Graph　convolutional　networks　(graph　models　for　short)　are　crucial　for　understanding　model　decisions　through　mathematical　white-box　interpretation,　which　can　radically　improve　the　performance　and　credibility　of　downstream　artificial　intelligence　applications.　To　address　the　limitations　of　existing　interpretability　of　over-　smoothing　and　over-squashing,　we　propose　an　explainable　graph　model　based　on　nonlinear　catastrophe　theory　and　apply　it　to　group　activity　recognition　to　validate　the　usefulness　of　interpretability.　(1)　We　introduce　catastrophe　mathematical　theory　to　explore　the　internal　processes　of　graph　models　and　construct　the　explainable　dynamical　equations　of　the　graph　convolutional　network;　(2)　When　graph　node　features　lose　uniqueness,　leading　to　over-smoothing,　which　reduces　the　discriminative　power　of　the　graph　model,　we　propose　a　mathematical　method　to　predict　over-smoothing;　(3)　In　response　to　the　over-squashing　of　the　node　feature　values　that　is　excessively　compressed,　we　design　a　channel　expansion　unit　to　extend　the　transmission　paths　of　graph　nodes　and　alleviate　the　over-squashing　in　the　graph　structure.　Finally,　we　apply　our　model　to　group　activity　recognition　tasks　to　capture　complex　interactions　within　groups.　We　obtain　the　competitive　results　on　five　publicly　available　graph　structure　datasets　(Actor,　Chameleon,　Texas,　Cornell,　Cora)　and　our　self-built　group　activity　dataset.　Our　model　can　effectively　capture　node　and　graph-level　features　with　stronger　generalization　capabilities.　For　complex　and　diverse　real-world　group　activity　data,　our　model　offers　intuitive　graph-level　explanations　for　group　activity　analysis.　Through　the　analysis　of　over-smoothing　and　over-squashing,　our　method　extends　new　theoretical　approaches　in　explainable　artificial　intelligence.