Consider a countably infinite collection of interacting queues, with a queue located at each point of the d-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type, with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.invariant. Conditional on the queue lengths {x i (t)} i∈Z d at time t, the instantaneous departure rate from any queue i at time t is given by, with 0/0 interpreted as being equal to 0. Note that since the interference sequence {a i } i∈Z d is non-negative, and a 0 = 1, for all t ∈ R and all i ∈ Z d , the instantaneous departure rate from queue i at time t is 1 j∈Z d a j x i−j (t) . This can be viewed as the low 'Signal-to-Noise-and-Interference-Ratio (SINR)' channel capacity of a point-to-point Gaussian channel (see [12]). Since there are x i (t) links simultaneously transmitting, and each of them has an independent unit mean exponentially distributed file, the rate at which a link departs is thenThe instantaneous rate of transmission of a link is lowered if it is in a 'crowded' area of space, due to interference, and hence it takes longer for this link to complete the transmission of its file. In the meantime, it is more likely that a new link will arrive at some point nearby before it finishes transmitting, further reducing the rate of transmission. Understanding how the network evolves due to such spatio temporal interference dynamics is crucial in designing and provisioning of wireless systems (see discussions in [29]).Corollary 1.4. If λ < 2 3 1+c j∈Z d a j , where c is given in Proposition 1.3, then {x i;∞ (0)} i∈Z d is the unique translation invariant stationary solution with finite second moment.Our next set of results assesses whether queue length process converges to any stationary solution when started from different st...

We introduce a novel decentralized, multi agent version of the classical Multi-Arm Bandit (MAB) problem, consisting of n agents, that collaboratively and simultaneously solve the same instance of K armed MAB to minimize individual regret. The agents can communicate and collaborate among each other only through a pairwise asynchronous gossip based protocol that exchange a limited number of bits. In our model, agents at each point decide on (i) which arm to play, (ii) whether to, and if so (iii) what and whom to communicate with.

We study the problem of community detection on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a R d valued location label which forms the support of a Poisson point process of intensity λ. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial and exact, using the random graph and the location labels. We establish phase transitions for both sparse and logarithmic degree regimes, and provide bounds on the location of the thresholds, conjectured to be tight in the case of exact recovery. We also show that the threshold of the distinguishability problem, i.e., the testing between our model and the null model without community labels exhibits no phase-transition and in particular, does not match the weak recovery threshold (in contrast to the SBM).

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We consider a decentralized multi-agent Multi Armed Bandit (MAB) setup consisting of N agents, solving the same MAB instance to minimize individual cumulative regret. In our model, agents collaborate by exchanging messages through pairwise gossip style communications on an arbitrary connected graph. We develop two novel algorithms, where each agent only plays from a subset of all the arms. Agents use the communication medium to recommend only arm-IDs (not samples), and thus update the set of arms from which they play. We establish that, if agents communicate Ω(log(T )) times through any connected pairwise gossip mechanism, then every agent's regret is a factor of order N smaller compared to the case of no collaborations. Furthermore, we show that the communication constraints only have a second order effect on the regret of our algorithm. We then analyze this second order term of the regret to derive bounds on the regret-communication tradeoffs. Finally, we empirically evaluate our algorithm and conclude that the insights are fundamental and not artifacts of our bounds. We also show a lower bound which gives that the regret scaling obtained by our algorithm cannot be improved even in the absence of any communication constraints. Our results demonstrate that even a minimal level of collaboration among agents greatly reduces regret for all agents.