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 $k$ 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 $G$ is an $n$-vertex $r$-regular graph whose transition matrix has second eigenvalue $\lambda$, then the COBRA cover time of $G$ is $\mathcal O(\log n )$, if $1-\lambda$ is greater than a positive constant, and $\mathcal O((\log n)/(1-\lambda)^3))$, if $1-\lambda \gg \sqrt{\log( n)/n}$. These bounds are independent of $r$ and hold for $3 \le r \le n-1$. They improve the previous bound of $O(\log^2 n)$ 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 $v$ is the source of an infection and remains permanently infected. At each step each vertex $u$ other than $v$ selects $k$ neighbours, independently and uniformly, and $u$ 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

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-$1$f) KCM on the discrete $d$-dimensional torus. In view of applications to the more complex FA-$j$f KCM, $j>1$, we also extend part of the analysis to an analogous process with a more general product state space.Comment: 19 pages, minor change

Randomised Algorithms on Networks

Networks form an indispensable part of our lives. In particular, computer networks have ranked amongst the most influential networks in recent times. In such an ever-evolving and fast growing network, the primary concern is to understand and analyse different aspects of the network behaviour, such as the quality of service and efficient information propagation. It is also desirable to predict the behaviour of a large computer network if, for example, one of the computers is infected by a virus. In all of the aforementioned cases, we need protocols that are able to make local decisions and handle the dynamic changes in the network topology. Here, randomised algorithms are preferred because many deterministic algorithms often require a central control. In this thesis, we investigate three network-based randomised algorithms, threshold load balancing with weighted tasks, the pull-Moran process and the coalescing-branching random walk. Each of these algorithms has extensive applicability within networks and computational complexity within computer science. In this thesis we investigate threshold-based load balancing protocols. We introduce a generalisation of protocols in [2, 3] to weighted tasks. This thesis also analyses an evolutionary-based process called the death-birth update, defined here as the Pull-Moran process. We show that a class of strong universal amplifiers does not exist for the Pull-Moran process. We show that any class of selective amplifiers in the (standard) Moran process is a class of selective suppressors under the Pull-Moran process. We then introduce a class of selective amplifiers called Punk graphs. Finally, we improve the broadcasting time of the coalescing-branching (COBRA) walk analysed in [4], for random regular graphs. Here, we look into the COBRA approach as a randomised rumour spreading protocol

An exposition to information percolation for the Ising model

Information percolation is a new method for analyzing stochastic spin systems through classifying and controlling the clusters of information-flow in the space-time slab. It yielded sharp mixing estimates (cutoff with an $O(1)$-window) for the Ising model on $Z^d$ up to the critical temperature, as well as results on the effect of initial conditions on mixing. In this expository note we demonstrate the method on lattices (more generally, on any locally-finite transitive graph) at very high temperatures.Comment: 11 page