Variance Analysis of Randomized Consensus in Switching Directed Networks

In this paper, we study the asymptotic properties of distributed consensus algorithms over switching directed random networks. More specifically, we focus on consensus algorithms over independent and identically distributed, directed Erdos-Renyi random graphs, where each agent can communicate with any other agent with some exogenously specified probability $p$. While it is well-known that consensus algorithms over Erdos-Renyi random networks result in an asymptotic agreement over the network, an analytical characterization of the distribution of the asymptotic consensus value is still an open question. In this paper, we provide closed-form expressions for the mean and variance of the asymptotic random consensus value, in terms of the size of the network and the probability of communication $p$. We also provide numerical simulations that illustrate our results.Comment: 6 pages, 3 figures, submitted to American Control Conference 201

Distribution of shortest cycle lengths in random networks

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdos-R\'enyi network), degenerate distribution (random regular graph) and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.Comment: 44 pages, 11 figure

Scaling behavior of the contact process in networks with long-range connections

We present simulation results for the contact process on regular, cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered, that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyper-scaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster spreading quantities are averaged over initial sites the hyper-scaling law is restored.Comment: 9 pages, 10 figure

Entanglement percolation in quantum complex networks

Quantum networks are essential to quantum information distributed applications, and communicating over them is a key challenge. Complex networks have rich and intriguing properties, which are as yet unexplored in the quantum setting. Here, we study the effect of entanglement percolation as a means to establish long-distance entanglement between arbitrary nodes of quantum complex networks. We develop a theory to analytically study random graphs with arbitrary degree distribution and give exact results for some models. Our findings are in good agreement with numerical simulations and show that the proposed quantum strategies enhance the percolation threshold substantially. Simulations also show a clear enhancement in small-world and other real-world networks

A network epidemic model with preventive rewiring: comparative analysis of the initial phase

This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate $\omega$ (and reconnect to non-infectious individuals with probability $\alpha$ or else simply drop the edge if $\alpha=0$), so-called preventive rewiring. The models are denoted SIR-$\omega$ and SEIR-$\omega$, and we focus attention on the early stages of an outbreak, where we derive expression for the basic reproduction number $R_0$ and the expected degree of the infectious nodes $E(D_I)$ using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-$\omega$ and SEIR-$\omega$ epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same $E(D_I)$ for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of $R_0$ computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with $\alpha > 0$ (perhaps unaware of being infected), the two approaches give different epidemic thresholds, with the branching process approximation being more in agreement with simulations.Comment: 25 pages, 7 figure

Random features and polynomial rules

Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite.Comment: 11 pages + appendix, 4 figures. Comments are welcom

Generalized two-body self-consistent theory of random linear dielectric composites: an effective-medium approach to clustering in highly-disordered media

Effects of two-body dipolar interactions on the effective permittivity/conductivity of a binary, symmetric, random dielectric composite are investigated in a self-consistent framework. By arbitrarily splitting the singularity of the Green tensor of the electric field, we introduce an additional degree of freedom into the problem, in the form of an unknown "inner" depolarization constant. Two coupled self-consistent equations determine the latter and the permittivity in terms of the dielectric contrast and the volume fractions. One of them generalizes the usual Coherent Potential condition to many-body interactions between single-phase clusters of polarizable matter elements, while the other one determines the effective medium in which clusters are embedded. The latter is in general different from the overall permittivity. The proposed approach allows for many-body corrections to the Bruggeman-Landauer (BL) scheme to be handled in a multiple-scattering framework. Four parameters are used to adjust the degree of self-consistency and to characterize clusters in a schematic geometrical way. Given these parameters, the resulting theory is "exact" to second order in the volume fractions. For suitable parameter values, reasonable to excellent agreement is found between theory and simulations of random-resistor networks and pixelwise-disordered arrays in two and tree dimensions, over the whole range of volume fractions. Comparisons with simulation data are made using an "effective" scalar depolarization constant that constitutes a very sensitive indicator of deviations from the BL theory.Comment: 14 pages, 7 figure