Rotor walks on general trees

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially towards the root. For i.i.d. random initial rotor directions on a regular tree, the limiting proportion of escapes is either zero or the random walk escape probability, and undergoes a discontinuous phase transition between the two as the distribution is varied. In the critical case there are no escapes, but the walker's maximum distance from the root grows doubly exponentially with the number of visits to the root. We also prove that there exist trees of bounded degree for which the proportion of escapes eventually exceeds the escape probability by arbitrarily large o(1) functions. No larger discrepancy is possible, while for regular trees the discrepancy is at most logarithmic.Comment: 32 page

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b

Information dissemination via random walks

Information dissemination is a fundamental task in distributed computing: How to deliver a piece of information from a node of a network to some or all other nodes? In the face of large and still growing modern networks, it is imperative that dissemination algorithms are decentralised and can operate under unreliable conditions. In the past decades, randomised rumour spreading algorithms have addressed these challenges. In these algorithms, a message is initially placed at a source node of a network, and, at regular intervals, each node contacts a randomly selected neighbour. A message may be transmitted in one or both directions during each of these communications, depending on the exact protocol. The main measure of performance for these algorithms is their broadcast time, which is the time until a message originating from a source node is disseminated to all nodes of the network. Apart from being extremely simple and robust to failures, randomised rumour spreading achieves theoretically optimal broadcast time in many common network topologies. In this thesis, we propose an agent-based information dissemination algorithm, called Visit-Exchange. In our protocol, a number of agents perform independent random walks in the network. An agent becomes informed when it visits a node that has a message, and later informs all future nodes it visits. Visit-Exchange shares many of the properties of randomised rumour spreading, namely, it is very simple and uses the same amount of communication in a unit of time. Moreover, the protocol can be used as a simple model of non-recoverable epidemic processes. We investigate the broadcast time of Visit-Exchange on a variety of network topologies, and compare it to traditional rumour spreading. On dense regular networks we show that the two types of protocols are equivalent, which means that in this setting the vast literature on randomised rumour spreading applies in our model as well. Since many networks of interest, including real-world ones, are very sparse, we also study agent-based broadcast for sparse networks. Our results include almost optimal or optimal bounds for sparse regular graphs, expanders, random regular graphs, balanced trees and grids. We establish that depending on the network topology, Visit-Exchange may be either slower or faster than traditional rumour spreading. In particular, in graphs consisting of hubs that are not well connected, broadcast using agents can be significantly faster. Our conclusion is that a combined broadcasting protocol that simultaneously uses both traditional rumour spreading and agent-based dissemination can be fast on a larger range of topologies than each of its components separately.Gates Cambridge Trust, St John's College Benefactors' Scholarshi

How to Spread a Rumor: Call Your Neighbors or Take a Walk?

We study the problem of randomized information dissemination in networks. We compare the now standard PUSH-PULL protocol, with agent-based alternatives where information is disseminated by a collection of agents performing independent random walks. In the VISIT-EXCHANGE protocol, both nodes and agents store information, and each time an agent visits a node, the two exchange all the information they have. In the MEET-EXCHANGE protocol, only the agents store information, and exchange their information with each agent they meet. We consider the broadcast time of a single piece of information in an $n$-node graph for the above three protocols, assuming a linear number of agents that start from the stationary distribution. We observe that there are graphs on which the agent-based protocols are significantly faster than PUSH-PULL, and graphs where the converse is true. We attribute the good performance of agent-based algorithms to their inherently fair bandwidth utilization, and conclude that, in certain settings, agent-based information dissemination, separately or in combination with PUSH-PULL, can significantly improve the broadcast time. The graphs considered above are highly non-regular. Our main technical result is that on any regular graph of at least logarithmic degree, PUSH-PULL and VISIT-EXCHANGE have the same asymptotic broadcast time. The proof uses a novel coupling argument which relates the random choices of vertices in PUSH-PULL with the random walks in VISIT-EXCHANGE. Further, we show that the broadcast time of MEET-EXCHANGE is asymptotically at least as large as the other two's on all regular graphs, and strictly larger on some regular graphs. As far as we know, this is the first systematic and thorough comparison of the running times of these very natural information dissemination protocols.The authors would like to thank Thomas Sauerwald and Nicol\'{a}s Rivera for helpful discussions. This research was undertaken, in part, thanks to funding from the ANR Project PAMELA (ANR-16-CE23-0016-01), the NSF Award Numbers CCF-1461559, CCF-0939370 and CCF-18107, the Gates Cambridge Scholarship programme, and the ERC grant DYNAMIC MARCH

Deterministic Extractors for Small-Space Sources

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space s sources on n{0,1} as sources generated by width s2 branching programs. Specifically, there is a constant η>0 such that for any ζ>n−η, our algorithm extracts m=(δ−ζ)n bits that are exponentially close to uniform (in variation distance) from space s sources with min-entropy δn, where s=Ω(ζ3n). Previously, nothing was known for δ≤1/2, even for space 0. Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of r independent smaller sources over ℓ{0,1}, where the total min-entropy over all the smaller sources is k. We give deterministic extractors for such sources when k is as small as polylog(r), for small enough ℓ.Engineering and Applied Science