John N. Tsitsiklis

Julien M. Hendrickx, John N. Tsitsiklis

We consider continuous-time consensus seeking systems whose time-dependent interactions are cut-balanced, in the following sense: if a group of agents influences the remaining ones, the former group is also influenced by the remaining ones by at least a proportional amount. Models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases. We prove that such systems always converge. We give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same. Conversely, we show that if our condition is not satisfied, then these limits are generically different. These results allow treating systems where the agent interactions are a priori unknown, e.g., random or determined endogenously by the agent values. We also derive corresponding results for discrete-time systems.

Julien M. Hendrickx, Alex Olshevsky, John N. Tsitsiklis

We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size. Furthermore, each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that satisfies this condition can at least be approximated. The problem of computing suitably rounded averages in a distributed manner plays a central role in our development; we provide an algorithm that solves it in time that grows quadratically with the size of the network.

Arsalan Sharifnassab, John N. Tsitsiklis, Jamaloddin Golestani

We consider a class of continuous-time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces, and which include the fluid-level dynamics of the Max-Weight scheduling policy as a special case. We study the effect of an external disturbance/perturbation on the state trajectory, and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation.

Arsalan Sharifnassab, John N. Tsitsiklis, S. Jamaloddin Golestani

Consider a partition of $R^n$ into finitely many polyhedral regions $D_i$ and associated drift vectors $μ_i\in R^n$. We study ``hybrid'' dynamical systems whose trajectories have a constant drift, $\dot x=μ_i$, whenever $x$ is in the interior of the $i$th region $D_i$, and behave consistently on the boundary between different regions. Our main result asserts that if such a system is nonexpansive (i.e., if the Euclidean distance between any pair of trajectories is a nonincreasing function of time), then the system must be conservative, i.e., its trajectories are the same as the trajectories of the negative subgradient flow associated with a potential function. Furthermore, this potential function is necessarily convex, and is linear on each of the regions $D_i$. We actually establish a more general version of this result, by making seemingly weaker assumptions on the dynamical system of interest.

Mihalis G. Markakis, Eytan Modiano, John N. Tsitsiklis

We carry out a delay stability analysis (i.e., determine conditions under which expected steady-state delays at a queue are finite) for a simple 3-queue system operated under the Max-Weight scheduling policy, for the case where one of the queues is fed by heavy-tailed traffic (i.e, when the number of arrivals at each time slot has infinite second moment). This particular system exemplifies an intricate phenomenon whereby heavy-tailed traffic at one queue may or may not result in the delay instability of another queue, depending on the arrival rates. While the ordinary stability region (in the sense of convergence to a steady-state distribution) is straightforward to determine, the determination of the delay stability region is more involved: (i) we use "fluid-type" sample path arguments, combined with renewal theory, to prove delay instability outside a certain region; (ii) we use a piecewise linear Lyapunov function to prove delay stability in the interior of that same region; (iii) as an intermediate step in establishing delay stability, we show that the expected workload of a stable M/GI/1 queue scales with time as $\mathcal{O}(t^{1/(1+γ)})$, assuming that service times have a finite $1+γ$ moment, where $γ\in (0,1)$.

Michael T. Schaub, Santiago Segarra, John N. Tsitsiklis

We consider a blind identification problem in which we aim to recover a statistical model of a network without knowledge of the network's edges, but based solely on nodal observations of a certain process. More concretely, we focus on observations that consist of single snapshots taken from multiple trajectories of a diffusive process that evolves over the unknown network. We model the network as generated from an independent draw from a latent stochastic block model (SBM), and our goal is to infer both the partition of the nodes into blocks, as well as the parameters of this SBM. We discuss some non-identifiability issues related to this problem and present simple spectral algorithms that provably solve the partition recovery and parameter estimation problems with high accuracy. Our analysis relies on recent results in random matrix theory and covariance estimation, and associated concentration inequalities. We illustrate our results with several numerical experiments.

John N. Tsitsiklis, Kuang Xu, Zhi Xu

We formulate a private learning model to study an intrinsic tradeoff between privacy and query complexity in sequential learning. Our model involves a learner who aims to determine a scalar value, $v^*$, by sequentially querying an external database and receiving binary responses. In the meantime, an adversary observes the learner's queries, though not the responses, and tries to infer from them the value of $v^*$. The objective of the learner is to obtain an accurate estimate of $v^*$ using only a small number of queries, while simultaneously protecting her privacy by making $v^*$ provably difficult to learn for the adversary. Our main results provide tight upper and lower bounds on the learner's query complexity as a function of desired levels of privacy and estimation accuracy. We also construct explicit query strategies whose complexity is optimal up to an additive constant.

Alex Olshevsky, John N. Tsitsiklis

We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.