Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

General theory for extended-range percolation on simple and multiplex networks

Extended-range percolation is a robust percolation process that has relevance for quantum communication problems. In extended-range percolation nodes can be trusted or untrusted. Untrusted facilitator nodes are untrusted nodes that can still allow communication between trusted nodes if they lie on a path of distance at most R between two trusted nodes. In extended-range percolation the extended-range giant component (ERGC) includes trusted nodes connected by paths of trusted and untrusted facilitator nodes. Here, based on a message-passing algorithm, we develop a general theory of extended-range percolation, valid for arbitrary values of R as long as the networks are locally treelike. This general framework allows us to investigate the properties of extended-range percolation on interdependent multiplex networks. While the extended-range nature makes multiplex networks more robust, interdependency makes them more fragile. From the interplay between these two effects a rich phase diagram emerges including discontinuous phase transitions and reentrant phases. The theoretical predictions are in excellent agreement with extensive Monte Carlo simulations. The proposed exactly solvable model constitutes a fundamental reference for the study of models defined through properties of extended-range paths.

Extended-range percolation in complex networks

Classical percolation theory underlies many processes of information transfer along the links of a network. In these standard situations, the requirement for two nodes to be able to communicate is the presence of at least one uninterrupted path of nodes between them. In a variety of more recent data transmission protocols, such as the communication of noisy data via error-correcting repeaters, both in classical and quantum networks, the requirement of an uninterrupted path is too strict: two nodes may be able to communicate even if all paths between them have interruptions or gaps consisting of nodes that may corrupt the message. In such a case a different approach is needed. We develop the theoretical framework for extended-range percolation in networks, describing the fundamental connectivity properties relevant to such models of information transfer. We obtain exact results, for any range R, for infinite random uncorrelated networks and we provide a message-passing formulation that works well in sparse real-world networks. The interplay of the extended range and heterogeneity leads to novel critical behavior in scale-free networks.

Random site percolation on honeycomb lattices with complex neighborhoods

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds _p follow the dependency _p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of _p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i _{z r}-where _z and _r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence _p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.

Cumulative merging percolation: A long-range percolation process in networks

Percolation on networks is a common framework to model a wide range of processes, from cascading failures to epidemic spreading. Standard percolation assumes short-range interactions, implying that nodes can merge into clusters only if they are nearest neighbors. Cumulative merging percolation (CMP) is a percolation process that assumes long-range interactions such that nodes can merge into clusters even if they are topologically distant. Hence, in CMP clusters do not coincide with the topologically connected components of the network. Previous work has shown that a specific formulation of CMP features peculiar mechanisms for the formation of the giant cluster and allows one to model different network dynamics such as recurrent epidemic processes. Here we develop a more general formulation of CMP in terms of the functional form of the cluster interaction range, showing an even richer phase transition scenario with competition of different mechanisms resulting in crossover phenomena. Our analytic predictions are confirmed by numerical simulations.

Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions

Extended-range percolation on various regular lattices, including all 11 Archimedean lattices in two dimensions and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds p_{c} for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/p_{c} and z versus -1/ln(1-p_{c}) using the data of d'Iribarne et al. [J. Phys. A 32, 2611 (1999)JPHAC50305-447010.1088/0305-4470/32/14/002] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zp_{c}∼4η_{c}=4.51235, where η_{c} is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zp_{c}∼8η_{c}=2.7351, where η_{c} is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior p_{c}=1/(z-1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zp_{c}-1∼a_{1}z^{-x}, with x=2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21, 56 (2016)1083-648910.1214/16-EJP6] and by Xu et al. [Phys. Rev. E 103, 022127 (2021)2470-004510.1103/PhysRevE.103.022127]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zp_{c} has universal properties, depending only on the dimension of the system and whether site or bond percolation but not on the type of lattice.

Percolation thresholds on a triangular lattice for neighborhoods containing sites up to the fifth coordination zone

We determine thresholds p_{c} for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_{c} on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ=∑_{i}z_{i}r_{i}^{2}/i (depending on the coordination zone number i, the neighborhood coordination number z, and the square distance r^{2} to sites in ith coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_{c} to ξ. The thresholds roughly follow a power law p_{c}∝ξ^{-γ} with γ≈0.710(19).

