Research Portfolio

On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence

This paper introduces a class of non-linear and continuous-time opinion dynamics model with additive noise and state dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed form expressions for this class of dynamics are obtained for the two agent case. However the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

An Analytical Framework for Modeling a Spatially Repulsive Cellular Network

We propose a new cellular network model that captures both deterministic and random aspects of base station (BS) deployments. Namely, the BS locations are modeled as the superposition of two independent stationary point processes: a random shifted grid with intensity λ _g and a Poisson point process (PPP) with intensity λ _p . Grid and PPP deployments are special cases with λ _p → 0 and λ _g → 0 , with actual deployments in between these two extremes, as we demonstrate with deployment data. Assuming that each user is associated with the BS that provides the strongest average received signal power, we obtain the probability that a typical user is associated with either a grid or PPP BS. Assuming Rayleigh fading channels, we derive the expression for the coverage probability of the typical user, resulting in the following observations. First, the association and the coverage probability of the typical user are fully characterized as functions of intensity ratio ρ _λ = λ _p /λ _g . Second, the user association is biased toward the BSs located on a grid. Finally, the proposed model predicts the coverage probability of the actual deployment with great accuracy.

Entropy and Mutual Information of Point Processes

This paper is focused on information theoretic properties of point processes. Firstly, we discuss the entropy of a point process and the entropy rate of a stationary point process. Then we give explicit formulas for these quantities in the Poisson case, as well as maximal entropy properties for homogeneous Poisson point processes. Secondly, we define the mutual information rate of two stationary point processes. We then give explicit formulas for the mutual information rate between a homogeneous Poisson point process and its displacement.

Entropy Inequalities for Sums in Prime Cyclic Groups

Lower bounds for the Rényi entropies of sums of independent random variables taking values in cyclic groups of prime order, or in the integers, are established. The main ingredients of our approach are extended rearrangement inequalities in prime cyclic groups building on Lev (2001), and notions of stochastic ordering. Several applications are developed, including to discrete entropy power inequalities, the Littlewood-Offord problem, and counting solutions of certain linear systems.

Majorization and Renyi Entropy Inequalities via Sperner Theory

A natural link between the notions of majorization and strongly Sperner posets is elucidated. It is then used to obtain a variety of consequences, including new Rényi entropy inequalities for sums of independent, integer-valued random variables.