Benjamin Doerr

Why rumors spread so quickly in social networks

A few hubs with many connections share with many individuals with few connections.

Social Networks Spread Rumors in Sublogarithmic Time

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 Karp et al. (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.

Crossover can provably be useful in evolutionary computation

We show that the natural evolutionary algorithm for the all-pairs shortest path problem is significantly faster with a crossover operator than without. This is the first theoretical analysis proving the usefulness of crossover for a non-artificial problem.

Optimal fixed and adaptive mutation rates for the leadingones problem

We reconsider a classical problem, namely how the (1+1) evolutionary algorithm optimizes the LEADINGONES function. We prove that if a mutation probability of p is used and the problem size is n, then the optimization time is1/2p2 ((1 - p)-n+1 - (1 - p)). For the standard value of p ≅ 1/n, this is approximately 0.86n2. As our bound shows, this mutation probability is not optimal: For p ≅ 1.59/n, the optimization time drops by more than 16% to approximately 0.77n2. Our method also allows to analyze mutation probabilities depending on the current fitness (as used in artificial immune systems). Again, we derive an exact expression. Analysing it, we find a fitness dependent mutation probability that yields an expected optimization time of approximately 0.68n2, another 12% improvement over the optimal mutation rate. In particular, this is the first example where an adaptive mutation rate provably speeds up the computation time. In a general context, these results suggest that the final word on mutation probabilities in evolutionary computation is not yet spoken.

Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness

Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erdős-Renyi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability f , then the runtime increases only by a factor of

On the runtime analysis of the 1-ANT ACO algorithm

\O(1/(1-f))

Stabilizing consensus with the power of two choices

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Multiplicative drift analysis

The runtime analysis of randomized search heuristics is a growing field where, in the last two decades, many rigorous results have been obtained. These results, however, apply particularly to classical search heuristics such as Evolutionary Algorithms (EAs) and Simulated Annealing. First runtime analyses of modern search heuristics have been conducted only recently w.r.t a simple Ant Colony Optimization (ACO) algorithm called 1-ANT. In particular, the influence of the evaporation factor in the pheromone update mechanism and the robustness of this parameter w.r.t the runtime behavior have been determined for the example function OneMax.This paper puts forward the rigorous runtime analysis of the 1-ANT on example functions, namely on the functions LeadingOnes and BinVal. With respect to EAs, such analyses have been essential to develop methods for the analysis on more complicated problems. The proof techniques required for the 1-ANT, unfortunately, differ significantly from those for EAs, which means that a new reservoir of methods has to be built up. Again, the influence of the evaporation factor is analyzed rigorously, and it is proved that its choice can be very crucial to allow efficient runtimes. Moreover, the analyses provide insight into the working principles of ACO algorithms and, in terms of their robustness, describe essential differences to other randomized search heuristics.

Social networks spread rumors in sublogarithmic time

In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem [2]. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common, stable value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Our main result is a simple randomized algorithm called median rule that, with high probability, just needs O(log m log log n + log n) time and work per process to arrive at an almost stable consensus for any set of m legal values as long as an adversary can corrupt the states of at most √n processes at any time. Without adversarial involvement, just O(log n) time and work is needed for a stable consensus, with high probability. As a by-product, we obtain a simple distributed algorithm for approximating the median of n numbers in time O(log m log log n + log n) under adversarial presence.

Drift analysis and linear functions revisited

Drift analysis is one of the strongest tools in the analysis of evolutionary algorithms. Its main weakness is that it is often very hard to find a good drift function. In this paper, we make progress in this direction. We prove a multiplicative version of the classical drift theorem. This allows easier analyses in those settings, where the optimization progress is roughly proportional to the current objective value. Our drift theorem immediately gives natural proofs for the best known run-time bounds for the (1+1) Evolutionary Algorithm computing minimum spanning trees and shortest paths, since here we may simply take the objective function as drift function. As a more challenging example, we give a relatively simple proof for the fact that any linear function is optimized in time O(n log n). In the multiplicative setting, a simple linear function can be used as drift function (without taking any logarithms). However, we also show that, both in the classical and the multiplicative setting, drift functions yielding good results for all linear functions exist only if the mutation probability is at most c/n for a small constant c.