arXiv subjects

Peter Grassberger

Publications and source records attributed to Peter Grassberger.

At least 37 records · Page 2Linked to original sources

arXiv2017

Universality and Asymptotic Scaling in Drilling Percolation

We present simulations of a 3-d percolation model studied recently by K.J. Schrenk et al. [Phys. Rev. Lett. 116, 055701 (2016)], obtained with a new and more efficient algorithm. They confirm most of their results in spite of larger systems and higher statistics used in the present paper, but we also find indications that the results do not yet represent the true asymptotic behavior. The model is obtained by replacing the isotropic holes in ordinary Bernoulli percolation by randomly placed and oriented cylinders, with the constraint that the cylinders are parallel to one of the three coordinate axes. We also speculate on possible generalizations.

cond-mat.stat-mech↗

arXiv2016

Immunization and targeted destruction of networks using explosive percolation

A new method (`explosive immunization' (EI)) is proposed for immunization and targeted destruction of networks. It combines the explosive percolation (EP) paradigm with the idea of maintaining a fragmented distribution of clusters. The ability of each node to block the spread of an infection (or to prevent the existence of a large cluster of connected nodes) is estimated by a score. The algorithm proceeds by first identifying low score nodes that should not be vaccinated/destroyed, analogously to the links selected in EP if they do not lead to large clusters. As in EP, this is done by selecting the worst node (weakest blocker) from a finite set of randomly chosen `candidates'. Tests on several real-world and model networks suggest that the method is more efficient and faster than any existing immunization strategy. Due to the latter property it can deal with very large networks.

physics.soc-ph↗

arXiv2016

The Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

We present simulations of the 1-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sandpile model is hyperuniform to reach system of sizes $> 10^7$. Most previous simulations were seriously flawed by important finite size corrections. We find that all critical exponents have values consistent with simple rationals: $ν=4/3$ for the correlation length exponent, $D =9/4$ for the fractal dimension of avalanche clusters, and $z=10/7 $ for the dynamical exponent. In addition we relate the hyperuniformity exponent to the correlation length exponent $ν$. Finally we discuss the relationship with the quenched Edwards-Wilkinson (qEW) model, where we find in particular that the local roughness exponent is $α_{\rm loc} = 1$.

cond-mat.stat-mech↗

arXiv2016

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS

The "SOS" in the title does not refer to the international distress signal, but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are those observed by Buldyrev {\it et al.} in interdependent networks, which we re-interpret as multiplex networks with agents that can only survive if they mutually support each other, and whose survival struggle we map onto an SOS type growth model. This mapping not only reveals non-trivial structures in the phase space of the model, but also leads to a new and extremely efficient simulation algorithm. We use this algorithm to study interdependent agents on duplex Erdös-Rényi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain new and surprising results in all these cases, and we correct statements in the literature for ER networks and for 2-d lattices. In particular, we find that $d=4$ is the upper critical dimension, that the percolation transition is continuous for $d\leq 4$ but -- at least for $d\neq 3$ -- not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean field theory, but find evidence that the cascade process is not. For $d=5$ we find a first order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for $d=2$ with intermediate range dependency links we propose a scenario different from that proposed in W. Li {\it et al.}, PRL {\bf 108}, 228702 (2012).

cond-mat.dis-nn↗

arXiv2015

Phase Transitions in Cooperative Coinfections: Simulation Results for Networks and Lattices

We study the spreading of two mutually cooperative diseases on different network topologies, and with two microscopic realizations, both of which are stochastic versions of an SIR type model studied by us recently in mean field approximation. There it had been found that cooperativity can lead to first-order spreading/extinction transitions. However, due to the rapid mixing implied by the mean field assumption, first order transitions required non-zero initial densities of sick individuals. For the stochastic model studied here the results depend strongly on the underlying network. First order transitions are found when there are few short but many long loops: (i) No first order transitions exist on trees and on 2-d lattices with local contacts (ii) They do exist on Erdos-Renyi (ER) networks, on d-dimensional lattices with d >= 4, and on 2-d lattices with sufficiently long-ranged contacts; (iii) On 3-d lattices with local contacts the results depend on the microscopic details of the implementation; (iv) While single infected seeds can always lead to infinite epidemics on regular lattices, on ER networks one sometimes needs finite initial densities of infected nodes; (v) In all cases the first order transitions are actually "hybrid", i.e. they display also power law scaling usually associated with second order transitions. On regular lattices, our model can also be interpreted as the growth of an interface due to cooperative attachment of two species of particles. Critically pinned interfaces in this model seem to be in different universality classes than standard critically pinned interfaces in models with forbidden overhangs. Finally, the detailed results mentioned above hold only when both diseases propagate along the same network of links. If they use different links, results can be rather different in detail, but are similar overall.

