# Backward and Forward Inference in Interacting Independent-Cascade Processes: A Scalable and Convergent Message-Passing Approach

arXiv: https://arxiv.org/abs/2310.19138 · HTML: https://mlsanity.com/paper/2310.19138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 35 references that are in the map.
- First posted on arXiv 29 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [Catch'Em All: Locating Multiple Diffusion Sources in Networks with Partial Observations](https://mlsanity.com/paper/1611.06963/) (2016)
- [Inferring the origin of an epidemic with a dynamic message-passing algorithm](https://mlsanity.com/paper/1303.5315/) (2013)
- [A Robust Information Source Estimator with Sparse Observations](https://mlsanity.com/paper/1309.4846/) (2013)
- [Low-Complexity Stochastic Generalized Belief Propagation](https://mlsanity.com/paper/1605.02046/) (2016)
- [Convex Combination Belief Propagation Algorithms](https://mlsanity.com/paper/2105.12815/) (2021)

## Abstract (arXiv)

We study the problems of estimating the past and future evolutions of two diffusion processes that spread concurrently on a network. Specifically, given a known network G=(V, overrightarrowE) and a (possibly noisy) snapshot On of its state taken at (a possibly unknown) time W, we wish to determine the posterior distributions of the initial state of the network and the infection times of its nodes. These distributions are useful in finding source nodes of epidemics and rumors -- textitbackward inference -- , and estimating the spread of a fixed set of source nodes -- textitforward inference. To model the interaction between the two processes, we study an extension of the independent-cascade (IC) model where, when a node gets infected with either process, its susceptibility to the other one changes. First, we derive the exact joint probability of the initial state of the network and the observation-snapshot On. Then, using the machinery of factor-graphs, factor-graph transformations, and the generalized distributive-law, we derive a Belief-Propagation (BP) based algorithm that is scalable to large networks and can converge on graphs of arbitrary topology (at a likely expense in approximation accuracy).