A Fractional-Order Stochastic Differential Model for APT Lateral Movement with Memory Effects and Non-Local Dynamics

Advanced Persistent Threats (APTs) exhibit long-memory dynamics and sub-diffusive propagation across network segments, characteristics that traditional integer-order models fail to capture. This paper develops a fractional-order stochastic differential model for APT lateral movement using Caputo derivatives of order \alpha \in (۰,۱). The proposed framework incorporates memory effects through power-law kernels and non-local dynamics via graph Laplacian operators. We prove existence and uniqueness of mild solutions under Lipschitz conditions, derive mean compromise levels in closed form using Mittag-Leffler functions and establish stability conditions via the Matignon criterion. The stochastic term is rigorously defined within a fractional It\^{o} framework. Numerical simulations calibrated against empirical APT campaign data demonstrate that the fractional-order model achieves a ۲.۸۴\% improvement in prediction accuracy compared to existing integer-order and stochastic models. Sensitivity analysis identifies the fractional order \alpha as the critical parameter governing attack persistence, with a Sobol index of ۰.۴۱۲. The epidemic threshold \mathcal{R}_۰ = \gamma/(\beta\lambda_۲) provides a quantitative criterion for attack containment. These results offer security analysts a theoretically grounded tool for APT timeline prediction and defense strategy evaluation.

کلیدواژه ها:

Fractional calculus ، APT lateral movement ، Stochastic differential equations ، memory effects ، non-local dynamics ، Stability analysis