[en] The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree). The network is built by (i) creation of new nodes which each immediately attach to a preexisting node, and (ii) creation of new links between preexisting nodes. This process naturally generates correlated in-degree and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained

[en] One of the major questions in complex network research is to identify the range of mechanisms by which a complex network can self organize into a scale-free state. In this paper we investigate the interplay between a fitness linking mechanism and both random and preferential attachment. In our models, each vertex is assigned a fitness x, drawn from a probability distribution ρ(x). In Model A, at each time step a vertex is added and joined to an existing vertex, selected at random, with probability p and an edge is introduced between vertices with fitnesses x and y, with a rate f(x,y), with probability 1-p. Model B differs from Model A in that, with probability p, edges are added with preferential attachment rather than randomly. The analysis of Model A shows that, for every fixed fitness x, the network's degree distribution decays exponentially. In Model B we recover instead a power-law degree distribution whose exponent depends only on p, and we show how this result can be generalized. The properties of a number of particular networks are examined

[en] We consider the random sequence x_n = x_n-1 + γx_q, with γ > 0, where q = 0, 1, ..., n - 1 is chosen randomly from a probability distribution P_n(q). When all q are chosen with equal probability, i.e. P_n(q) = 1/n, we obtain an exact solution for the mean < x_n> and the divergence of the second moment < x²_n> as functions of n and γ. For γ = 1 we examine the divergence of the mean value of x_n, as a function of n, for the random sequences generated by power law and exponential P_n(q) and for the non-random sequence P_n(q) = δ_q,β(n-1)

[en] An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability p or two edges with probability 1 − p. With each edge having a resistance equal to 1Ω, the total resistance R_n between the root vertex and a busbar connecting all the vertices at the nth level is considered. A dynamical system is presented which approximates R_n, it is shown that the mean value 〈R_n〉 for this system approaches (1 + p)/(1 − p) as n → ∞, the distribution of R_n at large n is also examined. Additionally, a random sequence construction akin to a random Fibonacci sequence is used to approximate R_n; this sequence is shown to be related to the Legendre polynomials and its mean is shown to converge with |〈R_n〉 − (1 + p)/(1 − p)| ∼ n^−1/2. (paper)

[en] In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For this purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree p. In the limit p → ∞, known asymptotic formulae are rederived. Moreover, the 1/p corrections to known results are analytically calculated by a perturbative method

[en] We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it

[en] We examine the eigenvalue spectrum, ρ(μ), of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p → ∞, one can obtain two relatively simple coupled equations whose solution yields ρ(μ) for an arbitrary complex network. For scale-free graphs, with degree distribution exponent λ, we obtain an exact expression for the eigenvalue spectrum when λ = 3 and show that ρ(μ) ∼ 1/μ^2λ-1 for large μ. In the limit λ → ∞ we recover known results for the Erdoes-Renyi random graph