35 research outputs found
Hamilton Cycles in a Class of Random Directed Graphs
AbstractWe prove that almost every 3-in, 3-out digraph is Hamiltonian
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
Discordant voting processes on finite graphs
We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n2/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = =(n log n), whereas the pull protocol has ET = =(2n). For the cycle Cn all three protocols have ET = =(n2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol is slower with ET = =(n2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders
The Power of Two Choices in Distributed Voting
Distributed voting is a fundamental topic in distributed computing. In pull
voting, in each step every vertex chooses a neighbour uniformly at random, and
adopts its opinion. The voting is completed when all vertices hold the same
opinion. On many graph classes including regular graphs, pull voting requires
expected steps to complete, even if initially there are only two
distinct opinions.
In this paper we consider a related process which we call two-sample voting:
every vertex chooses two random neighbours in each step. If the opinions of
these neighbours coincide, then the vertex revises its opinion according to the
chosen sample. Otherwise, it keeps its own opinion. We consider the performance
of this process in the case where two different opinions reside on vertices of
some (arbitrary) sets and , respectively. Here, is the
number of vertices of the graph.
We show that there is a constant such that if the initial imbalance
between the two opinions is ?, then with high probability two sample voting completes in a random
regular graph in steps and the initial majority opinion wins. We
also show the same performance for any regular graph, if where is the second largest eigenvalue of the transition
matrix. In the graphs we consider, standard pull voting requires
steps, and the minority can still win with probability .Comment: 22 page
Discordant Voting Processes on Finite Graphs
We consider an asynchronous voting process on graphs called discordant voting, which can be described as follows. Initially each vertex holds one of two opinions, red or blue. Neighboring vertices with different opinions interact pairwise along an edge. After an interaction both vertices have the same color. The quantity of interest is the time to reach consensus, i.e., the number of steps needed for all vertices have the same color. We show that for a given initial coloring of the vertices, the expected time to reach consensus depends strongly on the underlying graph and the update rule (i.e., push, pull, oblivious)
First Passage Properties of the Erdos-Renyi Random Graph
We study the mean time for a random walk to traverse between two arbitrary
sites of the Erdos-Renyi random graph. We develop an effective medium
approximation that predicts that the mean first-passage time between pairs of
nodes, as well as all moments of this first-passage time, are insensitive to
the fraction p of occupied links. This prediction qualitatively agrees with
numerical simulations away from the percolation threshold. Near the percolation
threshold, the statistically meaningful quantity is the mean transit rate,
namely, the inverse of the first-passage time. This rate varies
non-monotonically with p near the percolation transition. Much of this behavior
can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma