http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

Random-matrix theories in quantum physics : common concepts

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

The success of new scientific areas can be assessed by their potential for contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling, after an introduction to the main concepts and models. A diversity of phenomena are surveyed, which may be classified into no less than 22 areas, providing a clear indication of the impact of the field of complex networks.

Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications

Although being a very popular sport in many countries, soccer has not received much attention from the scientific community. In this paper, we study soccer from a complex network point of view. First, we consider a bipartite network with two kinds of vertices or nodes: the soccer players and the clubs. Real data were gathered from the 32 editions of the Brazilian soccer championship, in a total of $13\phantom{\rule{0.2em}{0ex}}411$ soccer players and 127 clubs. We find a lot of interesting and perhaps unsuspected results. The probability that a Brazilian soccer player has worked at $N$ clubs or played $M$ games shows an exponential decay while the probability that he has scored $G$ goals is power law. Now, if two soccer players who have worked at the same club at the same time are connected by an edge, then a new type of network arises (composed exclusively by soccer player nodes). Our analysis shows that for this network the degree distribution decays exponentially. We determine the exact values of the clustering coefficient, the assortativity coefficient and the average shortest path length and compare them with those of the Erd\"os-R\'enyi and configuration model. The time evolution of these quantities are calculated and the corresponding results discussed.

Complex network study of Brazilian soccer players

Experiments of water infiltration in a vertical plane through layered soils were carried out in the laboratory. We have followed the fingering dynamics by using image techniques. The fractal theory has been applied to describe the fingering phenomena. We also have simulated a modified site gradient invasion percolation model where most of the physical ingredients relevant to the problem have been incorporated. In contrast to what occurs with the standard gradient invasion percolation model, we have found the formation of many fingers. Taking into account the bond number, the number of fingers, and the wetted area of the experiments, we have fitted the parameters of our model. This opens the possibility of a theoretical prediction of the pore size distribution of the soil. The results are then compared with the experimental data.

Experimental studies of the fingering phenomena in two dimensions and simulation using a modified invasion percolation model

In recent years there has been considerable interest in the structure and dynamics of complex networks. One of the most studied networks is the linear Barabasi–Albert model. Here we investigate the nonlinear Barabasi–Albert growing network. In this model, a new node connects to a vertex of degree k with a probability proportional to k α ( α real). Each vertex adds m new edges to the network. We derive an analytic expression for the degree distribution P ( k ) which is valid for all values of m and α ⩽1. In the limit α →−∞ the network is homogeneous. If α >1 there is a gel phase with m super-connected nodes. It is proposed a formula for the clustering coefficient which is in good agreement with numerical simulations. The assortativity coefficient r is determined and it is shown that the nonlinear Barabasi–Albert network is assortative (disassortative) if α (α>1) and no assortative only when α =1. In the limit α →−∞ the assortativity coefficient can be exactly calculated. We find r= 7 13 when m =2. Finally, the minimum average shortest path length l min is numerically evaluated. Increasing the network size, l min diverges for α ⩽1 and it is equal to 1 when α >1.

Nonlinear Barabási–Albert network

Using a transfer-matrix technique the authors obtain extended series expansion of the percolation probability for the directed site percolation problem on the square lattice. Their approach reveals a previously unsuspected connection between his problem and the enumeration of the number of ways of dissecting a ball. They show that the method can also be used to determine a series expansion for the mean cluster size. An analysis based on Pade approximants gives estimates of the critical threshold and also of the critical exponent beta .

Series expansion of the directed percolation probability

Using a transfer matrix method the authors derive a series expansion for the percolation probability on the directed honeycomb lattice. The high-density series is obtained to order q13. A Pade approximant analysis of the series has been used to estimate the percolation threshold qc and the critical exponent beta .

https://iopscience.iop.org/article/10.1088/0305-4470/23/7/009/pdf

Roberto N. Onody

Papers

Complex network study of Brazilian soccer players

Experimental studies of the fingering phenomena in two dimensions and simulation using a modified invasion percolation model

Nonlinear Barabási–Albert network

Series expansion of the directed percolation probability

Directed bond percolation on the honeycomb lattice