Directed q-Analysis Overview

• Directed q-analysis is a formalism that extends q-connectivity by using oriented simplices and directed cliques to capture multi-node interactions in digraphs.
• It leverages algebraic topology tools, including incidence matrices, boundary operators, and Betti numbers, to quantify connectivity and identify homotopy invariants.
• The framework enables output-sensitive algorithms for large-scale computations, as demonstrated in applications like the C. elegans connectome and BBP cortex graph.

Searching arXiv for the cited topic and related papers to ground the article in current literature. Directed q-analysis is a higher-order network-analysis formalism for directed graphs in which ordered directed cliques are assembled into a directed flag complex and then related through orientation-sensitive notions of $q$-nearness, $q$-connectivity, and simplicial walks. In the literature, it appears both as an extension of Atkin’s $q$-connectivity to directed simplices, yielding preorders and associated finite topological spaces, and as a quantitative framework built on incidence operators, homology, structure vectors, and directed higher-order connectivity measures for digraphs (Riihimäki, 2022, Baldo et al., 13 May 2026).

1. Emergence and scope

Traditional graph analysis focuses on nodes and edges, that is, pairwise relationships. The motivation for directed q-analysis is that many real-world networks, including biological, social, and communication networks, involve higher-order relationships in which multiple nodes interact simultaneously, while many higher-order methods address only undirected networks. Directed q-analysis addresses this gap by working with directed cliques and the directed flag complex, thereby retaining orientation information that is discarded by undirected constructions (Baldo et al., 13 May 2026).

Riihimäki’s formulation extends Atkin’s theory of $q$-connectivity to directed simplices and produces a preorder in which simplices are related by sequences of simplices that share a $q$-face with respect to directions specified by chosen face maps. This preorder supports simplicial path analysis and, through the Alexandroff equivalence between preorders and finite topological spaces, yields new homotopy types that can differ from those detected by simplicial homology of the directed flag complex (Riihimäki, 2022).

A later quantitative development formalizes directed Q-analysis on digraphs by stressing the interrelations between directed cliques, described there as directed higher-order connectivities. In that framework, the objective is not only to represent higher-order structure, but to quantify, characterize, and compare similarities involving simplicial structures (Baldo et al., 13 May 2026).

2. Directed simplices and directed flag complexes

Let $G=(V,E)$ be a simple directed graph, with no loops and no parallel arcs. A directed $(q+1)$-clique is an ordered tuple of $q+1$ vertices

$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$

such that for all $0\le i<j\le q$, the arc $q$0. The order on the vertices induces the orientation of the simplex. Equivalently, a $q$1-simplex is an ordered $q$2-tuple $q$3 with $q$4 for all $q$5; in Riihimäki’s terminology, such a simplex has source $q$6 and sink $q$7 (Baldo et al., 13 May 2026, Riihimäki, 2022).

The directed flag complex, denoted $q$8 in the quantitative formulation and $q$9 in the algorithmic literature, is the graded collection of all directed simplices of $q$0. Its $q$1-simplices are precisely the directed $q$2-cliques. This construction is the directed analogue of the undirected flag complex, but it is strictly finer because the orientation and the vertex order are part of the simplex data (Baldo et al., 13 May 2026, Windisch et al., 21 Aug 2025).

For a directed $q$3-simplex $q$4, the $q$5-th face is obtained by deleting the $q$6-th vertex: $q$7 The same operation is written $q$8 in other sources. Dually, the coface map

$q$9

collects all simplices whose $q$0-th face is the simplex $q$1 (Baldo et al., 13 May 2026, Windisch et al., 21 Aug 2025).

3. Algebraic and topological formalism

Over a coefficient ring $q$2, the chain group $q$3 is the free $q$4-module generated by the oriented $q$5-simplices. The boundary operator is

$q$6

and the adjoint coboundary is $q$7. After fixing orderings of the $q$8- and $q$9-simplices, one obtains the incidence matrix

$q$0

with entries

$q$1

In matrix form, $q$2 and $q$3 (Baldo et al., 13 May 2026).

The standard algebraic-topological objects are then defined exactly as in simplicial homology: $q$4 When $q$5 is a field, $q$6 is a vector space of dimension $q$7, the $q$8-th Betti number. The Euler characteristic of $q$9 is

$G=(V,E)$0

This places directed q-analysis within the standard homological toolkit, while preserving orientation-sensitive simplex data at the combinatorial level (Baldo et al., 13 May 2026).

