  The causal dependence in data is often characterized by Directed Acyclic
Graphical (DAG) models, widely used in many areas. Causal discovery aims to
recover the DAG structure using observational data. This paper focuses on
causal discovery with multi-variate count data. We are motivated by real-world
web visit data, recording individual user visits to multiple websites. Building
a causal diagram can help understand user behavior in transitioning between
websites, inspiring operational strategy. A challenge in modeling is user
heterogeneity, as users with different backgrounds exhibit varied behaviors.
Additionally, social network connections can result in similar behaviors among
friends. We introduce personalized Binomial DAG models to address heterogeneity
and network dependency between observations, which are common in real-world
applications. To learn the proposed DAG model, we develop an algorithm that
embeds the network structure into a dimension-reduced covariate, learns each
node's neighborhood to reduce the DAG search space, and explores the
variance-mean relation to determine the ordering. Simulations show our
algorithm outperforms state-of-the-art competitors in heterogeneous data. We
demonstrate its practical usefulness on a real-world web visit dataset.
