Interaction-Based Anomalies in Complex Systems

• Complex interaction-based anomalies are atypical deviations arising from the interplay among system components rather than from isolated features.
• Detection methodologies leverage latent variable models, spatiotemporal graph analyses, and scene-graph constructions to quantify abnormal interdependencies.
• Applications span diverse fields such as network traffic analysis, cyber-physical systems, video surveillance, and theoretical physics, highlighting both challenges and practical insights.

Complex interaction-based anomalies are deviations or violations from expected regularities that emerge specifically from the interplay between components, agents, or features within complex systems. Rather than arising from isolated components, these anomalies are rooted in the structure, strength, or evolution of interactions themselves—whether in networks, multivariate signals, processes, or particle systems. This concept spans domains including cyber-physical systems, dynamic networks, network traffic, video surveillance, and even the foundations of physical theory, where the symmetry properties of interactions fundamentally shape or restrict possible anomalous phenomena.

1. Foundational Concepts and Definitions

Complex interaction-based anomalies refer to rare, atypical, or statistically unlikely patterns that result from disrupted, unexpected, or abnormal relationships among interacting entities in a system. In contrast to traditional point anomalies—or univariate outliers—these anomalies are undetectable when features or agents are considered independently. The anomaly becomes observable only through joint modeling or analysis of interaction patterns, dependencies, or causal pathways.

Key settings include:

• Networks: Anomalous dyads or subgraphs where observed interactions deviate from those predicted by community structure, reciprocity, latent space models, or temporal evolution (Safdari et al., 2022, Safdari et al., 2023, Safdari et al., 16 Apr 2024).
• Time series and CPS: Irregularities in spatiotemporal or causal linkages between nodes or processes, including failures propagating along unusual causal paths (Liu et al., 2018).
• Video: Multi-agent scenes where abnormality is defined by interaction violation (e.g., two people fighting, a skateboard without a rider), not by isolated appearance (Mumcu et al., 16 Jan 2025, Arad et al., 2023).
• Physical Theories: Anomalous terms in symmetry algebra commutators arising from the structure of allowable interactions, subject to constraints imposed by symmetry groups such as the Poincaré group or its subgroups (Asorey et al., 13 May 2025).

2. Statistical and Algorithmic Methodologies

Several classes of methodologies target detection and characterization of complex interaction-based anomalies:

1. Graphical and Network Latent Variable Models: Probabilistic generative models that incorporate domain knowledge such as community structure (Safdari et al., 2022, Safdari et al., 16 Apr 2024), or community plus reciprocity (Safdari et al., 2023) as a null model to define regular interaction patterns. Deviations from the expected rates—quantified via latent variables (e.g., $Z_{ij}$ for edge anomaly indicators)—characterize anomalous interactions.
2. Spatiotemporal and Correlation Structure Modeling: Frameworks such as Spatiotemporal Pattern Networks (STPN) (Liu et al., 2018) and correlation-graph-based VAE architectures (Qin et al., 2021) capture both spatial dependencies and time-evolving causal or correlation structures. Anomaly detection focuses on failures or disruptions in the usual interdependence patterns, as measured by inference metrics (e.g., $\Lambda^{ab}$) or collective reconstruction deficiencies.
3. Higher-order and Heterogeneous Interactions: Statistical models such as Decades (Larroche et al., 2021) factorize the joint probability of multi-entity events, accounting for combinatorial event structures (beyond dyads), event type heterogeneity, and temporal nonstationarity, enabling detection of anomalies in higher-order interaction contexts.
4. Temporal and Dynamic Network Approaches: Markovian models (Safdari et al., 16 Apr 2024), temporal graph neural networks (Lazebnik et al., 2023), and latent space models with dynamic updates (Lee et al., 2019) leverage the temporal evolution of networks, capturing anomalies that manifest only as deviations from expected temporal or structural transitions governing interactions.
5. Scene Graph and Object Interaction Modeling in Video: Techniques such as scene-graph construction per frame (Mumcu et al., 16 Jan 2025) or deep video encoding (Arad et al., 2023) explicitly model multi-object relationships and their spatio-temporal attributes, using graph distance metrics to match or flag outlier interaction patterns.
6. Process Mining and Object-centric Approaches: In settings like business process logs, object-centric methods build interaction feature maps, aggregate anomaly scores from both object attributes and interaction context, and propagate features through interaction networks to capture anomalies not evident in isolated objects (Berti et al., 12 Jul 2024).

