Directed Ordered Acyclic Graph (DOAG)

• Directed Ordered Acyclic Graphs (DOAGs) are DAGs enriched with a total order or per-vertex edge ordering that enforces acyclicity via local inequality constraints.
• They enable efficient algorithmic techniques such as optimal sorting, incremental topological ordering, and entropy-optimal random sampling through structured combinatorial frameworks.
• DOAGs underpin scalable structure learning in causal graphical models and distributed systems by integrating order constraints that simplify model estimation and ensure reliable convergence.

A Directed Ordered Acyclic Graph (DOAG) generalizes the directed acyclic graph (DAG) by enriching it with a compatible ordering structure—either as a total order (topological order) on the vertex set, or as a finer combinatorial structure such as a total order on the out-edges at each vertex. The DOAG formalism underpins a variety of algorithmic, combinatorial, and statistical frameworks, including sorting under partial orders, random combinatorial object generation, efficient incremental graph algorithms, and order-constrained structure learning in probabilistic graphical models and distributed systems.

1. Formal Definitions and Structural Properties

A DOAG can be characterized in several equivalent but distinct ways, each tailored to different research objectives.

• Canonical Definition:

A DOAG is a pair $(G,O)$, where $G = (V,E)$ is a DAG and $O$ is a total ordering of $V$ (a topological order) such that for each $(u,v) \in E$, $O(u) < O(v)$ (0803.0792, Zantedeschi et al., 2023, Shojaie et al., 2024). In this sense, a DOAG is a DAG together with an explicit compatible order, or equivalently, a DAG equipped with one of its topological orderings.

• Edge-Ordering Variant:

In an alternative approach, a DOAG is defined as a directed ordered graph, in which for each vertex $v$ the set of out-edges $\{ (v \rightarrow w) \}$ is totally ordered, and the set of sources may be totally ordered as well (Pépin et al., 2023). The underlying digraph must be acyclic.

• Partial/Layered Ordering:

The notion extends to partially ordered and layered constraints: a partition $V = V_1 \cup \cdots \cup V_R$, together with the rule that edges between $V_r$ and $V_s$ are only allowed from $V_r$ to $V_s$ when $r < s$ (Rahman et al., 2019, Shojaie et al., 2024).

• Order-Based Acyclicity Guarantee:

For $(V, E, \pi)$ with $\pi : V \to \{1, \ldots, |V|\}$ a bijection, requiring that for all $(i, j) \in E$, $\pi(i) < \pi(j)$ ensures every edge points strictly "forward" in the ordering, banning cycles by construction (Zantedeschi et al., 2023, Shojaie et al., 2024).

These definitions confer both notational clarity and algorithmic leverage: the explicit order transforms global acyclicity constraints into local inequalities, facilitating efficient model specification and manipulation.

2. Enumeration, Random Sampling, and Asymptotics

The combinatorics of DOAGs is sharply richer than for ordinary DAGs, due to the explicit incorporation of ordering structures.

• Enumeration via Variation Matrices:
  • An exact recursive counting formula for the number $D_{n,m,k}$ of (unlabelled) DOAGs with $n$ vertices, $m$ edges, $k$ sources,

  $$D_{n, m, k} = \sum_{p=0}^{n-k} \sum_{i=0}^p D_{n-1, m-p,\, k-1+p-i}\; \binom{n-k-p+i}{i}\; \binom{p}{i}\; i!$$ - Asymptotic analysis (Theorem 5.3):

  $D_n \sim C n^{-1/2} e^{n-1} \prod_{k=1}^{n-1} k!$

  and in terms of the Barnes G-function,

  $$D_n \sim c\, n^{-7/12}\, (e\sqrt{2\pi})^n\, e^{-\frac{3}{4} n^2}\, n^{n^2/2}$$ - In large $n$ limit, a random DOAG almost surely has a unique source, and the expected number of edges is $\binom{n}{2} - O(n)$.

• Efficient Random Samplers:

  • The recursive method samples DOAGs by dynamic programming over the recursion above, with $O(m^2)$ big-integer arithmetic complexity per sample.
  • The optimal rejection sampler constructs a candidate DOAG via stochastic filling of the variation matrix, rejecting invalid objects early, yielding expected $O(n^2)$ time and space complexity with entropy-optimal random bit usage (Pépin et al., 2023).
• Labelled DOAGs:

The same framework extends to labelled DAGs with prescribed edge counts and source constraints, with polynomial time random sampling and combinatorial enumeration (Pépin et al., 2023).

3. Algorithmics: Sorting and Dynamic Maintenance

Ordering structures in DOAGs afford algorithmic optimality for sorting and incremental updates.

• Sorting from a DOAG:
  • The van der Hoog–Rotenberg–Rutschmann algorithm sorts $G$ using $\Theta(n + m + \log e(P_G))$ time and $\Theta(\log e(P_G))$ oracle queries, matching the information-theoretic lower bounds.
  • The scheme removes a longest path $\pi$, then inserts remaining vertices via finger search on $\pi$, supported by dynamic 2–4 trees and dynamic LCA structures.
• Incremental Topological Ordering:
  • The Haeupler–Sen–Tarjan algorithm supports $O(m^{1/2})$ amortized time per arc insertion in sparse graphs, using balanced bidirectional search and dynamic ordered lists (0803.0792).
  • The method dynamically identifies the minimal region of the order requiring repair, eschewing priority queues, and supports strong component maintenance as well.

