arXiv subjects

Z. Burda

Publications and source records attributed to Z. Burda.

At least 37 records · Page 2Linked to original sources

arXiv2008

Adaptive networks of trading agents

Multi-agent models have been used in many contexts to study generic collective behavior. Similarly, complex networks have become very popular because of the diversity of growth rules giving rise to scale-free behavior. Here we study adaptive networks where the agents trade ``wealth'' when they are linked together while links can appear and disappear according to the wealth of the corresponding agents; thus the agents influence the network dynamics and vice-versa. Our framework generalizes a multi-agent model of Bouchand and Mezard, and leads to a steady state with fluctuating connectivities. The system spontaneously self-organizes into a critical state where the wealth distribution has a fat tail and the network is scale-free; in addition, network heterogeneities lead to enhanced wealth condensation.

q-fin.TR↗

arXiv2008

Power laws in zero-range processes on random networks

We study statistical properties of a zero-range process (ZRP) on random networks. We derive an analytic expression for the distribution of particles (also called node occupation distribution) in the steady state of the ZRP in the ensemble of uncorrelated random graphs. We analyze the dependence of this distribution on the node-degree distribution. In particular, we show that when the degree distribution is tuned properly, one can obtain scale-free fluctuations in the distribution of particles. Such fluctuations lead to a power law in the distribution of particles, just like in the ZRP with the hopping rate u(m)=1+b/m on homogeneous graphs.

cond-mat.stat-mech↗

arXiv2008

Causal and homogeneous networks

Growing networks have a causal structure. We show that the causality strongly influences the scaling and geometrical properties of the network. In particular the average distance between nodes is smaller for causal networks than for corresponding homogeneous networks. We explain the origin of this effect and illustrate it using as an example a solvable model of random trees. We also discuss the issue of stability of the scale-free node degree distribution. We show that a surplus of links may lead to the emergence of a singular node with the degree proportional to the total number of links. This effect is closely related to the backgammon condensation known from the balls-in-boxes model.

cond-mat.dis-nn↗

arXiv2008

Counting metastable states of Ising spin glasses on arbitrary graphs

Using a field-theoretical representation of the Tanaka-Edwards integral we develop a method to systematically compute the number N_s of 1-spin-stable states (local energy minima) of a glassy Ising system with nearest-neighbor interactions and random Gaussian couplings on an arbitrary graph. In particular, we use this method to determine N_s for K-regular random graphs and d-dimensional regular lattices for d=2,3. The method works also for other graphs. Excellent accuracy of the results allows us to observe that the number of local energy minima depends mainly on local properties of the graph on which the spin glass is defined.

cond-mat.dis-nn↗

arXiv2007

Network of inherent structures in spin glasses: scaling and scale-free distributions

The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic ``length scale'' grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, for spin glasses based on random graphs, the degree distribution of the network of inherent structures exhibits a non-trivial scale-free tail.

cond-mat.dis-nn↗

arXiv2007

Free zero-range processes on networks

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the current distribution of particles. The case (a) provides a very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate of particles on the node with maximal degree. The case (b) is very interesting since the averaging over typical ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point of view of the FZRP. In the case (c), the distribution of particles and the dynamics of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales: for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and show that it leads to an interesting phase diagram.

cond-mat.stat-mech↗

arXiv2007

Condensation in zero-range processes on inhomogeneous networks

We investigate the role of inhomogeneities in zero-range processes in condensation dynamics.We consider the dynamics of balls hopping between nodes of a network, and find that the condensation is triggered by the ratio k_1/k of the highest degree k_1 to the average degree k. Although the condensate takes on the average an extensive number of balls, its occupation can oscillate in a wide range. We show that in systems with strong inhomogeneity, the typical melting time of the condensate grows exponentially with the number of balls.

cond-mat.stat-mech↗

arXiv2007

Balls-in-boxes condensation on networks

We discuss two different regimes of condensate formation in zero-range processes on networks: on a q-regular network, where the condensate is formed as a result of a spontaneous symmetry breaking, and on an irregular network, where the symmetry of the partition function is explicitly broken. In the latter case we consider a minimal irregularity of the q-regular network introduced by a single Q-node with degree Q>q. The statics and dynamics of the condensation depends on the parameter log(Q/q), which controls the exponential fall-off of the distribution of particles on regular nodes and the typical time scale for melting of the condensate on the Q-node which increases exponentially with the system size $N$. This behavior is different than that on a q-regular network where log(Q/q)=0 and where the condensation results from the spontaneous symmetry breaking of the partition function, which is invariant under a permutation of particle occupation numbers on the q-nodes of the network. In this case the typical time scale for condensate melting is known to increase typically as a power of the system size.

