arXiv subjects

Z. Toroczkai

Publications and source records attributed to Z. Toroczkai.

At least 19 recordsLinked to original sources

arXiv2011

Constructing and sampling directed graphs with given degree sequences

The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existing methods first generates a restricted class of graphs, then uses a Markov Chain Monte-Carlo algorithm based on edge swaps to generate other realizations. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. The other class of methods is based on the Configuration Model that may lead to unacceptably many sample rejections due to self-loops and multiple edges. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. Our method is rejection free, guarantees the independence of the constructed samples, and provides their weight. The weights can then be used to compute statistical averages of network observables as if they were obtained from uniformly distributed sampling, or from any other chosen distribution.

physics.soc-ph↗

arXiv2010

Network Discovery by Generalized Random Walks

We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as function of time. We show that for STP walks site and edge exploration obey the same scaling $\sim n^λ$ as function of time $n$. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the Edge Explorer Model, which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks.

cond-mat.stat-mech↗

arXiv2008

Using relaxational dynamics to reduce network congestion

We study the effects of relaxational dynamics on congestion pressure in scale free networks by analyzing the properties of the corresponding gradient networks (Z. Toroczkai, K. E. Bassler, Nature {\bf 428}, 716 (2004)). Using the Family model (F. Family, J. Phys. A, {\bf 19}, L441 (1986)) from surface-growth physics as single-step load-balancing dynamics, we show that the congestion pressure considerably drops on scale-free networks when compared with the same dynamics on random graphs. This is due to a structural transition of the corresponding gradient network clusters, which self-organize such as to reduce the congestion pressure. This reduction is enhanced when lowering the value of the connectivity exponent $λ$ towards 2.

cond-mat.stat-mech↗

arXiv2007

Proximity Networks and Epidemics

Disease spread in most biological populations requires the proximity of agents. In populations where the individuals have spatial mobility, the contact graph is generated by the "collision dynamics" of the agents, and thus the evolution of epidemics couples directly to the spatial dynamics of the population. We first briefly review the properties and the methodology of an agent-based simulation (EPISIMS) to model disease spread in realistic urban dynamic contact networks. Using the data generated by this simulation, we introduce the notion of dynamic proximity networks which takes into account the relevant time scales for disease spread: contact duration, infectivity period and rate of contact creation. This approach promises to be a good candidate for a unified treatment of epidemic types that are driven by agent collision dynamics. In particular, using a simple model, we show that it can can account for the observed qualitative differences between the degree distributions of contact graphs of diseases with short infectivity period (such as air-transmitted diseases) or long infectivity periods (such as HIV).

physics.soc-ph↗

arXiv2007

Extreme fluctuations in noisy task-completion landscapes on scale-free networks

We study the statistics and scaling of extreme fluctuations in noisy task-completion landscapes, such as those emerging in synchronized distributed-computing networks, or generic causally-constrained queuing networks, with scale-free topology. In these networks the average size of the fluctuations becomes finite (synchronized state) and the extreme fluctuations typically diverge only logarithmically in the large system-size limit ensuring synchronization in a practical sense. Provided that local fluctuations in the network are short-tailed, the statistics of the extremes are governed by the Gumbel distribution. We present large-scale simulation results using the exact algorithmic rules, supported by mean-field arguments based on a coarse-grained description.

cond-mat.dis-nn↗

arXiv2006

Synchronization Landscapes in Small-World-Connected Computer Networks

Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two-dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by ``simulating the simulations", based on the exact algorithmic rules, supported by coarse-grained arguments.

cond-mat.stat-mech↗

arXiv2003

Virtual Time Horizon Control via Communication Network Design

We consider massively parallel discrete event simulations where the communication topology among the processing elements is a complex graph. In the case of regular topologies we review recent results on virtual time horizon management. First we analyze the computational scalability of the conservative massively parallel update scheme for discrete event simulations by using the analogy with a well-known surface growth model, then we show that a simple modification of the regular PE communication topology to a small-world topology will also ensure measurement scalability. This leads to a fully scalable parallel simulation for systems with asynchronous dynamics and short-range interactions. Finally, we present numerical results for the evolution of the virtual time horizon on scale-free Barabasi-Albert networks serving as communication topology among the processing elements.

