1,672 research outputs found
Convergence results for the linear consensus problem under Markovian random graphs
This note discusses the linear discrete and continuous time consensus problem for a network of
dynamic agents with directed information flows and random switching topologies. The switching is
determined by a Markov chain, each topology corresponding to a state of the Markov chain. We show
that, under doubly stochastic assumption on the matrices involved in the linear consensus scheme,
average consensus is achieved in the mean square sense and almost surely if and only if the graph resulted
from the union of graphs corresponding to the states of the Markov chain is strongly connected. The
aim of this note is to show how techniques from Markovian jump linear systems theory, in conjunction
with results inspired by matrix and graph theory, can be used to prove convergence results for stochastic
consensus problems.
How Homophily Affects Learning and Diffusion in Networks
We examine how three different communication processes operating through social networks are affected by homophily - the tendency of individuals to associate with others similar to themselves. Homophily has no effect if messages are broadcast or sent via shortest paths; only connection density matters. In contrast, homophily substantially slows learning based on repeated averaging of neighbors' information and Markovian diffusion processes such as the Google random surfer model. Indeed, the latter processes are strongly affected by homophily but completely independent of connection density, provided this density exceeds a low threshold. We obtain these results by establishing new results on the spectra of large random graphs and relating the spectra to homophily. We conclude by checking the theoretical predictions using observed high school friendship networks from the Adolescent Health dataset.Networks, Learning, Diffusion, Homophily, Friendships, Social Networks, Random Graphs, Mixing Time, Convergence, Speed of Learning, Speed of Convergence
Distributing the Kalman Filter for Large-Scale Systems
This paper derives a \emph{distributed} Kalman filter to estimate a sparsely
connected, large-scale, dimensional, dynamical system monitored by a
network of sensors. Local Kalman filters are implemented on the
(dimensional, where ) sub-systems that are obtained after
spatially decomposing the large-scale system. The resulting sub-systems
overlap, which along with an assimilation procedure on the local Kalman
filters, preserve an th order Gauss-Markovian structure of the centralized
error processes. The information loss due to the th order Gauss-Markovian
approximation is controllable as it can be characterized by a divergence that
decreases as . The order of the approximation, , leads to a lower
bound on the dimension of the sub-systems, hence, providing a criterion for
sub-system selection. The assimilation procedure is carried out on the local
error covariances with a distributed iterate collapse inversion (DICI)
algorithm that we introduce. The DICI algorithm computes the (approximated)
centralized Riccati and Lyapunov equations iteratively with only local
communication and low-order computation. We fuse the observations that are
common among the local Kalman filters using bipartite fusion graphs and
consensus averaging algorithms. The proposed algorithm achieves full
distribution of the Kalman filter that is coherent with the centralized Kalman
filter with an th order Gaussian-Markovian structure on the centralized
error processes. Nowhere storage, communication, or computation of
dimensional vectors and matrices is needed; only dimensional
vectors and matrices are communicated or used in the computation at the
sensors
Symmetrization for Quantum Networks: a continuous-time approach
In this paper we propose a continuous-time, dissipative Markov dynamics that
asymptotically drives a network of n-dimensional quantum systems to the set of
states that are invariant under the action of the subsystem permutation group.
The Lindblad-type generator of the dynamics is built with two-body subsystem
swap operators, thus satisfying locality constraints, and preserve symmetric
observables. The potential use of the proposed generator in combination with
local control and measurement actions is illustrated with two applications: the
generation of a global pure state and the estimation of the network size.Comment: submitted to MTNS 201
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