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

Cover Time and Broadcast Time

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows: begin{itemize} item For any graph $G=(V,E)$ of size $n$ and minimum degree $delta$, we have $mathcal{R}(G)= mathcal{O}(frac{|E|}{delta} cdot log n)$, where $mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. item For any $delta$-regular (or almost $delta$-regular) graph $G$ it holds that $mathcal{R}(G) = Omega(frac{delta^2}{n} cdot frac{1}{log n})$. Together with our upper bound on $mathcal{R}(G)$, this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree $Theta(n)$, since then the cover time equals the broadcast time multiplied by $n$ (neglecting logarithmic factors). item Conversely, for any $delta$ we construct almost $delta$-regular graphs that satisfy $mathcal{R}(G) = mathcal{O}(max { sqrt{n},delta } cdot log^2 n)$. Since any regular expander satisfies $mathcal{R}(G) = Theta(n)$, the strong relationship given above does not hold if $delta$ is polynomially smaller than $n$. end{itemize} Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)

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

Efficient randomised broadcasting in random regular networks with applications in peer-to-peer systems

We consider broadcasting in random d-regular graphs by using a simple modification of the random phone call model introduced by Karp et al. (Proceedings of the FOCS ’00, 2000). In the phone call model, in every time step, each node calls a randomly chosen neighbour to establish a communication channel to this node. The communication channels can then be used bi-directionally to transmit messages. We show that, if we allow every node to choose four distinct neighbours instead of one, then the average number of message transmissions per node required to broadcast a message efficiently decreases exponentially. Formally, we present an algorithm that has time complexity O(logn) and uses O(nloglogn) transmissions per message. In contrast, we show for the standard model that every distributed algorithm in a restricted address-oblivious model that broadcasts a message in time O(logn) requires Ω(nlogn/logd) message transmissions. Our algorithm efficiently handles limited communication failures, only requires rough estimates of the number of nodes, and is robust against limited changes in the size of the network. Our results have applications in peer-to-peer networks and replicated databases. Preliminary version published in the Proceedings of the 27th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2008)

Universal Protocols for Information Dissemination Using Emergent Signals

We consider a population of $n$ agents which communicate with each other in a decentralized manner, through random pairwise interactions. One or more agents in the population may act as authoritative sources of information, and the objective of the remaining agents is to obtain information from or about these source agents. We study two basic tasks: broadcasting, in which the agents are to learn the bit-state of an authoritative source which is present in the population, and source detection, in which the agents are required to decide if at least one source agent is present in the population or not.We focus on designing protocols which meet two natural conditions: (1) universality, i.e., independence of population size, and (2) rapid convergence to a correct global state after a reconfiguration, such as a change in the state of a source agent. Our main positive result is to show that both of these constraints can be met. For both the broadcasting problem and the source detection problem, we obtain solutions with a convergence time of $O(\log^2 n)$ rounds, w.h.p., from any starting configuration. The solution to broadcasting is exact, which means that all agents reach the state broadcast by the source, while the solution to source detection admits one-sided error on a $\varepsilon$-fraction of the population (which is unavoidable for this problem). Both protocols are easy to implement in practice and have a compact formulation.Our protocols exploit the properties of self-organizing oscillatory dynamics. On the hardness side, our main structural insight is to prove that any protocol which meets the constraints of universality and of rapid convergence after reconfiguration must display a form of non-stationary behavior (of which oscillatory dynamics are an example). We also observe that the periodicity of the oscillatory behavior of the protocol, when present, must necessarily depend on the number ^\\# X of source agents present in the population. For instance, our protocols inherently rely on the emergence of a signal passing through the population, whose period is \Theta(\log \frac{n}{^\\# X}) rounds for most starting configurations. The design of clocks with tunable frequency may be of independent interest, notably in modeling biological networks

Quasirandom Rumor Spreading

We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within O (log n ) rounds on complete graphs, hypercubes, random regular graphs, Erdős-Rényi random graphs, and Ramanujan graphs with probability 1 − o (1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown. </jats:p

