16 research outputs found
Distributed anonymous discrete function computation
We propose a model for deterministic distributed function computation by a
network of identical and anonymous nodes. In this model, each node has bounded
computation and storage capabilities that do not grow with the network size.
Furthermore, each node only knows its neighbors, not the entire graph. Our goal
is to characterize the class of functions that can be computed within this
model. In our main result, we provide a necessary condition for computability
which we show to be nearly sufficient, in the sense that every function that
satisfies this condition can at least be approximated. The problem of computing
suitably rounded averages in a distributed manner plays a central role in our
development; we provide an algorithm that solves it in time that grows
quadratically with the size of the network
An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs
We analyze a class of distributed quantized consensus algorithms for
arbitrary static networks. In the initial setting, each node in the network has
an integer value. Nodes exchange their current estimate of the mean value in
the network, and then update their estimation by communicating with their
neighbors in a limited capacity channel in an asynchronous clock setting.
Eventually, all nodes reach consensus with quantized precision. We analyze the
expected convergence time for the general quantized consensus algorithm
proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric
networks, random walks, and couplings of Markov chains to derive an upper bound for the expected convergence time on an arbitrary graph of size
, improving on the state of art bound of for quantized consensus
algorithms. Our result is not dependent on graph topology. Example of complete
graphs is given to show how to extend the analysis to graphs of given topology.Comment: to appear in IEEE Trans. on Automatic Control, January, 2015. arXiv
admin note: substantial text overlap with arXiv:1208.078
MAX-consensus in open multi-agent systems with gossip interactions
We study the problem of distributed maximum computation in an open
multi-agent system, where agents can leave and arrive during the execution of
the algorithm. The main challenge comes from the possibility that the agent
holding the largest value leaves the system, which changes the value to be
computed. The algorithms must as a result be endowed with mechanisms allowing
to forget outdated information. The focus is on systems in which interactions
are pairwise gossips between randomly selected agents. We consider situations
where leaving agents can send a last message, and situations where they cannot.
For both cases, we provide algorithms able to eventually compute the maximum of
the values held by agents.Comment: To appear in the proceedings of the 56th IEEE Conference on Decision
and Control (CDC 17). 8 pages, 3 figure
Tight bounds on the convergence rate of generalized ratio consensus algorithms
The problems discussed in this paper are motivated by general ratio consensus
algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as
the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the
name weighted gossip algorithm. We consider a communication protocol described
by a strictly stationary, ergodic, sequentially primitive sequence of
non-negative matrices, applied iteratively to a pair of fixed initial vectors,
the components of which are called values and weights defined at the nodes of a
network. The subject of ratio consensus problems is to study the asymptotic
properties of ratios of values and weights at each node, expecting convergence
to the same limit for all nodes. The main results of the paper provide upper
bounds for the rate of the almost sure exponential convergence in terms of the
spectral gap associated with the given sequence of random matrices. It will be
shown that these upper bounds are sharp. Our results complement previous
results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013)
Global Network Prediction from Local Node Dynamics
The study of dynamical systems on networks, describing complex interactive
processes, provides insight into how network structure affects global
behaviour. Yet many methods for network dynamics fail to cope with large or
partially-known networks, a ubiquitous situation in real-world applications.
Here we propose a localised method, applicable to a broad class of dynamical
models on networks, whereby individual nodes monitor and store the evolution of
their own state and use these values to approximate, via a simple computation,
their own steady state solution. Hence the nodes predict their own final state
without actually reaching it. Furthermore, the localised formulation enables
nodes to compute global network metrics without knowledge of the full network
structure. The method can be used to compute global rankings in the network
from local information; to detect community detection from fast, local
transient dynamics; and to identify key nodes that compute global network
metrics ahead of others. We illustrate some of the applications of the
algorithm by efficiently performing web-page ranking for a large internet
network and identifying the dynamic roles of inter-neurons in the C. Elegans
neural network. The mathematical formulation is simple, widely applicable and
easily scalable to real-world datasets suggesting how local computation can
provide an approach to the study of large-scale network dynamics
Know your audience
Distributed function computation is the problem, for a networked system of
autonomous agents, to collectively compute the value
of some input values, each initially private to one agent in the network. Here,
we study and organize results pertaining to distributed function computation in
anonymous networks, both for the static and the dynamic case, under a
communication model of directed and synchronous message exchanges, but with
varying assumptions in the degree of awareness or control that a single agent
has over its outneighbors.
Our main argument is three-fold. First, in the "blind broadcast" model, where
in each round an agent merely casts out a unique message without any knowledge
or control over its addressees, the computable functions are those that only
depend on the set of the input values, but not on their multiplicities or
relative frequencies in the input. Second, in contrast, when we assume either
that a) in each round, the agents know how many outneighbors they have; b) all
communications links in the network are bidirectional; or c) the agents may
address each of their outneighbors individually, then the set of computable
functions grows to contain all functions that depend on the relative
frequencies of each value in the input - such as the average - but not on their
multiplicities - thus, not the sum. Third, however, if one or several agents
are distinguished as leaders, or if the cardinality of the network is known,
then under any of the above three assumptions it becomes possible to recover
the complete multiset of the input values, and thus compute any function of the
distributed input as long as it is invariant under permutation of its
arguments. In the case of dynamic networks, we also discuss the impact of
multiple connectivity assumptions
The Power of Small Coalitions under Two-Tier Majority on Regular Graphs
In this paper, we study the following problem. Consider a setting where a
proposal is offered to the vertices of a given network , and the vertices
must conduct a vote and decide whether to accept the proposal or reject it.
Each vertex has its own valuation of the proposal; we say that is
``happy'' if its valuation is positive (i.e., it expects to gain from adopting
the proposal) and ``sad'' if its valuation is negative. However, vertices do
not base their vote merely on their own valuation. Rather, a vertex is a
\emph{proponent} of the proposal if the majority of its neighbors are happy
with it and an \emph{opponent} in the opposite case. At the end of the vote,
the network collectively accepts the proposal whenever the majority of its
vertices are proponents. We study this problem for regular graphs with loops.
Specifically, we consider the class of -regular graphs
of odd order with all loops and happy vertices. We are interested
in establishing necessary and sufficient conditions for the class
to contain a labeled graph accepting the proposal, as
well as conditions to contain a graph rejecting the proposal. We also discuss
connections to the existing literature, including that on majority domination,
and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied
Mathematic