arXiv subjects

Ali Jadbabaie

Publications and source records attributed to Ali Jadbabaie.

155 records · Page 9Linked to original sources

arXiv2011

A Distributed Newton Method for Network Utility Maximization

Most existing work uses dual decomposition and subgradient methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This work develops an alternative distributed Newton-type fast converging algorithm for solving network utility maximization problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner with limited information exchange. Similarly, the stepsize can be obtained via an iterative consensus-based averaging scheme. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing subgradient methods based on dual decomposition.

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arXiv2011

On Non-Bayesian Social Learning

We study a model of information aggregation and social learning recently proposed by Jadbabaie, Sandroni, and Tahbaz-Salehi, in which individual agents try to learn a correct state of the world by iteratively updating their beliefs using private observations and beliefs of their neighbors. No individual agent's private signal might be informative enough to reveal the unknown state. As a result, agents share their beliefs with others in their social neighborhood to learn from each other. At every time step each agent receives a private signal, and computes a Bayesian posterior as an intermediate belief. The intermediate belief is then averaged with the belief of neighbors to form the individual's belief at next time step. We find a set of minimal sufficient conditions under which the agents will learn the unknown state and reach consensus on their beliefs without any assumption on the private signal structure. The key enabler is a result that shows that using this update, agents will eventually forecast the indefinite future correctly.

cs.SI↗

arXiv2011

From Local Measurements to Network Spectral Properties: Beyond Degree Distributions

It is well-known that the behavior of many dynamical processes running on networks is intimately related to the eigenvalue spectrum of the network. In this paper, we address the problem of inferring global information regarding the eigenvalue spectrum of a network from a set of local samples of its structure. In particular, we find explicit relationships between the so-called spectral moments of a graph and the presence of certain small subgraphs, also called motifs, in the network. Since the eigenvalues of the network have a direct influence on the network dynamical behavior, our result builds a bridge between local network measurements (i.e., the presence of small subgraphs) and global dynamical behavior (via the spectral moments). Furthermore, based on our result, we propose a novel decentralized scheme to compute the spectral moments of a network by aggregating local measurements of the network topology. Our final objective is to understand the relationships between the behavior of dynamical processes taking place in a large-scale complex network and its local topological properties.

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arXiv2010

Spectral Control of Mobile Robot Networks

The eigenvalue spectrum of the adjacency matrix of a network is closely related to the behavior of many dynamical processes run over the network. In the field of robotics, this spectrum has important implications in many problems that require some form of distributed coordination within a team of robots. In this paper, we propose a continuous-time control scheme that modifies the structure of a position-dependent network of mobile robots so that it achieves a desired set of adjacency eigenvalues. For this, we employ a novel abstraction of the eigenvalue spectrum by means of the adjacency matrix spectral moments. Since the eigenvalue spectrum is uniquely determined by its spectral moments, this abstraction provides a way to indirectly control the eigenvalues of the network. Our construction is based on artificial potentials that capture the distance of the network's spectral moments to their desired values. Minimization of these potentials is via a gradient descent closed-loop system that, under certain convexity assumptions, ensures convergence of the network topology to one with the desired set of moments and, therefore, eigenvalues. We illustrate our approach in nontrivial computer simulations.

cs.MA↗

arXiv2010

On Asymptotic Consensus Value in Directed Random Networks

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 random graphs, where each agent can communicate with any other agent with some exogenously specified probability. While different aspects of consensus algorithms over random switching networks have been widely studied, a complete characterization of the distribution of the asymptotic value for general \textit{asymmetric} random consensus algorithms remains an open problem. In this paper, we derive closed-form expressions for the mean and an upper bound for the variance of the asymptotic consensus value, when the underlying network evolves according to an i.i.d. \textit{directed} random graph process. We also provide numerical simulations that illustrate our results.

cs.MA↗

arXiv2010

Spectral Analysis of Virus Spreading in Random Geometric Networks

In this paper, we study the dynamics of a viral spreading process in random geometric graphs (RGG). The spreading of the viral process we consider in this paper is closely related with the eigenvalues of the adjacency matrix of the graph. We deduce new explicit expressions for all the moments of the eigenvalue distribution of the adjacency matrix as a function of the spatial density of nodes and the radius of connection. We apply these expressions to study the behavior of the viral infection in an RGG. Based on our results, we deduce an analytical condition that can be used to design RGG's in order to tame an initial viral infection. Numerical simulations are in accordance with our analytical predictions.

cs.MA↗

arXiv2010

Moment-Based Analysis of Synchronization in Small-World Networks of Oscillators

In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distribution of the Laplacian matrix of the network. We introduce, for the first time, analytical expressions for the first three moments of the eigenvalue distribution of the Laplacian matrix as a function of the probability of shortcuts and the connectivity of the underlying nearest-neighbor coupled ring. We apply these expressions to estimate the spectral support of the Laplacian matrix in order to predict synchronization in small-world networks. We verify the efficiency of our predictions with numerical simulations.

cs.MA↗

arXiv2010

Distributed Control of the Laplacian Spectral Moments of a Network

It is well-known that the eigenvalue spectrum of the Laplacian matrix of a network contains valuable information about the network structure and the behavior of many dynamical processes run on it. In this paper, we propose a fully decentralized algorithm that iteratively modifies the structure of a network of agents in order to control the moments of the Laplacian eigenvalue spectrum. Although the individual agents have knowledge of their local network structure only (i.e., myopic information), they are collectively able to aggregate this local information and decide on what links are most beneficial to be added or removed at each time step. Our approach relies on gossip algorithms to distributively compute the spectral moments of the Laplacian matrix, as well as ensure network connectivity in the presence of link deletions. We illustrate our approach in nontrivial computer simulations and show that a good final approximation of the spectral moments of the target Laplacian matrix is achieved for many cases of interest.

cs.MA↗

arXiv2009

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.

cs.MA↗

arXiv2007

Approximation of the joint spectral radius using sum of squares

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.

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arXiv2005

On the stability of the Kuramoto model of coupled nonlinear oscillators

We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

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