4. $G=(V,E)$1-nearness, preorders, and corrected definitions

In Riihimäki’s construction, directed q-analysis is organized around orientation-sensitive nearness relations. Fix $G=(V,E)$2 and face-map indices $G=(V,E)$3. Two ordered simplices $G=(V,E)$4 of dimension at least $G=(V,E)$5 are $G=(V,E)$6-near if either $G=(V,E)$7, or there exists a $G=(V,E)$8-simplex $G=(V,E)$9 such that

$(q+1)$0

A directed $(q+1)$1-connection along $(q+1)$2 is then a finite chain

$(q+1)$3

in which each consecutive pair is $(q+1)$4-near. The resulting relation

$(q+1)$5

is reflexive and transitive, hence a preorder on simplices of dimension at least $(q+1)$6 (Riihimäki, 2022).

This preorder has a topological interpretation. By Alexandroff’s theorem, finite preorders correspond to finite topological spaces; after passing to the partial-order reflection by identifying strongly connected pairs, one obtains a finite $(q+1)$7 space $(q+1)$8 without changing homotopy type. Riihimäki showed that these spaces can yield new homotopy invariants that detect orientation in ways not visible in the directed flag complex alone (Riihimäki, 2022).

Subsequent work identified a hidden bias in the original formulation. The issue arises because face maps in the original condition may act on simplices of arbitrary dimension at least $(q+1)$9, requiring clamping of indices when $q+1$0 or $q+1$1 exceed the actual dimension. This clamping depends on simplex dimension and introduces an index bias. The corrected, bias-free definition applies face maps only to genuine $q+1$2-simplices and then uses upward-closure to propagate the relation to larger simplices (Windisch et al., 8 Jan 2025).

In the later algorithmic presentation, the original and novel $q+1$3-nearness definitions are stated side by side. The two definitions coincide on the level of $q+1$4-simplices, but differ in upward-closure properties. This distinction is central both conceptually and computationally, because the corrected definition supports output-sensitive construction of the associated directed $q+1$5-digraphs (Windisch et al., 21 Aug 2025).

5. Quantitative descriptors and directed higher-order connectivity

The quantitative framework defines the directed $q+1$6-vector by

$q+1$7

so that the full $q+1$8-vector is $q+1$9, where $\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$0. This records the simplex counts at each dimension and is the basic size descriptor of the directed flag complex (Baldo et al., 13 May 2026).

For a directed flag complex $\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$1, the paper defines five structure vectors in the Atkin/Andjelković sense:

Vector	Definition
$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$2	$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$3
$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$4	$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$5, where $\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$6 is the number of weakly $\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$7-connected components
$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$8	$\sigma^{(q)}=[v_0< v_1<\dots <v_q]\subseteq V$9, where $0\le i<j\le q$0 is the number of strongly $0\le i<j\le q$1-connected components
$0\le i<j\le q$2	$0\le i<j\le q$3
$0\le i<j\le q$4	$0\le i<j\le q$5

These vectors separate simplex abundance from connectivity organization and from normalized cohesion proxies, allowing comparisons across levels $0\le i<j\le q$6 and across digraphs (Baldo et al., 13 May 2026).

The same framework introduces directed $0\le i<j\le q$7-connectivity numbers. For $0\le i<j\le q$8-simplices $0\le i<j\le q$9, lower, upper, and bidirectional $q$00-nearness are defined in the paper’s Definition 6.6, together with maximal $q$01-$q$02-walks in Definition 6.16. The resulting quantities include

$q$03

$q$04

$q$05

Pathways through $q$06-simplices are represented by directed simplicial $q$07-walks of minimal length $q$08, and the corresponding length defines a quasi-distance $q$09 (Baldo et al., 13 May 2026).

This quantitative layer is complementary to the preorder-based viewpoint. The preorder captures reachability and induced finite spaces; the quantitative framework attaches explicit counts, component statistics, degree-like quantities, and walk-based distances to the directed higher-order connectivity structure.

