  Traces of user activities recorded in online social networks such as the
creation, viewing and forwarding/sharing of information over time open new
possibilities to quantitatively and systematically understand the information
diffusion process on social networks. From an online social network like
WeChat, we could collect a large number of information cascade trees, each of
which tells the spreading trajectory of a message/information such as which
user creates the information and which users view or forward the information
shared by which neighbors. In this work, we propose two heterogeneous
non-linear models. Both models are validated by the WeChat data in reproducing
and explaining key features of cascade trees. Specifically, we firstly apply
the Random Recursive Tree (RRT) to model the cascade tree topologies, capturing
key features, i.e. the average path length and degree variance of a cascade
tree in relation to the number of nodes (size) of the tree. The RRT model with
a single parameter $\theta$ describes the growth mechanism of a tree, where a
node in the existing tree has a probability $d_i^{\theta}$ of being connected
to a newly added node that depends on the degree $d_i$ of the existing node.
The identified parameter $\theta$ quantifies the relative depth or broadness of
the cascade trees, indicating that information propagates via a star-like
broadcasting or viral-like hop by hop spreading. The RRT model explains the
appearance of hubs, thus a possibly smaller average path length as the cascade
size increases, as observed in WeChat. We further propose the stochastic
Susceptible View Forward Removed (SVFR) model to depict the dynamic user
behaviors including creating, viewing, forwarding and ignoring a message on a
given social network.
