Transitivity Preserving Projection (TPP)

• Transitivity Preserving Projection (TPP) is a mechanism that maintains transitive relationships by enforcing idempotence in projected subspaces or reduced graphs.
• In Rot-Pro, TPP employs relation-specific idempotent linear operators and identity rotations on the projection subspace to ensure that transitive links remain inferable.
• In directed hypergraphs, TPP efficiently constructs a minimal, unique projection that preserves dominant irreducible metapaths while reducing computational complexity.

Transitivity Preserving Projection (TPP) is a term used in recent arXiv literature for projection mechanisms designed so that transitive structure survives reduction. In the knowledge-graph embedding model "Rot-Pro: Modeling Transitivity by Projection in Knowledge Graph Embedding" (Song et al., 2021), TPP is instantiated as a relation-specific idempotent linear operator used jointly with relational rotation so that transitive links remain inferable after projection. In "Transitivity Preserving Projection in Directed Hypergraphs" (Parsonage et al., 4 Sep 2025), TPP denotes a projection of a directed hypergraph onto a designated subset $X'$ that is minimal, complete, unique, and idempotent with respect to dominant irreducible metapaths. The two usages are mathematically distinct, but both make transitivity preservation the defining constraint.

1. Terminological scope

The expression "Transitivity Preserving Projection" does not denote a single standardized construction across all subfields. In the cited literature, it refers to two different objects: a linear operator on embedding space for link prediction, and a projection procedure on directed hypergraphs for structural reduction.

Context	Projected object	Preservation target
Rot-Pro	Entity embeddings in $\mathbb{C}^d \cong \mathbb{R}^{2d}$	Transitive relations in knowledge graphs
Directed hypergraphs	A hypergraph $H=(V,E)$ onto $X' \subseteq V$	Dominant irreducible metapaths on $X'$

A common misconception is that TPP is synonymous with ordinary transitive closure. The hypergraph formulation is explicitly minimal and keeps only irreducible dominant metapaths, while the Rot-Pro formulation is a relation-conditioned projection inside a scoring function rather than a graph-rewriting operation.

2. Algebraic formulation in Rot-Pro

In Rot-Pro, the embedding space is a real vector space $V$ with $V \cong \mathbb{R}^{2d}$ when $\mathbb{C}^d$ is identified with $\mathbb{R}^{2d}$. A linear operator

$P_r : V \to V$

is called a projection, or idempotent transformation, if and only if

$\mathbb{C}^d \cong \mathbb{R}^{2d}$0

equivalently

$\mathbb{C}^d \cong \mathbb{R}^{2d}$1

When $\mathbb{C}^d \cong \mathbb{R}^{2d}$2 holds in the knowledge graph, Rot-Pro enforces

$\mathbb{C}^d \cong \mathbb{R}^{2d}$3

where $\mathbb{C}^d \cong \mathbb{R}^{2d}$4 is an orthogonal rotation operator associated with $\mathbb{C}^d \cong \mathbb{R}^{2d}$5 (Song et al., 2021).

The parameterization is block-diagonal over complex coordinates. Each complex embedding $\mathbb{C}^d \cong \mathbb{R}^{2d}$6 is written as a real $\mathbb{C}^d \cong \mathbb{R}^{2d}$7-vector

$\mathbb{C}^d \cong \mathbb{R}^{2d}$8

For each relation $\mathbb{C}^d \cong \mathbb{R}^{2d}$9 and coordinate $H=(V,E)$0, Rot-Pro defines a small rotation $H=(V,E)$1 by an angle $H=(V,E)$2, a diagonal idempotent matrix $H=(V,E)$3 with $H=(V,E)$4, and the projection block

$H=(V,E)$5

Because $H=(V,E)$6 is idempotent and $H=(V,E)$7 is invertible, $H=(V,E)$8. The full projection operator is

$H=(V,E)$9

The relational rotation is likewise block-diagonal, with each $X' \subseteq V$0 block given by a rotation through $X' \subseteq V$1:

$X' \subseteq V$2

Given a triple $X' \subseteq V$3 with $X' \subseteq V$4, Rot-Pro defines the distance

$X' \subseteq V$5

and the score

$X' \subseteq V$6

3. Transitivity mechanism and expressive consequences in Rot-Pro

The transitivity argument is based on idempotence. Suppose $X' \subseteq V$7 and $X' \subseteq V$8 satisfy

$X' \subseteq V$9

Then

$X'$0

If, furthermore,

$X'$1

for example if $X'$2 acts as the identity on $X'$3, then

$X'$4

so by the same rule $X'$5 holds. The key algebraic requirements are therefore

$X'$6

(Song et al., 2021).

