Conflict Directed Graph (CDG) Overview

• Conflict Directed Graphs (CDGs) are directed graphs that encode conflict relationships between model components or agents via partitioning and directed edges.
• In Bayesian evidence synthesis, CDGs enable node-splitting to compare independent posterior inferences, providing quantitative conflict diagnostics through p-value assessments.
• For multi-agent scheduling, CDGs structure precedence and mutual exclusion constraints to facilitate safe and efficient vehicle coordination at intersections.

A Conflict Directed Graph (CDG) encodes pairwise conflict relationships in a directed graph structure, enabling systematic reasoning about model consistency, evidence conflict diagnostics in probabilistic graphical models, and conflict-avoiding coordination in multi-agent dynamic scheduling. The CDG formalism appears in both complex Bayesian evidence synthesis—where it supports local assessments of conflict between disparate evidence sources within Directed Acyclic Graphs (DAGs)—and in trajectory planning for multi-vehicle systems—where it structures precedence and concurrency constraints for conflict-free intersection management. The details of construction and use of CDGs vary substantially between these domains but share the central logic of formal partitioning, conflict encoding, and combinatorial analysis.

1. Definition and Formal Construction

In Bayesian evidence frameworks, the CDG is instantiated as a modified DAG containing one or multiple “separator” nodes θ, which partition the factors (e.g., priors, likelihoods, submodels) impacting inference about θ. The original θ is split into θₐ and θ_b, each attached to the subset of factors from one evidence partition. Edges in the modified graph reflect the flow of information within each partition and, if needed, “cut” functions block feedback through shared nuisance parameters. The approach enables direct comparison of independent inferences about θ from disjoint data/model partitions (Presanis et al., 2013).

In multi-agent scheduling problems, such as vehicle coordination at intersections, the CDG is a finite directed (and bidirected) graph $\mathcal{G}_{N+1}=(V_{N+1},E_{N+1})$ where $N$ is the number of agents or vehicles. An auxiliary vertex (often labeled $0$) acts as a virtual leader or reference. Vertices $1,\ldots,N$ correspond to the real agents. Edges are of two main types:

• Unidirectional edges (Eᵤ): Encode strict precedence demands, e.g., due to lane-following (diverging) or reachability constraints.
• Bidirectional edges (Eᵦ): Encode mutual exclusion for simultaneous occupation of the same geometric or temporal conflict point, but with a flexible crossing order.

The edge sets are formally:

$$E^{u}_{N+1} = \{(i,j) : i < j,\, i \in D_j \cup R_j\}$$

$$E^{b}_{N+1} = \{(i,j),(j,i): i < j,\, i \in C_j \cup V_j\}$$

where $D_j$, $C_j$, $V_j$, $R_j$ represent diverging, crossing, converging, and reachability conflict sets, respectively (Chen et al., 2021, Chen et al., 2021).

2. CDG Construction Algorithms and Examples

Bayesian Evidence Synthesis

The construction sequence involves:

1. Identifying separator node(s) θ with incoming information grouped into partitions $(\lambda_a, y_a)$ and $(\lambda_b, y_b)$.
2. Creating duplicated nodes θₐ and θ_b, each linked solely to their partition’s factors and data.
3. Assigning a non-informative prior (such as Jeffreys’) to θ_b if the subgraph is otherwise improper.
4. Applying “cuts” to block cross-partition feedback through shared nuisance parameters if strict independence is required.

This leads to two localized submodels whose posterior inferences about θ are compared via a conflict diagnostic (Presanis et al., 2013).

Multi-Vehicle Scheduling

The construction follows:

1. Sort vehicles by arrival. Assign virtual leader $0$.
2. For each vehicle $j$:
   • Compute $D_j$ (diverging conflicts: same incoming lane).
   • Compute $C_j$ (crossing conflicts: path intersection).
   • Optionally, compute $V_j$ (converging: same exit) and $R_j$ (reachability).
3. Add appropriate edges to $E^u$ and $E^b$ as per formal rules.

A worked example with four vehicles demonstrates the resulting adjacency matrix and explains how the sparse, structured edge set completely encodes safety- and order-related constraints (Chen et al., 2021).

3. Scheduling and Conflict Diagnosis via CDG

CDGs serve as algorithmic substrates for both probabilistic conflict detection and multi-agent scheduling.

Model Criticism in DAGs

Posterior distributions of the split nodes are sampled independently, and then a test statistic δ is computed (e.g., difference in means after suitable transformation). Several p-value diagnostics are available, including:

• One-sided and two-sided tail tests for univariate θ,
• Mahalanobis-type test or kernel density estimates for multivariate θ.

