arXiv subjects

Géza Ódor

Publications and source records attributed to Géza Ódor.

At least 37 records · Page 2Linked to original sources

arXiv2021

Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions

In $d > 2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power-laws, thus it is called mixed-order or hybrid type. It has recently been shown that the introduction of quenched disorder rounds the discontinuity and second order phase transition and Griffiths phases appear. Here we provide numerical evidence, that even in case of high graph dimensional hierarchical modular networks the Griffiths phase of the $K=2$ threshold model is present below the hybrid phase transition. This is due to the fragmentation of the activity propagation by modules, which are connected via single links. This provides a widespread mechanism in case of threshold type of heterogeneous systems, modeling the brain or epidemics for the occurrence of dynamical criticality in extended Griffiths phase parameter spaces. We investigate this in synthetic modular networks with and without inhibitory links as well as in the presence of refractory states.

cond-mat.dis-nn↗

arXiv2020

Power-law distributions of dynamic cascade failures in power-grid models

Power-law distributed cascade failures are well known in power-grid systems. Understanding this phenomena has been done by various DC threshold models, self-tuned at their critical point. Here we attempt to describe it using an AC threshold model, with a second-order Kuramoto type equation of motion of the power-flow. We have focused on the exploration of network heterogeneity effects, starting from homogeneous 2D lattices to the US power-grid, possessing identical nodes and links, to a realistic electric power-grid obtained from the Hungarian electrical database. The last one exhibits node dependent parameters, topologically marginally on the verge of robust networks. We show that too weak quenched heterogeneity, coming solely from the probabilistic self-frequencies of nodes (2D lattice) is not sufficient to find power-law distributed cascades. On the other hand too strong heterogeneity destroys the synchronization of the system. We found agreement with the empirically observed power-law failure size distributions on the US grid, as well as on the Hungarian networks near the synchronization transition point. We have also investigated the consequence of replacing the usual Gaussian self-frequencies to exponential distributed ones, describing renewable energy sources. We found a drop in the steady state synchronization averages, but the cascade size distribution both for the US and Hungarian systems remained insensitive and have kept the universal tails, characterized by the exponent $τ\simeq 1.8$. We have also investigated the effect of an instantaneous feedback mechanism in case of the Hungarian power-grid.

nlin.AO↗

arXiv2020

The effect of noise on the synchronization dynamics of the Kuramoto model on a large human connectome graph

We have extended the study of the Kuramoto model with additive Gaussian noise running on the KKI-18 large human connectome graph. We determined the dynamical behavior of this model by solving it numerically in an assumed homeostatic state, below the synchronization crossover point we determined previously. The de-synchronization duration distributions exhibit power-law tails, characterized by the exponent in the range $1.1 < τ_t < 2$, overlapping the in vivo human brain activity experiments by Palva et al. We show that these scaling results remain valid, by a transformation of the ultra-slow eigen-frequencies to Gaussian with unit variance. We also compare the connectome results with those, obtained on a regular cube with $N=10^6$ nodes, related to the embedding space, and show that the quenched internal frequencies themselves can cause frustrated synchronization scaling in an extended coupling space.

nlin.AO↗

arXiv2019

Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 804092 nodes, in an assumed homeostatic state. Since this graph has a topological dimension $d < 4$, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law--tailed synchronization durations, with $τ_t \simeq 1.2(1)$, away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: $τ_t \simeq 1.6(1)$. However, below the transition of the connectome we found global coupling control-parameter dependent exponents $1 < τ_t \le 2$, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

cond-mat.dis-nn↗

arXiv2019

Robustness of Griffiths effects in homeostatic connectome models

I provide numerical evidence for the robustness of the Griffiths phase (GP) reported previously in dynamical threshold model simulations on a large human brain network with N=836733 connected nodes. The model, with equalized network sensitivity, is extended in two ways: introduction of refractory states or by randomized time dependent thresholds. The non-universal power-law dynamics in an extended control parameter region survives these modifications for a short refractory state and weak disorder. In case of temporal disorder the GP shrinks and for stronger heterogeneity disappears, leaving behind a mean-field type of critical transition. Activity avalanche size distributions below the critical point decay faster than in the original model, but the addition of inhibitory interactions sets it back to the range of experimental values.