Rumor spreading: robustness and limiting distributions

In this thesis, we study mathematical aspects of information dissemination. The four collected works investigate randomized rumor spreading with regard to its robustness and asymptotic runtime as well as adversarial effects on opinion forming. In the first contribution, Robustness of Randomized Rumor Spreading, we investigate the popular randomized rumor spreading algorithms push, pull and pushpull. These are used to spread information quickly through large networks, typically modelled by graphs. Starting with one informed vertex and depending on the used algorithm the information is spread in a round based manner. Using push, every informed vertex chooses a random neighbour and passes the information forward. With pull, each vertex yet uninformed connects to a randomly chosen neighbor and receives the information, if the vertex it connected to is informed. pushpull is a combination of push and pull. Every vertex chooses a random neighbour, if one of them is informed then the other will be informed as well. Their advantages over deterministic algorithms are, that they are easy to implement, fast and very robust against failures. However, there is only sporadic information available to substantiate the claimed robustness. The aim of this work is to close this gap. To that end, three orthogonal properties and their effects on the speed of the dissemination are studied. First, we show that the density of the graph does not play an important role. For fast dissemination it is not relevant how many edges there are, but how evenly they are distributed in the graph. Thus, a network could have many faulty connections, but as long as the remaining ones are spread evenly the speed of the dissemination is not significantly impacted. This begs the question how evenly the remaining edges need to be spread to guarantee a fast dissemination. Surprisingly, the answer to this question is not the same for all three rumor spreading algorithms. pull and pushpull are very robust. Starting from a graph with evenly distributed edges and thus fast dissemination one may introduce irregularities by deleting up to one half of all edges at each node and the dissemination remains fast. However, for push the dissemination already slows down significantly if only few irregularities are introduced. Lastly, we additionally consider random message transmission failures. From previous works, we know that on "nice" graphs all three algorithms only slow down proportionally to the failure probability. However, when considering the effect of density and irregularities together with transmission failures, the picture changes once more. pull alone retains its fast dissemination. With a suitable choice of parameters, pushpull similar to \push can be slowed down significantly. Thus, we can not unconditionally confirm the claimed robustness for all three rumor spreading algorithms, only pull proved to be robust against all introduced challenges, push and pushpull, however, did not. In the second contribution, Asymptotics for Push on the Complete Graph, we move from the general approach of quantifying the robustness of all three randomized rumor spreading algorithms on a broad range of networks to very precisely describing the runtime of push on complete graphs only. Thereby, the runtime is defined as the time until the information is disseminated to all vertices in the graph. In this work, we completely describe the limiting distribution of the runtime of push on the complete graph in terms of a Gumbel distributed random variable. We made a surprising observation, the asymptotic distribution does not converge everywhere, only on suitable subsequences. This results in the phenomena, that the expected runtime is not constant either but infimum and supremum over all n differ by about 10^-4. After successfully solving push on the complete graph, a natural question is to ask whether the same can be achieved for other rumor spreading algorithms. The third contribution, Asymptotics for Pull on the Complete Graph, answers this question for pull, describing the asymptotic distribution of the runtime of pull on the complete graph in terms of a martingale limit. Again we observed that the limiting distribution only exists on suitable subsequences. We study the expected runtime numerically, finding strong evidence that it is not constant either. The last contribution, The Effect of Iterativity on Adversarial Opinion Forming, deviates from the previously considered model and introduces a second competing piece of information. We interpret them as opinions and assume one to be the truth and the other one to be a falsehood. The opinions are spread through the network by a simple majority rule, i.e. uninformed vertices take the majority opinion of their informed neighbours. Known properties that guarantee robustness are the degree being sufficiently bounded or the edges being evenly distributed. The question considered in this contribution is whether an alternative iterative dissemination process influences robustness. Alon et al. conjecture that iterativity is always beneficial for the adversary. We refute that conjecture by giving a graph where iterativity benefits robustness.In dieser Arbeit beschäftigen wir uns mit mathematischen Aspekten der Informationsverbreitung in Netzwerken. Die vier gesammelten Beiträge untersuchen randomisierte Gerüchteverbreitungsalgorithmen hinsichtlich ihrer Robustheit und asymptotischen Laufzeit, sowie gegnerische Auswirkungen auf die Meinungsbildung. Der erste Beitrag, Robustness of Randomized Rumor Spreading, befasst sich mit den populären randomisierten