The paper also states a lemma for transitive chains. A transitive chain for relation $X'$7 is a sequence of distinct entities $X'$8 such that $X'$9 holds for $V$0, and its transitive closure is the set of all ordered pairs $V$1 whenever $V$2. Any linear projection $V$3 satisfying $V$4 can represent such a chain by assigning embeddings with

$V$5

and choosing $V$6 so that $V$7. Because $V$8 is constant along the chain, all derived edges $V$9 with $V \cong \mathbb{R}^{2d}$0 satisfy the same rule.

For transitive relations in Rot-Pro, the paper’s proof sketch sets for each coordinate $V \cong \mathbb{R}^{2d}$1

$V \cong \mathbb{R}^{2d}$2

so that $V \cong \mathbb{R}^{2d}$3 projects onto a single $V \cong \mathbb{R}^{2d}$4-D subspace, and forces $V \cong \mathbb{R}^{2d}$5 so that $V \cong \mathbb{R}^{2d}$6 is the identity on the projected line. Then for any chain $V \cong \mathbb{R}^{2d}$7 under $V \cong \mathbb{R}^{2d}$8,

$V \cong \mathbb{R}^{2d}$9

hence $\mathbb{C}^d$0 and $\mathbb{C}^d$1 is scored perfectly, with zero distance. Concretely, all projected points lie on the same line given by

$\mathbb{C}^d$2

The paper further states that Rot-Pro can simultaneously model symmetry, asymmetry, inversion, composition (as in RotatE) and transitivity. The toy example uses the transitive relation isLocatedIn with the chain

$\mathbb{C}^d$3

Rot-Pro can embed these four entities in one complex coordinate so that they lie on a line in the complex plane; the chosen projection maps that line to a single real coordinate $\mathbb{C}^d$4, and $\mathbb{C}^d$5 so that $\mathbb{C}^d$6 acts as the identity. Thus $\mathbb{C}^d$7 is recovered even though it was not seen at training time.

Empirically, on the synthetic Countries tasks $\mathbb{C}^d$8, each requiring one-, two- and three-hop transitive inference, Rot-Pro achieves AUC-PR of $\mathbb{C}^d$9 respectively, whereas RotatE drops to $\mathbb{R}^{2d}$0 on the three-hop task. On YAGO3-10’s specialized transitivity test sets $\mathbb{R}^{2d}$1, Rot-Pro outperforms RotatE by over $\mathbb{R}^{2d}$2 MRR points on $\mathbb{R}^{2d}$3. Visualizations of the learned relational rotation phases show that non-trivial projection dimensions settle at $\mathbb{R}^{2d}$4 or $\mathbb{R}^{2d}$5, confirming that $\mathbb{R}^{2d}$6 becomes the identity on the projection subspace as required for transitivity.

4. Directed-hypergraph TPP: formal objects and guarantees

In the directed-hypergraph setting, a directed hypergraph is a pair

$\mathbb{R}^{2d}$7

where $\mathbb{R}^{2d}$8 is a set of vertices and $\mathbb{R}^{2d}$9 is a set of edges, each edge $P_r : V \to V$0 being a pair

$P_r : V \to V$1

with $P_r : V \to V$2 its tail (invertex) and $P_r : V \to V$3 its head (outvertex). A metapath from $P_r : V \to V$4 to $P_r : V \to V$5 is any set of edges $P_r : V \to V$6 satisfying three conditions: the source inputs are all covered, the target outputs are all produced, and each edge in $P_r : V \to V$7 lies on some simple-path connecting some $P_r : V \to V$8 to some $P_r : V \to V$9 (Parsonage et al., 4 Sep 2025).