A summary table contrasting algorithmic perspectives:

Application	Complexity Bound	Essential Structure
Sorting from DOAG	$\Theta(n + m + \log e(P_G))$	Linear extension/order queries, entropy considerations
Incremental topo. ordering	$O(m^{1/2})$ (amort.)	Explicit topological order with $O(1)$ compare/insert
Uniform random DOAG gen.	$O(n^2)$ (expected)	Variation matrix, Poisson statistics

4. Structure Learning with Order Constraints

DOAGs provide the formalism for order-aware or partially ordered structure discovery in causal graphical models.

• Total Ordering (Topological):
  • The permutahedron framework leverages the convex hull of all order vectors, enabling continuous optimization over orderings for structure learning (Zantedeschi et al., 2023).
  • The explicit ordering maps the acyclicity constraint into a family of local linear inequalities, rendering every solution automatically acyclic.
• Partial/LAYERED Ordering:
  • The Partition-DAG algorithm solves penalized likelihood estimation of Gaussian graphical models subject to block lower-triangular constraints, with parallelizable block coordinate descent and theoretical guarantees on stationary points (Rahman et al., 2019).
  • The two-stage "screening + search" algorithm by Shojaie–Chen (Shojaie et al., 2024) exploits known orderings (layer-wise or total), intersecting sets of candidate parents, and iteratively eliminating spurious links by Markov blanket-restricted conditioning.
• Statistical Guarantees and Empirical Findings:

Algorithms exploiting DOAG structure increase both computational tractability and statistical accuracy. Finer ordering information shrinks the search space and improves recovery rates, as demonstrated on DREAM3 Yeast networks and real eQTL/biological datasets (Rahman et al., 2019, Shojaie et al., 2024).

5. Directed Ordered Acyclic Graphs in Distributed and Asynchronous Algorithms

In distributed computing, the DOAG (or $\prec$-DAG) formalism captures the state transitions of asynchronous systems, particularly those tolerating non-synchronized execution.

• DAG-Inducing Problems/Algorithms:

A problem is "DAG-inducing" if there exists a predicate $\mathcal{P}$ and node-wise local partial orders $\prec_l$ such that the global state space forms a $\prec$-DAG, where every edge increases the system rank and sinks correspond to optimal solutions (Gupta et al., 2023).

• Key theorem: If an algorithm induces a global $\prec$-DAG, it remains correct under arbitrary asynchrony; conversely, convergence under asynchrony implies the extended transition system is acyclic and thus the algorithm is DAG-inducing (Gupta et al., 2023).
• The $\prec$-DAG is defined on global states $s, s'$ by $s \prec s'$ if $s[i] \preceq_l s'[i]$ for all $i$ and $s[i] \prec_l s'[i]$ for some $i$.
  • Monotonicity and Self-stabilization:

These properties support the resilience of "guarded" distributed algorithms; local moves always raise the node’s local rank, global rank decreases monotonically, and once a sink is reached, the system becomes silent and stable (Gupta et al., 2023).

6. Connections, Misconceptions, and Research Directions

• Not all DOAGs Are Merely Labelled DAGs:

Explicit ordering (total or per-vertex out-edge orderings) matters significantly in combinatorial enumeration, random generation, and learning contexts, as it introduces constraints and symmetries absent in simple labelled DAGs (Pépin et al., 2023).

• Orderings Versus Partial Orders:

A DOAG’s structure is defined by its base partial order (the DAG) and the auxiliary ordering(s). The presence of a topological order, an ordering of out-edges, or layered partitions can greatly simplify algorithms or make inferences tractable that are hard for arbitrary partial orders (Shojaie et al., 2024, Rahman et al., 2019).

• Open Problems and Algorithmic Frontiers:

Research continues on refining enumeration asymptotics (Pépin et al., 2023), developing fully polynomial time random samplers for labelled/parameterized DOAG variants, and understanding the extremal/limit behaviors of DOAGs under various constraints.

• Impact in Structural Graphical Models and Causal Discovery:

The modular, order-aware frameworks enable scalable estimation and improved model identifiability in high-dimensional or partially specified domains, with applications in genomics, systems biology, and agroecology (Shojaie et al., 2024, Rahman et al., 2019).

7. Summary Table of Key Constructions and Algorithms

Structural Viewpoint	Ordering Used	Key Algorithmic Result	Reference
Standard DOAG (topological)	Total order on $V$	Optimal sorting from DAG	(Hoog et al., 2024)
Edge-ordered DOAG	Per-vertex total order on out-edges	Enumeration, uniform random gen	(Pépin et al., 2023)
Layered/partitioned DOAG	Partial (layered) order, block-triangular	Efficient structure learning, Partition-DAG	(Rahman et al., 2019, Shojaie et al., 2024)
Distributed $\prec$-DAG	Local partial orders on node state	Asynchrony tolerance, convergence	(Gupta et al., 2023)

The DOAG formalism unifies a broad class of structures where explicit ordering—global or local—enables efficient computation, precise enumeration, and tractable learning, while also supporting distributed and asynchronous algorithm design via monotonic progress guarantees.