We provide a systematic study of the problem of finding the source of a rumor in a network. We model rumor spreading in a network with a variant of the popular SIR model and then construct an estimator for the rumor source. This estimator is based upon a novel topological quantity which we term rumor centrality. We establish that this is an ML estimator for a class of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has non-trivial detection probability, whereas on trees that grow like a line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops of the true source across different network topologies. We compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding rumor sources in networks which are not tree-like. arXiv:0909.4370v2 [stat.ML]

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We predict the popularity of short messages called tweets created in the micro-blogging site known as Twitter. We measure the popularity of a tweet by the time-series path of its retweets, which is when people forward the tweet to others. We develop a probabilistic model for the evolution of the retweets using a Bayesian approach, and form predictions using only observations on the retweet times and the local network or "graph" structure of the retweeters. We obtain good step ahead forecasts and predictions of the final total number of retweets even when only a small fraction (i.e., less than one tenth) of the retweet path is observed. This translates to good predictions within a few minutes of a tweet being posted, and has potential implications for understanding the spread of broader ideas, memes or trends in social networks.

Waveguide Faraday rotation was demonstrated in an InP waveguide with magnetic dopants. The modal Verdet coefficient was 10°∕mmT at a wavelength of 1.55μm. It was found that the Verdet coefficient of the Fe:In1−xGaxAsyP1−y (x=0.290, y=0.628) core was 12.5°∕mmT at a wavelength of 1.55μm. The Verdet coefficient dependence on wavelength is in agreement with theory. The loss of the waveguide was 4.34dB∕cm, giving a magneto-optic figure of merit of 23 at a magnetic field of 1T and a wavelength of 1.55μm. This semiconductor waveguide shows promise for monolithically integrated optical isolators.