physics.soc-ph↗

arXiv2015

Asymmetry of cross correlations between intra-day and overnight volatilities

We point out a stunning time asymmetry in the short time cross correlations between intra-day and overnight volatilities (absolute values of log-returns of stock prices). While overnight volatility is significantly (and positively) correlated with the intra-day volatility during the \textit{following} day (allowing thus non-trivial predictions), it is much less correlated with the intra-day volatility during the \textit{preceding} day. While the effect is not unexpected in view of previous observations, its robustness and extreme simplicity are remarkable.

q-fin.ST↗

arXiv2013

Outbreaks of coinfections: the critical role of cooperativity

Modeling epidemic dynamics plays an important role in studying how diseases spread, predicting their future course, and designing strategies to control them. In this letter, we introduce a model of SIR (susceptible-infected-removed) type which explicitly incorporates the effect of {\it cooperative coinfection}. More precisely, each individual can get infected by two different diseases, and an individual already infected with one disease has an increased probability to get infected by the other. Depending on the amount of this increase, we observe different threshold scenarios. Apart from the standard continuous phase transition for single disease outbreaks, we observe continuous transitions where both diseases must coexist, but also discontinuous transitions are observed, where a finite fraction of the population is already affected by both diseases at the threshold. All our results are obtained in a mean field model using rate equations, but we argue that they should hold also in more general frameworks.

q-bio.PE↗

arXiv2013

On the Continuum Time Limit of Reaction-Diffusion Systems

The parity conserving branching-annihilating random walk (pc-BARW) model is a reaction-diffusion system on a lattice where particles can branch into $m$ offsprings with even $m$ and hop to neighboring sites. If two or more particles land on the same site, they immediately annihilate pairwise. In this way the number of particles is preserved modulo two. It is well known that the pc-BARW with $m=2$ in 1 spatial dimension has no phase transition (it is always subcritical), if the hopping is described by a continuous time random walk. In contrast, the $m=2$ 1-d pc-BARW has a phase transition when formulated in discrete time, but we show that the continuous time limit is non-trivial: When the time step $δt\to 0$, the branching and hopping probabilities at the critical point scale with different powers of $δt$. These powers are different for different microscopic realizations. Although this phenomenon is not observed in some other reaction-diffusion systems like, e.g. the contact process, we argue that it should be generic and not restricted to the 1-d pc-BARW model.

cond-mat.stat-mech↗

arXiv2013

Polymer collapse and crystallization in bond fluctuation models

While the $Θ$-collapse of single long polymers in bad solvents is usually a continuous (tri-critical) phase transition, there are exceptions where it is preempted by a discontinuous crystallization (liquid $\leftrightarrow$ solid) transition. For a version of the bond-fluctuation model (a model where monomers are represented as $2\times 2\times 2$ cubes, and bonds can have lengths between 2 and $\sqrt{10}$) it was recently shown by F. Rampf {\it et al.} that there exist distinct collapse and crystallization transitions for long but {\it finite} chains. But as the chain length goes to infinity, both transition temperatures converge to the same $T^*$, i.e. infinitely long polymers collapse immediately into a solid state. We explain this by the observation that polymers crystallize in the Rampf {\it et al.} model into a non-trivial cubic crystal structure (the `A15' or `Cr$_3$Si' Frank-Kasper structure) which has many degenerate ground states and, as a consequence, Bloch walls. If one controlls the polymer growth such that only one ground state is populated and Bloch walls are completely avoided, the liquid-solid transition is a smooth cross-over without any sharp transition at all.

