Nonbacktracking Bounds on the Influence in Independent Cascade Models

This paper develops upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model. In particular, our bounds exploit nonbacktracking walks, Fortuin-Kasteleyn-Ginibre (FKG) type inequalities, and are computed by message passing implementation. Nonbacktracking walks have recently allowed for headways in community detection, and this paper shows that their use can also impact the influence computation. Further, we provide a knob to control the trade-off between the efficiency and the accuracy of the bounds. Finally, the tightness of the bounds is illustrated with simulations on various network models

Scalable Influence Estimation Without Sampling

In a diffusion process on a network, how many nodes are expected to be influenced by a set of initial spreaders? This natural problem, often referred to as influence estimation, boils down to computing the marginal probability that a given node is active at a given time when the process starts from specified initial condition. Among many other applications, this task is crucial for a well-studied problem of influence maximization: finding optimal spreaders in a social network that maximize the influence spread by a certain time horizon. Indeed, influence estimation needs to be called multiple times for comparing candidate seed sets. Unfortunately, in many models of interest an exact computation of marginals is #P-hard. In practice, influence is often estimated using Monte-Carlo sampling methods that require a large number of runs for obtaining a high-fidelity prediction, especially at large times. It is thus desirable to develop analytic techniques as an alternative to sampling methods. Here, we suggest an algorithm for estimating the influence function in popular independent cascade model based on a scalable dynamic message-passing approach. This method has a computational complexity of a single Monte-Carlo simulation and provides an upper bound on the expected spread on a general graph, yielding exact answer for treelike networks. We also provide dynamic message-passing equations for a stochastic version of the linear threshold model. The resulting saving of a potentially large sampling factor in the running time compared to simulation-based techniques hence makes it possible to address large-scale problem instances

Network Diffusions via Neural Mean-Field Dynamics

We propose a novel learning framework based on neural mean-field dynamics for inference and estimation problems of diffusion on networks. Our new framework is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators, resulting in a highly structured and interpretable RNN. Directly using cascade data, our framework can jointly learn the structure of the diffusion network and the evolution of infection probabilities, which are cornerstone to important downstream applications such as influence maximization. Connections between parameter learning and optimal control are also established. Empirical study shows that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.Comment: Accepted by NIPS2020, 21 pages, 5 figure

Sharp Thresholds for a SIR Model on One-Dimensional Small-World Networks

We study epidemic spreading according to a \emph{Susceptible-Infectious-Recovered} (for short, \emph{SIR}) network model known as the {\em Reed-Frost} model, and we establish sharp thresholds for two generative models of {\em one-dimensional small-world graphs}, in which graphs are obtained by adding random edges to a cycle. In $3$-regular graphs obtained as the union of a cycle and a random perfect matching, we show that there is a sharp threshold at $.5$ for the contagion probability along edges. In graphs obtained as the union of a cycle and of a $\mathcal{G}_{n,c/n}$ Erd\H{o}s-R\'enyi random graph with edge probability $c/n$, we show that there is a sharp threshold $p_c$ for the contagion probability: the value of $p_c$ turns out to be $\sqrt 2 -1\approx .41$ for the sparse case $c=1$ yielding an expected node degree similar to the random $3$-regular graphs above. In both models, below the threshold we prove that the infection only affects $\mathcal{O}(\log n)$ nodes, and that above the threshold it affects $\Omega(n)$ nodes. These are the first fully rigorous results establishing a phase transition for SIR models (and equivalent percolation problems) in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative phenomena emerge from our analysis, including the spread of the disease through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.Comment: 28 pages, 0 figure