Spherical Model in a Random Field

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations $\sim r^{4-d}$.Comment: 29 page

Path finding strategies in scale-free networks

We numerically investigate the scale-free network model of Barab{\'a}si and Albert [A. L. Barab{\'a}si and R. Albert, Science {\bf 286}, 509 (1999)] through the use of various path finding strategies. In real networks, global network information is not accessible to each vertex, and the actual path connecting two vertices can sometimes be much longer than the shortest one. A generalized diameter depending on the actual path finding strategy is introduced, and a simple strategy, which utilizes only local information on the connectivity, is suggested and shown to yield small-world behavior: the diameter $D$ of the network increases logarithmically with the network size $N$, the same as is found with global strategy. If paths are sought at random, $D \sim N^{0.5}$ is found.Comment: 4 pages, final for

Scaling exponents and clustering coefficients of a growing random network

The statistical property of a growing scale-free network is studied based on an earlier model proposed by Krapivsky, Rodgers, and Redner [Phys. Rev. Lett. 86, 5401 (2001)], with the additional constraints of forbidden of self-connection and multiple links of the same direction between any two nodes. Scaling exponents in the range of 1-2 are obtained through Monte Carlo simulations and various clustering coefficients are calculated, one of which, $C_{\rm out}$, is of order $10^{-1}$, indicating the network resembles a small-world. The out-degree distribution has an exponential cut-off for large out-degree.Comment: six pages, including 5 figures, RevTex 4 forma

Giant Clusters in Random Ad Hoc Networks

The present paper introduces ad hoc communication networks as examples of large scale real networks that can be prospected by statistical means. A description of giant cluster formation based on the single parameter of node neighbor numbers is given along with the discussion of some asymptotic aspects of the giant cluster sizes.Comment: 6 pages, 5 figures; typos and correction

Structure comparison of binary and weighted niche-overlap graphs

In ecological networks, niche-overlap graphs are considered as complex systems. They represent the competition between two predators that share common resources. The purpose of this paper is to investigate the structural properties of these graphs considered as weighted networks and compare their measures with the ones calculated for the binary networks. To conduct this study, we select four classical network measures : the degree of nodes, the clustering coefficient, the assortativity, and the betweenness centrality. These measures were used to analyse different type of networks such as social networks, biological networks, world wide web, etc. Interestingly, we identify significant differences between the structure of the binary and the weighted niche-overlap graphs. This study indicates that weight information reveals different features that may provide other implications on the dynamics of these networks

Transition from fractal to non-fractal scalings in growing scale-free networks

Real networks can be classified into two categories: fractal networks and non-fractal networks. Here we introduce a unifying model for the two types of networks. Our model network is governed by a parameter $q$. We obtain the topological properties of the network including the degree distribution, average path length, diameter, fractal dimensions, and betweenness centrality distribution, which are controlled by parameter $q$. Interestingly, we show that by adjusting $q$, the networks undergo a transition from fractal to non-fractal scalings, and exhibit a crossover from `large' to small worlds at the same time. Our research may shed some light on understanding the evolution and relationships of fractal and non-fractal networks.Comment: 7 pages, 3 figures, definitive version accepted for publication in EPJ

The spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.Comment: 12 pages, 3 figure

World-Wide Web scaling exponent from Simon's 1955 model

Recently, statistical properties of the World-Wide Web have attracted considerable attention when self-similar regimes have been observed in the scaling of its link structure. Here we recall a classical model for general scaling phenomena and argue that it offers an explanation for the World-Wide Web's scaling exponent when combined with a recent measurement of internet growth.Comment: 1 page RevTeX, no figure

Review of Viola, L.A. (2020) The closure of the international system: how institutions create political equalities and hierarchies

History and International Relation