Gerüchteverbreitungsalgorithmen Push, Pull und Push&Pull. Diese werden dazu verwendet, um Informationen schnell durch große, als Graphen modellierte Netzwerke zu verteilen. Beginnend mit einem informierten Knoten und in Runden verfahrend, werden die Informationen abhängig vom verwendeten Algorithmus verteilt. Wird \push benutzt, so wählt jeder informierte Knoten einen zufälligen Nachbarn und gibt die Information weiter. Mit Pull wählen uninformierte Knoten zufällige Nachbarn und werden informiert, falls der gewählte Nachbar informiert ist. Push&Pull ist eine Kombination aus Push und Pull. Jeder Knoten wählt einen zufälligen Nachbarn aus, ist einer der beiden informiert, so wird auch der andere informiert. Mit einer einfachen Implementierung, hohen Geschwindigkeit und einer starken Robustheit heben sich die randomisierten Gerüchteverbreitungsalgorithmen positiv von deterministischen Algorithmen ab. Bisher liegen jedoch nur sporadische Informationen vor, um die beobachtete Robustheit auch rigoros zu belegen. Ziel dieser Arbeit ist es, diese Lücke zu schließen. Dafür betrachten wir drei verschiedene, strukturelle Eigenschaften der Graphen, um deren Auswirkungen auf die Geschwindigkeit der Verbreitung zu studieren. Als erstes Ergebnis zeigen wir, dass die Dichte des Netzwerks keinen nennenswerten Einfluss hat. Für eine schnelle Verbreitung der Informationen ist nicht die Anzahl der Kanten relevant, sondern deren gleichmäßige Verteilung. Ein Netzwerk könnte folglich viele fehlerhafte Verbindungen haben, aber solange die verbleibenden Verbindungen gleichmäßig verteilt sind, wird die Verbreitung nicht wesentlich verlangsamt. Dies regt die Untersuchung an, wie gleichmäßig die verbleibenden Kanten sein müssen, um eine schnelle Verbreitung zu gewährleisten. Wider Erwarten konnten wir Unterschiede in Abhängigkeit des gewählten Gerüchteverbreitungsalgorithmus aufzeigen. Pull und Push&Pull sind sehr widerstandsfähig. Denn ausgehend von einem „schönen“ Graph mit gleichmäßig verteilten Kanten können durch Löschen von Kanten Unregelmäßigkeiten eingebracht werden durch die sich die Geschwindigkeit der Gerüchteverbreitung nicht nennenswert verändert. Im Gegensatz dazu verlangsamt sich die Verbreitung mit Push bereits erheblich, wenn nur wenige Unregelmäßigkeiten auftreten. Abschließend befassen wir uns ergänzend mit zufällig auftretenden Übertragungsfehlern. Aus früheren Arbeiten wissen wir, dass sich bei „schönen“ Graphen alle drei Algorithmen nur proportional zur Ausfallswahrscheinlichkeit verlangsamen. Betrachten wir hingegen die Auswirkungen der Dichte und der Unregelmäßigkeiten mit Übertragungsfehlern zusammen, entsteht eine neue Sachlage. Dabei behält nur Pull seine schnelle Verbreitung bei, Push&Ppull kann bei einer entsprechenden Wahl der Parameter ähnlich wie Push verlangsamt werden. Somit ist eine Bestätigung der behaupteten Robustheit der drei Gerüchteverbreitungsalgorithmen nicht bedingungslos möglich. Lediglich Pull erwies sich als widerstandsfähig gegenüber allen betrachteten Problemen, Push und Push&Pull jedoch nicht. Im zweiten Beitrag, Asymptotics for Push on the Complete Graph, gehen wir vom allgemeinen Ansatz der Beschreibung der Robustheit aller drei randomisierten Gerüchteverbreitungsalgorithmen auf einem breiten Spektrum von Netzwerken zu einer sehr präzise Beschreibung der Laufzeit von Push auf vollständigen Graphen über. Dabei definiert sich die Laufzeit als die Zeit, in der die Information an alle Knoten im Graph verteilt wird. In dieser Arbeit beschreiben wir die Grenzverteilung der Laufzeit von Push auf dem vollständigen Graph. Dabei haben wir eine überraschende Beobachtung gemacht, denn die asymptotische Verteilung konvergiert nicht überall, sondern nur auf geeigneten Teilfolgen. Dies resultiert in dem Phänomen, dass die erwartete Laufzeit nicht konstant ist, vielmehr unterscheiden sich Supremum und Infimum über alle n um ungefähr 10^-4. Nach dieser erkenntnisreichen Arbeit stellt sich die natürliche Frage, ob dasselbe für die anderen Gerüchteverbreitungsalgorithmen gilt. Die daran anschließende Arbeit Asymptotics for Pull on the Complete Graph bejaht die aufgeworfene Frage für Pull, indem die asymptotische Verteilung der Laufzeit von Pull auf vollständigen Graph mit Hilfe eines Martingalgrenzwertes beschrieben wird. Ferner wird beobachtet, dass die Grenzverteilung nur auf geeigneten Teilfolgen existiert. Die erwartete Laufzeit wird mit Hilfe dieser Beschreibungen empirisch untersucht, wobei es eine starke Evidenz gibt, dass auch diese nicht konstant ist. Der letzte Beitrag, The Effect of Iterativity on Adversarial Opinion Forming, weicht vom bisher betrachteten Modell ab und führt eine zweite, konkurrierende Information ein. Diese interpretieren wir als Meinungen und nehmen eine davon als wahr an. Die Meinungen werden durch eine einfache Mehrheitsregel im Netzwerk verbreitet, d.h. uninformierte Knoten nehmen die Mehrheitsmeinung ihrer informierten Nachbarn an. Dabei sehen wir ein Netzwerk als robust an, wenn selbst ein Kontrahent die anfangs informierten Knoten nur so wählen kann, dass am Ende der Verbreitung stets die Mehrheit der Knoten von der Wahrheit überzeugt ist. Bekannte Beispiele robuster Netzwerke sind solche mit hinreichend beschränkten Knotengraden oder mit ausreichend gleichmäßig verteilten Kanten. In unserem Beitrag betrachten wir die Frage, inwiefern Robustheit durch einen alternativen, iterativen Verbreitungsprozess beeinflusst wird. Alon et al. vermuten eine negative Auswirkung von Iteration auf Robustheit. Wir widerlegen diese Vermutung durch Konstruktion eines Graphen, auf welchem ein iterativer Prozess die Verbreitung der Wahrheit begünstigt