Critical polynomials in the nonplanar and continuum percolation models

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

By means of extensive Monte Carlo simulation, we study extended-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors up to the eighth nearest neighbors for the square lattice and the ninth nearest neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods of connected sites can be mapped to problems of lattice percolation of extended objects of a given shape, such as disks and spheres, and the thresholds can be related to the continuum thresholds η_{c} for objects of those shapes. This mapping implies zp_{c}∼4η_{c}=4.51235 in two dimensions and zp_{c}∼8η_{c}=2.7351 in three dimensions for large z for circular and spherical neighborhoods, respectively, where z is the coordination number. Fitting our data for compact neighborhoods to the form p_{c}=c/(z+b) we find good agreement with this prediction, c=2^{d}η_{c}, with the constant b representing a finite-z correction term. We also examined results from other studies using this fitting formula. A good fit of the large but finite-z behavior can also be made using the formula p_{c}=1-exp(-2^{d}η_{c}/z), a generalization of a formula of Koza, Kondrat, and Suszcayński [J. Stat. Mech.: Theor. Exp. (2014) P110051742-546810.1088/1742-5468/2014/11/P11005]. We also study power-law fits which are applicable for the range of values of z considered here.

Site percolation thresholds on triangular lattice with complex neighborhoods

We determine thresholds p_c for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are p_c(4NN)=0.192410(43), p_c(3NN+2NN)=0.232008(38), p_c(5NN+4NN)=0.140286(5), p_c(3NN+2NN+NN)=0.215484(19), p_c(5NN+4NN+NN)=0.131792(58), p_c(5NN+4NN+3NN+2NN)=0.117579(41), and p_c(5NN+4NN+3NN+2NN+NN)=0.115847(21). The method is tested on the standard case of site percolation on the triangular lattice, where p_c(NN)=p_c(2NN)=p_c(3NN)=p_c(5NN)=12 is recovered with five digits accuracy p_c(NN)=0.500029(46) by averaging over one thousand lattice realizations only.

Noise induced unanimity and disorder in opinion formation

We propose an opinion dynamics model based on Latané’s social impact theory. Actors in this model are heterogeneous and, in addition to opinions, are characterised by their varying levels of persuasion and support. The model is tested for two and three initial opinions randomly distributed among actors. We examine how the noise (randomness of behaviour) and the flow of information among actors affect the formation and spread of opinions. Our main research involves the process of opinion formation and finding phases of the system in terms of parameters describing noise and flow of the information for two and three opinions available in the system. The results show that opinion formation and spread are influenced by both (i) flow of information among actors (effective range of interactions among actors) and (ii) noise (randomness in adopting opinions). The noise not only leads to opinions disorder but also it promotes consensus under certain conditions. In disordered phase and when the exchange of information is spatially effectively limited, various faces of disorder are observed, including system states, where the signatures of self-organised criticality manifest themselves as scale-free probability distribution function of cluster sizes. Then increase of noise level leads system to disordered random state. The critical noise level above which histograms of opinion clusters’ sizes loose their scale-free character increases with increase of the easy of information flow.

Bond percolation on simple cubic lattices with extended neighborhoods

We study bond percolation on the simple cubic lattice with various combinations of first, second, third, and fourth nearest neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result p_{c}=1/(z-1). Fitting our data and data for additional nearest neighbors, we find p_{c}(z-1)=1+1.224z^{-1/2}.

Random sequential adsorption of lattice animals on a three-dimensional cubic lattice

The properties of the random sequential adsorption of objects of various shapes on simple three-dimensional (3D) cubic lattice are studied numerically by means of Monte Carlo simulations. Depositing objects are "lattice animals," made of a certain number of nearest-neighbor sites on a lattice. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). We analyzed all lattice animals of size n=1, 2, 3, 4, and 5. A significant number of objects of size n⩾6 were also used to confirm our findings. Approach of the coverage θ(t) to the jamming limit θ_{J} is found to be exponential, θ_{J}-θ(t)∼exp(-t/σ), for all lattice animals. It was shown that the relaxation time σ increases with the number of different orientations m that lattice animals can take when placed on a cubic lattice. Orientations of the lattice animal deposited in two randomly chosen places on the lattice are different if one of them cannot be translated into the other. Our simulations performed for large collections of 3D objects confirmed that σ≅m∈{1,3,4,6,8,12,24}. The presented results suggest that there is no correlation between the number of possible orientations m of the object and the corresponding values of the jamming density θ_{J}. It was found that for sufficiently large objects, changing of the shape has considerably more influence on the jamming density than increasing of the object size.

Dimer covering and percolation frustration

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.