  Inferring causal links or subgraphs corresponding to a specific phenotype or
label based solely on measured data is an important yet challenging task, which
is also different from inferring causal nodes. While Graph Neural Network (GNN)
Explainers have shown potential in subgraph identification, existing methods
with GNN often offer associative rather than causal insights. This lack of
transparency and explainability hinders our understanding of their results and
also underlying mechanisms. To address this issue, we propose a novel method of
causal link/subgraph inference, called CIDER: Counterfactual-Invariant
Diffusion-based GNN ExplaineR, by implementing both counterfactual and
diffusion implementations. In other words, it is a model-agnostic and
task-agnostic framework for generating causal explanations based on a
counterfactual-invariant and diffusion process, which provides not only causal
subgraphs due to counterfactual implementation but reliable causal links due to
the diffusion process. Specifically, CIDER is first formulated as an inference
task that generatively provides the two distributions of one causal subgraph
and another spurious subgraph. Then, to enhance the reliability, we further
model the CIDER framework as a diffusion process. Thus, using the causal
subgraph distribution, we can explicitly quantify the contribution of each
subgraph to a phenotype/label in a counterfactual manner, representing each
subgraph's causal strength. From a causality perspective, CIDER is an
interventional causal method, different from traditional association studies or
observational causal approaches, and can also reduce the effects of unobserved
confounders. We evaluate CIDER on both synthetic and real-world datasets, which
all demonstrate the superiority of CIDER over state-of-the-art methods.
