79 research outputs found
Products of Generalized Stochastic Sarymsakov Matrices
In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class
of stochastic Sarymsakov matrices is the largest known subset (i) that is
closed under matrix multiplication and (ii) the infinitely long left-product of
the elements from a compact subset converges to a rank-one matrix. In this
paper, we show that a larger subset with these two properties can be derived by
generalizing the standard definition for Sarymsakov matrices. The
generalization is achieved either by introducing an "SIA index", whose value is
one for Sarymsakov matrices, and then looking at those stochastic matrices with
larger SIA indices, or by considering matrices that are not even SIA. Besides
constructing a larger set, we give sufficient conditions for generalized
Sarymsakov matrices so that their products converge to rank-one matrices. The
new insight gained through studying generalized Sarymsakov matrices and their
products has led to a new understanding of the existing results on consensus
algorithms and will be helpful for the design of network coordination
algorithms
Random Asynchronous Iterations in Distributed Coordination Algorithms
Distributed coordination algorithms (DCA) carry out information processing
processes among a group of networked agents without centralized information
fusion. Though it is well known that DCA characterized by an SIA (stochastic,
indecomposable, aperiodic) matrix generate consensus asymptotically via
synchronous iterations, the dynamics of DCA with asynchronous iterations have
not been studied extensively, especially when viewed as stochastic processes.
This paper aims to show that for any given irreducible stochastic matrix, even
non-SIA, the corresponding DCA lead to consensus successfully via random
asynchronous iterations under a wide range of conditions on the transition
probability. Particularly, the transition probability is neither required to be
independent and identically distributed, nor characterized by a Markov chain
How to decide consensus? A combinatorial necessary and sufficient condition and a proof that consensus is decidable but NP-hard
A set of stochastic matrices is a consensus set if for every
sequence of matrices whose elements belong to
and every initial state , the sequence of states defined by converges to a vector whose entries are all identical.
In this paper, we introduce an "avoiding set condition" for compact sets of
matrices and prove in our main theorem that this explicit combinatorial
condition is both necessary and sufficient for consensus. We show that several
of the conditions for consensus proposed in the literature can be directly
derived from the avoiding set condition. The avoiding set condition is easy to
check with an elementary algorithm, and so our result also establishes that
consensus is algorithmically decidable. Direct verification of the avoiding set
condition may require more than a polynomial time number of operations. This is
however likely to be the case for any consensus checking algorithm since we
also prove in this paper that unless , consensus cannot be decided in
polynomial time
Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation
This paper presents new sufficient conditions for convergence and asymptotic
or exponential stability of a stochastic discrete-time system, under which the
constructed Lyapunov function always decreases in expectation along the
system's solutions after a finite number of steps, but without necessarily
strict decrease at every step, in contrast to the classical stochastic Lyapunov
theory. As the first application of this new Lyapunov criterion, we look at the
product of any random sequence of stochastic matrices, including those with
zero diagonal entries, and obtain sufficient conditions to ensure the product
almost surely converges to a matrix with identical rows; we also show that the
rate of convergence can be exponential under additional conditions. As the
second application, we study a distributed network algorithm for solving linear
algebraic equations. We relax existing conditions on the network structures,
while still guaranteeing the equations are solved asymptotically.Comment: 14 pages, 1 figur
Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks
In-depth understanding of the spectral properties of grounded Laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks, such as opinion dynamics in social networks with stubborn agents. We focus on grounded Laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real. Power and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory. For those eigenvectors corresponding to such eigenvalues, we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component. We then discuss an application in leader-follower social networks where the grounded Laplacian matrices arise naturally. With the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue, we prove that by removing or weakening specic directed couplings pointing to the vertex having the smallest eigenvector component, all the states of the other vertices converge faster to that of the leading vertex. This result is in sharp contrast to the well-known fact that when the vertices are connected together through undirected links, removing or weakening links does not accelerate and in general decelerates the converging process
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