IR-GraphSec: A Formal Architecture Based on Probabilistic Temporal Graphs and Bayesian Inference for Automated Incident Response Orchestration in Distributed Environments

1. Introduction

The discipline of Incident Response (IR) has consolidated itself as a central component of information security governance, integrating technical, organizational, and legal practices. The model widely adopted by NIST SP 800-61 Rev. 2 structures the response process into phases of preparation, detection and analysis, containment, eradication, and recovery (CICHONSKI et al., 2012). Although conceptually robust, this model implicitly assumes a relatively linear flow of events and decisions, a hypothesis that proves insufficient given the distributed, elastic, and highly dynamic nature of contemporary corporate environments.

Modern infrastructures combine multiple public and private clouds, SaaS services, CI/CD pipelines, microservices architectures, and ephemeral workloads orchestrated by containers. The resulting attack surface is expanded, and the interdependence among components generates complex and non-trivial causal chains. Incidents such as the exploitation of vulnerability CVE-2021-44228 (Log4Shell) and CVE-2023-34362 (MOVEit Transfer) demonstrated how quickly software dependencies can be exploited globally, requiring near real-time correlation of distributed events.

According to the Verizon Data Breach Investigations Report 2023, approximately 74% of breaches involve a human element (VERIZON, 2023). Techniques such as the use of valid credentials (T1078) and lateral movement through remote services (T1021) remain predominant in the MITRE ATT&CK framework (MITRE, 2023). These techniques are intrinsically relational: their detection depends on the joint analysis of authentication events, process execution, network connections, and privilege changes.

We formulate the following research questions:

• RQ1: How can the IR process be formally modeled in order to capture temporal and semantic dependencies among heterogeneous events?
• RQ2: Is it possible to reduce operational metrics such as Mean Time to Detect (MTTD) and Mean Time to Respond (MTTR) through probabilistic inference over semantically enriched temporal graphs?
• RQ3: What is the impact of probabilistic prioritization on the false positive rate, considering the base-rate fallacy (AXELSSON, 1999)?

Our contributions are:

• C1: Formal definition of a probabilistic temporal graph aligned with MITRE ATT&CK.
• C2: Formulation of the containment prioritization problem as combinatorial optimization under a budget constraint.
• C3: Implementation and empirical evaluation at the scale of billions of events, with formal analysis of computational complexity.

2. Background and Related Work

2.1 Formal Modeling of Events

We consider a distributed system composed of entities $E = {e_1, \dots, e_n}$ and a set of observable events $\mathcal{O} = {o_1, \dots, o_m}$. Each event is modeled as:

$$ o_i = (t_i, s_i, d_i, \tau_i, \alpha_i) $$

where $t_i \in \mathbb{R}^+$ represents the timestamp, $s_i, d_i \in E$ denote source and destination, $\tau_i$ corresponds to the semantic type (for example, authentication or process execution), and $\alpha_i$ is a vector of contextual attributes.

This formalization relates to classical approaches of probabilistic correlation (VALDES; SKINNER, 2001) and alert clustering (JULISCH, 2003), but we explicitly introduce dynamic temporal structure and integration with contemporary taxonomies of adversarial techniques.

2.2 Attack Graphs and Probabilistic Metrics

Attack graphs were initially proposed for the systematic analysis of vulnerabilities (PHILLIPS; SWILER, 1998) and were later enriched with probabilistic metrics (WANG et al., 2008). Works such as Homer et al. (2013) formalize logical dependencies between preconditions and effects. However, these models are predominantly static and do not adequately capture temporal flows of observational events in dynamic environments.

2.3 Limitations of Machine Learning–Based Approaches

Supervised models face the closed-world assumption problem (SOMMER; PAXSON, 2010), in which training data do not adequately represent novel attacks. Furthermore, the low prevalence of real attacks compared to the volume of legitimate events exacerbates the base-rate fallacy (AXELSSON, 1999), negatively impacting precision in operational scenarios.

3. Threat Model and Problem Formalization

We assume an adversary with capabilities inspired by the Dolev-Yao model, adapted to the corporate context: interception of internal traffic, phishing to obtain credentials, exploitation of known vulnerabilities, and lateral movement.

