54 research outputs found
Distributed Agreement on Activity Driven Networks
In this paper, we investigate asymptotic properties of a consensus protocol
taking place in a class of temporal (i.e., time-varying) networks called the
activity driven network. We first show that a standard methodology provides us
with an estimate of the convergence rate toward the consensus, in terms of the
eigenvalues of a matrix whose computational cost grows exponentially fast in
the number of nodes in the network. To overcome this difficulty, we then derive
alternative bounds involving the eigenvalues of a matrix that is easy to
compute. Our analysis covers the regimes of 1) sparse networks and 2)
fast-switching networks. We numerically confirm our theoretical results by
numerical simulations
On the mean square error of randomized averaging algorithms
This paper regards randomized discrete-time consensus systems that preserve
the average "on average". As a main result, we provide an upper bound on the
mean square deviation of the consensus value from the initial average. Then, we
apply our result to systems where few or weakly correlated interactions take
place: these assumptions cover several algorithms proposed in the literature.
For such systems we show that, when the network size grows, the deviation tends
to zero, and the speed of this decay is not slower than the inverse of the
size. Our results are based on a new approach, which is unrelated to the
convergence properties of the system.Comment: 11 pages. to appear as a journal publicatio
When gossip meets consensus : convergence in correlated random WSNs
Peer ReviewedPostprint (author’s final draft
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
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
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