arXiv subjects

Peter Grassberger

Publications and source records attributed to Peter Grassberger.

At least 55 records · Page 3Linked to original sources

arXiv2011

Agglomerative Percolation in Two Dimensions

We study a process termed "agglomerative percolation" (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation.

cond-mat.stat-mech↗

arXiv2011

Percolation Theory on Interdependent Networks Based on Epidemic Spreading

We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025 (2010), and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory -- which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.

physics.data-an↗

arXiv2011

Are Percolation Transitions always Sharpened by Making Networks Interdependent?

We study a model for coupled networks introduced recently by Buldyrev et al., Nature 464, 1025 (2010), where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erdos-Renyi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimension, the order parameter exponent $β$ is larger than in ordinary percolation, showing that the transition is less sharp, i.e. further from discontinuity, than for isolated networks. Possible consequences for spatially embedded networks are discussed.

physics.data-an↗

arXiv2011

Exact solutions for mass-dependent irreversible aggregations

We consider the mass-dependent aggregation process (k+1)X -> X, given a fixed number of unit mass particles in the initial state. One cluster is chosen proportional to its mass and is merged into one either with k-neighbors in one dimension, or -- in the well-mixed case -- with k other clusters picked randomly. We find the same combinatorial exact solutions for the probability to find any given configuration of particles on a ring or line, and in the well-mixed case. The mass distribution of a single cluster exhibits scaling laws and the finite size scaling form is given. The relation to the classical sum kernel of irreversible aggregation is discussed.

physics.data-an↗

arXiv2011

A review of Monte Carlo simulations of polymers with PERM

In this review, we describe applications of the pruned-enriched Rosenbluth method (PERM), a sequential Monte Carlo algorithm with resampling, to various problems in polymer physics. PERM produces samples according to any given prescribed weight distribution, by growing configurations step by step with controlled bias, and correcting "bad" configurations by "population control". The latter is implemented, in contrast to other population based algorithms like e.g. genetic algorithms, by depth-first recursion which avoids storing all members of the population at the same time in computer memory. The problems we discuss all concern single polymers (with one exception), but under various conditions: Homopolymers in good solvents and at the $Θ$ point, semi-stiff polymers, polymers in confining geometries, stretched polymers undergoing a forced globule-linear transition, star polymers, bottle brushes, lattice animals as a model for randomly branched polymers, DNA melting, and finally -- as the only system at low temperatures, lattice heteropolymers as simple models for protein folding. PERM is for some of these problems the method of choice, but it can also fail. We discuss how to recognize when a result is reliable, and we discuss also some types of bias that can be crucial in guiding the growth into the right directions.

cond-mat.soft↗

arXiv2011

Random Sequential Renormalization of Networks I: Application to Critical Trees

We introduce the concept of Random Sequential Renormalization (RSR) for arbitrary networks. RSR is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network. It is analogous to the (quasi-)parallel renormalization schemes introduced by C. Song {\it et al.} (Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it et al.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easier to implement. In this first paper we apply RSR to critical trees and derive analytical results consistent with numerical simulations. Critical trees exhibit three regimes in their evolution under RSR: (i) An initial regime $N_0^ν\lesssim N<N_0$, where $N$ is the number of nodes at some step in the renormalization and $N_0$ is the initial size. RSR in this regime is described by a mean field theory and fluctuations from one realization to another are small. The exponent $ν=1/2$ is derived using random walk arguments. The degree distribution becomes broader under successive renormalization -- reaching a power law, $p_k\sim 1/k^γ$ with $γ=2$ and a variance that diverges as $N_0^{1/2}$ at the end of this regime. Both of these results are derived based on a scaling theory. (ii) An intermediate regime for $N_0^{1/4}\lesssim N \lesssim N_0^{1/2}$, in which hubs develop, and fluctuations between different realizations of the RSR are large. Crossover functions exhibiting finite size scaling, in the critical region $N\sim N_0^{1/2} \to \infty$, connect the behaviors in the first two regimes. (iii) The last regime, for $1 \ll N\lesssim N_0^{1/4}$, is characterized by the appearance of star configurations with a central hub surrounded by many leaves. The distribution of sizes where stars first form is found numerically to be a power law up to a cutoff that scales as $N_0^{ν_{star}}$ with $ν_{star}\approx 1/4$.

