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Cross-institutional collaborative fraud detection is essential for combating increasingly sophisticated financial fraud, yet privacy regulations and data silos severely constrain knowledge sharing among institutions. This study aims to develop a privacy-preserving framework that enables effective collaborative fraud detection while protecting raw data, with particular emphasis on exploiting symmetry properties in federated architectures and graph topology analysis. We propose an Adaptive Federated Graph Attention Network (FedGAT), which employs spatio-temporal graph attention mechanisms to capture topological structures and dynamic fraud patterns within institutional transaction networks. The framework introduces a symmetric similarity matrix derived from graph topological features, where the symmetry property (sij=sji) ensures consistent and unbiased measurement of structural relationships between any pair of institutions. Based on this symmetric similarity metric, an adaptive weighted aggregation mechanism is designed for cross-institutional parameter fusion, enabling balanced knowledge transfer that respects the symmetric collaborative relationship among participating institutions. The symmetric information exchange protocol between local institutions and the central server further guarantees equitable contribution and benefit distribution throughout the federated learning process. The framework is evaluated on the Elliptic Bitcoin transaction dataset and the IEEE-CIS fraud detection dataset, with recall rate and false positive rate as primary performance metrics. Results show that FedGAT achieves a recall of 0.85 and a false-positive rate of 0.038 in single-institution detection, representing approximately 40% and 70% improvements over existing methods, respectively. In collaborative detection across five virtual institutions, the symmetry-aware adaptive aggregation mechanism enables all participants to achieve performance gains exceeding 15% while completely eliminating negative transfer effects observed in simple averaging approaches. This work contributes a novel symmetry-based federated learning framework that balances privacy protection with detection performance, advancing the literature on cross-institutional financial risk management.