Géza Ódor

Istitute of Technical Physics and Materials Science, Centre for Energy Research · Complex Systems

We present a synchronization transition study of the locally coupled Kuramoto model on extremely large graphs. We compare regular 405 and 1004 lattice results with those of 120002 lattice substrates with power-law decaying long links (ll). The latter heterogeneous network exhibits ds>4 spectral dimensions. We show strong corrections to scaling and...

A two-dimensional lattice of oscillators with identical (zero) intrinsic frequencies and Kuramoto type of interactions with randomly frustrated couplings is considered. Starting the time evolution from slightly perturbed synchronized states, we study numerically the relaxation properties, as well as properties at the stable fixed point which can al...

The dynamics of electric power systems are widely studied through the phase synchronization of oscillators, typically with the use of the Kuramoto equation. While there are numerous well-known order parameters to characterize these dynamics, shortcoming of these metrics are also recognized. To capture all transitions from phase disordered states ov...

To determine the precise link between form and function, brain studies primarily concentrate on the anatomical wiring of the brain and its topological properties. In this work, we investigate the weighted degree and connection length distributions of the KKI-113 and KKI-18 human connectomes, the fruit fly, and of the mouse retina. We found that the...

Partial, frustrated synchronization, and chimera-like states are expected to occur in Kuramoto-like models if the spectral dimension of the underlying graph is low: d s < 4. We provide numerical evidence that this really happens in the case of the high-voltage power grid of Europe ( d s < 2), a large human connectome (KKI113) and in the case of the...

Cascade failures in power grids occur when the failure of one component or subsystem causes a chain reaction of failures in other components or subsystems, ultimately leading to a widespread blackout or outage. Controlling cascade failures on power grids is important for many reasons like economic impact, national security, public safety and even r...

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field...

While weak, tuned asymmetry can improve, strong heterogeneity destroys synchronization in the electric power system. We study the level of heterogeneity, by comparing large high voltage (HV) power-grids of Europe and North America. We provide an analysis of power capacities and loads of various energy sources from the databases and found heavy tail...

The size distributions of planned and forced outages and their restoration times in power systems have been studied for almost two decades and have drawn great interest as they display heavy tails. Understanding heavy tails has been provided by various threshold models, which are self-tuned at their critical points, but as many papers pointed out,...

Partial, frustrated synchronization and chimera states are expected to occur in Kuramoto-like models if the spectral dimension of the underlying graph is low: $d_s < 4$. We provide numerical evidence that this really happens in case of the high-voltage power grid of Europe ($d_s < 2$) and in case of the largest, exactly known brain network correspo...

The size distribution of planned and forced outages in power systems have been studied for almost two decades and has drawn great interest as they display heavy tails. Understanding of this phenomenon has been done by various threshold models, which are self-tuned at their critical points, but as many papers pointed out, explanations are intuitive,...

We investigate the synchronization transition of the Shinomoto-Kuramoto model on networks of the fruit-fly and two large human connectomes. This model contains a force term, thus is capable of describing critical behavior in the presence of external excitation. By numerical solution we determine the crackling noise durations with and without therma...

The critical behavior of the nondiffusive susceptible-infected-recovered model on lattices had been well established in virtue of its duality symmetry. By performing simulations and scaling analyses for the diffusive variant on the two-dimensional lattice, we show that diffusion for all agents, while rendering this symmetry destroyed, constitutes a...

The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the c...

We investigate the synchronization transition of the Shinomoto-Kuramoto model on networks of the fruit-fly and two large human connectomes. This model contains a force term, thus is capable of describing critical behavior in the presence of external excitation. By numerical solution we determine the crackling noise durations with and without therma...

The second-order Kuramoto equation describes synchronization of coupled oscillators with inertia, which occur in power grids for example. Contrary to the first-order Kuramoto equation it's synchronization transition behavior is much less known. In case of Gaussian self-frequencies it is discontinuous, in contrast to the continuous transition for th...

Dynamical simulation of the cascade failures on the Europe and United States (U.S.) high-voltage power grids has been done via solving the second-order Kuramoto equation. We show that synchronization transition happens by increasing the global coupling parameter K with metasatble states depending on the initial conditions so that hysteresis loops o...

The critical behavior of the non-diffusive susceptible-infected-recovered model on lattices had been well established in virtue of its duality symmetry. By performing simulations and scaling analyses, we show that diffusion for all agents, while rendering this symmetry destroyed, constitutes a singular perturbation that induces asymptotically disti...

This corrects the article DOI: 10.1103/PhysRevE.103.062112.

Dynamical simulation of the cascade failures on the EU and USA high-voltage power grids has been done via solving the second-order Kuramoto equation. We show that synchronization transition happens by increasing the global coupling parameter $K$ with metasatble states depending on the initial conditions so that hysteresis loops occur. We provide an...

Previous simulation studies on human connectomes suggested that critical dynamics emerge subcritically in the so-called Griffiths phases. Now we investigate this on the largest available brain network, the 21662 node fruit-fly connectome, using the Kuramoto synchronization model. As this graph is less heterogeneous, lacking modular structure and ex...

Previous simulation studies on human connectomes suggested, that critical dynamics emerge subcrititcally in the so called Griffiths Phases. %This is the consequence of the strong heterogeneity of the graphs. Now we investigate this on the largest available brain network, the $21.662$ node fruit-fly connectome, using the Kuramoto synchronization mod...

In this review, we discuss critical dynamics of simple nonequilibrium models on large connectomes, obtained by diffusion MRI, representing the white matter of the human brain. In the first chapter, we overview graph theoretical and topological analysis of these networks, pointing out that universality allows selecting a representative network, the...

