Modeling Viral Information Spreading via Directed Acyclic Graph Diffusion

Viral information like rumors or fake news is spread over a communication network like a virus infection in a unidirectional manner: entity
$i$
conveys information to a neighbor
$j$
, resulting in two equally informed (infected) parties. Existing graph diffusion processes focus only on bidirectional diffusion on an undirected graph. Instead, leveraging recent research in graph signal processing (GSP), we propose a new directed acyclic graph (DAG) diffusion process to estimate the probability
$x_{i}(t)$
of node
$i$
's infection at time
$t$
given an initial infected source node
$s$
, where
$x_{i}(\infty)=1$
. Specifically, given an undirected positive graph modeling node-to-node communication, we first estimate its graph embedding: a latent coordinate for each graph node in an assumed low-dimensional manifold space via extreme eigenvectors computed using LOBPCG. Next, we construct a DAG based on Euclidean distances between latent coordinates. Spectrally, we prove that the asymmetric DAG Laplacian matrix contains real non-negative eigenvalues, and that the DAG diffusion converges to the all-infection vector
$\mathbf{x}(\infty)=1$
as
$t\rightarrow\infty$
. Simulations show that our DAG diffusion process accurately estimates the probabilities of node infection over a variety of graph structures at different time instants.