Motif Parameters over Categories

• Motif parameters over categories are counting functions defined as finite linear combinations of subobject counts using designated embeddings and purity conditions.
• They generalize traditional graph motif counting by replacing induced subgraphs with M-subobjects, thus extending the framework to various mathematical structures.
• The dichotomy theorem highlights that combinatorial interpretability is achieved if and only if all coefficients in these linear combinations are nonnegative integers.

Searching arXiv for papers on motif parameters over categories and related graph motif parameters. Motif parameters over categories are counting functions defined by taking finite linear combinations of subobject-counting basis functions in a category equipped with a designated class of morphisms that play the role of embeddings. In the formulation introduced in "Which graph motif parameters count?" (Bläser et al., 16 Jul 2025), if $C$ is a category with a chosen class $\mathcal M$ of morphisms, then for objects $a,b\in C$, the $\mathcal M$-subobjects of $a$ under $b$ are morphisms $a\to b$ in $\mathcal M$, modulo isomorphism of the domain, and a motif parameter has the form

$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$

with pairwise non-isomorphic pattern objects $a_i$. This abstracts the graph-theoretic notion of graph motif parameters from induced-subgraph counting to a categorical setting, and it yields a general dichotomy: under structural hypotheses on the category and its pure objects, combinatorial interpretability is equivalent to having nonnegative integer coefficients (Bläser et al., 16 Jul 2025). Related work on labeled graphs studies graph motif parameters as finite linear combinations of homomorphism counts and shows that their Weisfeiler–Leman dimension is exactly the maximum treewidth of the support patterns (Lanzinger et al., 2023).

1. From graph motif parameters to categorical motif parameters

In the graph case, the basis function associated with a fixed finite graph $\mathcal M$0 is

$\mathcal M$1

and a graph motif parameter is a finite rational linear combination

$\mathcal M$2

where the $\mathcal M$3 are pairwise non-isomorphic patterns (Bläser et al., 16 Jul 2025). The coefficients are unique, integer-valuedness forces all coefficients to be integers, and motif parameters are described as the natural linear span of induced-subgraph counts because of a linearization phenomenon: polynomials in such counts can be rewritten uniquely as linear combinations of basis counts (Bläser et al., 16 Jul 2025).

The categorical generalization retains this linear-combination viewpoint while replacing induced subgraphs by $\mathcal M$4-subobjects. For a fixed target object $\mathcal M$5, the class of all $\mathcal M$6-subobjects under $\mathcal M$7 is denoted $\mathcal M$8, and the counting basis functions are $\mathcal M$9 for pattern objects $a,b\in C$0 (Bläser et al., 16 Jul 2025). The “pure” condition is the categorical analogue of the graph restriction “no isolated vertices”: one fixes a subcategory $a,b\in C$1 of pure objects and only allows patterns $a,b\in C$2 (Bläser et al., 16 Jul 2025).

A related but distinct line of work defines a labeled graph motif parameter as any graph parameter expressible as a finite linear combination of homomorphism counts,

$a,b\in C$3

for fixed labeled graphs $a,b\in C$4, with support

$a,b\in C$5

thereby subsuming subgraph counting and induced subgraph counting in the labeled setting (Lanzinger et al., 2023). This suggests two complementary notions of motif parameter: one centered on induced-subobject basis functions and combinatorial interpretability, the other on homomorphism-count expansions and WL/GNN expressivity.

2. Categorical formulation and the role of purity

The general theory is formulated for a category $a,b\in C$6 together with a chosen class of morphisms $a,b\in C$7, intended to model subobject embeddings (Bläser et al., 16 Jul 2025). For objects $a,b\in C$8, $a,b\in C$9-subobjects of $\mathcal M$0 under $\mathcal M$1 are morphisms $\mathcal M$2 in $\mathcal M$3, modulo isomorphism of the domain. A motif parameter over $\mathcal M$4 is then

$\mathcal M$5

with pairwise non-isomorphic pattern objects $\mathcal M$6 (Bläser et al., 16 Jul 2025).

The categorical framework used for the dichotomy theorem imposes explicit structural assumptions. The category $\mathcal M$7 and the pure subcategory $\mathcal M$8 are taken to be locally small and finitely $\mathcal M$9-well-powered, equipped with a compatible proper factorization system $a$0, with $a$1 a full subcategory of $a$2 closed under isomorphism (Bläser et al., 16 Jul 2025). In addition, the theorem assumes the joint $a$3-embedding property, an $a$4-Ramsey property, a blowup property, and suitable set-instantiators for the oracle encoding under consideration (Bläser et al., 16 Jul 2025).