3. Examples and Theoretical Insights Across Domains

Network Science and Dynamic Graphs

Community-based models (Safdari et al., 2022, Safdari et al., 16 Apr 2024) use edge probabilities parameterized by latent memberships ($u$, $v$) and affinity matrices ($w$). Edges with intensities not predicted by the inferred community structure or with anomalous persistence/transition rates are marked as interaction-based anomalies. In (Safdari et al., 2023), reciprocity is explicitly included via a joint edge distribution—nonreciprocal or atypically reciprocal patterns are treated as anomalous compared to the prevailing structure.

Cyber-Physical Systems and Sensor Networks

In distributed CPS, spatiotemporal symbolic dynamics are modeled by constructing atomic and relational patterns among nodes (Liu et al., 2018). Anomaly detection requires identifying failures in expected causal or correlational linkages, where root-cause analysis searches for patterns whose perturbation most reduces distributional divergence between anomalous and nominal inference metrics.

Video Analytics

Complex interaction-based anomalies in video are delineated by modeling the joint behavior of multiple objects. Scene-graph-based methods encode each frame as a graph of tracked objects; graphs are compared against exemplars using specialized distances over locations, trajectories, and poses (Mumcu et al., 16 Jan 2025). Anomalies are defined as those graphs for which no similar nominal exemplar exists—highlighting out-of-norm interactions rather than isolated motion or appearance.

Physical Theories: Symmetry and the Non-Interaction Theorem

Interaction anomalies rooted in symmetry are elucidated through the paper of the algebraic structures of the symmetry group generators, e.g., Poincaré or Galilei (Asorey et al., 13 May 2025). For fully Poincaré-invariant systems, the so-called World Line Conditions (WLC) impose that acceleration-dependent anomaly terms in the symmetry generators’ commutators must vanish, leading to the classical no-interaction theorem for single-particle dynamics. For Lorentz-violating systems (e.g., Very Special Relativity/VSR), where only a proper subgroup is preserved (e.g., generated by $\{P_i, H, K_1-\beta J_2, K_2+\beta J_1, J_3\}$), the symmetry-imposed cancellation conditions are relaxed, allowing for controlled interaction anomalies. In Galilei-invariant multiparticle systems, the anomaly constraints permit nontrivial interaction via relative positions and velocities.

4. Mathematical Formulations and Metrics

Methodologies formalize interaction-based anomalies via statistical tests, optimization, and algebraic constraints:

• Latent variable edge model:

$$A_{ij} \sim \text{Poisson}(M_{ij}) \quad \text{(regular)}, \qquad A_{ij} \sim \text{Poisson}(\pi) \quad \text{(anomalous)}$$

$M_{ij} = \sum_k u_{ik} v_{jk} w_k$

with latent indicator $Z_{ij} \sim \operatorname{Bern}(\mu)$.

• Inference-based causal pattern metric in CPS:

$$\Lambda^{ab}(\tilde{Q}, \tilde{S}) = K \prod_{m=1}^{|Q^a|} \frac{(\tilde{N}_m^{ab})! (N_m^{ab} + |\Sigma^b| - 1)!}{(\tilde{N}_m^{ab} + N_m^{ab} + |\Sigma^b| - 1)!} \prod_{n=1}^{|\Sigma^b|} \frac{(\tilde{N}_{mn}^{ab} + N_{mn}^{ab})!}{(\tilde{N}_{mn}^{ab})! (N_{mn}^{ab})!}$$

• SCENE-GRAPH interaction anomaly score and distance (video):

$$AS(n,\mathcal{E}_{iso}) = \min_{n_e \in \mathcal{E}_{iso}} D(n, n_e)$$

$$D(n_1, n_2) = \max \bigg( \frac{L(n_1, n_2) - \mu_L}{\sigma_L}, ... \bigg)$$

with $L(n_1, n_2) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$ over attributes.

