We perform a thorough study of various characteristics of the asynchronous push–pull protocol for spreading a rumor on Erdős–Renyi random graphs (G_ n,p ), for any (p>c (n) n ) with (c>1 ). In particular, we provide a simple strategy for analyzing the asynchronous push–pull protocol on arbitrary graph topologies and apply this strategy to (G_ n,p ). We prove tight bounds of logarithmic order for the total time that is needed until the information has spread to all nodes. Surprisingly, the time required by the asynchronous push–pull protocol is asymptotically almost unaffected by the average degree of the graph. Similarly tight bounds for Erdős–Renyi random graphs have previously only been obtained for the synchronous push protocol, where it has been observed that the total running time increases significantly for sparse random graphs. Finally, we quantify the robustness of the protocol with respect to transmission and node failures. Our analysis suggests that the asynchronous protocols are particularly robust with respect to these failures compared to their synchronous counterparts.
We analyze the popular push-pull protocol for spreading a rumor in networks. Initially, a single node knows of a rumor. In each succeeding round, every node chooses a random neighbor, and the two nodes share the rumor if one of them is already aware of it. We present the first theoretical analysis of this protocol on random graphs that have a power law degree distribution with an arbitrary exponent β > 2. Our main findings reveal a striking dichotomy in the performance of the protocol that depends on the exponent of the power law. More specifically, we show that if 2 3, then Ω(log n) rounds are necessary. We also investigate the asynchronous version of the push-pull protocol, where the nodes do not operate in rounds, but exchange information according to a Poisson process with rate 1. Surprisingly, we are able to show that, if 2 < β < 3, the rumor spreads even in constant time, which is much smaller than the typical distance of two nodes. To the best of our knowledge, this is the first result that establishes a gap between the synchronous and the asynchronous protocol. Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. Abstract It has been observed that information spreads extremely fast in social networks. We model social networks with the preferential attachment model of Barabasi and Albert (Science 1999) and information spreading with the random phone call model of (FOCS 2000). In a recent paper (STOC 2011), we prove the following two results. (i) The random phone call model delivers a message to all nodes of graphs in the preferential attachment model within Θ ( log n ) rounds with high probability. The best known bound so far was O ( log 2 n ) . (ii) If we slightly modify the protocol so that contacts are chosen uniformly from all neighbors but the one Θ ( log n log log n ) , which is the diameter of the graph. This is the first time that a sublogarithmic broadcast time is proven for a natural setting. Also, this is the first time that avoiding doublecontacts reduces the run-time to a smaller order of magnitude.