We present an efficient algorithm for simulating percolation transitions of mutually supporting viable clusters on multiplex networks (also known as "catastrophic cascades on interdependent networks"). This algorithm maps the... more
We present an efficient algorithm for simulating percolation transitions of mutually supporting viable clusters on multiplex networks (also known as "catastrophic cascades on interdependent networks"). This algorithm maps the problem onto a solid-on-solid-type model. 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 surprising results in all these cases, and we correct statements in the literature for ER networks and for two-dimensional lattices. In particular, we find that d=4 is the upper critical dimension and that the percolation transition is continuous for d≤4 but-at least for d≠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 clust...
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Lei Yang and Peter Grassberger John-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany (Dated: February 2, 2008) Recently, it has been claimed (OV Gendelman and AV Savin, Phys. Rev. Lett. 84, 2381... more
Lei Yang and Peter Grassberger John-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany (Dated: February 2, 2008) Recently, it has been claimed (OV Gendelman and AV Savin, Phys. Rev. Lett. 84, 2381 (2000); AVSavin and OVGendelman, ...
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We study a plant population model introduced recently by J. Wallinga [OIKOS {\bf 74}, 377 (1995)]. It is similar to the contact process (`simple epidemic', `directed percolation'), but instead of using an infection or recovery... more
We study a plant population model introduced recently by J. Wallinga [OIKOS {\bf 74}, 377 (1995)]. It is similar to the contact process (`simple epidemic', `directed percolation'), but instead of using an infection or recovery rate as control parameter, the population size is controlled directly and globally by removing excess plants. We show that the model is very closely related
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Networks to infer causal structure from spatiotemporal data are constructed making minimal a priori assumptions about the underlying dynamics. The elementary concept of recurrence for a point process in time is generalized to recurrent... more
Networks to infer causal structure from spatiotemporal data are constructed making minimal a priori assumptions about the underlying dynamics. The elementary concept of recurrence for a point process in time is generalized to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than
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Alignment of biological sequences such as DNA, RNA or proteins is one of the most widely used tools in computational bioscience. While the accomplishments of sequence alignment algorithms are undeniable the fact remains that these... more
Alignment of biological sequences such as DNA, RNA or proteins is one of the most widely used tools in computational bioscience. While the accomplishments of sequence alignment algorithms are undeniable the fact remains that these algorithms are based upon heuristic scoring schemes. Therefore, these algorithms do not provide model independent and objective measures for how similar two (or more) sequences
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We present a method for hierarchical clustering of data called {\it mutual information clustering} (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects... more
We present a method for hierarchical clustering of data called {\it mutual information clustering} (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects $X, Y,$ and $Z$ is equal to the sum of the MI between $X$ and $Y$, plus the MI between $Z$ and the combined object
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Motivation: Clustering is a frequently used con- cept in variety of bioinformatical applications. We present a new method for hierarchical clustering of data called mutual information clustering (MIC) al- gorithm. It uses mutual... more
Motivation: Clustering is a frequently used con- cept in variety of bioinformatical applications. We present a new method for hierarchical clustering of data called mutual information clustering (MIC) al- gorithm. It uses mutual information (MI) as a sim- ilarity measure and exploits its grouping property: The MI between three objects X,Y, and Z is equal to the sum of the
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Obtaining the most independent components from a mixture (under a chosen model) is only the first part of an ICA analysis. After that, it is necessary to measure the actual dependency between the components and the reliability of the... more
Obtaining the most independent components from a mixture (under a chosen model) is only the first part of an ICA analysis. After that, it is necessary to measure the actual dependency between the components and the reliability of the decomposition. We have to identify one- and multidimensional components (i.e., clusters of mutually dependent components) or channels which are too close
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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... more
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: Th...
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In experiments involving deterministic chaotic signals, contamination by random noise is unavoidable. A practical method that disentangles the deterministic chaos from the random part is discussed. The method yields a characterization of... more
In experiments involving deterministic chaotic signals, contamination by random noise is unavoidable. A practical method that disentangles the deterministic chaos from the random part is discussed. The method yields a characterization of the strange attractors together with an estimate of the size of random noise.