Power-law (PL) time-dependent infection growth has been reported in many COVID-19 statistics. In simple susceptible infected recovered (SIR) models, the number of infections grows at the outbreak as I(t)∝td−1 on d-dimensional Euclidean lattices in the endemic phase, or it follows a slower universal PL at the critical point, until finite sizes cause...

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 distributi...

In this review, we discuss critical dynamics of simple nonequilibrium models on large connectomes, obtained by diffusion MRI, representing the white matter of the human brain. In the first chapter, we overview graph theoretical and topological analysis of these networks, pointing out that universality allows selecting a representative network, the...

Power-law (PL) time dependent infection growth has been reported in many Covid statistics. In simple SIR models the number of infections grows at the outbreak as $I(t) \propto t^{d-1}$ on $d$-dimensional Euclidean lattices in the endemic phase or follow a slower universal PL at the critical point, until finite sizes cause immunity and a crossover t...

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, and 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 an...

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...

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 explor...

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 explora...

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...

We have extended the study of the Kuramoto model with additive Gaussian noise running on the {\it 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 dist...

We consider the Kuramoto model on sparse random networks such as the Erdős–Rényi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical coupling...

The time dependent behavior of the Kuramoto model, describing synchronization, has been studied numerically on small-world graphs. We determined the desynchronziation behavior, by solving this model via the 4th order Runge-Kutta algorithm on a large, weighted human connectome network and compared the results with those of a two-dimensional lattice,...

We consider the Kuramoto model on sparse random networks such as the Erd\H{o}s-R\'enyi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical cou...

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. T...

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. T...

We investigate how the dimensionality of the embedding space affects the microscopic crackling dynamics and the macroscopic response of heterogeneous materials. Using a fiber bundle model with localized load sharing, computer simulations are performed from one to eight dimensions slowly increasing the external load up to failure. Analyzing the cons...

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 pos...

Many critical phenomena are drastically altered by quenched disorder, leading to Griffiths phases (GPs). In the GPs, there is dynamical criticality and high sensitivity to external stimuli in an extended parameter space. It was conjectured that systems with modular structures can sustain GPs if they are hierarchically organized and possess finite d...

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 hyster...

Large scale, dynamical simulations have been performed for the two dimensional octahedron model, describing Kardar-Parisi-Zhang (KPZ) for nonlinear, or Edwards-Wilkinson 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...

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...

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...

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 Kardar-Parisi-Zhang surface growth scaling, irrespective of the step heights N. We show that by increasing N the correc...

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 automato...

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 s...

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...

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 c...

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...

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 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 c...

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 linea...

Bursty dynamics of agents is shown to appear at criticality or in extended Griffiths phases, even in case of Poisson processes. I provide numerical evidence for a power-law type of intercommunication time distributions by simulating the contact process and the susceptible-infected-susceptible model. This observation suggests that in the case of non...

Extended dynamical simulations have been performed on a (2+1)-dimensional driven dimer lattice-gas model to estimate aging properties. The autocorrelation and the autoresponse 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 m...

Bursty dynamics of agents is shown to appear at criticality or in extended Griffiths phases, even in case of Poisson processes. I provide numerical evidence for power-law type of inter-communication time distributions by simulating the Contact Process and the Susceptible-Infected-Susceptible model. This observation suggests that in case of non-stat...

The Susceptible-Infected-Susceptible (SIS) model is one of the simplest memoryless system for describing information/epidemic spreading phenomena with competing creation and spontaneous annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-fr...

I extend a previous work to susceptible-infected-susceptible (SIS) models on weighted Barabási-Albert scale-free networks. Numerical evidence is provided that phases with slow, power-law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process. I compare simulation results with spectral...

The Contact Process has been studied on complex networks exhibiting different kinds of quenched disorder. Numerical evidence is found for Griffiths phases and other rare region effects, in Erd\H os R\'enyi networks, leading rather generically to anomalously slow (algebraic, logarithmic,...) relaxation. More surprisingly, it turns out that Griffiths...

We study coarsening phenomena in three different simple exclusion processes with quenched disordered jump rates. In the case of the totally asymmetric process, an earlier phenomenological description is improved, yielding for the time dependence of the length scale $\xi(t)\sim t/(\ln t)^2$, which is found to be in agreement with results of Monte Ca...

We show that generic, slow dynamics can occur in the contact process on complex networks with a tree-like structure and a superimposed weight pattern, in the absence of additional (non-topological) sources of quenched disorder. The slow dynamics is induced by rare-region effects occurring on correlated subspaces of vertices connected by large weigh...

We show that efficient simulations of the Kardar-Parisi-Zhang interface growth in 2 + 1 dimensions and of the 3-dimensional Kinetic Monte Carlo of thermally activated diffusion can be realized both on GPUs and modern CPUs. In this article we present results of different implementations on GPUs using CUDA and OpenCL and also on CPUs using OpenCL and...

We show that the emergence of different surface patterns (ripples, dots) can be well understood by a suitable mapping onto the simplest nonequilibrium lattice gases and cellular automata.Using this efficient approach difficult, unanswered questions of surface growth and its scaling can be studied. The mapping onto binary variables facilitates effec...

The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large-scale simulations via binary lattice gases and bit-coded algorithms. We confirm scaling behavior belonging to the two-dimensional Kardar-Parisi-Zhang universality class and find a surface growth exponent: β = 0.2415(15) on 2^17 × 2^17 sy...

Networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems. Studying the role of quenched disorder, both intrinsic to nodes and topological, is a key challenge. With this in mind, here we analyse the contact process, i.e. the simplest model for propagation phenomena, with node-dependent infection...

The critical 3-state two-dimensional Potts model with one line of defect couplings has been studied. Interpolating the coupling strength between the free and periodic boundary conditions renormalization group arguments suggest that the algebraic content should change discontinuously. Monte Carlo data on different sized and shaped lattice systems ha...