14,967 research outputs found
Randomized Consensus with Attractive and Repulsive Links
We study convergence properties of a randomized consensus algorithm over a
graph with both attractive and repulsive links. At each time instant, a node is
randomly selected to interact with a random neighbor. Depending on if the link
between the two nodes belongs to a given subgraph of attractive or repulsive
links, the node update follows a standard attractive weighted average or a
repulsive weighted average, respectively. The repulsive update has the opposite
sign of the standard consensus update. In this way, it counteracts the
consensus formation and can be seen as a model of link faults or malicious
attacks in a communication network, or the impact of trust and antagonism in a
social network. Various probabilistic convergence and divergence conditions are
established. A threshold condition for the strength of the repulsive action is
given for convergence in expectation: when the repulsive weight crosses this
threshold value, the algorithm transits from convergence to divergence. An
explicit value of the threshold is derived for classes of attractive and
repulsive graphs. The results show that a single repulsive link can sometimes
drastically change the behavior of the consensus algorithm. They also
explicitly show how the robustness of the consensus algorithm depends on the
size and other properties of the graphs
Lyapunov Approach to Consensus Problems
This paper investigates the weighted-averaging dynamic for unconstrained and
constrained consensus problems. Through the use of a suitably defined adjoint
dynamic, quadratic Lyapunov comparison functions are constructed to analyze the
behavior of weighted-averaging dynamic. As a result, new convergence rate
results are obtained that capture the graph structure in a novel way. In
particular, the exponential convergence rate is established for unconstrained
consensus with the exponent of the order of . Also, the
exponential convergence rate is established for constrained consensus, which
extends the existing results limited to the use of doubly stochastic weight
matrices
Dynamics over Signed Networks
A signed network is a network with each link associated with a positive or
negative sign. Models for nodes interacting over such signed networks, where
two different types of interactions take place along the positive and negative
links, respectively, arise from various biological, social, political, and
economic systems. As modifications to the conventional DeGroot dynamics for
positive links, two basic types of negative interactions along negative links,
namely the opposing rule and the repelling rule, have been proposed and studied
in the literature. This paper reviews a few fundamental convergence results for
such dynamics over deterministic or random signed networks under a unified
algebraic-graphical method. We show that a systematic tool of studying node
state evolution over signed networks can be obtained utilizing generalized
Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie
Variance Analysis of Randomized Consensus in Switching Directed Networks
In this paper, we study the asymptotic properties of distributed consensus
algorithms over switching directed random networks. More specifically, we focus
on consensus algorithms over independent and identically distributed, directed
Erdos-Renyi random graphs, where each agent can communicate with any other
agent with some exogenously specified probability . While it is well-known
that consensus algorithms over Erdos-Renyi random networks result in an
asymptotic agreement over the network, an analytical characterization of the
distribution of the asymptotic consensus value is still an open question. In
this paper, we provide closed-form expressions for the mean and variance of the
asymptotic random consensus value, in terms of the size of the network and the
probability of communication . We also provide numerical simulations that
illustrate our results.Comment: 6 pages, 3 figures, submitted to American Control Conference 201
Emergent Behaviors over Signed Random Networks in Dynamical Environments
We study asymptotic dynamical patterns that emerge among a set of nodes that
interact in a dynamically evolving signed random network. Node interactions
take place at random on a sequence of deterministic signed graphs. Each node
receives positive or negative recommendations from its neighbors depending on
the sign of the interaction arcs, and updates its state accordingly. Positive
recommendations follow the standard consensus update while two types of
negative recommendations, each modeling a different type of antagonistic or
malicious interaction, are considered. Nodes may weigh positive and negative
recommendations differently, and random processes are introduced to model the
time-varying attention that nodes pay to the positive and negative
recommendations. Various conditions for almost sure convergence, divergence,
and clustering of the node states are established. Some fundamental
similarities and differences are established for the two notions of negative
recommendations
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