6. Algorithms, complexity, and large-scale computation

At the foundational level, directed clique enumeration extends Bron–Kerbosch to oriented cliques by recursively building all vertex lists $q$10 such that all $q$11 exist. In worst-case graphs, the number of directed cliques is exponential, $q$12. Once all $q$13- and $q$14-simplices are known, building the incidence matrix $q$15 requires computing the $q$16 faces of each $q$17-simplex and has complexity $q$18. For each level $q$19, one may form the set $q$20 of maximal $q$21-simplices, build the $q$22-adjacency matrix $q$23 by checking shared $q$24-faces in $q$25, and compute weak or strong connectivity by BFS or DFS in $q$26. Overall, constructing all levels up to maximal dimension runs in time polynomial in $q$27 plus the exponential clique enumeration term (Baldo et al., 13 May 2026).

Later work reformulated the computation of directed q-analysis to avoid the output-insensitive top-down strategy. In the top-down approach, all simplices in $q$28 are enumerated and every ordered pair is tested for $q$29-nearness, giving runtime $q$30 when $q$31. The hybrid, output-sensitive algorithm instead inverts the order of computation: it first collects inclusion edges, then enumerates local face-sharing edges

$q$32

and finally lifts these by upward-closure through the inclusion graph. The resulting runtime is

$q$33

and if $q$34 and $q$35 are treated as constants, this becomes

$q$36

which is time-optimal because any algorithm must at least write down all edges (Windisch et al., 21 Aug 2025).

The practical effect is substantial. Windisch and Unger report a Rust implementation and show, for the C. elegans connectome at $q$37, Hybrid runtime of approximately $q$38 ms versus approximately $q$39 s for Top-Down; for the BBP cortex graph, with $q$40 and $q$41 K simplices at $q$42, the corresponding times are $q$43 s versus more than $q$44 h. The same line of work emphasizes that the corrected definition both removes the bias in the original formulation and enables these algorithmic gains (Windisch et al., 21 Aug 2025).

These computational improvements made large-scale applications feasible. In connectome studies, directed q-analysis was compared with null models having the same undirected graph and simplex statistics. Reported findings include a maximal total-degree $q$45-score of $q$46 versus null for C. elegans at $q$47 and $q$48, an approximate longest directed path more than $q$49 above null for C. elegans at $q$50, and a $q$51 deviation for the number of weakly connected components in the BBP setting, indicating structural differences beyond chance in the corresponding directed $q$52-graphs (Windisch et al., 8 Jan 2025).

7. Canonical example and relation to undirected Q-analysis

A worked example in the quantitative paper considers the digraph on $q$53 with arcs

$q$54

Its directed flag complex has

$q$55

with $q$56-simplices $q$57, $q$58-simplices $q$59, and $q$60-simplices $q$61 and $q$62. There is no $q$63-clique, so both $q$64-simplices are maximal (Baldo et al., 13 May 2026).

Let

$q$65

At level $q$66, the strict lower $q$67-adjacency vanishes: $q$68 and $q$69 share $q$70 as a $q$71-face, so they are $q$72-near but also $q$73-near, hence there is no strict $q$74-adjacency and $q$75. At level $q$76, however, $q$77 because they share the $q$78-face $q$79 but not a $q$80-face, giving one arc $q$81 and

$q$82

For this level-$q$83 digraph, the paper reports $q$84, $q$85, $q$86, global efficiency $q$87, energy $q$88, reaching centrality $q$89, and harmonic centralities $q$90, $q$91 (Baldo et al., 13 May 2026).

This example also clarifies the relation to undirected Q-analysis. In the undirected flag complex, one ignores arc orientation and the order of vertices; a directed simplex $q$92 collapses to the undirected simplex $q$93. Lower and upper $q$94-adjacencies become symmetric, and there is a single notion of $q$95-adjacency as face-sharing. Directed Q-analysis refines this by splitting adjacency into incoming and outgoing face-sharing and by using directed $q$96-walks to capture flow of influence. On maximal simplices, the paper proves that strictly lower $q$97-adjacency equals maximal $q$98-adjacency, a proposition with no undirected counterpart (Baldo et al., 13 May 2026).

A common simplification is to treat the directed theory as merely the undirected theory plus arrow directions. The literature does not support that reduction. In the preorder-based formulation, the orientation-sensitive connectivity relations can produce finite spaces $q$99 whose homotopy types differ from those of the directed flag complex itself; examples include cases where a directed flag complex has the homotopy type of a $q$00-sphere while the induced $q$01 is a wedge of circles or a single circle (Riihimäki, 2022).