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

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

Who Started This Rumor? Quantifying the Natural Differential Privacy of Gossip Protocols

Gossip protocols (also called rumor spreading or epidemic protocols) are widely used to disseminate information in massive peer-to-peer networks. These protocols are often claimed to guarantee privacy because of the uncertainty they introduce on the node that started the dissemination. But is that claim really true? Can the source of a gossip safely hide in the crowd? This paper examines, for the first time, gossip protocols through a rigorous mathematical framework based on differential privacy to determine the extent to which the source of a gossip can be traceable. Considering the case of a complete graph in which a subset of the nodes are curious, we study a family of gossip protocols parameterized by a "muting" parameter s: nodes stop emitting after each communication with a fixed probability 1-s. We first prove that the standard push protocol, corresponding to the case s = 1, does not satisfy differential privacy for large graphs. In contrast, the protocol with s = 0 (nodes forward only once) achieves optimal privacy guarantees but at the cost of a drastic increase in the spreading time compared to standard push, revealing an interesting tension between privacy and spreading time. Yet, surprisingly, we show that some choices of the muting parameter s lead to protocols that achieve an optimal order of magnitude in both privacy and speed. Privacy guarantees are obtained by showing that only a small fraction of the possible observations by curious nodes have different probabilities when two different nodes start the gossip, since the source node rapidly stops emitting when s is small. The speed is established by analyzing the mean dynamics of the protocol, and leveraging concentration inequalities to bound the deviations from this mean behavior. We also confirm empirically that, with appropriate choices of s, we indeed obtain protocols that are very robust against concrete source location attacks (such as maximum a posteriori estimates) while spreading the information almost as fast as the standard (and non-private) push protocol

Minimizing Message Size in Stochastic Communication Patterns: Fast Self-Stabilizing Protocols with 3 bits

This paper considers the basic $\mathcal{PULL}$ model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of $n$ agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast. We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property that uses $\ell$-bits messages to one that uses only $\log \ell$-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer $T$, within $\tilde O(\log n\log T)$ rounds w.h.p., and using messages that contain $3$ bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in $\tilde O(\log n)$ time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.Comment: 28 pages, 4 figure

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

Rational Fair Consensus in the GOSSIP Model

The \emph{rational fair consensus problem} can be informally defined as follows. Consider a network of $n$ (selfish) \emph{rational agents}, each of them initially supporting a \emph{color} chosen from a finite set $\Sigma$. The goal is to design a protocol that leads the network to a stable monochromatic configuration (i.e. a consensus) such that the probability that the winning color is $c$ is equal to the fraction of the agents that initially support $c$, for any $c \in \Sigma$. Furthermore, this fairness property must be guaranteed (with high probability) even in presence of any fixed \emph{coalition} of rational agents that may deviate from the protocol in order to increase the winning probability of their supported colors. A protocol having this property, in presence of coalitions of size at most $t$, is said to be a \emph{whp\,-$t$-strong equilibrium}. We investigate, for the first time, the rational fair consensus problem in the GOSSIP communication model where, at every round, every agent can actively contact at most one neighbor via a \emph{push$/$pull} operation. We provide a randomized GOSSIP protocol that, starting from any initial color configuration of the complete graph, achieves rational fair consensus within $O(\log n)$ rounds using messages of $O(\log^2n)$ size, w.h.p. More in details, we prove that our protocol is a whp\,-$t$-strong equilibrium for any $t = o(n/\log n)$ and, moreover, it tolerates worst-case permanent faults provided that the number of non-faulty agents is $\Omega(n)$. As far as we know, our protocol is the first solution which avoids any all-to-all communication, thus resulting in $o(n^2)$ message complexity.Comment: Accepted at IPDPS'1