Graph　Convolutional　Networks　(GCNs)　have　demonstrated　remarkable　success　in　various　graph-based　tasks.　However,　their　performance　can　be　severely　compromised　by　perturbations　or　adversarial　attacks　on　graph　structures,　and　research　into　understanding　the　vulnerability　of　GCNs　remains　in　its　infancy.　To　demystify　this　issue,　this　paper　reexamines　GCNs　from　a　novel　perspective,　revealing　that　the　recursive　neighborhood　fusion　process　in　the　core　mechanism　of　GCNs　is　intrinsically　linked　to　graph　Laplacian　regularization.　Through　this　lens,　we　identify　that　the　neighborhood　fusion　process　in　GCNs　suffers　from　insufficient　exploration　of　the　feature　space　and　is　driven　by　an　2-norm-based　graph　regularizer,　which　significantly　amplifies　their　vulnerability　to　anomalous　edges.　This　insight　motivates　us　to　design　a　more　robust　objective　by　introducing　a　feature　fitting　term　and　an　2,p-norm-based　graph　regularizer,　thereby　leading　to　a　GCN　with　a　stabilized　fusion　process.　Consequently,　we　propose　a　Stable　Fusion-based　Graph　Convolutional　Network　(SFGCN)　and　its　enhanced　variant　SFGCN+,　which　implement　a　stable　neighborhood　fusion　mechanism　that　dynamically　adjusts　edge　weights　based　on　their　suspiciousness.　Extensive　experiments　under　both　benign　and　adversarial　settings　demonstrate　that　SFGCN　and　SFGCN+　outperform　state-of-the-art　methods.　©　2025　Elsevier　B.V.