Community analysis in social networks

We present an empirical study of different social networks obtained from digital repositories. Our analysis reveals the community structure and provides a useful visualising technique. We investigate the scaling properties of the community size distribution, and find that all the networks exhibit power law scaling in the community size distributions with exponent either -0.5 or -1. Finally we find that the networks community structure is topologically self-similar using the Horton-Strahler index.Received: 3 November 2003, Published online: 14 May 2004PACS: 89.75.Fb Structures and organization in complex systems - 89.75.Da Systems obeying scaling laws - 89.75.Hc Networks and genealogical trees     

Scale-free network provides an optimal pattern for knowledge transfer

We study numerically the knowledge innovation and diffusion process on four representative network models, such as regular networks, small-world networks, random networks and scale-free networks. The average knowledge stock level as a function of time is measured and the corresponding growth diffusion time, τ is defined and computed. On the four types of networks, the growth diffusion times all depend linearly on the network size N as τN, while the slope for scale-free network is minimal indicating the fastest growth and diffusion of knowledge. The calculated variance and spatial distribution of knowledge stock illustrate that optimal knowledge transfer performance is obtained on scale-free networks. We also investigate the transient pattern of knowledge diffusion on the four networks, and a qualitative explanation of this finding is proposed.     

Complex networks

We briefly describe the toolkit used for studying complex systems: nonlinear dynamics, statistical physics, and network theory. We place particular emphasis on network theory--the topic of this special issue--and its importance in augmenting the framework for the quantitative study of complex systems. In order to illustrate the main issues, we briefly review several areas where network theory has led to significant developments in our understanding of complex systems. Specifically, we discuss changes, arising from network theory, in our understanding of (i) the Internet and other communication networks, (ii) the structure of natural ecosystems, (iii) the spread of diseases and information, (iv) the structure of cellular signalling networks, and (v) infrastructure robustness. Finally, we discuss how complexity requires both new tools and an augmentation of the conceptual framework--including an expanded definition of what is meant by a quantitative prediction.Received: 12 November 2003, Published online: 14 May 2004PACS: 89.75.Fb Structures and organization in complex systems - 89.75.Da Systems obeying scaling laws     

Dynamics of multi-frequency minority games

The dynamics of minority games with agents trading on different time scales is studied via dynamical mean-field theory. We analyze the case where the agents decision-making process is deterministic and its stochastic generalization with finite heterogeneous learning rates. In each case, we characterize the macroscopic properties of the steady states resulting from different frequency and learning rate distributions and calculate the corresponding phase diagrams. Finally, the different roles played by regular and occasional traders, as well as their impact on the systems global efficiency, are discussed.Received: 4 August 2003, Published online: 22 September 2003PACS: 87.23.Ge Dynamics of social systems - 05.65.+b Self-organized systems - 02.50.Le Decision theory and game theory - 05.10.Gg Stochastic analysis methods     

Neighborhood models of minority opinion spreading

We study the effect of finite size population in Galams model [Eur. Phys. J. B 25, 403 (2002)] of minority opinion spreading and introduce neighborhood models that account for local spatial effects. For systems of different sizes N, the time to reach consensus is shown to scale as in the original version, while the evolution is much slower in the new neighborhood models. The threshold value of the initial concentration of minority supporters for the defeat of the initial majority, which is independent of N in Galams model, goes to zero with growing system size in the neighborhood models. This is a consequence of the existence of a critical size for the growth of a local domain of minority supporters.Received: 13 March 2004, Published online: 23 July 2004PACS: 87.23.Ge Dynamics of social systems - 0.5.50. + q Lattice theory and statisticsH.S. Wio: Present address: Instituto de Física de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain.     

