# Physics-Informed Graph Neural Networks for Attack Path Prediction
## 1 Introduction  
Information Systems (ISs) are socio-technical structures integrating organizational and technological components [1, 2] to support business operations, decision-making, and management [3]. IS security can be compromised by vulnerabilities arising from software flaws and misconfigurations. Consequently, organizations must assess and manage risks associated with IS [3]. Risk-based IS management has been a key focus since the 1990s [4, 5, 6, 7], providing a structured approach to identifying, analyzing, and mitigating potential threats [8]. A key aspect of IS risk management is risk assessment [4, 8], which consists of three phases: risk identification, risk analysis, and risk evaluation. Attack Path Analysis (APA) is a method for the risk assessment of attack scenarios applied in complex environments [9, 10, 11]. A typical APA framework consists of an adversary model, an environment model, and an optimization algorithm [11, 12]. Early APA frameworks used graphical approaches to model adversarial behavior and systems [13, 14], later evolving into probabilistic models [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Modern APA frameworks generally fall intoadversary-oriented [29; 30; 31; 32; 33; 34; 35; 36; 37; 38] or system-oriented methods [39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53] and can vary in their level of abstraction. Higher-abstraction models keep scenario to tractable [29; 30; 31], while lower abstraction offers more detailed analysis but is computationally intensive [43; 44; 45; 46; 47; 48; 49]. For a given environment, APA frameworks provide risk assessment decision support in three main directions: (1) deductive reasoning, i.e., identifying the impact of an attack path (consequence) given its starting point (cause), (2) inductive reasoning, i.e., determining the initial access point of the attack path given its impact, and (3) a combination of both, i.e., identifying the full path when both the cause and consequence are fixed (see Section 2.3 for formal notation). Yet, as infrastructures grow more complex and adversary techniques evolve, uncertainty makes these settings increasingly challenging and existing model-based methods impractical and prone to imprecisions [54; 55; 56]. To address this issue, state-of-the-art (SoTA) data-driven methods like Exploit Prediction Scoring System (EPSS) [57] leverage machine learning (ML) algorithms to predict exploitation likelihood in a model-agnostic manner, outperforming model-based equivalents [57; 58; 59; 60; 61; 62; 63]. This research focuses on a subclass of ML algorithms known as deep learning (DL) [64], which uses Deep Neural Networks (DNNs) for function approximation, motivated by the \"universal approximation theorem\" [65]. A DNN designed to predict full attack paths in any given environment must learn the joint distribution of adversary behavior and environmental characteristics across various combinations. However, the feasibility of generalizing over such a high-dimensional space remains unclear, as no public dataset currently supports this learning task. Existing APA datasets typically provide either detailed attack sequences or environmental properties but not both [54; 66]. For instance, datasets derived from cyber threat intelligence (CTI) report mining offer rich APA details but lack broader environmental descriptions, focusing only on the affected scope [67]. Conversely, macro-level datasets [68] aggregate public breach reports across diverse environments but do not include attack path details. Reinforcement Learning (RL) environments could be considered an alternative. However, RL-based approaches suffer from a similar trade-off: high-fidelity emulated frameworks demand significant computational resources, while lightweight simulated frameworks can only represent limited environments [69].