cond-mat.stat-mech↗

arXiv2011

Explosive Percolation is Continuous, but with Unusual Finite Size Behavior

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size $s_{\rm max}/N$ of the largest cluster, are double-humped. But -- in contrast to first order phase transitions -- the distance between the two peaks decreases with system size $N$ as $N^{-η}$ with $η> 0$. We find different positive values of $β$ (defined via $< s_{\rm max}/N > \sim (p-p_c)^β$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $Θ$ (defined such that observables are homogeneous functions of $(p-p_c)N^Θ$) is close to -- or even equal to -- 1/2 for all models.

cond-mat.dis-nn↗

arXiv2011

Irreversible Aggregation and Network Renormalization

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process $(k+1)X\to X$, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar.

physics.data-an↗

arXiv2011

Clustering Drives Assortativity and Community Structure in Ensembles of Networks

Clustering, assortativity, and communities are key features of complex networks. We probe dependencies between these attributes and find that ensembles with strong clustering display both high assortativity by degree and prominent community structure, while ensembles with high assortativity are much less biased towards clustering or community structure. Further, clustered networks can amplify small homophilic bias for trait assortativity. This marked asymmetry suggests that transitivity, rather than homophily, drives the standard nonsocial/social network dichotomy.

physics.soc-ph↗

arXiv2010

Edge direction and the structure of networks

Directed networks are ubiquitous and are necessary to represent complex systems with asymmetric interactions---from food webs to the World Wide Web. Despite the importance of edge direction for detecting local and community structure, it has been disregarded in studying a basic type of global diversity in networks: the tendency of nodes with similar numbers of edges to connect. This tendency, called assortativity, affects crucial structural and dynamic properties of real-world networks, such as error tolerance or epidemic spreading. Here we demonstrate that edge direction has profound effects on assortativity. We define a set of four directed assortativity measures and assign statistical significance by comparison to randomized networks. We apply these measures to three network classes---online/social networks, food webs, and word-adjacency networks. Our measures (i) reveal patterns common to each class, (ii) separate networks that have been previously classified together, and (iii) expose limitations of several existing theoretical models. We reject the standard classification of directed networks as purely assortative or disassortative. Many display a class-specific mixture, likely reflecting functional or historical constraints, contingencies, and forces guiding the system's evolution.

physics.soc-ph↗

arXiv2010

The Interacting Branching Process as a Simple Model of Innovation

We describe innovation in terms of a generalized branching process. Each new invention pairs with any existing one to produce a number of offspring, which is Poisson distributed with mean p. Existing inventions die with probability p/τat each generation. In contrast to mean field results, no phase transition occurs; the chance for survival is finite for all p > 0. For τ= \infty, surviving processes exhibit a bottleneck before exploding super-exponentially - a growth consistent with a law of accelerating returns. This behavior persists for finite τ. We analyze, in detail, the asymptotic behavior as p \to 0.

physics.soc-ph↗

arXiv2010

Sequence alignment, mutual information, and dissimilarity measures for constructing phylogenies

Existing sequence alignment algorithms use heuristic scoring schemes which cannot be used as objective distance metrics. Therefore one relies on measures like the p- or log-det distances, or makes explicit, and often simplistic, assumptions about sequence evolution. Information theory provides an alternative, in the form of mutual information (MI) which is, in principle, an objective and model independent similarity measure. MI can be estimated by concatenating and zipping sequences, yielding thereby the "normalized compression distance". So far this has produced promising results, but with uncontrolled errors. We describe a simple approach to get robust estimates of MI from global pairwise alignments. Using standard alignment algorithms, this gives for animal mitochondrial DNA estimates that are strikingly close to estimates obtained from the alignment free methods mentioned above. Our main result uses algorithmic (Kolmogorov) information theory, but we show that similar results can also be obtained from Shannon theory. Due to the fact that it is not additive, normalized compression distance is not an optimal metric for phylogenetics, but we propose a simple modification that overcomes the issue of additivity. We test several versions of our MI based distance measures on a large number of randomly chosen quartets and demonstrate that they all perform better than traditional measures like the Kimura or log-det (resp. paralinear) distances. Even a simplified version based on single letter Shannon entropies, which can be easily incorporated in existing software packages, gave superior results throughout the entire animal kingdom. But we see the main virtue of our approach in a more general way. For example, it can also help to judge the relative merits of different alignment algorithms, by estimating the significance of specific alignments.