cond-mat.stat-mech↗

arXiv2003

Suppressing Roughness of Virtual Times in Parallel Discrete-Event Simulations

In a parallel discrete-event simulation (PDES) scheme, tasks are distributed among processing elements (PEs), whose progress is controlled by a synchronization scheme. For lattice systems with short-range interactions, the progress of the conservative PDES scheme is governed by the Kardar-Parisi-Zhang equation from the theory of non-equilibrium surface growth. Although the simulated (virtual) times of the PEs progress at a nonzero rate, their standard deviation (spread) diverges with the number of PEs, hindering efficient data collection. We show that weak random interactions among the PEs can make this spread nondivergent. The PEs then progress at a nonzero, near-uniform rate without requiring global synchronizations.

cond-mat.stat-mech↗

arXiv2001

Going through Rough Times: from Non-Equilibrium Surface Growth to Algorithmic Scalability

Efficient and faithful parallel simulation of large asynchronous systems is a challenging computational problem. It requires using the concept of local simulated times and a synchronization scheme. We study the scalability of massively parallel algorithms for discrete-event simulations which employ conservative synchronization to enforce causality. We do this by looking at the simulated time horizon as a complex evolving system, and we identify its universal characteristics. We find that the time horizon for the conservative parallel discrete-event simulation scheme exhibits Kardar-Parisi-Zhang-like kinetic roughening. This implies that the algorithm is asymptotically scalable in the sense that the average progress rate of the simulation approaches a non-zero constant. It also implies, however, that there are diverging memory requirements associated with such schemes.

cond-mat.stat-mech↗

arXiv2001

On the topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index $r=\max{(i,j)}+δ_{i,j}$ has been shown to be relevant to a number of physical (such at diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees) and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of $n$ leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of non-linear functional equations, we are able to give a double-exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.

cond-mat.stat-mech↗

arXiv2001

Competing Populations in Flows with Chaotic Mixing

We investigate the effects of spatial heterogeneity on the coexistence of competing species in the case when the heterogeneity is dynamically generated by environmental flows with chaotic mixing properties. We show that one of the effects of chaotic advection on the passively advected species (such as phytoplankton, or self-replicating macromolecules) is the possibility of coexistence of more species than that limited by the number of niches they occupy. We derive a novel set of dynamical equations for competing populations.

nlin.CD↗

arXiv2001

Universality Class of Discrete Solid-on-Solid Limited Mobility Nonequilibrium Growth Models for Kinetic Surface Roughening

We investigate, using the noise reduction technique, the asymptotic universality class of the well-studied nonequilibrium limited mobility atomistic solid-on-solid surface growth models introduced by Wolf and Villain (WV) and Das Sarma and Tamborenea (DT) in the context of kinetic surface roughening in ideal molecular beam epitaxy. We find essentially all the earlier conclusions regarding the universality class of DT and WV models to be severely hampered by slow crossover and extremely long lived transient effects. We identify the correct asymptotic universality class(es) which differs from earlier conclusions in several instances.

cond-mat.stat-mech↗

arXiv2001

Autocatalytic Reactions in Systems with Hyperbolic Mixing: Exact Results for the Active Baker Map

We investigate the effects of hyperbolic hydrodynamical mixing on the reaction kinetics of autocatalytic systems. Exact results are derived for the two dimensional open baker map as an underlying mixing dynamics for a two-component autocatalytic system, $A+B \to 2B$. We prove that the hyperboliticity exponentially enhances the productivity of the reaction which is due the fact that the reaction kinetics is catalyzed by the fractal unstable manifold of the chaotic set of the reaction-free dynamics. The results are compared with phenomenological theories of active advection.

