22 research outputs found
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
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
Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability
Distributed consensus and other linear systems with system stochastic
matrices emerge in various settings, like opinion formation in social
networks, rendezvous of robots, and distributed inference in sensor networks.
The matrices are often random, due to, e.g., random packet dropouts in
wireless sensor networks. Key in analyzing the performance of such systems is
studying convergence of matrix products . In this paper, we
find the exact exponential rate for the convergence in probability of the
product of such matrices when time grows large, under the assumption that
the 's are symmetric and independent identically distributed in time.
Further, for commonly used random models like with gossip and link failure, we
show that the rate is found by solving a min-cut problem and, hence, easily
computable. Finally, we apply our results to optimally allocate the sensors'
transmission power in consensus+innovations distributed detection
Eminence Grise Coalitions: On the Shaping of Public Opinion
We consider a network of evolving opinions. It includes multiple individuals
with first-order opinion dynamics defined in continuous time and evolving based
on a general exogenously defined time-varying underlying graph. In such a
network, for an arbitrary fixed initial time, a subset of individuals forms an
eminence grise coalition, abbreviated as EGC, if the individuals in that subset
are capable of leading the entire network to agreeing on any desired opinion,
through a cooperative choice of their own initial opinions. In this endeavor,
the coalition members are assumed to have access to full profile of the
underlying graph of the network as well as the initial opinions of all other
individuals. While the complete coalition of individuals always qualifies as an
EGC, we establish the existence of a minimum size EGC for an arbitrary
time-varying network; also, we develop a non-trivial set of upper and lower
bounds on that size. As a result, we show that, even when the underlying graph
does not guarantee convergence to a global or multiple consensus, a generally
restricted coalition of agents can steer public opinion towards a desired
global consensus without affecting any of the predefined graph interactions,
provided they can cooperatively adjust their own initial opinions. Geometric
insights into the structure of EGC's are given. The results are also extended
to the discrete time case where the relation with Decomposition-Separation
Theorem is also made explicit.Comment: 35 page