The dominantness conditions are central. A metapath $\mathbb{C}^d \cong \mathbb{R}^{2d}$00 is edge-dominant if no proper subset $\mathbb{C}^d \cong \mathbb{R}^{2d}$01 is a metapath from $\mathbb{C}^d \cong \mathbb{R}^{2d}$02 to $\mathbb{C}^d \cong \mathbb{R}^{2d}$03. It is input-dominant if there is no metapath $\mathbb{C}^d \cong \mathbb{R}^{2d}$04 with $\mathbb{C}^d \cong \mathbb{R}^{2d}$05. It is dominant if it is both edge-dominant and input-dominant. An irreducible metapath over $\mathbb{C}^d \cong \mathbb{R}^{2d}$06 is a dominant metapath $\mathbb{C}^d \cong \mathbb{R}^{2d}$07 with $\mathbb{C}^d \cong \mathbb{R}^{2d}$08 and $\mathbb{C}^d \cong \mathbb{R}^{2d}$09 that cannot be factored within $\mathbb{C}^d \cong \mathbb{R}^{2d}$10:

$\mathbb{C}^d \cong \mathbb{R}^{2d}$11

with both factors nonempty metapaths satisfying the dominance conditions.

The TPP problem is: given $\mathbb{C}^d \cong \mathbb{R}^{2d}$12 and a designated subset $\mathbb{C}^d \cong \mathbb{R}^{2d}$13, construct a hypergraph

$\mathbb{C}^d \cong \mathbb{R}^{2d}$14

such that: $\mathbb{C}^d \cong \mathbb{R}^{2d}$15 $\mathbb{C}^d \cong \mathbb{R}^{2d}$16 contains only edges whose source and target lie in $\mathbb{C}^d \cong \mathbb{R}^{2d}$17, $\mathbb{C}^d \cong \mathbb{R}^{2d}$18 $\mathbb{C}^d \cong \mathbb{R}^{2d}$19 preserves all dominant transitive relationships among $\mathbb{C}^d \cong \mathbb{R}^{2d}$20, and $\mathbb{C}^d \cong \mathbb{R}^{2d}$21 $\mathbb{C}^d \cong \mathbb{R}^{2d}$22 is minimal—no edge can be removed without losing one of those derived relationships.

Definition 2.1 gives the formal TPP conditions. A directed hypergraph $\mathbb{C}^d \cong \mathbb{R}^{2d}$23 is the TPP of $\mathbb{C}^d \cong \mathbb{R}^{2d}$24 over $\mathbb{C}^d \cong \mathbb{R}^{2d}$25 if: for every edge $\mathbb{C}^d \cong \mathbb{R}^{2d}$26 and every $\mathbb{C}^d \cong \mathbb{R}^{2d}$27 there exists in $\mathbb{C}^d \cong \mathbb{R}^{2d}$28 a dominant metapath $\mathbb{C}^d \cong \mathbb{R}^{2d}$29 which is irreducible over $\mathbb{C}^d \cong \mathbb{R}^{2d}$30; if in $\mathbb{C}^d \cong \mathbb{R}^{2d}$31 there is any dominant irreducible metapath $\mathbb{C}^d \cong \mathbb{R}^{2d}$32 with $\mathbb{C}^d \cong \mathbb{R}^{2d}$33 and $\mathbb{C}^d \cong \mathbb{R}^{2d}$34, then $\mathbb{C}^d \cong \mathbb{R}^{2d}$35 contains an edge $\mathbb{C}^d \cong \mathbb{R}^{2d}$36 with $\mathbb{C}^d \cong \mathbb{R}^{2d}$37; and no two distinct edges in $\mathbb{C}^d \cong \mathbb{R}^{2d}$38 share the same tail set.

The theoretical guarantees are stated as theorems. Theorem 2.2 gives uniqueness and idempotence: the TPP over $\mathbb{C}^d \cong \mathbb{R}^{2d}$39 is unique, and applying TPP twice yields the same result. Theorem 2.3 gives minimality: no proper subset of $\mathbb{C}^d \cong \mathbb{R}^{2d}$40 preserves all dominant irreducible metapaths on $\mathbb{C}^d \cong \mathbb{R}^{2d}$41. Theorem 2.4 gives completeness: every dominant metapath in $\mathbb{C}^d \cong \mathbb{R}^{2d}$42 whose source and target lie in $\mathbb{C}^d \cong \mathbb{R}^{2d}$43 can be decomposed into a sequence of irreducible metapaths, each realized by an edge in $\mathbb{C}^d \cong \mathbb{R}^{2d}$44.

