4,695 research outputs found
Convergence of type-symmetric and cut-balanced consensus seeking systems (extended version)
We consider continuous-time consensus seeking systems whose time-dependent
interactions are cut-balanced, in the following sense: if a group of agents
influences the remaining ones, the former group is also influenced by the
remaining ones by at least a proportional amount. Models involving symmetric
interconnections and models in which a weighted average of the agent values is
conserved are special cases. We prove that such systems always converge. We
give a sufficient condition on the evolving interaction topology for the limit
values of two agents to be the same. Conversely, we show that if our condition
is not satisfied, then these limits are generically different. These results
allow treating systems where the agent interactions are a priori unknown, e.g.,
random or determined endogenously by the agent values. We also derive
corresponding results for discrete-time systems.Comment: update of the file following publication of journal version,
including a minor correction in the proof of theorem 1(b). 12 pages, 12 tex
files, no figur
Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling
Distributed algorithms of multi-agent coordination have attracted substantial
attention from the research community; the simplest and most thoroughly studied
of them are consensus protocols in the form of differential or difference
equations over general time-varying weighted graphs. These graphs are usually
characterized algebraically by their associated Laplacian matrices. Network
algorithms with similar algebraic graph theoretic structures, called being of
Laplacian-type in this paper, also arise in other related multi-agent control
problems, such as aggregation and containment control, target surrounding,
distributed optimization and modeling of opinion evolution in social groups. In
spite of their similarities, each of such algorithms has often been studied
using separate mathematical techniques. In this paper, a novel approach is
offered, allowing a unified and elegant way to examine many Laplacian-type
algorithms for multi-agent coordination. This approach is based on the analysis
of some differential or difference inequalities that have to be satisfied by
the some "outputs" of the agents (e.g. the distances to the desired set in
aggregation problems). Although such inequalities may have many unbounded
solutions, under natural graphic connectivity conditions all their bounded
solutions converge (and even reach consensus), entailing the convergence of the
corresponding distributed algorithms. In the theory of differential equations
the absence of bounded non-convergent solutions is referred to as the
equation's dichotomy. In this paper, we establish the dichotomy criteria of
Laplacian-type differential and difference inequalities and show that these
criteria enable one to extend a number of recent results, concerned with
Laplacian-type algorithms for multi-agent coordination and modeling opinion
formation in social groups.Comment: accepted to Automatic
Continuous-time consensus under persistent connectivity and slow divergence of reciprocal interaction weights
In this paper, we present new results on consensus for continuous-time multi-
agent systems. We introduce the assumptions of persistent connectivity of the
interaction graph and of slow divergence of reciprocal interaction weights.
Persistent connectivity can be considered as the counterpart of the notion of
ultimate connectivity used in discrete- time consensus protocols. Slow
divergence of reciprocal interaction weights generalizes the assumption of
cut-balanced interactions. We show that under these two assumptions, the
continuous-time consensus protocol succeeds: the states of all the agents
converge asymptotically to a common value. Moreover, our proof allows us to
give an estimate of the rate of convergence towards the consensus. We also
provide two examples that make us think that both of our assumptions are tight
Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization
Most of the distributed protocols for multi-agent consensus assume that the
agents are mutually cooperative and "trustful," and so the couplings among the
agents bring the values of their states closer. Opinion dynamics in social
groups, however, require beyond these conventional models due to ubiquitous
competition and distrust between some pairs of agents, which are usually
characterized by repulsive couplings and may lead to clustering of the
opinions. A simple yet insightful model of opinion dynamics with both
attractive and repulsive couplings was proposed recently by C. Altafini, who
examined first-order consensus algorithms over static signed graphs. This
protocol establishes modulus consensus, where the opinions become the same in
modulus but may differ in signs. In this paper, we extend the modulus consensus
model to the case where the network topology is an arbitrary time-varying
signed graph and prove reaching modulus consensus under mild sufficient
conditions of uniform connectivity of the graph. For cut-balanced graphs, not
only sufficient, but also necessary conditions for modulus consensus are given.Comment: scheduled for publication in IEEE Transactions on Automatic Control,
2016, vol. 61, no. 7 (accepted in August 2015
Continuous-Time Consensus under Non-Instantaneous Reciprocity
We consider continuous-time consensus systems whose interactions satisfy a
form or reciprocity that is not instantaneous, but happens over time. We show
that these systems have certain desirable properties: They always converge
independently of the specific interactions taking place and there exist simple
conditions on the interactions for two agents to converge to the same value.
This was until now only known for systems with instantaneous reciprocity. These
result are of particular relevance when analyzing systems where interactions
are a priori unknown, being for example endogenously determined or random. We
apply our results to an instance of such systems.Comment: 12 pages, 4 figure
On Endogenous Random Consensus and Averaging Dynamics
Motivated by various random variations of Hegselmann-Krause model for opinion
dynamics and gossip algorithm in an endogenously changing environment, we
propose a general framework for the study of endogenously varying random
averaging dynamics, i.e.\ an averaging dynamics whose evolution suffers from
history dependent sources of randomness. We show that under general assumptions
on the averaging dynamics, such dynamics is convergent almost surely. We also
determine the limiting behavior of such dynamics and show such dynamics admit
infinitely many time-varying Lyapunov functions
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