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Register a new userSpectral properties of complex networks
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Ginestra Bianconi
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We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian distributions, the present theory provides a solution for the non linear integral equations for the spectra density in random matrix theory of the spectra of sparse random matrices making a step forward with respect to the effective medium approximation (EMA) . We extend these results also to weighted networks with weight-degree correlations
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In this chapter we discuss how the results developed within the theory of fractals and Self-Organized Criticality (SOC) can be fruitfully exploited as ingredients of adaptive network models. In order to maintain the presentation self-contained, we first review the basic ideas behind fractal theory and SOC. We then briefly review some results in the field of complex networks, and some of the models that have been proposed. Finally, we present a self-organized model recently proposed by Garlaschelli et al. [Nat. Phys. 3, 813 (2007)] that couples the fitness network model defined by Caldarelli et al. [Phys. Rev. Lett. 89, 258702 (2002)] with the evolution model proposed by Bak and Sneppen [Phys. Rev. Lett. 71, 4083 (1993)] as a prototype of SOC. Remarkably, we show that the results obtained for the two models separately change dramatically when they are coupled together. This indicates that self-organized networks may represent an entirely novel class of complex systems, whose properties cannot be straightforwardly understood in terms of what we have learnt so far.
We present a thorough inspection of the dynamical behavior of epidemic phenomena in populations with complex and heterogeneous connectivity patterns. We show that the growth of the epidemic prevalence is virtually instantaneous in all networks characterized by diverging degree fluctuations, independently of the structure of the connectivity correlation functions characterizing the population network. By means of analytical and numerical results, we show that the outbreak time evolution follows a precise hierarchical dynamics. Once reached the most highly connected hubs, the infection pervades the network in a progressive cascade across smaller degree classes. Finally, we show the influence of the initial conditions and the relevance of statistical results in single case studies concerning heterogeneous networks. The emerging theoretical framework appears of general interest in view of the recently observed abundance of natural networks with complex topological features and might provide useful insights for the development of adaptive strategies aimed at epidemic containment.
We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path $ell_{rm opt}$ in a disordered ErdH{o}s-Renyi (ER) random network and scale-free (SF) network. Each link $i$ is associated with a weight $tau_iequivexp(a r_i)$, where $r_i$ is a random number taken from a uniform distribution between 0 and 1 and the parameter $a$ controls the strength of the disorder. We find that for any finite $a$, there is a crossover network size $N^*(a)$ at which the transition occurs. For $N ll N^*(a)$ the scaling behavior of $ell_{rm opt}$ is in the strong disorder regime, with $ell_{rm opt} sim N^{1/3}$ for ER networks and for SF networks with $lambda ge 4$, and $ell_{rm opt} sim N^{(lambda-3)/(lambda-1)}$ for SF networks with $3 < lambda < 4$. For $N gg N^*(a)$ the scaling behavior is in the weak disorder regime, with $ell_{rm opt}simln N$ for ER networks and SF networks with $lambda > 3$. In order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder. We propose a scaling ansatz for this measure and demonstrate its validity. We proceed to derive the scaling relation between $N^*(a)$ and $a$. We find that $N^*(a)sim a^3$ for ER networks and for SF networks with $lambdage 4$, and $N^*(a)sim a^{(lambda-1)/(lambda-3)}$ for SF networks with $3 < lambda < 4$.
We studied, both analytically and numerically, complex excitable networks, in which connections are time dependent and some of the nodes remain silent at each time step. More specifically, (a) there is a heterogenous distribution of connection weights and, depending on the current degree of order, some connections are reinforced/weakened with strength Phi on short-time scales, and (b) only a fraction rho of nodes are simultaneously active. The resulting dynamics has attractors which, for a range of Phi values and rho exceeding a threshold, become unstable, the instability depending critically on the value of rho. We observe that (i) the activity describes a trajectory in which the close neighborhood of some of the attractors is constantly visited, (ii) the number of attractors visited increases with rho, and (iii) the trajectory may change from regular to chaotic and vice versa as rho is, even slightly modified. Furthermore, (iv) time series show a power-law spectra under conditions in which the attractors space is most efficiently explored. We argue on the possible qualitative relevance of this phenomenology to networks in several natural contexts.
Subsystems of strongly disordered, interacting quantum systems can fail to thermalize because of the phenomenon of many-body localization (MBL). In this article, we explore a tensor network description of the eigenspectra of such systems. Specifically, we will argue that the presence of a complete set of local integrals of motion in MBL implies an efficient representation of the entire spectrum of energy eigenstates with a single tensor network, a emph{spectral} tensor network. Our results are rigorous for a class of idealized systems related to MBL with integrals of motion of finite support. In one spatial dimension, the spectral tensor network allows for the efficient computation of expectation values of a large class of operators (including local operators and string operators) in individual energy eigenstates and in ensembles.
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