Mohammad Amin Rahimian

Minimal Actuator Placement With Bounds on Control Effort

We address the problem of minimal actuator placement in a linear system subject to an average control energy bound. First, following the recent work of Olshevsky, we prove that this is NP-hard. Then, we provide an efficient algorithm which, for a given range of problem parameters, approximates up to a multiplicative factor of O(log n), with n being the network size, any optimal actuator set that meets the same energy criteria; this is the best approximation factor one can achieve in polynomial time in the worst case. Moreover, the algorithm uses a perturbed version of the involved control energy metric, which we prove to be supermodular. Next, we focus on the related problem of cardinality-constrained actuator placement for minimum control effort, where the optimal actuator set is selected so that an average input energy metric is minimized. While this is also an NP-hard problem, we use our proposed algorithm to efficiently approximate its solutions as well. Finally, we run our algorithms over large random networks to illustrate their efficiency.

Learning without recall by random walks on directed graphs

We consider a network of agents that aim to learn some unknown state of the world using private observations and exchange of beliefs. At each time, agents observe private signals generated based on the true unknown state. Each agent might not be able to distinguish the true state based only on her private observations. This occurs when some other states are observationally equivalent to the true state from the agents perspective. To overcome this shortcoming, agents must communicate with each other to benefit from local observations. We propose a model where each agent selects one of her neighbors randomly at each time. Then, she refines her opinion using her private signal and the prior of that particular neighbor. The proposed rule can be thought of as a Bayesian agent who cannot recall the priors based on which other agents make inferences. This learning without recall approach preserves some aspects of the Bayesian inference while being computationally tractable. By establishing a correspondence with a random walk on the network graph, we prove that under the described protocol, agents learn the truth exponentially fast in the almost sure sense. The asymptotic rate is expressed as the sum of the relative entropies between the signal structures of every agent weighted by the stationary distribution of the random walk.

Minimal actuator placement with optimal control constraints

We introduce the problem of minimal actuator placement in a linear control system so that a bound on the minimum control effort for a given state transfer is satisfied while controllability is ensured. We first show that this is an NP-hard problem following the recent work of Olshevsky [1]. Next, we prove that this problem has a supermodular structure. Afterwards, we provide an efficient algorithm that approximates up to a multiplicative factor of O(log n), where n is the size of the multi-agent network, any optimal actuator set that meets the specified energy criterion. Moreover, we show that this is the best approximation factor one can achieve in polynomial-time for the worst case. Finally, we test this algorithm over large Erdös-Rényi random networks to further demonstrate its efficiency.

Switching to learn

A network of agents attempt to learn some unknown state of the world drawn by nature from a finite set. Agents observe private signals conditioned on the true state, and form beliefs about the unknown state accordingly. Each agent may face an identification problem in the sense that she cannot distinguish the truth in isolation. However, by communicating with each other, agents are able to benefit from side observations to learn the truth collectively. Unlike many distributed algorithms which rely on all-time communication protocols, we propose an efficient method by switching between Bayesian and non-Bayesian regimes. In this model, agents exchange information only when their private signals are not informative enough; thence, by switching between the two regimes, agents efficiently learn the truth using only a few rounds of communications. The proposed algorithm preserves learnability while incurring a lower communication cost. We also verify our theoretical findings by simulation examples.

Detection and Isolation of Failures in Directed Networks of LTI Systems

We propose a methodology to detect and isolate link failures in a weighted and directed network of identical multi-input multioutput linear time-invariant (LTI) systems when only the output responses of a subset of nodes are available. Our method is based on the observation of jump discontinuities in the output derivatives, which can be explicitly related to the occurrence of link failures. The order of the derivative at which the jump is observed is given by r(d+1), where r is the relative degree of each systems transfer matrix, and d denotes the distance from the location of the failure to the observation point. We then propose detection and isolation strategies based on this relation. Furthermore, we propose an efficient algorithm for sensor placement to detect and isolate any possible link failure using a small number of sensors. Available results from the theory of submodular set functions provide us with performance guarantees that bound the size of the chosen sensor set within a logarithmic factor of the smallest feasible set of sensors. These results are illustrated through elaborative examples and supplemented by computer experiments.

Controllability and fraction of leaders in infinite networks

In this paper, we study controllability of a network of linear single-integrator agents when the network size goes to infinity. We first investigate the effect of increasing size by injecting an input at every node and requiring that network controllability Gramian remain well-conditioned with the increasing dimension. We provide theoretical justification to the intuition that high degree nodes pose a challenge to network controllability. In particular, the controllability Gramian for the networks with bounded maximum degrees is shown to remain well-conditioned even as the network size goes to infinity. In the canonical cases of star, chain and ring networks, we also provide closed-form expressions which bound the condition number of the controllability Gramian in terms of the network size. We next consider the effect of the choice and number of leader nodes by actuating only a subset of nodes and considering the least eigenvalue of the Gramian as the network size increases. Accordingly, while a directed star topology can never be made controllable for all sizes by injecting an input just at a fraction f <; 1 of nodes; for path or cycle networks, the designer can actuate a non-zero fraction of nodes and spread them throughout the network in such way that the least eigenvalue of the Gramians remain bounded away from zero with the increasing size. The results offer interesting insights on the challenges of control in large networks and with high-degree nodes.

Naive social learning in Ising networks

We analyze a model of learning and belief formation in networks where agents attempt to maximize their state-dependent utilities by their choice of actions, while being unaware of the true state. They do so by making rational inferences about their observations which include a sequence of independent and identically distributed private signals as well as the decisions of their neighboring agents at each time. Successive applications of Bayes rule to the entire history of past observations lead to forebodingly complex inferences due to lack of knowledge about the global network structure that is causing those observations. To address these complexities, we consider a Bayesian without Recall (BWR) model of inference, which in addition to providing a tractable framework for analyzing the behavior of rational agents in social networks, can also provide a behavioral foundation for the variety of non-Bayesian update rules in the literature. We specialize the model to the case of binary state and action spaces and show that the action updates in this case take the form of a weighted majority and threshold function leading to an Ising model. We analyze the evolution of action profiles under the derived rules and investigate behavioral implications that are of interest in our model, including consensus, learning and emergence of experts and opinion leaders.

Failure detection and isolation in integrator networks

Detection and isolation of link failures in directed networks of LTI systems have been the focus of our previous study. Our results relate the failure of links in the network to jump discontinuities in the derivatives of the output responses of the nodes and exploit this relation to propose failure detection and isolation (FDI) techniques, accordingly. In this work, we consider these results in the context of single-integrator networked dynamics, and show that with the additional niceties of the integrator networks and the enhanced proofs, one is able to incorporate both unidirectional and bidirectional link failures in our FDI algorithms. Computer experiments with large networks and both directed and undirected topologies provide interesting insights as to the role of directionality, as well as the scalability of the proposed FDI techniques with the network size.