Purity is essential in the concrete graph theorem. For graphs, “pure” means no isolated vertices (Bläser et al., 16 Jul 2025). For relational structures, the analogue is defined via a padding notion $a$5 excluding “padding vertices,” and for colored graphs the theorem applies when the pattern graphs avoid one designated padding color (Bläser et al., 16 Jul 2025). By contrast, for finite-dimensional vector spaces over a fixed finite field $a$6 and for parameter sets, $a$7, so no additional purity restriction is needed (Bläser et al., 16 Jul 2025).

3. Combinatorial interpretability and the general dichotomy

The evaluation problem $a$8 is formulated as a type-2 or promise counting problem: one computes $a$9 on oracle-coded objects (Bläser et al., 16 Jul 2025). For graphs and relational structures, the oracle encodes the object’s relations on a large universe. A nondeterministic oracle Turing machine computes $b$0 by having its accepting computation paths correspond to witnesses for the counted subobjects, and the resulting class of functions is denoted $b$1, a promise-version of $b$2 adapted to objects that may not be easily verifiable from the oracle alone (Bläser et al., 16 Jul 2025). Combinatorial interpretability is defined by the equivalence

$b$3

The core theorem is a categorical dichotomy. Under the structural assumptions above, if $b$4 is a $b$5-pure motif parameter and $b$6, then $b$7 must be good, meaning that all coefficients are nonnegative integers (Bläser et al., 16 Jul 2025). Equivalently, any $b$8-pure motif parameter with a negative coefficient is not in $b$9 (Bläser et al., 16 Jul 2025). When the obvious upper bound for positive coefficients is also available, this becomes a full characterization: $a\to b$0

This result formalizes a sharp distinction between being nonnegative-valued and genuinely “counting something.” The paper emphasizes that graph motif parameters can be nonnegative for all inputs even when some coefficients are negative (Bläser et al., 16 Jul 2025). The obstruction is therefore not pointwise negativity, but failure of combinatorial interpretability in the oracle-$a\to b$1 sense.

4. Proof architecture: set-instantiators, Ramsey theory, and linear independence

The proof framework in the categorical paper has three stated components (Bläser et al., 16 Jul 2025). The first is the construction of set-instantiators. For a fixed polynomial-time nondeterministic oracle Turing machine $a\to b$2 computing $a\to b$3, one builds, for every finite object $a\to b$4, an encoding $a\to b$5 such that each accepting path of $a\to b$6 on a subobject instance perceives only some subobject of $a\to b$7, and this perception respects subobject inclusion (Bläser et al., 16 Jul 2025). Formally, a set-instantiator consists of a size parameter $a\to b$8, an instantiation map $a\to b$9, and a perception map $\mathcal M$0, with accepting paths corresponding exactly to perceived subobjects and preserving the parameter value (Bläser et al., 16 Jul 2025).

The second component is Ramsey theory. The accepting-path counts induce a coloring of subobjects, and a Ramsey theorem yields a sufficiently large object $\mathcal M$1 in which one can find a copy $\mathcal M$2 of a target object $\mathcal M$3 on which the coloring depends only on isomorphism type (Bläser et al., 16 Jul 2025). In categorical language, the relevant Ramsey property says that for any $\mathcal M$4 and number of colors $\mathcal M$5, there is a $\mathcal M$6 such that every coloring of $\mathcal M$7 has a monochromatic copy of $\mathcal M$8 on all $\mathcal M$9-subobjects (Bläser et al., 16 Jul 2025). This forces the machine’s local behavior to become good.

The third component is linear independence and witness extraction. On a finite subobject poset, the counting functions $$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$0 are linearly independent, and the evaluation matrix is triangular with $$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$1’s on the diagonal (Bläser et al., 16 Jul 2025). Consequently, any bad linear combination differs from every good one on some witness object, contradicting the existence of a nondeterministic machine that computes it in $$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$2 (Bläser et al., 16 Jul 2025).

5. Concrete settings and specialized dichotomies

The general theorem is instantiated in several concrete categories (Bläser et al., 16 Jul 2025).

Setting	Basis counts	Criterion
Graphs	induced copies of pure graphs	$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$3
Relational structures	induced substructures of $$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$4-pure patterns	$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$5 all coefficients are nonnegative integers
Colored graphs	induced colored subgraphs avoiding the padding color	$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$6 all coefficients are nonnegative integers
Finite vector spaces over $$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$7	subspace counts	$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$8 coefficients are nonnegative integers
Parameter sets	subobject counts in the parameter-set category	$$\varphi(b)=\sum_{i=1}^s \alpha_i \cdot Sub_{a_i}(b),$$9 is good

For graphs, the proof first establishes an ordered-graph version and then reduces unordered graphs by symmetrization,

$a_i$0

because the Ramsey property required in the argument fails for unordered graphs (Bläser et al., 16 Jul 2025). For relational structures, the same theorem holds with induced substructures replacing induced subgraphs, and the proof uses the Ramsey theorem for relational structures with forbidden irreducible substructures together with the corresponding padding operation (Bläser et al., 16 Jul 2025).