• Symmetry-induced constraint for absence of interaction-induced anomalies (physics):

$$[H, K_i] = P_i + \sum_a \big( (a)_i (a)_l + (a)_l (a)_i \big) \partial_{v^{(a)}_l}$$

For Poincaré invariance, cancellation requires $(a)_i (a)_l + (a)_l (a)_i = 0$ for all $i, l$.

5. Practical Applications and Use Cases

Interaction-based anomaly detection frameworks have been implemented and validated in a variety of real-world and synthetic scenarios:

• Enterprise and Social Networks: Detection of fraudulent connections, bot networks, or anomalous message traffic not explainable by inferred community structure (Safdari et al., 2022, Safdari et al., 16 Apr 2024, Safdari et al., 2023).
• Surveillance and Video Analytics: Tracking multi-agent behaviors to identify anomalous group motion, object co-occurrence, or violations of semantic interaction regularities (Mumcu et al., 16 Jan 2025, Arad et al., 2023).
• Industrial and Sensor Networks: Identification of faults not in individual sensors but manifesting as breakdowns in normal correlation or causal network structure; early detection of process anomalies, cyber-physical sabotage, or coordinated sensor attacks (Liu et al., 2018, Qin et al., 2021).
• Process Mining: Detection of compliance violations, fraud, or inefficiencies arising purely from the interlocking behavior of objects, orders, invoices, and workflows (Berti et al., 12 Jul 2024).
• Critical Infrastructure: Bayesian network models detect and localize leakages or sensor faults based on how anomalies disrupt the usual propagation or drift of observable variables across interconnected components (Vaquet et al., 2023).

6. Challenges, Limitations, and Future Directions

Persistent challenges in detecting and characterizing complex interaction-based anomalies include:

• Interpretability and domain knowledge integration: Many models require auxiliary diagnostic tools (e.g., feature importance scores, explanation models, or LLMs) to make anomalies actionable and understandable by experts (Berti et al., 12 Jul 2024).
• Sensitivity to model assumptions: As in centrality anomaly studies, oversimplified network models may artificially produce anomalies; richer models—incorporating weights, multilayers, or accurate null models—are required to avoid misattribution (Alves et al., 2019).
• Scalability and efficiency: Large, high-dimensional interaction spaces (e.g., in video or dynamic network settings) necessitate efficient algorithms, memory management, and sometimes pruning heuristics (Foorthuis, 2020).
• Data complexity and sparsity: Highly noisy, sparse, or incomplete interaction data complicates inference, especially in unsupervised or semi-supervised settings.
• Theoretical limits imposed by symmetry: As shown in (Asorey et al., 13 May 2025), the scope of allowable dynamic or interactive anomalies is heavily constrained by the specific symmetries preserved in the system, whether Poincaré, Galilei, or VSR subgroups, with implications for both physical modeling and data-driven systems.

Planned advances include deeper integration of domain knowledge (e.g., via LLMs, process mining), more robust explanation and alert aggregation frameworks, improved handling of temporal dynamics and concept drift, and the extension of algebraic and probabilistic characterizations to increasingly heterogeneous and higher-order interaction systems.

7. Summary Table: Model Classes and Domains

Approach / Model	Anomaly Target Domain	Key Interaction Principle
Latent Variable Community Models	Social, infrastructure networks	Community-driven edge regularity
Spatiotemporal Graphical Models	Cyber-physical, sensor networks	Symbolic dynamics and causal link failures
Deep Scene Graphs	Video surveillance	Object-object spatio-temporal relationships
World Line Condition Analyses	Theoretical physics	Algebraic constraints from symmetry groups
Intuitionistic Fuzzy Networks	Network traffic	Non-deterministic correlations in metrics

This synthesis incorporates formalisms, methodologies, domain-specific findings, and theoretical constraints as described in the literature, and establishes the central role of complex interactions in shaping, enabling, or suppressing anomaly phenomena across science and engineering.