cond-mat.soft↗

arXiv2013

Two-dimensional SIR epidemics with long range infection

We extend a recent study of susceptible-infected-removed epidemic processes with long range infection (referred to as I in the following) from 1-dimensional lattices to lattices in two dimensions. As in I we use hashing to simulate very large lattices for which finite size effects can be neglected, in spite of the assumed power law $p({\bf x})\sim |{\bf x}|^{-σ-2}$ for the probability that a site can infect another site a distance vector ${\bf x}$ apart. As in I we present detailed results for the critical case, for the supercritical case with $σ= 2$, and for the supercritical case with $0< σ< 2$. For the latter we verify the stretched exponential growth of the infected cluster with time predicted by M. Biskup. For $σ=2$ we find generic power laws with $σ-$dependent exponents in the supercritical phase, but no Kosterlitz-Thouless (KT) like critical point as in 1-d. Instead of diverging exponentially with the distance from the critical point, the correlation length increases with an inverse power, as in an ordinary critical point. Finally we study the dependence of the critical exponents on $σ$ in the regime $0<σ<2$, and compare with field theoretic predictions. In particular we discuss in detail whether the critical behavior for $σ$ slightly less than 2 is in the short range universality class, as conjectured recently by F. Linder {\it et al.}. As in I we also consider a modified version of the model where only some of the contacts are long range, the others being between nearest neighbors. If the number of the latter reaches the percolation threshold, the critical behavior is changed but the supercritical behavior stays qualitatively the same.

cond-mat.stat-mech↗

arXiv2013

SIR epidemics with long range infection in one dimension

We study epidemic processes with immunization on very large 1-dimensional lattices, where at least some of the infections are non-local, with rates decaying as power laws p(x) ~ x^{-sigma-1} for large distances x. When starting with a single infected site, the cluster of infected sites stays always bounded if $σ>1$ (and dies with probability 1, of its size is allowed to fluctuate down to zero), but the process can lead to an infinite epidemic for sigma <1. For sigma <0 the behavior is essentially of mean field type, but for 0 < sigma <= 1 the behavior is non-trivial, both for the critical and for supercritical cases. For critical epidemics we confirm a previous prediction that the critical exponents controlling the correlation time and the correlation length are simply related to each other, and we verify detailed field theoretic predictions for sigma --> 1/3. For sigma = 1 we find generic power laws with continuously varying exponents even in the supercritical case, and confirm in detail the predicted Kosterlitz-Thouless nature of the transition. Finally, the mass N(t) of supercritical clusters seems to grow for 0 < sigma < 1 like a stretched exponential. The latter implies that networks embedded in 1-d space with power-behaved link distributions have infinite intrinsic dimension (based on the graph distance), but are not small world.

cond-mat.stat-mech↗

arXiv2012

Information theoretic aspects of the two-dimensional Ising model

We present numerical results for various information theoretic properties of the square lattice Ising model. First, using a bond propagation algorithm, we find the difference $2H_L(w) - H_{2L}(w)$ between entropies on cylinders of finite lengths $L$ and 2L with open end cap boundaries, in the limit $L\to\infty$. This essentially quantifies how the finite length correction for the entropy scales with the cylinder circumference $w$. Secondly, using the transfer matrix, we obtain precise estimates for the information needed to specify the spin state on a ring encircling an infinite long cylinder. Combining both results we obtain the mutual information between the two halves of a cylinder (the "excess entropy" for the cylinder), where we confirm with higher precision but for smaller systems results recently obtained by Wilms et al. -- and we show that the mutual information between the two halves of the ring diverges at the critical point logarithmically with $w$. Finally we use the second result together with Monte Carlo simulations to show that also the excess entropy of a straight line of $n$ spins in an infinite lattice diverges at criticality logarithmically with $n$. We conjecture that such logarithmic divergence happens generically for any one-dimensional subset of sites at any 2-dimensional second order phase transition. Comparing straight lines on square and triangular lattices with square loops and with lines of thickness 2, we discuss questions of universality.

