Spectral alignment of correlated Gaussian random matrices

In this paper we analyze a simple method ($EIG1$) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given $A$ and $B$, we compute $v_1$ and $v'_1$ two leading eigenvectors of $A$ and $B$. The algorithm returns a permutation $\hat{\Pi}$ such that the rank of the coordinate $\hat{\Pi}(i)$ in $v_1$ is the rank of the coordinate $i$ in $v'_1$ (up to the sign of $v'_1$). We consider a model where $A$ belongs to the Gaussian Orthogonal Ensemble (GOE), and $B= \Pi^T (A+\sigma H) \Pi$, where $\Pi$ is a permutation matrix and $H$ is an independent copy of $A$. We show the following 0-1 law: under the condition $\sigma N^{7/6+\epsilon} \to 0$, the $EIG1$ method recovers all but a vanishing part of the underlying permutation $\Pi$. When $\sigma N^{7/6-\epsilon} \to \infty$, this algorithm cannot recover more than $o(N)$ correct matches. This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.Comment: 29 pages, 4 figure

Exploiting Semantic Proximity in Peer-to-Peer Content Searching

A lot of recent work has dealt with improving performance of content searching in peer-to-peer file sharing systems. In this paper we attack this problem by modifying the overlay topology describing the peer relations in the system. More precisely, we create a semantic overlay, linking nodes that are "semantically close", by which we mean that they are interested in similar documents. This semantic overlay provides the primary search mechanism, while the initial peer-to-peer system provides the fail-over search mechanism. We focus on implicit approaches for discovering semantic proximity. We evaluate and compare three candidate methods, and review open questions

Stability Properties of Networks with Interacting TCP Flows

The equilibrium distributions of a Markovian model describing the interaction of several classes of permanent connections in a network are analyzed. It has been introduced by Graham and Robert. For this model each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. It has been shown that the invariant distributions are determined by the solutions of a fixed point equation in a finite dimensional space. In this paper, several examples of these fixed point equations are studied. The topologies investigated are rings, trees and a linear network, with various sets of routes through the nodes

On the flow-level stability of data networks without congestion control: the case of linear networks and upstream trees

In this paper, flow models of networks without congestion control are considered. Users generate data transfers according to some Poisson processes and transmit corresponding packet at a fixed rate equal to their access rate until the entire document is received at the destination; some erasure codes are used to make the transmission robust to packet losses. We study the stability of the stochastic process representing the number of active flows in two particular cases: linear networks and upstream trees. For the case of linear networks, we notably use fluid limits and an interesting phenomenon of "time scale separation" occurs. Bounds on the stability region of linear networks are given. For the case of upstream trees, underlying monotonic properties are used. Finally, the asymptotic stability of those processes is analyzed when the access rate of the users decreases to 0. An appropriate scaling is introduced and used to prove that the stability region of those networks is asymptotically maximized

The impact of reneging in processor sharing queues

We investigate an overloaded processor sharing queue with renewal arrivals and generally distributed service times. Impatient customers may abandon the queue, or renege, before completing service. The random time representing a customer’s patience has a general distribution and may be dependent on his initial service time requirement. We propose a scaling procedure that gives rise to a fluid model, with nontrivial yet tractable steady state behavior. This fluid model captures many essential features of the underlying stochastic model, and we use it to analyze the impact of impatience in processor sharing queues. We show that this impact can be substantial compared with FCFS, and we propose a simple admission control policy to overcome these negative impacts

Gibbsian Method for the Self-Optimization of Cellular Networks

In this work, we propose and analyze a class of distributed algorithms performing the joint optimization of radio resources in heterogeneous cellular networks made of a juxtaposition of macro and small cells. Within this context, it is essential to use algorithms able to simultaneously solve the problems of channel selection, user association and power control. In such networks, the unpredictability of the cell and user patterns also requires distributed optimization schemes. The proposed method is inspired from statistical physics and based on the Gibbs sampler. It does not require the concavity/convexity, monotonicity or duality properties common to classical optimization problems. Besides, it supports discrete optimization which is especially useful to practical systems. We show that it can be implemented in a fully distributed way and nevertheless achieves system-wide optimality. We use simulation to compare this solution to today's default operational methods in terms of both throughput and energy consumption. Finally, we address concrete issues for the implementation of this solution and analyze the overhead traffic required within the framework of 3GPP and femtocell standards.Comment: 25 pages, 9 figures, to appear in EURASIP Journal on Wireless Communications and Networking 201