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 $G=(V, \overrightarrow{E})$ and a (possibly noisy) snapshot $\mathcal{O}_n$ 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 -- $\textit{backward inference}$ -- , and estimating the spread of a fixed set of source nodes -- $\textit{forward inference}$. 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 $\mathcal{O}_n$. 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).

翻译：本文研究在网络上同时传播的两个扩散过程的过去与未来演化估计问题。具体而言，给定一个已知网络 $G=(V, \overrightarrow{E})$ 及其在（可能未知的）时间 $W$ 处采集的（可能含噪声的）快照 $\mathcal{O}_n$，我们希望确定网络初始状态及其节点感染时间的后验分布。这些分布可用于寻找流行病和谣言的源头节点（即反向推理），以及估计固定源节点集的传播范围（即正向推理）。为模拟两个过程间的交互作用，我们研究了独立级联（IC）模型的一种扩展形式，在该扩展中，当节点被任一过程感染时，其对另一过程的易感性会发生改变。首先，我们推导出网络初始状态与观测快照 $\mathcal{O}_n$ 的精确联合概率。随后，利用因子图、因子图变换和广义分配律的框架，我们提出一种基于置信传播（BP）的算法，该算法可扩展到大规模网络，并且能够在任意拓扑结构的图上收敛（但可能牺牲近似精度）。