arXiv subjects

Gergely Palla

Publications and source records attributed to Gergely Palla.

51 records · Page 3Linked to original sources

arXiv2007

Weighted network modules

The inclusion of link weights into the analysis of network properties allows a deeper insight into the (often overlapping) modular structure of real-world webs. We introduce a clustering algorithm (CPMw, Clique Percolation Method with weights) for weighted networks based on the concept of percolating k-cliques with high enough intensity. The algorithm allows overlaps between the modules. First, we give detailed analytical and numerical results about the critical point of weighted k-clique percolation on (weighted) Erdos-Renyi graphs. Then, for a scientist collaboration web and a stock correlation graph we compute three-link weight correlations and with the CPMw the weighted modules. After reshuffling link weights in both networks and computing the same quantities for the randomised control graphs as well, we show that groups of 3 or more strong links prefer to cluster together in both original graphs.

cond-mat.stat-mech↗

arXiv2007

Spectral transitions in networks

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erdos-Renyi (E-R) random graph is determined by the average degree , and p(s) undergoes a dramatic change when is varied around the critical point of the percolation transition, =1. When > 1, the p(s) is described by the statistics of the Gaussian Orthogonal Ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at =1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularised Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the Internet, a word association network and a protein protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E-R graph.

physics.data-an↗

arXiv2006

The critical point of k-clique percolation in the Erdos-Renyi graph

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdos-Renyi graph. When the probability p of two nodes being connected is above a certain threshold p_c(k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p_c(k).

cond-mat.dis-nn↗

arXiv2006

CFinder: Locating cliques and overlapping modules in biological networks

Summary: Most cellular tasks are performed not by individual proteins, but by groups of functionally associated proteins, often referred to as modules. In a protein assocation network modules appear as groups of densely interconnected nodes, also called communities or clusters. These modules often overlap with each other and form a network of their own, in which nodes (links) represent the modules (overlaps). We introduce CFinder, a fast program locating and visualizing overlapping, densely interconnected groups of nodes in undirected graphs, and allowing the user to easily navigate between the original graph and the web of these groups. We show that in gene (protein) association networks CFinder can be used to predict the function(s) of a single protein and to discover novel modules. CFinder is also very efficient for locating the cliques of large sparse graphs. Availability: CFinder (for Windows, Linux, and Macintosh) and its manual can be downloaded from http://angel.elte.hu/clustering. Contact: cfinder@angel.elte.hu

q-bio.MN↗

arXiv2006

Preferential attachment of communities: the same principle, but a higher level

The graph of communities is a network emerging above the level of individual nodes in the hierarchical organisation of a complex system. In this graph the nodes correspond to communities (highly interconnected subgraphs, also called modules or clusters), and the links refer to members shared by two communities. Our analysis indicates that the development of this modular structure is driven by preferential attachment, in complete analogy with the growth of the underlying network of nodes. We study how the links between communities are born in a growing co-authorship network, and introduce a simple model for the dynamics of overlapping communities.

cond-mat.stat-mech↗

arXiv2005

Uncovering the overlapping community structure of complex networks in nature and society

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.

physics.soc-ph↗

arXiv2005

Clique percolation in random networks

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Renyi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold pc(k)=[(k-1)N]^{-1/(k-1)}. At the transition point the scaling of the giant component with N is highly non-trivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.

cond-mat.dis-nn↗

arXiv2004

Reverse engineering of linking preferences from network restructuring

We provide a method to deduce the preferences governing the restructuring dynamics of a network from the observed rewiring of the edges. Our approach is applicable for systems in which the preferences can be formulated in terms of a single-vertex energy function with f(k) being the contribution of a node of degree k to the total energy, and the dynamics obeys the detailed balance. The method is first tested by Monte-Carlo simulations of restructuring graphs with known energies, then it is used to study variations of real network systems ranging from the co-authorship network of scientific publications to the asset graphs of the New York Stock Exchange. The empirical energies obtained from the restructuring can be described by a universal function f(k) -k ln(k), which is consistent with and justifies the validity of the preferential attachment rule proposed for growing networks.

cond-mat.stat-mech↗

arXiv2004

Statistical mechanics of topological phase transitions in networks

We provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. The associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge-dual graph of a fully connected network. In our studies -- based on an exact enumeration method, Monte-Carlo simulations, and theoretical considerations -- we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure of the network. Depending on the energy function chosen, the observed transitions can be best monitored using the order parameters Phi_s=s_{max}/M, i.e., the size of the largest connected component divided by the number of edges, or Phi_k=k_{max}/M, the largest degree in the network divided by the number of edges. If, for example the energy is chosen to be E=-s_{max}, the observed transition is analogous to the percolation phase transition of random graphs. For this choice of the energy, the phase-diagram in the [ ,T] plane is constructed. Single vertex energies of the form E=sum_i f(k_i), where k_i is the degree of vertex i, are also studied. Depending on the form of f(k_i), first order and continuous phase transitions can be observed. In case of f(k_i)=-(k_i+c)ln(k_i), the transition is continuous, and at the critical temperature scale-free graphs can be recovered.

cond-mat.stat-mech↗

arXiv2003

Topological phase transitions of random networks

To provide a phenomenological theory for the various interesting transitions in restructuring networks we employ a statistical mechanical approach with detailed balance satisfied for the transitions between topological states. This enables us to establish an equivalence between the equilibrium rewiring problem we consider and the dynamics of a lattice gas on the edge-dual graph of a fully connected network. By assigning energies to the different network topologies and defining the appropriate order parameters, we find a rich variety of topological phase transitions, defined as singular changes in the essential feature(s) of the global connectivity as a function of a parameter playing the role of the temperature. In the ``critical point'' scale-free networks can be recovered.

cond-mat.stat-mech↗

arXiv2001

Noise corrections to stochastic trace formulas

We review studies of an evolution operator L for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of L yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator L in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators L^n. Using an integral representation of the evolution operator L, we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula.

nlin.CD↗

arXiv2001

Trace formula for noise corrections to trace formulas

We consider an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and Gaussian noise. Using an integral representation of the evolution operator we investigate the high order corrections to the trace of arbitary power of the operator. The asymptotic behaviour is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion. We show that a trace formula can be derived to describe the high order noise corrections.

nlin.CD↗

arXiv1999

Asymptotics of high order noise corrections

We consider an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and noise with finite moments. Using a perturbative expansion of the evolution operator we calculate high order corrections to its trace in the case of a quartic map and Gaussian noise. The leading contributions come from the period one orbits of the map. The asymptotic behaviour is investigated and is found to be independent up to a multiplicative constant of the distribution of noise.

chao-dyn↗

arXiv1999

Spectrum of stochastic evolution operators: Local matrix representation approach

A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits. The evaluation of perturbative corrections is easier to implement in this framework than in the standard Feynman diagram perturbation theory. The result are perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum-mechanical applications.

chao-dyn↗

arXiv1999

Stochastic trace formulas

The spectrum of the evolution Operator associated with a nonlinear stochastic flow with additive noise is evaluated by diagonalization in a polynomial basis. The method works for arbitrary noise strength. In the weak noise limit we formulate a new perturbative expansion for the spectrum of the stochastic evolution Operator in terms of expansions around the classical periodic orbits. The diagonalization of such operators is easier to implement than the standard Feynman diagram perturbation theory. The result is a stochastic analog of the Gutzwiller semiclassical spectral determinant with the ``$\hbar$'' corrections computed to at least two orders more than what has so far been attainable in stochastic and quantum-mechanical applications, supplemented by the estimate for the late terms in the asymptotic saddlepoint expansions.

math.NA↗