considerable interest. Following these first announcements new work emerged [12,13] that doubted the result of the Achlioptas problem being a first-order phase transition, thus making this problem a hot contentious issue. Recently, applications of the "explosive" percolation transition for "real" systems have been proposed [14,15]. Up to now the systems investigated include networks and lattice bond percolation.In this paper we extend these investigations to include now site explosive percolation and its finite-size hysteresis pro...

In this paper we review the recent advances on explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (Science, 2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in "Achlioptas processes". Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdös networks), but with very unusual finite size scaling. We present a review of the field, indicate open "problems" and propose directions for future research.

In phase transition phenomena, the estimation of the critical point is crucial for the calculation of the various critical exponents and the determination of the universality class they belong to. However, this is not an easy task, since a huge amount of realizations is needed to eliminate the noise in the data. In this paper, we introduce a novel method for the simultaneous estimation of the critical point pc and the critical exponent β/ν, applied for the case of "explosive" bond percolation on 2D square lattices and ER networks. The results show that with only a few hundred of realizations, it is possible to acquire accurate values for these quantities. Guidelines are given at the end for the applicability of the method to other cases as well.

-It has recently been suggested [1] that in a two-level multiplex network, a gradual change in the value of the"interlayer" strength p can provoke an abrupt structural transition. The critical point p * at which this happens is system-dependent. In this article, we show in a similar way as in [2] that this is a consequence of the graph Laplacian formalism used in [1]. We calculate the evolution of p * as a function of system size for ER and RR networks. We investigate the behavior of structural measures and dynamical processes of a two-level system as a function of p, by Monte-Carlo simulations, for simple particle diffusion and for reaction-diffusion systems. We find that as p increases there is a smooth transition from two separate networks to a single one. We cannot find any abrupt change in static or dynamic behavior of the underlying system.Introduction. -In the past several years, single networks have been extensively studied [3-8] both regarding their structure and also regarding different interactions between their nodes. In real world systems, however, there may be more than one type of relationship for the same collection of objects constituting the network. Consider, for example, the communication and power networks in a given geographical region. In this case, a failure in some power station will affect not only the functionality of the power grid, but also the routing system for all computers that uses electrical power to sustain its functionality. Thus, more recently [9], effort has been applied to the so called "interdependent" or "interconnected" networks, meaning a system of two or more networks that are linked together. Several articles have been published investigating the properties of these systems [9,10] as well as the evolution of dynamical processes on them [11,12]. In their simplest form, they consist of two single networks having their nodes connected in a one-to-one configuration with each "interlink" having the same strength p (in the range of 0 < p < ∞). A first question of interest is how does the variation of the parameter p affect the global structural properties of the entire system. To this end, one can make use of the Laplacian matrix. According to graph theory, given a graph of N nodes, the adjacency matrix A is defined as:

Given the increasing occurrence of deviant activities in online platforms, it is of paramount importance to develop methods and tools that allow in-depth analysis and understanding to then develop effective countermeasures. This work proposes a framework towards detecting statistically significant change points in terrorism-related time series, which may indicate the occurrence of events to be paid attention to. These change points may reflect changes in the attitude towards and/or engagement with terrorism-related activities and events, possibly signifying, for instance, an escalation in the radicalization process. In particular, the proposed framework involves: (i) classification of online textual data as terrorism- and hate speech-related, which can be considered as indicators of a potential criminal or terrorist activity; and (ii) change point analysis in the time series generated by these data. The use of change point detection (CPD) algorithms in the produced time series of the aforementioned indicators—either in a univariate or two-dimensional case—can lead to the estimation of statistically significant changes in their structural behavior at certain time locations. To evaluate the proposed framework, we apply it on a publicly available dataset related to jihadist forums. Finally, topic detection on the estimated change points is implemented to further assess its effectiveness.