Robust Cascade Reconstruction by Steiner Tree Sampling

Abstract

We consider a network where an infection has taken place and a subset of infected nodes has been partially observed. Our goal is to reconstruct the underlying cascade that is likely to have generated these observations. We reduce this cascadereconstruction problem to computing the marginal probability that a node is infected given the partial observations, which is a #P-hard problem. To circumvent this issue, we resort to estimating infection probabilities by generating a sample of probable cascades, which span the nodes that have already been observed to be infected, and avoid the nodes that have been observed to be uninfected. The sampling problem corresponds to sampling directed Steiner trees with a given set of terminals, which is a problem of independent interest and has received limited attention in the literature. For the latter problem we propose two novel algorithms with provable guarantees on the sampling distribution of the returned Steiner trees.The resulting method improves over state-of-the-art approaches that often make explicit assumptions about the infection-propagation model, or require additional parameters. Our method provides a more robust approach to the cascadereconstruction problem, which makes weaker assumptions about the infection model, requires fewer additional parameters, and can be used to estimate node infection probabilities. We experimentally validate the proposed reconstruction algorithm on realworld graphs with both synthetic and real cascades. We show that our method outperforms all other baseline strategies in most cases.

Predicting partially observed processes on temporal networks by Dynamics-Aware Node Embeddings (DyANE)

“…This issue has been addressed in the specific case of spreading processes, under various hypothesis. For instance, some works have put forward methods to recover the state of all nodes and the seeds of a spread from a partial observation of nodes at a given time [7,8], without attempting to recover the whole temporal evolution of the process. Methods to recover the state evolution of all nodes have also been proposed, using as input snapshots of the whole system, i.e., the knowledge of the state of all the nodes at a certain time [9][10][11].…”

“…Methods to recover the state evolution of all nodes have also been proposed, using as input snapshots of the whole system, i.e., the knowledge of the state of all the nodes at a certain time [9][10][11]. Finally, several methods using partially observed snapshots have also been proposed [8,[12][13][14], typically based on strong assumptions on the nature of the underlying diffusion process.…”

Inferring the Hidden Cascade Infection over Erdös-Rényi (ER) Random Graph

“…For this, they proposed an algorithm called NETFILL, which solves these two inference problems simultaneously. In addition, Xiao et al [6] considered a problem for reconstructing the underlying cascade that is likely to generate observations of infection in the graph. They estimated the infection probabilities by generating a sample of the probable cascades, and proposed several algorithms for sampling directed Steiner trees with a given set of terminals.…”

“…They estimated the infection probabilities by generating a sample of the probable cascades, and proposed several algorithms for sampling directed Steiner trees with a given set of terminals. However, it is known [6] that inferring a hidden infected node is extremely difficult, even if the network structure is given. This is because it is necessary to first identify the path through which an arbitrary node is infected in the graph, despite the potential of an exponential number of paths.…”

“…It is also known [7] that finding a hidden infected node that has not yet been observed is still a difficult problem even if some information regarding which node is infected and when such infection occurred is given. Hence, many prior studies have considered some of the greedy approaches to sampling a candidate infection graph using Steiner tree sampling [6,7]. An Expectation-Maximization (EM) approach applied under a partial observation scenario was also investigated in Reference [8].…”