5. Algorithms, complexity, and empirical behavior in directed hypergraphs

The algorithmic contribution of the hypergraph formulation is framed against the Basu and Blanning projection (BBP). Rather than enumerating all $\mathbb{C}^d \cong \mathbb{R}^{2d}$45 subsets $\mathbb{C}^d \cong \mathbb{R}^{2d}$46, TPP only needs, for each $\mathbb{C}^d \cong \mathbb{R}^{2d}$47, one call to enumerate all edge-dominant metapaths from $\mathbb{C}^d \cong \mathbb{R}^{2d}$48 to $\mathbb{C}^d \cong \mathbb{R}^{2d}$49 and then applies a polynomial-time filter via a set-trie to retain only those that are irreducible (Parsonage et al., 4 Sep 2025).

A set-trie stores key-to-value associations where each key is a subset of a ground universe $\mathbb{C}^d \cong \mathbb{R}^{2d}$50. It supports two critical queries: given $\mathbb{C}^d \cong \mathbb{R}^{2d}$51, retrieve all values whose key $\mathbb{C}^d \cong \mathbb{R}^{2d}$52 satisfies $\mathbb{C}^d \cong \mathbb{R}^{2d}$53; and given $\mathbb{C}^d \cong \mathbb{R}^{2d}$54, retrieve all values whose key $\mathbb{C}^d \cong \mathbb{R}^{2d}$55 satisfies $\mathbb{C}^d \cong \mathbb{R}^{2d}$56. The main algorithm initializes $\mathbb{C}^d \cong \mathbb{R}^{2d}$57, builds a set-trie $\mathbb{C}^d \cong \mathbb{R}^{2d}$58 of inverter-to-edge-index over all $\mathbb{C}^d \cong \mathbb{R}^{2d}$59, and for each $\mathbb{C}^d \cong \mathbb{R}^{2d}$60 lets $\mathbb{C}^d \cong \mathbb{R}^{2d}$61. It then calls

$\mathbb{C}^d \cong \mathbb{R}^{2d}$62

to enumerate all edge-dominant metapaths from $\mathbb{C}^d \cong \mathbb{R}^{2d}$63 to $\mathbb{C}^d \cong \mathbb{R}^{2d}$64. The result is filtered to keep only those metapaths whose inverting set $\mathbb{C}^d \cong \mathbb{R}^{2d}$65 is minimal among all enumerated paths, that is, irreducible; this filter inserts each $\mathbb{C}^d \cong \mathbb{R}^{2d}$66 into a set-trie $\mathbb{C}^d \cong \mathbb{R}^{2d}$67 and rejects any $\mathbb{C}^d \cong \mathbb{R}^{2d}$68 that has a strict subset already in $\mathbb{C}^d \cong \mathbb{R}^{2d}$69. For each surviving metapath, the algorithm adds an edge $\mathbb{C}^d \cong \mathbb{R}^{2d}$70 to $\mathbb{C}^d \cong \mathbb{R}^{2d}$71, and finally merges any edges with the same tail into a single edge.

The AllPaths procedure performs a breadth-first search on the “power-graph” of edges, indexes candidate edges via subset-queries in $\mathbb{C}^d \cong \mathbb{R}^{2d}$72, and prunes any partial path whose tail-set is a superset of an already-found path. This process calls a subset-query on $\mathbb{C}^d \cong \mathbb{R}^{2d}$73 exactly once per $\mathbb{C}^d \cong \mathbb{R}^{2d}$74, not $\mathbb{C}^d \cong \mathbb{R}^{2d}$75 times. The irreducibility filter in step 3(c) runs in

$\mathbb{C}^d \cong \mathbb{R}^{2d}$76

where $\mathbb{C}^d \cong \mathbb{R}^{2d}$77 and $\mathbb{C}^d \cong \mathbb{R}^{2d}$78 is average invertex-set size, both polynomial in $\mathbb{C}^d \cong \mathbb{R}^{2d}$79.

