Agreement over random networks

Submodularity in Dynamics and Control of Networked Systems Andrew ClarkCo-Chairs of the Supervisory Committee:Radha Poovendran Electrical Engineering Linda Bushnell Electrical EngineeringControlling a networked dynamical system to reach a desired state is a fundamental challenge in applications including transportation, energy, social, and biological systems. One scalable approach to controlling such systems is to directly control the states of a subset of leader nodes, while relying on local interactions to steer the remaining nodes towards their desired states. The choice of leader nodes is known to affect system metrics including robustness to noise, rate of convergence to a desired state, and controllability of the system.Selecting an optimal subset of leader nodes, however, is inherently a combinatorial problem, making optimal leader node selection intractable in general.This thesis presents a submodular optimization framework to selecting leader nodes for control of networked systems. We investigate the problem of selecting a subset of leader nodes in order to minimize node state errors due to noise in the communication links between nodes. We prove that the error due to link noise is a supermodular function of the set of leader nodes, leading to the first polynomial-time algorithms for minimizing error due to link noise with provable optimality bounds. We develop our approach for networks with static topologies, as well as dynamic topologies due to random link failures, switching between predefined topologies, and arbitrary mobility.We study selecting leader nodes in order to minimize convergence error, defined as the error in the intermediate node states prior to reaching their desired values. We derive upper bounds for a class of convergence error metrics based on the hitting time of random walk on the network, which we prove to be a supermodular function of the set of input nodes.We present polynomial-time algorithms for minimizing convergence error with provable optimality bounds, for static as well as dynamic networks.Efficient algorithms have recently been proposed for selecting leader nodes to satisfy controllability, defined as the ability to drive the non-input nodes from any initial state to any desired state in finite time. These algorithms, however, do not incorporate performance criteria including robustness to noise and convergence rate. We study the problem of leader selection for joint performance and controllability, and prove that controllability can be formulated as a matroid constraint on the set of leader nodes. We propose efficient algorithms with provable optimality gap for selecting leader nodes for joint performance and controllability, and characterize the submodular structure of the largest controllable subgraph of a network.We also investigate selecting input nodes for guaranteeing synchronization in networked systems. Using the widely-studied Kuramoto model of nonlinear phase-coupled oscillators, we develop novel threshold-based conditions for a set of input nodes to ...

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“…First, the nodes may experience link or device failures [59] . Second, the network may switch between prespecified topologies [101] .…”

Section: Leader Selection In Dynamic Networkmentioning

confidence: 99%

Submodularity in Dynamics and Control of Networked Systems

Clark

Alomair

Bushnell

et al. 2016

Communications and Control Engineering

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“…We begin this section by describing the stability matrix and the associated linear stability condition, i.e the analogue of (19), that arises by linearizing (2) around the two-dimensional simplex. We may then state the general d-dimensional problem as a straightforward generalization of the two-dimensional case.…”

Section: Problem Statement In Higher Dimensionsmentioning

confidence: 99%

“…The purely linear case g(s) ⌘ 1 of the system (2) appears in studies of consensus and synchronization algorithms on a random graph [24,36,19,37]. Related models also frequently demand a theoretical understanding how a (possibly dynamic) random network interaction structure a↵ects well-understood, deterministic behaviors such as phase transitions [1], consensus and synchronization [35,43], and the emergence of collective behavior in locust swarms [21].…”

Section: Introductionmentioning

confidence: 99%

Swarming on Random Graphs II

Brecht¹,

Sudakov

Bertozzi³

2014

J Stat Phys

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We consider an individual-based model where agents interact over a random network via firstorder dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents in R d this system has a lowest energy state in which an equal number of agents occupy the vertices of the d-dimensional simplex. The purpose of this paper is to study the behavior of this model when the interaction between the N agents occurs according to an Erdős-Rényi random graph G(N, p) instead of all-to-all coupling. In particular, we study the e↵ect of randomness on the stability of these simplicial solutions, and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than Np = O(log N ). In other words, only a relatively small number of interactions are required to maintain stability of the state. The results rely on basic probability arguments together with spectral properties of random graphs.

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“…Multi-vehicle consensus with time-varying reference state was discussed in [11]. Some other issues on consensus problem can be found in [12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Average Consensus in Networks of Multi-Agent with Multiple Time-Varying Delays

Zhang¹,

2010

IJCNS

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The average consensus in undirected networks of multi-agent with both fixed and switching topology coupling multiple time-varying delays is studied. By using orthogonal transformation techniques, the original system can be turned into a reduced dimensional system and then LMI-based method can be applied conveniently. Convergence analysis is conducted by constructing Lyapunov-Krasovskii function. Sufficient conditions on average consensus problem with multiple time-varying delays in undirected networks are obtained via linear matrix inequality (LMI) techniques. In particular, the maximal admissible upper bound of time-varying delays can be easily obtained by solving several simple and feasible LMIs. Finally, simulation examples are given to demonstrate the effectiveness of the theoretical results.

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Agreement over random networks

Cited by 332 publications

References 26 publications

Submodularity in Dynamics and Control of Networked Systems

Submodularity in Dynamics and Control of Networked Systems

Swarming on Random Graphs II

Average Consensus in Networks of Multi-Agent with Multiple Time-Varying Delays

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