Venkat Anantharam

Barlas Oğuz, Venkat Anantharam, Ilkka Norros · meta-ai

P2P systems provide a scalable solution for distributing large files in a network. The file is split into many chunks, and peers contact other peers to collect missing chunks to eventually complete the entire file. The so-called `rare chunk' phenomenon, where a single chunk becomes rare and prevents peers from completing the file, is a threat to the stability of such systems. Practical systems such as BitTorrent overcome this issue by requiring a global search for the rare chunk, which necessitates a centralized mechanism. We demonstrate a new system based on an approximate rare-chunk rule, allowing for completely distributed file sharing while retaining scalability and stability. We assume non-altruistic peers and the seed is required to make only a minimal contribution.

Qinghua, Ding, Venkat Anantharam

Reinforced random walks are random walks on graphs whose transition probabilities along edges from a vertex are proportional to the weights of those edges, but where the weight of an edge evolves in a way that depends on the past traversals across it. In an edge-reinforced random walk (ERRW), the weight of an edge increases by $1$ whenever that edge is traversed, in either direction. On a finite graph, an ERRW admits a remarkable representation as a random walk in a random environment. The law of the environment is given by the so-called {\em magic formula}, with this law depending on the initial edge weights. This representation provides a natural route for studying statistical properties of ERRWs. This work focuses on various information-theoretic quantities associated with ERRWs on finite graphs, motivated in part by the problem of statistically distinguishing between different ERRW models from observed trajectories. In particular, we study the entropy rate of an ERRW. We also study the Kullback--Leibler divergence (KL divergence) between two ERRW environment laws, and the KL divergence between the corresponding finite-trajectory distributions. Leveraging structural properties of the underlying random environment, we derive an annealed representation of the entropy rate, a closed-form formula for the environment-level KL divergence, and quantitative bounds on the convergence of trajectory-level KL divergence toward environment-level KL divergence. These information-theoretic quantities are motivated by the two-point hypothesis testing problem for ERRW trajectories, and in particular by the associated Stein exponent. We also expect them to play a fundamental role in the study of other testing problems for ERRWs, including identity testing and closeness testing.

Venkat Anantharam, Francois Baccelli

We study an N-player game where a pure action of each player is to select a non-negative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of its open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and comprised of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.