154 research outputs found
Controllability Metrics, Limitations and Algorithms for Complex Networks
This paper studies the problem of controlling complex networks, that is, the
joint problem of selecting a set of control nodes and of designing a control
input to steer a network to a target state. For this problem (i) we propose a
metric to quantify the difficulty of the control problem as a function of the
required control energy, (ii) we derive bounds based on the system dynamics
(network topology and weights) to characterize the tradeoff between the control
energy and the number of control nodes, and (iii) we propose an open-loop
control strategy with performance guarantees. In our strategy we select control
nodes by relying on network partitioning, and we design the control input by
leveraging optimal and distributed control techniques. Our findings show
several control limitations and properties. For instance, for Schur stable and
symmetric networks: (i) if the number of control nodes is constant, then the
control energy increases exponentially with the number of network nodes, (ii)
if the number of control nodes is a fixed fraction of the network nodes, then
certain networks can be controlled with constant energy independently of the
network dimension, and (iii) clustered networks may be easier to control
because, for sufficiently many control nodes, the control energy depends only
on the controllability properties of the clusters and on their coupling
strength. We validate our results with examples from power networks, social
networks, and epidemics spreading
Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs
We study a general framework for broadcast gossip algorithms which use
companion variables to solve the average consensus problem. Each node maintains
an initial state and a companion variable. Iterative updates are performed
asynchronously whereby one random node broadcasts its current state and
companion variable and all other nodes receiving the broadcast update their
state and companion variable. We provide conditions under which this scheme is
guaranteed to converge to a consensus solution, where all nodes have the same
limiting values, on any strongly connected directed graph. Under stronger
conditions, which are reasonable when the underlying communication graph is
undirected, we guarantee that the consensus value is equal to the average, both
in expectation and in the mean-squared sense. Our analysis uses tools from
non-negative matrix theory and perturbation theory. The perturbation results
rely on a parameter being sufficiently small. We characterize the allowable
upper bound as well as the optimal setting for the perturbation parameter as a
function of the network topology, and this allows us to characterize the
worst-case rate of convergence. Simulations illustrate that, in comparison to
existing broadcast gossip algorithms, the approaches proposed in this paper
have the advantage that they simultaneously can be guaranteed to converge to
the average consensus and they converge in a small number of broadcasts.Comment: 30 pages, submitte
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