Modelling one-dimensional driven diffusive systems by the Zero-Range Process

In order to characterize networks in the scale-free network class we study the frequency of cycles of length h that indicate the ordering of network structure and the multiplicity of paths connecting two nodes. In particular we focus on the scaling of the number of cycles with the system size in off-equilibrium scale-free networks. We observe that each off-equilibrium network model is characterized by a particular scaling in general not equal to the scaling found in equilibrium scale-free networks. We claim that this anomalous scaling can occur in real systems and we report the case of the Internet at the Autonomous System Level.Received: 15 January 2004, Published online: 14 May 2004PACS: 89.75.-k Complex systems - 89.75.Hc Networks and genealogical trees     

Spin-Peierls lattice fluctuations and disorders in CuGeO$_\mathsf{3}$ and its solid solutions

The inorganic quasi-one dimensional (1D) S = 1/2 antiferromagnetic (AF) system CuGeO₃ undergoes a 2nd order spin-Peierls (SP) phase transition at T _SP = 14.2 K. In this study we present an X-ray synchrotron radiation investigation which confirms that the SP instability is announced by an important regime of pretransitional structural fluctuations which have been detected until 36 K. Furthermore we show that these fluctuations are 1D above 24 K, a feature expected for a structural instability triggered by the Cu^{2 +} chains of spin 1/2. By extrapolating the thermal dependence of the correlation length in the chain direction, we show that formation of singlet dimers begins at about 50 K, a temperature that we identify as the mean field temperature of the SP chain. The critical nature of the pretransitional fluctuations does not change when low amounts (<1%) of non-magnetic dopants substitute either the Cu site (case of Zn and Mg) or the Ge site (case of Si and Al) of CuGeO₃. However, the spatial extension of the fluctuations is considerably reduced when the magnetic dopant Ni substitutes the Cu site. In the SP ground state of doped materials we have been able to detect, in addition to the superlattice SP reflections previously observed, a very weak anisotropic diffuse scattering. We give evidences that this scattering originates from dopant-induced quasi-1D domains in which the dimerisation is perturbed. If we assume that each domain is limited by a soliton-antisoliton pair, pinned either on the substituent of the Cu site or by the deformation field induced by the substituent of the Ge site, we deduce that the soliton and antisoliton are separated by a distance of about Å, and that the soliton half width amounts to about Å. With these numbers we are able to account for the rate of decrease of T _SP as a function of the dopant concentration, and to deduce the critical concentration above which the long-range SP order vanishes. The overall size of the perturbed domains thus obtained, Å, is comparable with the size of the magnetic inhomogeneities determined by muon spin spectroscopy in the AF phase of doped CuGeO₃.Received: 22 December 2003, Published online: 8 June 2004PACS: 71.27. + a Strongly correlated electron systems; heavy fermions - 61.72.Dd Experimental determination of defects by diffraction and scattering - 75.45. + j Macroscopic quantum phenomena in magnetic systems     

Boson fields with bounded interaction densities

We consider interaction densities of the formV((x)), where (x) is a scalar boson field andV() is a bounded real continuous function. We define the cut-off interaction by , where _E(x) is the momentum cut-off field. We prove that the scattering operator S_r(V) corresponding to the cut-off interaction exists, and we study the behavior of the scattering operator as well as the Heisenberg picture fields, as the cut-off is removed.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.At leave from Mathematical Institute, Oslo University.     

A general class of quantum fields without cut-offs in two space-time dimensions

We consider a selfinteracting boson field in two space-time dimensions, with interaction densities of the form:V((x)): where (x) is a scalar boson field, andV() is a real positive function of exponential type. We define the space cut-off interaction by and prove thatH _r =H ₀+V _r , whereH ₀ is the free energy, is essentially self adjoint. This permits us to take away the space cut-off and we obtain a quantum field free of cut-offs.At leave from Mathematical Institute, Oslo University.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.     