cond-mat.dis-nn↗

arXiv2018

Heterogeneity effects in power-grid network models

We have compared the phase synchronization transition of the second order Kuramoto model on 2D lattices and on large, synthetic power grid networks, generated from real data. The latter are weighted, hierarchical modular networks. Due to the inertia the synchronization transitions are of first order type, characterized by fast relaxation and hysteresis by varying the global coupling parameter K. Finite size scaling analysis shows that there is no real phase transition in the thermodynamic limit, unlike in the mean-field model. The order parameter and its fluctuations depend on the network size without any real singular behavior. In case of power grids the phase synchronization breaks down at lower global couplings, than in case of 2D lattices of the same sizes, but the hysteresis is much narrower or negligible due to the low connectivity of the graphs. The temporal behavior of de-synchronization avalanches after a sudden quench to low K values, has been followed and duration distributions with power-law tails have been detected. This suggests rare region effects, caused by frozen disorder, resulting in heavy tailed distributions, even without a self organization mechanism as a consequence of a catastrophic drop event in the couplings.

physics.soc-ph↗

arXiv2018

Griffiths phases in infinite-dimensional, non-hierarchical modular networks

Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.

physics.soc-ph↗

arXiv2017

Dynamical universality classes of simple growth and lattice gas models

Large scale, dynamical simulations have been performed for the two dimensional octahedron model, describing the Kardar-Parisi-Zhang (KPZ) for nonlinear, or the Edwards-Wilkinson (EW) class for linear surface growth. The autocorrelation functions of the heights and the dimer lattice gas variables are determined with high precision. Parallel random-sequential (RS) and two-sub-lattice stochastic dynamics (SCA) have been compared. The latter causes a constant correlation in the long time limit, but after subtracting it one can find the same height functions as in case of RS. On the other hand the ordered update alters the dynamics of the lattice gas variables, by increasing (decreasing) the memory effects for nonlinear (linear) models with respect to random-sequential. Additionally, we support the KPZ ansatz and the Kallabis-Krug conjecture in $2+1$ dimensions and provide a precise growth exponent value $β=0.2414(2)$. We show the emergence of finite size corrections, which occur long before the steady state roughness is reached.

cond-mat.stat-mech↗

arXiv2017

Suppressing correlations in massively parallel simulations of lattice models

For lattice Monte Carlo simulations parallelization is crucial to make studies of large systems and long simulation time feasible, while sequential simulations remain the gold-standard for correlation-free dynamics. Here, various domain decomposition schemes are compared, concluding with one which delivers virtually correlation-free simulations on GPU Extensive simulations of the octahedron model for $2+1$ dimensional Karda--Parisi--Zhang surface growth, which is very sensitive to correlation in the site-selection dynamics, were performed to show self-consistency of the parallel runs and agreement with the sequential algorithm. We present a GPU implementation providing a speedup of about $30\times$ over a parallel CPU implementation on a single socket and at least $180\times$ with respect to the sequential reference.

physics.comp-ph↗

arXiv2017

Local Scale-Invariance of the 2+1 dimensional Kardar-Parisi-Zhang model

Local Scale-Invariance theory is tested by extensive dynamical simulations of the driven dimer lattice gas model, describing the surface growth of the 2+1 dimensional Kardar-Parisi-Zhang surfaces. Very precise measurements of the universal autoresponse function enabled us to perform nonlinear fitting with the scaling forms, suggested by local scale-invariance (LSI). While the simple LSI ansatz does not seem to work, forms based on logarithmic extension of LSI provide satisfactory description of the full (measured) time evolution of the autoresponse function.

cond-mat.stat-mech↗

arXiv2016

Critical dynamics on a large human Open Connectome network

Extended numerical simulations of threshold models have been performed on a human brain network with N=836733 connected nodes available from the Open Connectome project. While in case of simple threshold models a sharp discontinuous phase transition without any critical dynamics arises, variable thresholds models exhibit extended power-law scaling regions. This is attributed to fact that Griffiths effects, stemming from the topological/interaction heterogeneity of the network, can become relevant if the input sensitivity of nodes is equalized. I have studied the effects effects of link directness, as well as the consequence of inhibitory connections. Non-universal power-law avalanche size and time distributions have been found with exponents agreeing with the values obtained in electrode experiments of the human brain. The dynamical critical region occurs in an extended control parameter space without the assumption of self organized criticality.