q-bio.GN↗

arXiv2010

Lower Bounds on Mutual Information

We correct claims about lower bounds on mutual information (MI) between real-valued random variables made in A. Kraskov {\it et al.}, Phys. Rev. E {\bf 69}, 066138 (2004). We show that non-trivial lower bounds on MI in terms of linear correlations depend on the marginal (single variable) distributions. This is so in spite of the invariance of MI under reparametrizations, because linear correlations are not invariant under them. The simplest bounds are obtained for Gaussians, but the most interesting ones for practical purposes are obtained for uniform marginal distributions. The latter can be enforced in general by using the ranks of the individual variables instead of their actual values, in which case one obtains bounds on MI in terms of Spearman correlation coefficients. We show with gene expression data that these bounds are in general non-trivial, and the degree of their (non-)saturation yields valuable insight.

physics.data-an↗

arXiv2009

Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.

cond-mat.stat-mech↗

arXiv2009

Local persistence in directed percolation

We reconsider the problem of local persistence in directed site percolation. We present improved estimates of the persistence exponent in all dimensions from 1+1 to 7+1, obtained by new algorithms and by improved implementations of existing ones. We verify the strong corrections to scaling for 2+1 and 3+1 dimensions found in previous analyses, but we show that scaling is much better satisfied for very large and very small dimensions. For d > 4 (d is the spatial dimension), the persistence exponent depends non-trivially on d, in qualitative agreement with the non-universal values calculated recently by Fuchs {\it et al.} (J. Stat. Mech.: Theor. Exp. P04015 (2008)). These results are mainly based on efficient simulations of clusters evolving under the time reversed dynamics with a permanently active site and a particular survival condition discussed in Fuchs {\it et al.}. These simulations suggest also a new critical exponent $ζ$ which describes the growth of these clusters conditioned on survival, and which turns out to be the same as the exponent, η+δin standard notation, of surviving clusters under the standard DP evolution.

cond-mat.stat-mech↗

arXiv2009

Scaling of loop-erased walks in 2 to 4 dimensions

We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2,3, and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in $d=4$, simulations in lower dimensions were done for completeness and in order to test the algorithm. In $d=2$, we verify with high precision the prediction $D=5/4$, where the number of steps $n$ after erasure scales with the number $N$ of steps before erasure as $n\sim N^{D/2}$. In $d=3$ we again find a power law, but with an exponent different from the one found in the most precise previous simulations: $D = 1.6236\pm 0.0004$. Finally, we see clear deviations from the naive scaling $n\sim N$ in $d=4$. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.

cond-mat.stat-mech↗

arXiv2009

Logarithmic corrections in (4+1)-dimensional directed percolation

We simulate directed site percolation on two lattices with 4 spatial and 1 time-like dimensions (simple and body-centered hypercubic in space) with the standard single cluster spreading scheme. For efficiency, the code uses the same ingredients (hashing, histogram re-weighing, and improved estimators) as described in Phys. Rev. {\bf E 67}, 036101 (2003). Apart from providing the most precise estimates for $p_c$ on these lattices, we provide a detailed comparison with the logarithmic corrections calculated by Janssen and Stenull [Phys. Rev. {\bf E 69}, 016125 (2004)]. Fits with the leading logarithmic terms alone would give estimates of the powers of these logarithms which are too big by typically 50%. When the next-to-leading terms are included, each of the measured quantities (the average number of sites wetted at time $t$, their average distance from the seed, and the probability of cluster survival) can be fitted nearly perfectly. But these fits would not be mutually consistent. With a consistent set of fit parameters, one obtains still much improvement over the leading log - approximation. In particular we show that there is one combination of these three observables which seems completely free of logarithmic terms.

cond-mat.stat-mech↗

arXiv2008

Sequence alignment and mutual information

Background: Alignment of biological sequences such as DNA, RNA or proteins is one of the most widely used tools in computational bioscience. All existing alignment algorithms rely on heuristic scoring schemes based on biological expertise. Therefore, these algorithms do not provide model independent and objective measures for how similar two (or more) sequences actually are. Although information theory provides such a similarity measure -- the mutual information (MI) -- previous attempts to connect sequence alignment and information theory have not produced realistic estimates for the MI from a given alignment. Results: Here we describe a simple and flexible approach to get robust estimates of MI from {\it global} alignments. For mammalian mitochondrial DNA, our approach gives pairwise MI estimates for commonly used global alignment algorithms that are strikingly close to estimates obtained by an entirely unrelated approach -- concatenating and zipping the sequences. Conclusions: This remarkable consistency may help establish MI as a reliable tool for evaluating the quality of global alignments, judging the relative merits of different alignment algorithms, and estimating the significance of specific alignments. We expect that our approach can be extended to establish further connections between information theory and sequence alignment, including applications to local and multiple alignment procedures.

q-bio.GN↗