Lie algebra on weighted function spaces for modeling network anomalies

The real-time analysis of dynamic and high-dimensional data streams, such as network traffic, raises a number of mathematical and structural difficulties that are not adequately addressed by purely statistical approaches. This study proposes a theoretical framework for anomaly detection based on the construction of topological Lie algebras on weighted spaces of continuous functions. From a mathematical perspective, network states are modeled as elements of a weighted function space $\operatorname{CV}(X, \mathfrak{g})$, where the underlying set $ X $ represents the network domain (for instance, nodes or time indices), the Lie algebra $ \mathfrak{g} $ encodes admissible state transitions, and a Nachbin family of weights $ V $ reflects the nonuniform relevance of different regions of the network. The core contribution of this paper is the identification of a concrete condition on the weight system $ V $, most notably the multiplicative stability condition $ V \leqslant V \cdot V $, which ensures that the pointwise Lie bracket induces a jointly continuous Lie algebra structure on $\operatorname{CV}(X, \mathfrak{g})$. Such an algebraic setting provides a principled mechanism for describing deviations from normal traffic behavior in terms of Lie brackets. In addition, it is shown that the subspace $\operatorname{CV}_0(X, \mathfrak{g})$, corresponding to traffic patterns that vanish at infinity, forms a closed Lie ideal. The closed Lie ideal allows for a natural mathematical distinction between localized anomalies and global, systemwide events. The developed theory is incorporated into an algebraically structured anomaly detection engine (AADE), designed to complement conventional detection techniques by capturing structural aspects of coordinated or evolving attack patterns. The framework provides explicit mathematical characterizations of diverse attack vectors, including distributed denial of service, lateral movement, data exfiltration, and advanced persistent threats, establishing a direct correspondence between algebraic structures and cybersecurity phenomena. As a proof-of-concept validation, experiments on stratified subsets of the NSL-KDD and CIC-IDS-2017 datasets demonstrate that the proposed approach achieves competitive performance compared to both classical machine learning and deep learning baselines. Notably, the algebraic classification mechanism achieves an accuracy of $ 87.4\% $ in distinguishing between localized and systemic threats, with a false localization rate of approximately $ 12\% $. Although the overall detection performance is slightly lower than that of bidirectional long short-term memory (Bi-LSTM) ($ 96.0\% $ vs $ 96.4\% $ F1-score), the approach demonstrates notable strength in detecting sophisticated attack patterns such as low-and-slow exfiltration, achieving a detection rate of $ 68.9\% $ compared to $ 61.8\% $ for Bi-LSTM. The observed performance suggests that the algebraic structure captures complementary structural information in specific scenarios.