nlin.CD↗

arXiv2001

Epitaxial mounding in limited mobility models of surface growth

We study, through large scale stochastic simulations using the noise reduction technique, a large number of simple nonequilibrium limited mobility solid-on-solid growth models. We find that d=2+1 dimensional surface growth in several noise reduced models (most notably the Wolf-Villain and the Larger-Curvature model) exhibits spectacular quasi-regular mound formation with slope selection in their dynamical surface morphology. The mounding instability in these epitaxial growth models does not involve the Ehrlich-Schwoebel step edge diffusion barrier. The mounded morphology in these growth models arises from the interplay between the line tension along step edges in the plane parallel to the average surface and the suppression of noise and island nucleation. The line tension tends to stabilize some of the step orientations that coincide with in-plane high symmetry crystalline directions, and thus the mounds that are formed assume quasi-regular structures. The noise reduction technique developed originally for Eden type models can be used to control the stochastic noise and enhance diffusion along the step edge, which ultimately leads to the formation of quasi-regular mounds during growth. We show that by increasing the diffusion surface length together with supression of nucleation and deposition noise, one can obtain a self-organization of the pyramids in quasi-regular patterns.

cond-mat.stat-mech↗

arXiv2000

Non-equilibrium Surface Growth and Scalability of Parallel Algorithms for Large Asynchronous Systems

The scalability of massively parallel algorithms is a fundamental question in computer science. We study the scalability and the efficiency of a conservative massively parallel algorithm for discrete-event simulations where the discrete events are Poisson arrivals. The parallel algorithm is applicable to a wide range of problems, including dynamic Monte Carlo simulations for large asynchronous systems with short-range interactions. The evolution of the simulated time horizon is analogous to a growing and fluctuating surface, and the efficiency of the algorithm corresponds to the density of local minima of this surface. In one dimension we find that the steady state of the macroscopic landscape is governed by the Edwards-Wilkinson Hamiltonian, which implies that the algorithm is scalable. Preliminary results for higher-dimensional logical topologies are discussed.

cond-mat.stat-mech↗

arXiv2000

Nonuniversal mound formation in nonequilibrium surface growth

We demonstrate, using well-established nonequilibrium limited-mobility solid-on-solid growth models, that mound formation in the dynamical surface growth morphology does not necessarily imply the existence of a surface edge diffusion bias (the Schwoebel barrier). We find mounded morphologies in several nonequilibrium growth models which incorporate no Schwoebel barrier. Our numerical results indicate that mounded morphologies in nonequilibrium surface growth may arise from a number of distinct physical mechanisms, with the Schwoebel instability being one of them.

cond-mat.stat-mech↗

arXiv2000

Extremal-point Densities of Interface Fluctuations

We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of non-equilibrium surface fluctuations. We give a number of exact, analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface growth models. We show that in spite of the non-universal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic to the macroscopic observables of the system. The quantities investigated here present us with tools that give an entirely un-orthodox approach to the dynamics of surface morphologies: a statistical analysis from the short wavelength end of the Fourier decomposition spectrum. In addition to surface growth applications, our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete event simulations, which are extensively used in Monte-Carlo type simulations on parallel architectures.

cond-mat.stat-mech↗

arXiv2000

From Massively Parallel Algorithms and Fluctuating Time Horizons to Non-equilibrium Surface Growth

We study the asymptotic scaling properties of a massively parallel algorithm for discrete-event simulations where the discrete events are Poisson arrivals. The evolution of the simulated time horizon is analogous to a non-equilibrium surface. Monte Carlo simulations and a coarse-grained approximation indicate that the macroscopic landscape in the steady state is governed by the Edwards-Wilkinson Hamiltonian. Since the efficiency of the algorithm corresponds to the density of local minima in the associated surface, our results imply that the algorithm is asymptotically scalable.

cond-mat.stat-mech↗