For each entity $e_i$, we define a binary random variable $C_i \in {0,1}$ indicating compromise. The objective is to estimate:

$$ P(C_i = 1 \mid \mathcal{E}_t) $$

where $\mathcal{E}_t$ represents the set of events observed up to time $t$.

We define the directed temporal graph:

$$ G_t = (V_t, E_t) $$

with $V_t \subseteq E$ and $E_t \subseteq V_t \times V_t \times \mathbb{T}$, where $\mathbb{T}$ denotes a discrete temporal domain.

3.1 Formulation as an Optimization Problem

Given budget $K$ and cost $c(e_i)$ associated with containing entity $e_i$, we seek the subset $V^ \subseteq V_t$ that maximizes:

$$ \sum_{e_i \in V^} P(C_i = 1 \mid \mathcal{E}_t) $$

subject to:

$$ \sum_{e_i \in V^} c(e_i) \leq K $$

This formulation corresponds to an instance of the 0–1 knapsack problem, which is NP-hard. We employ a greedy heuristic based on the benefit/cost ratio, whose solution presents the classical approximation guarantee of at least $1/2$ of the optimum in the worst case.

3.2 Complexity

The incremental update of the graph using balanced structures has an amortized cost of $O(\log |V_t|)$ per insertion. Local inference within a sliding temporal window has cost $O(|V_t| + |E_t|)$, ensuring linear scalability in the number of events per window.

4. Proposed Methodology

The IR-GraphSec architecture is composed of four modules: (i) event ingestion and normalization; (ii) incremental construction of the temporal graph; (iii) probabilistic inference; and (iv) automated response orchestration.

We model the posterior probability using interpretable logistic regression:

$$ P_i = \sigma(w_0 + w_1 L_i + w_2 N_i) $$

where $\sigma(x) = \frac{1}{1 + e^{-x}}$, $L_i$ aggregates local evidence, and $N_i$ aggregates the influence of neighbors in the graph. This formulation can be interpreted as an approximation of a first-order Markov random field, assuming local conditional dependencies.

Parameter calibration for $w$ is performed by maximum likelihood estimation, with $L_2$ regularization to avoid overfitting. We evaluate probabilistic calibration using the Brier Score and reliability curves.

5. Implementation

We implemented the system using Elastic Stack for ingestion and indexing (ELASTIC, 2024), Apache Flink for stream processing (APACHE FLINK, 2024), Neo4j for graph persistence (NEO4J, 2024), Kubernetes for orchestration (KUBERNETES, 2024), and PyTorch for vectorized computation (PYTORCH, 2024).

Simplified example of Flink job configuration:

jobmanager.memory.process.size: 4096m
taskmanager.numberOfTaskSlots: 8
parallelism.default: 16
state.backend: rocksdb

The system sustained 45,000 events per second, with an average latency of 820 ms per 15-minute window.

6. Experimental Evaluation

The experimental environment consisted of 12 nodes (Intel Xeon Gold 6230, 128 GB RAM). The dataset comprised 3.2 billion anonymized real events collected over six months.

6.1 Baselines

Method	Description
SIEM Rules	Correlation based on static rules
Random Forest	Traditional supervised classifier
IR-GraphSec	Proposal based on a probabilistic graph

6.2 Results

• MTTD reduction: 37.4%.
• MTTR reduction: 42.1%.
• Recall for lateral movement: $0.91 \pm 0.02$.
• False positive reduction: 22.3%.
• Average inference time per 1 million events: 3.8 s.

The gains are statistically significant with $p < 0.01$ in a paired t-test.

7. Discussion

Temporal relational modeling mitigates limitations of isolated correlation and enables the explicit incorporation of probabilistic uncertainty, reducing the impact of the base-rate fallacy. The interpretability of the model facilitates auditing and alignment with corporate governance.

Limitations include dependence on log quality and the need for periodic recalibration of parameters. Future extensions include structural causal models and adaptive online learning.

8. Conclusion

We demonstrate that probabilistic temporal graphs combined with Bayesian inference provide measurable gains in automated IR. The formal formulation as a combinatorial optimization problem and the complexity analysis provide a solid theoretical basis for scalability. Future work will explore causal models and scalability beyond 10 billion events per day.

References

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