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

1. Introduction

The discipline of Incident Response (IR) has become a central component of information security governance, integrating technical, organizational, and legal practices. The widely adopted NIST SP 800-61 Rev. 2 model structures the response process into phases of preparation, detection and analysis, containment, eradication, and recovery (CICHONSKI et al., 2012). Although conceptually robust, such a model implicitly assumes a relatively linear flow of events and decisions, an assumption that proves insufficient in the face of the distributed, elastic, and highly dynamic nature of contemporary enterprise 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 enlarged, and interdependence among components generates complex and non-trivial causal chains. Incidents such as the exploitation of CVE-2021-44228 (Log4Shell) and CVE-2023-34362 (MOVEit Transfer) highlighted 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 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 inherently 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 temporal probabilistic graph aligned with MITRE ATT&CK.
• C2: Formulation of the containment prioritization problem as a combinatorial optimization problem under budget constraints.
• 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 Event Modeling

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 dialogues with classical approaches to 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 later enriched with probabilistic metrics (WANG et al., 2008). Works such as Homer et al. (2013) formalize logical dependencies between preconditions and effects. However, such 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 problem of the closed-world assumption (SOMMER; PAXSON, 2010), in which training data do not adequately represent novel attacks. Moreover, 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 adopt an adversary with capabilities inspired by the Dolev-Yao model, adapted to the corporate context: interception of internal traffic, phishing for credential acquisition, 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 observed events 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 the 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 a 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 has the classical guarantee of at least $1/2$ of the optimum in the worst case.

3.2 Complexity

Incremental graph updating with balanced structures has amortized cost $O(\log |V_t|)$ per insertion. Local inference in 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 consists of four modules: (i) event ingestion and normalization; (ii) incremental construction of the temporal graph; (iii) probabilistic inference; (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 neighbor influence 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, with $L_2$ regularization to avoid overfitting. We evaluate probabilistic calibration using 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 a 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 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 real anonymized events over six months.

6.1 Baselines

Method	Description
SIEM Rules	Rule-based correlation using static rules
Random Forest	Traditional supervised classifier
IR-GraphSec	Proposed graph-based probabilistic approach

6.2 Results

• MTTD reduction: 37.4%.
• MTTR reduction: 42.1%.
• Recall for lateral movement: $0{,}91 \pm 0{,}02$.
• Reduction in false positives: 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 the limitations of isolated correlation and allows explicit incorporation of probabilistic uncertainty, reducing the impact of the base-rate fallacy. The model’s interpretability 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 demonstrated that probabilistic temporal graphs combined with Bayesian inference offer 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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