cond-mat.dis-nn↗

arXiv2016

Universality of (2+1)-dimensional restricted solid-on-solid models

Extensive dynamical simulations of Restricted Solid on Solid models in $D=2+1$ dimensions have been done using parallel multisurface algorithms implemented on graphics cards. Numerical evidence is presented that these models exhibit KPZ surface growth scaling, irrespective of the step heights $N$. We show that by increasing $N$ the corrections to scaling increase, thus smaller step-sized models describe better the asymptotic, long-wave-scaling behavior.

cond-mat.stat-mech↗

arXiv2016

Bit-Vectorized GPU Implementation of a Stochastic Cellular Automaton Model for Surface Growth

Stochastic surface growth models aid in studying properties of universality classes like the Kardar--Paris--Zhang class. High precision results obtained from large scale computational studies can be transferred to many physical systems. Many properties, such as roughening and some two-time functions can be studied using stochastic cellular automaton (SCA) variants of stochastic models. Here we present a highly efficient SCA implementation of a surface growth model capable of simulating billions of lattice sites on a single GPU. We also provide insight into cases requiring arbitrary random probabilities which are not accessible through bit-vectorization.

cs.DC↗

arXiv2016

The topology of large Open Connectome networks for the human brain

The structural human connectome (i.e.\ the network of fiber connections in the brain) can be analyzed at ever finer spatial resolution thanks to advances in neuroimaging. Here we analyze several large data sets for the human brain network made available by the Open Connectome Project. We apply statistical model selection to characterize the degree distributions of graphs containing up to $\simeq 10^6$ nodes and $\simeq 10^8$ edges. A three-parameter generalized Weibull (also known as a stretched exponential) distribution is a good fit to most of the observed degree distributions. For almost all networks, simple power laws cannot fit the data, but in some cases there is statistical support for power laws with an exponential cutoff. We also calculate the topological (graph) dimension $D$ and the small-world coefficient $σ$ of these networks. While $σ$ suggests a small-world topology, we found that $D < 4$ showing that long-distance connections provide only a small correction to the topology of the embedding three-dimensional space.

q-bio.NC↗

arXiv2016

Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks

We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-size random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects of outliers in both highly fluctuating (natural cutoff) and non-fluctuating (hard cutoff) most connected vertices. Logarithmic and power-law decays in time were found for natural and hard cutoffs, respectively. This happens in extended regions of the control parameter space $λ_1<λ<λ_2$, suggesting Griffiths effects, induced by the topological inhomogeneities. Optimal fluctuation theory considering sample-to-sample fluctuations of the pseudo thresholds is presented to explain the observed slow dynamics. A quasistationary analysis shows that response functions remain bounded at $λ_2$. We argue these to be signals of a smeared transition. However, in the thermodynamic limit the Griffiths effects loose their relevancy and have a conventional critical point at $λ_c=0$. Since many real networks are composed by heterogeneous and weakly connected modules, the slow dynamics found in our analysis of independent and finite networks can play an important role for the deeper understanding of such systems.

physics.soc-ph↗

arXiv2015

Griffiths phases and localization in hierarchical modular networks

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [Frontiers in Neuroinformatics, 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. Such networks can be embedded in two-dimensional Euclidean space. We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. Localization in the steady state is also shown via QMF. We investigate the effects of link asymmetry and coupling disorder, and show that localization can occur even in small-world networks with high connectivity in case of link disorder.

cond-mat.dis-nn↗

arXiv2014

Localization transition, Lifschitz tails and rare-region effects in network models

Effects of heterogeneity in the suspected-infected-susceptible model on networks are investigated using quenched mean-field theory. The emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare-region effects appearing in simulations and in the Lifschitz tails. The latter, in the linear approximation, is related to the spectral density of the Laplacian matrix and to the time dependent order parameter. I show that these approximations indicate correctly Griffiths Phases both on regular one-dimensional lattices and on small world networks exhibiting purely topological disorder. I discuss the localization transition that occurs on scale-free networks at $γ=3$ degree exponent.

cond-mat.stat-mech↗

arXiv2014

Ageing of the 2+1 dimensional Kardar-Parisi-Zhang model

Extended dynamical simulations have been performed on a 2+1 dimensional driven dimer lattice gas model to estimate ageing properties. The auto-correlation and the auto-response functions are determined and the corresponding scaling exponents are tabulated. Since this model can be mapped onto the 2+1 dimensional Kardar-Parisi-Zhang surface growth model, our results contribute to the understanding of the universality class of that basic system.

cond-mat.stat-mech↗