110 research outputs found
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
Fast plurality consensus in regular expanders
Pull voting is a classic method to reach consensus among vertices with
differing opinions in a distributed network: each vertex at each step takes on
the opinion of a random neighbour. This method, however, suffers from two
drawbacks. Even if there are only two opposing opinions, the time taken for a
single opinion to emerge can be slow and the final opinion is not necessarily
the initially held majority.
We refer to a protocol where 2 neighbours are contacted at each step as a
2-sample voting protocol. In the two-sample protocol a vertex updates its
opinion only if both sampled opinions are the same. Not much was known about
the performance of two-sample voting on general expanders in the case of three
or more opinions. In this paper we show that the following performance can be
achieved on a -regular expander using two-sample voting. We suppose there
are opinions, and that the initial size of the largest and second
largest opinions is respectively.
We prove that, if ,
where is the absolute second eigenvalue of matrix and
is a suitable constant, then the largest opinion wins in steps with high probability.
For almost all -regular graphs, we have for some
constant . This means that as increases we can separate an opinion
whose majority is , whereas majority is required for
constant.
This work generalizes the results of Becchetti et. al (SPAA 2014) for the
complete graph
Fast Low-Cost Estimation of Network Properties Using Random Walks
Abstract. We study the use of random walks as an efficient estimator of global properties of large undirected graphs, for example the number of edges, vertices, triangles, and generally, the number of small fixed subgraphs. We consider two methods based on first returns of random walks: the cycle formula of regenerative processes and weighted random walks with edge weights defined by the property under investigation. We review the theoretical foundations for these methods, and indicate how they can be adapted for the general non-intrusive investigation of large online networks. The expected value and variance of first return time of a random walk decrease with increasing vertex weight, so for a given time budget, re-turns to high weight vertices should give the best property estimates. We present theoretical and experimental results on the rate of convergence of the estimates as a function of the number of returns of a random walk to a given start vertex. We made experiments to estimate the number of vertices, edges and triangles, for two test graphs.
Discrete Incremental Voting
We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 (âdisagree stronglyâ), 2 (âdisagreeâ), . . ., 5 (âagree stronglyâ). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbourâs opinion, if different. For opinions drawn from a set {1, 2, . . ., k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by â1, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i †c †i + 1 for some integer i, then the asymptotic probability that opinion i wins is i + 1 â c, and the probability that opinion i + 1 wins is c â i. With high probability, the winning opinion cannot be other than i or i + 1. To contrast this, we show that for a path and opinions 0, 1, 2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.</p
Bamboo Garden Trimming Problem (Perpetual Maintenance of Machines with Different Attendance Urgency Factors)
International audienc
Robustness of the Rotor-Router Mechanism
International audienceThe rotor-router model, also called the Propp machine, was first considered as a deter-ministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer Ï(v) which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the "next exit port"). The rotor-router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In [Yanovski et al., Algorithmica 37(3), 165â186 (2003)], it was proved that, independently of the initial configuration of the rotor-router mechanism in G, the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. Our analysis is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values Ï(v). We show, for example, that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is O(m·min{log m, D}). We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers Ï(v) or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps
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