cond-mat.stat-mech↗

arXiv2012

Sampling properties of directed networks

For many real-world networks only a small "sampled" version of the original network may be investigated; those results are then used to draw conclusions about the actual system. Variants of breadth-first search (BFS) sampling, which are based on epidemic processes, are widely used. Although it is well established that BFS sampling fails, in most cases, to capture the IN-component(s) of directed networks, a description of the effects of BFS sampling on other topological properties are all but absent from the literature. To systematically study the effects of sampling biases on directed networks, we compare BFS sampling to random sampling on complete large-scale directed networks. We present new results and a thorough analysis of the topological properties of seven different complete directed networks (prior to sampling), including three versions of Wikipedia, three different sources of sampled World Wide Web data, and an Internet-based social network. We detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks. Most notably, we find that sampling method and coverage affect both the bow-tie structure, as well as the number and structure of strongly connected components in sampled networks. In addition, at low sampling coverage (i.e. less than 40%), the values of average degree, variance of out-degree, degree auto-correlation, and link reciprocity are overestimated by 30% or more in BFS-sampled networks, and only attain values within 10% of the corresponding values in the complete networks when sampling coverage is in excess of 65%. These results may cause us to rethink what we know about the structure, function, and evolution of real-world directed networks.

physics.soc-ph↗

arXiv2012

Randomness, Information, and Complexity

We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that one should not expect a unique observable of complexity. One of the main problems is to distinguish complex from disordered systems. This and the fact that complexity is closely related to information requires that we also give a review of information measures. We finally concentrate on quantities which measure in some way or other the difficulty of classifying and forecasting sequences of discrete symbols, and study them in simple examples.

physics.data-an↗

arXiv2012

Agglomerative Percolation on Bipartite Networks: A Novel Type of Spontaneous Symmetry Breaking

Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined `virtual' links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a `target cluster' and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdos-Renyi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -- both of which are bipartite -- are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z_2 symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k-partite graphs.

cond-mat.stat-mech↗

arXiv2012

Discontinuous Percolation Transitions in Epidemic Processes, Surface Depinning in Random Media and Hamiltonian Random Graphs

Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and non-equilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first order behaviors in two different classes of models: The first are generalized epidemic processes (GEP) that describe in their spatially embedded version - either on or off a regular lattice - compact or fractal cluster growth in random media at zero temperature. A random graph version of GEP is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian", or formally equilibrium) random graph models and includes the Strauss and the 2-star model, where 'chemical potentials' control the densities of links, triangles or 2-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second order behavior for decreasing link fugacity, and a jump (first order) when it increases.

cond-mat.dis-nn↗

arXiv2012

PageRank and rank-reversal dependence on the damping factor

PageRank (PR) is an algorithm originally developed by Google to evaluate the importance of web pages. Considering how deeply rooted Google's PR algorithm is to gathering relevant information or to the success of modern businesses, the question of rank-stability and choice of the damping factor (a parameter in the algorithm) is clearly important. We investigate PR as a function of the damping factor d on a network obtained from a domain of the World Wide Web, finding that rank-reversal happens frequently over a broad range of PR (and of d). We use three different correlation measures, Pearson, Spearman, and Kendall, to study rank-reversal as d changes, and show that the correlation of PR vectors drops rapidly as d changes from its frequently cited value, $d_0=0.85$. Rank-reversal is also observed by measuring the Spearman and Kendall rank correlation, which evaluate relative ranks rather than absolute PR. Rank-reversal happens not only in directed networks containing rank-sinks but also in a single strongly connected component, which by definition does not contain any sinks. We relate rank-reversals to rank-pockets and bottlenecks in the directed network structure. For the network studied, the relative rank is more stable by our measures around $d=0.65$ than at $d=d_0$.

physics.soc-ph↗

arXiv2011

Random Sequential Renormalization and Agglomerative Percolation in Networks: Application to Erd"os-R'enyi and Scale-free Graphs

We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R'enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more fine-grained analysis of the renormalization group (RG) flow, and unravels new features, that were not discussed in the previous analyses. In particular we find that all networks exhibit a second order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling laws seen there. For critical trees it happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a star-like structure in agreement with the results of Radicchi et. al. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering 'supernodes' as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

cond-mat.stat-mech↗