cond-mat.stat-mech↗

arXiv2006

From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity Ns of the inherent structures generically has a lognormal distribution. In addition, the large volume limit of ln / differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

cond-mat.dis-nn↗

arXiv2005

Eigenvalue density of empirical covariance matrix for correlated samples

We describe a method to determine the eigenvalue density of empirical covariance matrix in the presence of correlations between samples. This is a straightforward generalization of the method developed earlier by the authors for uncorrelated samples. The method allows for exact determination of the experimental spectrum for a given covariance matrix and given correlations between samples in the limit of large N and N/T=r=const with N being the number of degrees of freedom and T being the number of samples. We discuss the effect of correlations on several examples.

cond-mat.stat-mech↗

arXiv2005

A program generating homogeneous random graphs with given weights

We present a program package which generates homogeneous random graphs with probabilities prescribed by the user. The statistical weight of a labeled graph $α$ is given in the form $W(α)=\prod_{i=1}^N p(q_i)$, where $p(q)$ is an arbitrary user function and $q_i$ are the degrees of the graph nodes. The program can be used to generate two types of graphs (simple graphs and pseudo-graphs) from three types of ensembles (micro-canonical, canonical and grand-canonical).

cond-mat.dis-nn↗

arXiv2004

Perturbing General Uncorrelated Networks

This paper is a direct continuation of an earlier work, where we studied Erdös-Rényi random graphs perturbed by an interaction Hamiltonian favouring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erdös-Rényi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behaviour, studied thoroughly in this paper, both analytically and numerically.

cond-mat.dis-nn↗

arXiv2004

Signal and Noise in Correlation Matrix

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance (correlation) matrix. Results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom, like in statistical physics, lattice measurements of field theory, genetics, quantitative finance and other applications of multivariate statistics.

cond-mat.stat-mech↗

arXiv2003

Network Transitivity and Matrix Models

This paper is a step towards a systematic theory of the transitivity (clustering) phenomenon in random networks. A static framework is used, with adjacency matrix playing the role of the dynamical variable. Hence, our model is a matrix model, where matrices are random, but their elements take values 0 and 1 only. Confusion present in some papers where earlier attempts to incorporate transitivity in a similar framework have been made is hopefully dissipated. Inspired by more conventional matrix models, new analytic techniques to develop a static model with non-trivial clustering are introduced. Computer simulations complete the analytic discussion.

cond-mat↗

arXiv2003

Fermions in 2D Lorentzian Quantum Gravity

We implement Wilson fermions on 2D Lorentzian triangulation and determine the spectrum of the Dirac-Wilson operator. We compare it to the spectrum of the corresponding operator in the Euclidean background. We use fermionic particle to probe the fractal properties of Lorentzian gravity coupled to $c=1/2$ and $c=4$ matter. We numerically determine the scaling exponent of the mass gap $M \sim N^{-1/d_H}$ to be $d_H=2.11(5)$, and $d_H=1.77(3)$ for $c=1/2$ and $c=4$, respectively

hep-lat↗

arXiv2003

Uncorrelated Random Networks

We define a statistical ensemble of non-degenerate graphs, i.e. graphs without multiple- and self-connections between nodes. The node degree distribution is arbitrary, but the nodes are assumed to be uncorrelated. This completes our earlier publication \cite{bck}, where trees and degenerate graphs were considered. An efficient algorithm generating non-degenerate graphs is constructed. The corresponding computer code is available on request. Finite-size effects in scale-free graphs, i.e. those where the tail of the degree distribution falls like $n^{-β}$, are carefully studied. We find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like $N^γ$, with $γ= \min[1/2, 1/(β-1)]$, where $N$ is the total number of nodes. The consequence is that, independently of any specific model, the inter-node correlations seem to be a necessary ingredient of the physics of scale-free networks observed in nature.

cond-mat.stat-mech↗

arXiv2003

Tree Networks with Causal Structure

Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree graphs, an analytically solvable case. General formulae are derived, describing the degree distribution, the ancestor-descendant correlation and the probability a randomly chosen node lives at a given geodesic distance from the root. It is shown that the Hausdorff dimension $d_H$ of the causal networks is generically infinite, in contrast to the maximally random trees, where it is generically finite.

cond-mat.stat-mech↗

arXiv2003

Is Econophysics a Solid Science?

Econophysics is an approach to quantitative economy using ideas, models, conceptual and computational methods of statistical physics. In recent years many of physical theories like theory of turbulence, scaling, random matrix theory or renormalization group were successfully applied to economy giving a boost to modern computational techniques of data analysis, risk management, artificial markets, macro-economy, {\it etc.} Econophysics became a regular discipline covering a large spectrum of problems of modern economy. It is impossible to review the whole field in a short paper. Here we shall instead attempt to give a flavor of how econophysics approaches economical problems by discussing one particular issue as an example: the emergence and consequences of large scale regularities, which in particular occur in the presence of fat tails in probability distributions in macro-economy and quantitative finance.

cond-mat↗