Statistical properties of corporate board and director networks

The boards of directors of the largest corporations of a country together with the directors form a dense bipartite network. The board network consists of boards connected through common directors. The director network is obtained taking the directors as nodes, and a membership in the same board as a link. These networks are involved in the decision making processes relevant to the macro-economy of a country. We present an extensive and comparative analysis of the statistical properties of the board network and the director network for the first 1000 US corporations ranked by revenue (Fortune 1000) in the year 1999 and for the corporations of the Italian Stock Market. We find several common statistical properties across the data sets, despite the fact that they refer to different years and countries. This suggests an underlying universal formation mechanism which is not captured in a satisfactory way by the existent network models. In particular we find that all the considered networks are Small Worlds, assortative, highly clustered and dominated by a giant component. Several other properties are examined. The presence of a lobby in a board, a feature relevant to decision making dynamics, turns out to be a macroscopic phenomenon in all the data sets.Received: 21 February 2004, Published online: 14 May 2004PACS: 87.23.Ge Dynamics of social systems - 89.65.-s Social and economic systems - 89.65.Gh Economics, econophysics, financial markets, business and management     

Temperature scaling of quantum Hall plateau transition in bilayer systems

We have investigated the scaling behavior of the quantum Hall plateau transition in double quantum well systems with different interlayer tunneling strengths. The scaling behavior of the localization property is found to be similar between the case when the relevant Landau level (LL) is non-degenerate and the case when two LLs associated with the two layers are accidentally degenerate. In both cases, the scaling exponent κ0.4 close to the canonical value is obtained, and it is unaffected by the in-plane magnetic field which changes the interlayer tunneling strength.     

Efficiency and robustness in ant networks of galleries

Recent theoretical and empirical studies have focused on the topology of large networks of communication/interactions in biological, social and technological systems. Most of them have been studied in the scope of the small-world and scale-free networks theory. Here we analyze the characteristics of ant networks of galleries produced in a 2-D experimental setup. These networks are neither small-worlds nor scale-free networks and belong to a particular class of network, i.e. embedded planar graphs emerging from a distributed growth mechanism. We compare the networks of galleries with both minimal spanning trees and greedy triangulations. We show that the networks of galleries have a path system efficiency and robustness to disconnections closer to the one observed in triangulated networks though their cost is closer to the one of a tree. These networks may have been prevented to evolve toward the classes of small-world and scale-free networks because of the strong spatial constraints under which they grow, but they may share with many real networks a similar trend to result from a balance of constraints leading them to achieve both path system efficiency and robustness at low cost.Received: 16 July 2004, Published online: 26 November 2004PACS: 89.75.Fb Structures and organization in complex systems - 89.75.Hc Networks and genealogical trees - 87.23.Ge Dynamics of social systems     

Non-translation invariant Gibbs states with coexisting phases

We investigate the spatially inhomogeneous states of two component,A - B, Widom-Rowlinson type lattice systems. When the fugacity of the two components are equal and large, these systems can exist in two different homogeneous (translation invariant) pure phases oneA-rich and oneB-rich. We consider now the system in a box with boundaries favoring the segregation of these two phases into top and bottom parts of the box. Utilizing methods due to Dobrushin we prove the existence, in three or more dimensions, of a sharp interface for the system which persists in the limit of the size of the box going to infinity. We also give some background on rigorous results for the interface problem in Ising spin systems.Supported in part by NSF Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Thermodynamic properties of an alternating-spin (1/2,1) two-leg ladder

Most complex networks serve as conduits for various dynamical processes, ranging from mass transfer by chemical reactions in the cell to packet transfer on the Internet. We collected data on the time dependent activity of five natural and technological networks, finding evidence of orders of magnitude differences in the fluxes of individual nodes. This dynamical inhomogeneity reflects the emergence of localized high flux regions or hot spots, carrying an overwhelming fraction of the networks activity. We find that each system is characterized by a unique scaling law, coupling the flux fluctuations with the total flux on individual nodes, a result of the competition between the systems internal collective dynamics and changes in the external environment. We propose a method to separate these two components, allowing us to predict the relevant scaling exponents. As high fluctuations can lead to dynamical bottlenecks and jamming, these findings have a strong impact on the predictability and failure prevention of complex transportation networks.Received: 25 October 2003, Published online: 17 February 2004PACS: 89.75.-k Complex systems - 89.75.Da Systems obeying scaling laws - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion     