A low p-value (e.g., $<0.05$) signifies substantive conflict between evidence partitions at θ. The algorithmic outline is:

1. Partition predictor set and data.
2. Build partition models (with cut if needed).
3. Sample pairs $(\theta_a^{(m)}, \theta_b^{(m)})$ from posteriors.
4. Compute δ for each MCMC sample.
5. Calculate p-value and visualize the result (Presanis et al., 2013).

Vehicle Scheduling

The CDG is traversed to optimize layer-based crossing order:

• Improved Depth-First Spanning Tree (iDFST): Iteratively constructs a spanning tree rooted at the virtual leader, assigning depth/layer within the tree according to precedence and conflict constraints—ensuring maximally concurrent (but conflict-free) scheduling. Vehicles at the same tree depth are guaranteed to be mutually conflict-free and may proceed as a group. This method runs in $O(N^2)$ time per scheduling instance.
• Minimum Clique Cover (MCC) / Maximum Matching: The coexisting undirected graph (complement of CDG restricted to real vehicles) is used. Clique covers or matchings correspond to maximal conflict-free sets (layers). MCC—NP-complete—achieves parallelism-optimal scheduling but at higher computational cost ($O(N^4)$ for a maximum matching approach) (Chen et al., 2021, Chen et al., 2021).

4. Application Domains and Representative Results

Bayesian Evidence Synthesis

Example applications include:

• HIV prevalence estimation: Node-splitting revealed specific data sources in conflict with model structure or other evidence, quantitatively via small two-sided p-values.
• Influenza severity analysis: Partitioning seroprevalence vs. agency estimates, with kernel density-based p-values indicating conflict in child subpopulations.
• Hierarchical regression on rat weights: Multivariate node-splits flagged individual rats as outliers using χ² and Mahalanobis diagnostics (Presanis et al., 2013).

Intelligent Vehicle Coordination

• At unsignalized intersections, CDGs enable both real-time, distributed, and globally optimal (when feasible) scheduling.
• Simulation results confirm the effectiveness of iDFST and MCC methods for reducing intersection delay and enabling safe, conflict-free cooperative maneuvering among connected and automated vehicles (CAVs) (Chen et al., 2021, Chen et al., 2021).

5. Properties, Complexity, and Theoretical Insights

CDGs in scheduling contexts exhibit several key graph-theoretic properties:

• Spanning trees rooted at the virtual leader always exist under general geometric/practical conditions.
• Vehicles (nodes) at the same depth in the spanning tree schedule are never mutually connected by a CDG edge, ensuring their simultaneous operation is conflict-free.
• The global minimum evacuation time equals the minimum clique cover number in the coexisting undirected graph; finding this minimum is NP-complete.

Algorithmic complexities:

Method	Complexity	Context
CDG Construction	$O(N^2)$	Vehicle scheduling
iDFST/OPT-DFST	$O(N^2)$	Local scheduling optimum
Maximum Matching/MCC	$O(N^4)$	Global scheduling optimum

Implementation in Bayesian model checking requires separate or joint MCMC sampling from the partitioned submodels, which can be computationally intensive for many nodes (each node-split is essentially a cross-validation per separator) (Presanis et al., 2013).

6. Extensions, Limitations, and Practical Considerations

• In Bayesian models, partition choices (which factors/data to assign to each partition) affect the validity and sensitivity of results; improper configuration may reduce the power or interpretability of diagnostics.
• Multiple node-splits in large DAGs require careful adjustment for multiple hypothesis testing (e.g., Benjamini–Hochberg) and may need diagnostic prioritization.
• In vehicle scheduling, complexity limits practical MCC computation to moderate N; for large N, distributed greedy heuristics using iDFST or graph coloring yield near-optimal performance.
• In both domains, visualizing posterior distributions or schedule layers facilitates practitioner interpretation and detection of substantive disagreements.

A plausible implication is that CDG-based approaches offer a unifying formalism for localizing, quantifying, and mitigating conflict in both probabilistic inference and cooperative scheduling, though computational scalability and partition design remain active areas for research.

7. Selected References

• "Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis" (Presanis et al., 2013)
• "Conflict-free Cooperation Method for Connected and Automated Vehicles at Unsignalized Intersections: Graph-based Modeling and Optimality Analysis" (Chen et al., 2021)
• "A Graph-based Conflict-free Cooperation Method for Intelligent Electric Vehicles at Unsignalized Intersections" (Chen et al., 2021)