The complexity discussion is explicit. Let $\mathbb{C}^d \cong \mathbb{R}^{2d}$80, $\mathbb{C}^d \cong \mathbb{R}^{2d}$81, and $\mathbb{C}^d \cong \mathbb{R}^{2d}$82. Basu and Blanning must, for each $\mathbb{C}^d \cong \mathbb{R}^{2d}$83, consider every $\mathbb{C}^d \cong \mathbb{R}^{2d}$84, that is, $\mathbb{C}^d \cong \mathbb{R}^{2d}$85 subsets, and for each run an NP-hard dominant-metapath enumeration, yielding worst-case time

$\mathbb{C}^d \cong \mathbb{R}^{2d}$86

TPP runs AllPaths exactly $\mathbb{C}^d \cong \mathbb{R}^{2d}$87 times. AllPaths itself may in the worst case enumerate exponentially many paths, but in practice the set-trie filter and edge-combining cut down the search space dramatically; after enumeration it filters in $\mathbb{C}^d \cong \mathbb{R}^{2d}$88, and the final edge-merge is $\mathbb{C}^d \cong \mathbb{R}^{2d}$89. Empirically this runs in near-linear time in $\mathbb{C}^d \cong \mathbb{R}^{2d}$90 on real data.

The reported case studies are concrete. In the aircraft landing-gear supply chain, with $\mathbb{C}^d \cong \mathbb{R}^{2d}$91 and $\mathbb{C}^d \cong \mathbb{R}^{2d}$92, the BBP path-finder exhausted $\mathbb{C}^d \cong \mathbb{R}^{2d}$93 GB RAM in $\mathbb{C}^d \cong \mathbb{R}^{2d}$94 h and produced no projection in $\mathbb{C}^d \cong \mathbb{R}^{2d}$95 h, whereas TPP enumerated all $\mathbb{C}^d \cong \mathbb{R}^{2d}$96 edge-dominant paths in $\mathbb{C}^d \cong \mathbb{R}^{2d}$97 s and completed projection over $\mathbb{C}^d \cong \mathbb{R}^{2d}$98 in $\mathbb{C}^d \cong \mathbb{R}^{2d}$99 s. In the university network security metagraph, with $H=(V,E)$00 and $H=(V,E)$01, BBP ran out of memory and did not terminate in $H=(V,E)$02 h, whereas TPP completed projection over the relevant zone-subset in $H=(V,E)$03 s. The paper states that TPP always terminates in seconds on moderate graphs versus BBP failing beyond dozens of vertices, and that the edge count in TPP is $H=(V,E)$04 BBP’s and typically far smaller.

The application domains listed are network security, supply-chain modeling, and visualization and analysis. In network security, policy-reachability projections highlight firewall-zone relationships while preserving “can-reach-via-transitivity” semantics, and TPP enables administrators to explore only irreducible trust paths in large campus or cloud networks. In supply-chain modeling, TPP yields minimal supplier-component projections that preserve all essential multi-tier dependencies and supports faster “what if supplier G fails?” analysis by recomputing TPP on the subset without G. For visualization, TPP yields far fewer edges than naive transitive-closure projections or BBP, aiding human interpretation.

6. Comparative interpretation

A second common misconception is that the two TPP constructions are interchangeable. They are not. In Rot-Pro, TPP is an operator-level constraint inside a knowledge-graph embedding model: the central object is the relation-specific projection $H=(V,E)$05 with $H=(V,E)$06, and transitivity depends on the additional requirement that $H=(V,E)$07 act trivially on $H=(V,E)$08, implemented by $H=(V,E)$09 on the projection subspace (Song et al., 2021). In the directed-hypergraph formulation, TPP is a projection from $H=(V,E)$10 to $H=(V,E)$11 whose defining guarantees are existence, completeness, uniqueness of inverter, uniqueness, minimality, and idempotence (Parsonage et al., 4 Sep 2025).

This suggests an abstract commonality: both constructions use idempotence to ensure that once the relevant projection has been applied, reapplying it does not generate a different reduced object. A plausible implication is that, in both settings, idempotence serves as the formal boundary between irreducible structure and redundant transitive derivation. The shared phrase therefore marks a family resemblance rather than a single formalism.

The distinction from simpler baselines is also sharp in both cases. In Rot-Pro, transitivity is not obtained by rotation alone; the model’s transitive behavior arises from projection together with a rotation that becomes the identity on the projected subspace. In directed hypergraphs, TPP is not a brute-force transitive closure and not the Basu and Blanning projection; it keeps only dominant irreducible metapaths, and its output is minimal, complete, unique, and idempotent. Across both literatures, the phrase "Transitivity Preserving Projection" therefore denotes a design principle: preserve inferential consequences of transitivity while restricting representation to a structured subspace or a minimal projected subgraph.