Multi-component static model for social networks

The static model was introduced to generate a scale-free network. In the model, N number of vertices are present from the beginning. Each vertex has its own weight, representing how much the vertex is influential in a system. The static model, however, is not relevant, when a complex network is composed of many modules such as communities in social networks. An individual may belong to more than one community and has distinct weights for each community. Thus, we generalize the static model by assigning a q-component weight on each vertex. We first choose a component among the q components at random and a pair of vertices is linked with a color according to their weights of the component as in the static model. A (1-f) fraction of the entire edges is connected following this way. The remaining fraction f is added with (q + 1)-th color as in the static model but using the maximum weights among the q components each individual has. The social activity with such maximum weights is an essential ingredient to enhance the assortativity coefficient as large as the ones of real social networks.Received: 27 October 2003, Published online: 17 February 2004PACS: 89.65.-s Social and economic systems - 89.75.Hc Networks and genealogical trees - 89.75.Da Systems obeying scaling laws     

Invariance Implies Gibbsian: Some New Results

We investigate stationary distributions of stochastic gradient systems in Riemannian manifolds and prove that, under certain assumptions, such distributions are symmetric. These results are extended to countable products of finite dimensional manifolds and applied to Gibbs distributions in the case where the single spin spaces are Riemannian manifolds. In particular, we obtain a new result concerning the question whether all invariant measures are Gibbsian. Actually, we consider a more general object: weak elliptic equations for measures, which, on the one hand, yields the results obtained stronger than the above mentioned statements, and, on the other hand, enables us to give simpler proofs of more general than previously known facts. Applications to concrete models of lattice systems over ^d with not necessarily compact spin space are presented (also in the case d 3 under certain assumptions of decay of interaction).     

On the spectrum of the space cut-off :P(ϕ):Hamiltonian in two space-time dimensions

We use the method of the asymptotic fields to study the spectrum of the space cut-off :P(): Hamiltonian in two space time dimensions, whereP is a semibounded polynomial.This research is partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.On leave from Matematisk Institutt, Oslo University, Norway.     

Entropy Production: From Open Volume-Preserving to Dissipative Systems

We generalize Gaspard's method for computing the -entropy production rate in Hamiltonian systems to dissipative systems with attractors considered earlier by Tél, Vollmer, and Breymann. This approach leads to a natural definition of a coarse-grained Gibbs entropy which is extensive, and which can be expressed in terms of the SRB measures and volumes of the coarse-graining sets which cover the attractor. One can also study the entropy and entropy production as functions of the degree of resolution of the coarse-graining process, and examine the limit as the coarse-graining size approaches zero. We show that this definition of the Gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems. We apply the method to the case of a two-dimensional map, based upon a model considered by Vollmer, Tél, and Breymann, that is a deterministic version of a biased-random walk. We treat both volume-preserving and dissipative versions of the basic map, and make a comparison between the two cases. We discuss the -entropy production rate as a function of the size of the coarse-graining cells for these biased-random walks and, for an open system with flux boundary conditions, show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases. This work describes in some detail the relation between the results of Gaspard, those of of Tél, Vollmer, and Breymann, and those of Ruelle, on entropy production in various systems described by Anosov or Anosov-like maps.     

Perturbation of Singular Equilibria of Hyperbolic Two-Component Systems: A Universal Hydrodynamic Limit

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under the Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws:with where is a convex compact polygon in ². The system is typically strictly hyperbolic in the interior of with possible non-hyperbolic degeneracies on the boundary . We consider the case of an isolated singular (i.e. non-hyperbolic) point on the interior of one of the edges of , call it (₀,u₀). We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities (₀,u₀) of the conserved quantities. We prove that for a very rich class of systems, under a proper hydrodynamic limit the propagation of these small perturbations are universally driven by the two-by-two systemwhere the parameter is the only trace of the microscopic structure.The proof relies on the relative entropy method and thus, it is valid only in the regime of smooth solutions of the pde. But there are essential new elements: in order to control the fluctuations of the terms with Poissonian (rather than Gaussian) decay coming from the low density approximations we have to apply refined pde estimates. In particular Lax entropies of these pde systems play a not merely technical key role in the main part of the proof.