On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$ knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the rumour (a push operation), and if $x$ does not know the rumour and $y$ knows it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1-1/n$, after time $t$ all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n\log n)$. In the asynchronous version, both the average and guaranteed spread times are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is $O(n^{2/3})$.Comment: 25 page

Robustness of Randomized Rumour Spreading

In this work we consider three well-studied broadcast protocols: Push, Pull and Push&Pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized, and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of Local Resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary has to be allowed to delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. It turns out that Pull is robust with respect to all parameters that we consider. On the other hand, Push may slow down significantly, even if the adversary is allowed to modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, Push&Pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemer\`edi's Regularity Lemma.Comment: version 2: more thorough literature revie

Gossip in a Smartphone Peer-to-Peer Network

In this paper, we study the fundamental problem of gossip in the mobile telephone model: a recently introduced variation of the classical telephone model modified to better describe the local peer-to-peer communication services implemented in many popular smartphone operating systems. In more detail, the mobile telephone model differs from the classical telephone model in three ways: (1) each device can participate in at most one connection per round; (2) the network topology can undergo a parameterized rate of change; and (3) devices can advertise a parameterized number of bits about their state to their neighbors in each round before connection attempts are initiated. We begin by describing and analyzing new randomized gossip algorithms in this model under the harsh assumption of a network topology that can change completely in every round. We prove a significant time complexity gap between the case where nodes can advertise $0$ bits to their neighbors in each round, and the case where nodes can advertise $1$ bit. For the latter assumption, we present two solutions: the first depends on a shared randomness source, while the second eliminates this assumption using a pseudorandomness generator we prove to exist with a novel generalization of a classical result from the study of two-party communication complexity. We then turn our attention to the easier case where the topology graph is stable, and describe and analyze a new gossip algorithm that provides a substantial performance improvement for many parameters. We conclude by studying a relaxed version of gossip in which it is only necessary for nodes to each learn a specified fraction of the messages in the system.Comment: Extended Abstract to Appear in the Proceedings of the ACM Conference on the Principles of Distributed Computing (PODC 2017

The Capacity of Smartphone Peer-To-Peer Networks

We study three capacity problems in the mobile telephone model, a network abstraction that models the peer-to-peer communication capabilities implemented in most commodity smartphone operating systems. The capacity of a network expresses how much sustained throughput can be maintained for a set of communication demands, and is therefore a fundamental bound on the usefulness of a network. Because of this importance, wireless network capacity has been active area of research for the last two decades. The three capacity problems that we study differ in the structure of the communication demands. The first problem is pairwise capacity, where the demands are (source, destination) pairs. Pairwise capacity is one of the most classical definitions, as it was analyzed in the seminal paper of Gupta and Kumar on wireless network capacity. The second problem we study is broadcast capacity, in which a single source must deliver packets to all other nodes in the network. Finally, we turn our attention to all-to-all capacity, in which all nodes must deliver packets to all other nodes. In all three of these problems we characterize the optimal achievable throughput for any given network, and design algorithms which asymptotically match this performance. We also study these problems in networks generated randomly by a process introduced by Gupta and Kumar, and fully characterize their achievable throughput. Interestingly, the techniques that we develop for all-to-all capacity also allow us to design a one-shot gossip algorithm that runs within a polylogarithmic factor of optimal in every graph. This largely resolves an open question from previous work on the one-shot gossip problem in this model

Tight bounds for rumor spreading in graphs of a given conductance

We study the connection between the rate at which a rumor spreads throughout a graph and the conductance of the graph -- a standard measure of a graph\u27s expansion properties. We show that for any n-node graph with conductance phi, the classical PUSH-PULL algorithm distributes a rumor to all nodes of the graph in O(phi^(-1) log(n)) rounds with high probability (w.h.p.). This bound improves a recent result of Chierichetti, Lattanzi, and Panconesi [STOC 2010], and it is tight in the sense that there exist graphs where Omega(phi^(-1)log(n)) rounds of the PUSH-PULL algorithm are required to distribute a rumor w.h.p. We also explore the PUSH and the PULL algorithms, and derive conditions that are both necessary and sufficient for the above upper bound to hold for those algorithms as well. An interesting finding is that every graph contains a node such that the PULL algorithm takes O(phi^(-1) log(n)) rounds w.h.p. to distribute a rumor started at that node. In contrast, there are graphs where the PUSH algorithm requires significantly more rounds for any start node

Strong Robustness of Randomized Rumor Spreading Protocols

Randomized rumor spreading is a classical protocol to disseminate information across a network. At SODA 2008, a quasirandom version of this protocol was proposed and competitive bounds for its run-time were proven. This prompts the question: to what extent does the quasirandom protocol inherit the second principal advantage of randomized rumor spreading, namely robustness against transmission failures? In this paper, we present a result precise up to $(1 \pm o(1))$ factors. We limit ourselves to the network in which every two vertices are connected by a direct link. Run-times accurate to their leading constants are unknown for all other non-trivial networks. We show that if each transmission reaches its destination with a probability of $p \in (0,1]$, after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n) rounds the quasirandom protocol has informed all $n$ nodes in the network with probability at least 1-n^{-p\e/40}. Note that this is faster than the intuitively natural $1/p$ factor increase over the run-time of approximately $\log_2 n + \ln n$ for the non-corrupted case. We also provide a corresponding lower bound for the classical model. This demonstrates that the quasirandom model is at least as robust as the fully random model despite the greatly reduced degree of independent randomness.Comment: Accepted for publication in "Discrete Applied Mathematics". A short version appeared in the proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009). Minor typos fixed in the second version. Proofs of Lemma 11 and Theorem 12 fixed in the third version. Proof of Lemma 8 fixed in the fourth versio