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Estimating diffusion network structures: Recovery conditions, sample complexity & softthresholding algorithm
 In Proc. of the 31st International Conference on Machine Learning (ICML
, 2014
"... Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and ho ..."
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Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascadedata and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuoustime diffusion models using an `1regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe O(d3 logN) cascades, where d is the maximum number of parents of a node and N is the total number of nodes. Moreover, we develop a simple and efficient softthresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice.
Sketchbased influence maximization and computation: Scaling up with guarantees
 In International Conference on Information and Knowledge Management (ICIKM
, 2014
"... Propagation of contagion through networks is a fundamental process. It is used to model the spread of information, influence, or a viral infection. Diffusion patterns can be specified by a probabilistic model, such as Independent Cascade (IC), or captured by a set of representative traces. Basic co ..."
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Propagation of contagion through networks is a fundamental process. It is used to model the spread of information, influence, or a viral infection. Diffusion patterns can be specified by a probabilistic model, such as Independent Cascade (IC), or captured by a set of representative traces. Basic computational problems in the study of diffusion are influence queries (determining the potency of a specified seed set of nodes) and Influence Maximization (identifying the most influential seed set of a given size). Answering each influence query involves many edge traversals, and does not scale when there are many queries on very large graphs. The gold standard for Influence Maximization is the greedy algorithm, which iteratively adds to the seed set a node maximizing the marginal gain in influence. Greedy has a guaranteed approximation ratio of at least (1 − 1/e) and actually produces a sequence of nodes, with each prefix having approximation guarantee with respect to the samesize optimum. Since Greedy does not scale well beyond a few million edges, for larger inputs one must currently use either heuristics or alternative algorithms designed for a prespecified small seed set size. We develop a novel sketchbased design for influence computation. Our greedy Sketchbased Influence Maximization (SKIM) algorithm scales to graphs with billions of edges, with one to two orders of magnitude speedup over the best greedy methods. It still has a guaranteed approximation ratio, and in practice its quality nearly matches that of exact greedy. We also present influence oracles, which use lineartime preprocessing to generate a small sketch for each node, allowing the influence of any seed set to be quickly answered from the sketches of its nodes. 1
Timed Influence: Computation and Maximization
"... We consider a cost model for diffusion in a network that captures both the scope of infection and its propagation time: The edges of the network have associated lengths which model transmission times, and influence scores are higher for faster propagation. We propose an intuitive measure of timed in ..."
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We consider a cost model for diffusion in a network that captures both the scope of infection and its propagation time: The edges of the network have associated lengths which model transmission times, and influence scores are higher for faster propagation. We propose an intuitive measure of timed influence, which extends and unifies several classic measures, including the wellstudied “binary” influence [Richardson and Domingos 2002; Kempe et al. 2003] (which only measures scope), a recentlystudied threshold model of timed influence [GomezRodriguez et al. 2011] (which considers a node influenced only within a fixed time horizon), and closeness centrality (which is extended from being defined for a single node to multiple seed nodes and from a fixed network to distributions). Finally, we provide the first highly scalable algorithms for timed influence computation and maximization. In particular, we improve by orders of magnitude the scalability of stateoftheart threshold timed influence computation. Moreover, our design provides robust guarantees and is novel also as a theoretical contribution. 1.
Learning from Contagion (Without Timestamps)
"... We introduce and study new models for learning from contagion processes in a network. A learning algorithm is allowed to either choose or passively observe an initial set of seed infections. This seed set then induces a final set of infections resulting from the underlying stochastic contagion dyn ..."
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We introduce and study new models for learning from contagion processes in a network. A learning algorithm is allowed to either choose or passively observe an initial set of seed infections. This seed set then induces a final set of infections resulting from the underlying stochastic contagion dynamics. Our models differ from prior work in that detailed vertexbyvertex timestamps for the spread of the contagion are not observed. The goal of learning is to infer the unknown network structure. Our main theoretical results are efficient and provably correct algorithms for exactly learning trees. We provide empirical evidence that our algorithm performs well more generally on realistic sparse graphs. 1.
References
"... Abstract. Prediction tasks in machine learning usually require deducing a latent variable, or structure, from observed traces of activity — sometimes, these tasks can be carried out with a significant precision and statistical significance, while sometimes getting any useful prediction requires an ..."
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Abstract. Prediction tasks in machine learning usually require deducing a latent variable, or structure, from observed traces of activity — sometimes, these tasks can be carried out with a significant precision and statistical significance, while sometimes getting any useful prediction requires an unrealistically large number of traces. In this talk, we will study the trace complexity of (that is, the number of traces needed for carrying out) two prediction tasks in social networks: the network inference problem and the number of signers problem. The first problem [1] consists of reconstructing the edge set of a network given traces representing the chronology of infection times as epidemics spread through the network. The second problem’s [2] goal is to guess the unknown number of signers of some emailbased petitions, when only a small subset of the emails that circulated is available. These two examples will allow us to make some general remarks about