For finite-dimensional vector spaces over a fixed finite field $a_i$1, subobjects are subspaces and the corresponding subobject counts are Gaussian binomial coefficients (Bläser et al., 16 Jul 2025). The relevant Ramsey theorem is the Graham–Leeb–Rothschild theorem, and the upper bound is obtained by nondeterministically guessing a basis and checking containment in the oracle subspace (Bläser et al., 16 Jul 2025). For parameter sets, the relevant structural input is the Graham–Rothschild theorem (Bläser et al., 16 Jul 2025).

6. Relation to WL-dimension, treewidth, and GNN expressivity

In the labeled-graph setting, the expressive-power question for motif parameters is addressed in a different formalism. A labeled graph motif parameter is any finite linear combination of homomorphism counts from fixed labeled graphs, and its WL-dimension is defined as the least $a_i$2 such that $a_i$3-WL distinguishes it (Lanzinger et al., 2023). The principal theorem states

$a_i$4

so the WL-dimension is exactly the maximum treewidth among the support graphs (Lanzinger et al., 2023).

This theorem immediately yields the subgraph and induced-subgraph special cases. For subgraph counting with labeled pattern $a_i$5, the support is the spasm,

$a_i$6

and therefore

$a_i$7

(Lanzinger et al., 2023). For induced subgraph counting, the support consists of graphs obtained from $a_i$8 by adding edges, so

$a_i$9

(Lanzinger et al., 2023).

A further consequence is that whenever a motif parameter lies within the expressive power of $\mathcal M$00-WL, its exact value can be recovered from the final stable $\mathcal M$01-WL colors: $\mathcal M$02 for a suitable function $\mathcal M$03 on stable colors (Lanzinger et al., 2023). The paper explicitly notes the relevance of this statement for GNNs: under the standard correspondence between message-passing GNNs and $\mathcal M$04-WL, the last layer already contains all local information needed for exact motif counting, provided the network is appropriately designed (Lanzinger et al., 2023). It also gives a polynomial-time algorithm, for fixed $\mathcal M$05, to decide whether the WL-dimension of subgraph counting for a labeled pattern $\mathcal M$06 is at most $\mathcal M$07 (Lanzinger et al., 2023).

7. Misconceptions, examples, and conceptual scope

A central misconception addressed by the categorical theory is that a parameter that is nonnegative on every input should automatically admit a counting interpretation. The graph example

$\mathcal M$08

is nonnegative for all graphs even though the coefficient of $\mathcal M$09 is $\mathcal M$10 (Bläser et al., 16 Jul 2025). The paper also gives a more substantial graph motif parameter with a negative coefficient that is nevertheless nonnegative for all graphs, and concludes that because all patterns are pure, it is not combinatorially interpretable (Bläser et al., 16 Jul 2025). The theorem therefore separates nonnegativity of values from membership in the oracle counting class $\mathcal M$11.

The graph case is historically anchored in the counting-complexity framework inaugurated by the seminal paper of Curticapean, Dell and Marx (STOC’17), and the categorical extension is described as a vast generalization from graphs to relational structures, colored graphs, finite vector spaces, and parameter sets (Bläser et al., 16 Jul 2025). The WL-theoretic line of work, by contrast, places motif parameters in the landscape of graph isomorphism methods and GNN expressivity, using labeled versions of homomorphism-count characterizations, homomorphism-distinguishing closed classes, and a linear-algebraic lemma from Seppelt (Lanzinger et al., 2023).

Taken together, these developments identify motif parameters as a unifying formalism for local pattern counting across several mathematical settings. In one direction, the categorical theory characterizes when such linear combinations genuinely count subobjects in the oracle-$\mathcal M$12 sense: exactly when the coefficients are nonnegative integers on pure patterns (Bläser et al., 16 Jul 2025). In another, the WL theory characterizes when labeled graph motif parameters are visible to the $\mathcal M$13-WL hierarchy: exactly when their support patterns have treewidth at most $\mathcal M$14 (Lanzinger et al., 2023). This combination of counting complexity, structural Ramsey theory, and WL/treewidth analysis is the defining conceptual profile of motif parameters over categories.