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We propose a simple method to measure synchronization and time-delay patterns between signals. It is based on the relative timings of events in the time series, defined, e.g., as local maxima. The degree of synchronization is obtained... more
We propose a simple method to measure synchronization and time-delay patterns between signals. It is based on the relative timings of events in the time series, defined, e.g., as local maxima. The degree of synchronization is obtained from the number of quasisimultaneous appearances of events, and the delay is calculated from the precedence of events in one signal with respect to the other. Moreover, we can easily visualize the time evolution of the delay and synchronization level with an excellent resolution. We apply the algorithm to short rat electroencephalogram (EEG) signals, some of them containing spikes. We also apply it to an intracranial human EEG recording containing an epileptic seizure, and we propose that the method might be useful for the detection of epileptic foci. It can be easily extended to other types of data and it is very simple and fast, thus being suitable for on-line implementations.
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Recently, "renormalized entropy" was proposed as a novel measure of relative entropy [P. Saparin et al., Chaos, Solitons and Fractals 4, 1907 (1994)] and applied to several physiological time sequences, including... more
Recently, "renormalized entropy" was proposed as a novel measure of relative entropy [P. Saparin et al., Chaos, Solitons and Fractals 4, 1907 (1994)] and applied to several physiological time sequences, including electroencephalograms (EEGs) of patients with epilepsy. We show here that this measure is just a modified Kullback-Leibler (KL) relative entropy, and it gives similar numerical results to the standard KL entropy. The latter better distinguishes frequency contents of, e.g., seizure and background EEGs than renormalized entropy. We thus propose that renormalized entropy might not be as useful as claimed by its proponents. In passing, we also make some critical remarks about the implementation of these methods.
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We test recent claims that causal (driver-response) relationships can be deduced from interdependencies between simultaneously measured time series. We apply two recently proposed interdependence measures that should give results similar... more
We test recent claims that causal (driver-response) relationships can be deduced from interdependencies between simultaneously measured time series. We apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors. The systems that we study are asymmetrically coupled simple models (Lorenz, Roessler, and Hénon models), the couplings being such that they lead to generalized synchronization. If the data were perfect (noise-free, infinitely long), we should be able to detect, at least in some cases, which of the coupled systems is the driver and which the response. This might no longer be true if the time series has finite length. Instead, estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them, and causal relationships are more difficult to detect. We also show that slightly different variants of the interdepende...
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... It was found that the scaling function 0(x) is monotonic and that D > 2 for all values of a ... sizes are large enough to refute also another claim of [21], namely that P(s) shows ordinary finite-size scal-ing. ... While the... more
... It was found that the scaling function 0(x) is monotonic and that D > 2 for all values of a ... sizes are large enough to refute also another claim of [21], namely that P(s) shows ordinary finite-size scal-ing. ... While the distribution of the stress difference between neighbors (shown in Fig. ...
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ABSTRACT Four-pomeron cutting rules are studied in cut reggeon field theory (CRFT). Without any microscopic model, CRFT allows for three different 4-pomeron couplings. Demanding that CRFT is interpretable as a Markov process, only one of... more
ABSTRACT Four-pomeron cutting rules are studied in cut reggeon field theory (CRFT). Without any microscopic model, CRFT allows for three different 4-pomeron couplings. Demanding that CRFT is interpretable as a Markov process, only one of these couplings remains. The cutting rules for the 4-pomeron vertex thus become unique, disagreeing with those found in weak coupling phi3 theory.
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ABSTRACT Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature Tc of the undiluted Ising model, but above the critical temperature of the diluted... more
ABSTRACT Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature Tc of the undiluted Ising model, but above the critical temperature of the diluted system. Using damage spreading (or rather damage "healing"), we are able to measure down to much lower final magnetizations (10-9) and to much larger times than previous authors. Nevertheless, we do not yet find the predicted asymptotic behavior in the Griffiths phase T < Tc. But we can at least exclude a stretched exponential decay as found in previous papers, for T < Tc. Finally, we discuss the case T = Tc where a stretched exponential decay can be proven to hold, at least for p < pc. We indeed do see a stretched exponential for p = pc (and T = Tc), but we show that it cannot describe the asymptotic behavior either.
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Unitarity in the s-channel is invoked to derive an upper bound for the inelastic diffractive cross-section as a function of impact parameter. The application of this bound to high-energy proton-proton scattering strongly suggests that... more
Unitarity in the s-channel is invoked to derive an upper bound for the inelastic diffractive cross-section as a function of impact parameter. The application of this bound to high-energy proton-proton scattering strongly suggests that inelastic diffraction should be peripheral in the impact parameter space.
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We show that the persistance length of two-dimensional self-avoiding random walks is infinite, persistency being a critical phenomenon: For a walk of N steps starting in the positive x-direction, the mean end-point position scales like... more
We show that the persistance length of two-dimensional self-avoiding random walks is infinite, persistency being a critical phenomenon: For a walk of N steps starting in the positive x-direction, the mean end-point position scales like <xN> ~ N0.063 +/- 0.01.
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We study four Achlioptas-type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite... more
We study four Achlioptas-type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entirely holomorphic. The distributions of the order parameter, i.e., the relative size s(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(max)/N)∼(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.
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ABSTRACT A Reply to the Comment by Y. Berezin et al.
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A Comment on the Letter by Albert C.-C. Yang et al., Phys. Rev. Lett.PRLTAO0031-9007 90, 108103 (2003)10.1103/PhysRevLett.90.108103. The authors of the Letter offer a Reply.
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ABSTRACT A Comment on the Letter by P. M. Binder and V. Privman, Phys. Rev. Lett. 68, 3830 (1992).
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We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model related to ER graphs. In RSR the network is locally... more
We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model 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 [Song et al., Nature (London) 433, 392 (2005)], 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 behavior seen there. For critical trees it happens as N/N(0) → 0 in the limit of large systems (where N(0) is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N(0) in sparse ER graphs and in the annealed model, while it happens for N/N(0) → 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 starlike structure in agreement with the results of Radicchi et al. [Phys. Rev. Lett. 101, 148701 (2008)]. 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.
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Lattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of two-dimensional site animals on square and triangular lattices in nontrivial geometries. The... more
Lattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of two-dimensional site animals on square and triangular lattices in nontrivial geometries. The simulations are done with the pruned-enriched Rosenbluth method (PERM) algorithm, which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent theta (Z(N) approximately micro(N)N(-theta)). In particular, we studied animals grafted to the tips of wedges with a wide range of angles alpha, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have k sheets and no boundary, generalizing in this way cones to angles alpha>360 degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, theta approximately 1/alpha , only for small angles (alpha < 2pi) , while theta approximately = const-alpha/2pi for alpha < 2pi. These scalings hold both for wedges and cones. A heuristic (nonconformal) argument for the behavior at large alpha is given, and comparison is made with critical percolation.
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... Using this to extrapolate ton o, we obtain D = 1.28 0.01 in disagreement with previous boxcounting estimates, but in agreement with a recent in direct evaluation. One of the most basic properties of a strange attrac tor is its fractal... more
... Using this to extrapolate ton o, we obtain D = 1.28 0.01 in disagreement with previous boxcounting estimates, but in agreement with a recent in direct evaluation. One of the most basic properties of a strange attrac tor is its fractal dimension (or capacity [1]) D, de fined via the ...
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It is shown that the fluctuations in the divergence of near-by trajectories on (strictly deterministic) strange attractors can be modelled by stochastic concepts. In particular, we propose Kramers-Moyal type equations for correlation... more
It is shown that the fluctuations in the divergence of near-by trajectories on (strictly deterministic) strange attractors can be modelled by stochastic concepts. In particular, we propose Kramers-Moyal type equations for correlation functions between points on the attractor. The drift terms are the Lyapunov exponents, the diffusion terms depend on the above fluctuations. From this, we obtain bounds on generalized
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ABSTRACT A one-dimensional Lorentz-type model is studied where a point particle is reflected with some given probability p off randomly located fixed scatterers. The diffusion constant is calculated exactly, and the velocity... more
ABSTRACT A one-dimensional Lorentz-type model is studied where a point particle is reflected with some given probability p off randomly located fixed scatterers. The diffusion constant is calculated exactly, and the velocity autocorrelation is shown to decay like t-3/2, for 0
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We consider standard percolation processes such as epidemic processes with or without immunization. We show that their dynamics can be formulated so that they mimic self-organized critical phenomena: the wetting probability p needs not to... more
We consider standard percolation processes such as epidemic processes with or without immunization. We show that their dynamics can be formulated so that they mimic self-organized critical phenomena: the wetting probability p needs not to be fine tuned to its critical value pc in order to arrive at criticality, but it rather emerges as a singularity in some time-dependent distribution. On the one hand, this casts doubts on the significance of self-organized as opposed to ordinary criticality. On the other hand, it suggests very efficient ...