---
title: Transitivity Preserving Projection (TPP)
url: https://www.emergentmind.com/topics/transitivity-preserving-projection-tpp
type: topic
---

# Transitivity Preserving Projection (TPP)

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" [2110.14450], 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" [2509.04543], 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
$$
P_r \circ P_r = P_r,
$$
equivalently
$$
P_r^2 = P_r.
$$
When $(h,r,t)$ holds in the knowledge graph, Rot-Pro enforces
$$
R_r(P_r e_h)=P_r e_t,
$$
where $R_r:V\to V$ is an orthogonal rotation operator associated with $r$ [2110.14450].

The parameterization is block-diagonal over complex coordinates. Each complex embedding $e\in\mathbb{C}^d$ is written as a real $2d$-vector
$$
e \mapsto ([\mathrm{Re}(e(1)),\mathrm{Im}(e(1))]^T,\ldots,[\mathrm{Re}(e(d)),\mathrm{Im}(e(d))]^T).
$$
For each relation $r$ and coordinate $k=1,\ldots,d$, Rot-Pro defines a small rotation $S_r(k)\in SO(2)$ by an angle $\theta_p(k)$, a diagonal idempotent matrix $\mathrm{diag}(a_r(k),b_r(k))$ with $a_r(k),b_r(k)\in\{0,1\}$, and the projection block
$$
M_r(k)=S_r(k)^{-1}\cdot \mathrm{diag}(a_r(k),b_r(k))\cdot S_r(k).
$$
Because $\mathrm{diag}(a,b)$ is idempotent and $S_r(k)$ is invertible, $M_r(k)^2=M_r(k)$. The full projection operator is
$$
P_r=\mathrm{block\text{-}diag}(M_r(1),\ldots,M_r(d)).
$$

The relational rotation is likewise block-diagonal, with each $2\times 2$ block given by a rotation through $\theta_r(k)$:
$$
R_r(k)=
\begin{bmatrix}
\cos \theta_r(k) & -\sin \theta_r(k) \\
\sin \theta_r(k) & \cos \theta_r(k)
\end{bmatrix}.
$$
Given a triple $(h,r,t)$ with $e_h,e_t\in\mathbb{C}^d$, Rot-Pro defines the distance
$$
d_r(e_h,e_t)=\|R_r(P_r e_h)-P_r e_t\|_2,
$$
and the score
$$
S(h,r,t)=-d_r(e_h,e_t).
$$

## 3. Transitivity mechanism and expressive consequences in Rot-Pro

The transitivity argument is based on idempotence. Suppose $(e_1,r,e_2)$ and $(e_2,r,e_3)$ satisfy
$$
R_r(P_r e_1)=P_r e_2,\qquad R_r(P_r e_2)=P_r e_3.
$$
Then
$$
P_r e_3 = R_r(P_r e_2)=R_r(R_r(P_r e_1))=R_r^2(P_r e_1).
$$
If, furthermore,
$$
R_r^2 P_r = R_r P_r,
$$
for example if $R_r$ acts as the identity on $\mathrm{Range}(P_r)$, then
$$
P_r e_3 = R_r(P_r e_1),
$$
so by the same rule $(e_1,r,e_3)$ holds. The key algebraic requirements are therefore
$$
P_r^2=P_r,\qquad R_r^2P_r=R_rP_r
$$
[2110.14450].

The paper also states a lemma for transitive chains. A transitive chain for relation $r$ is a sequence of distinct entities $e_1,\ldots,e_m$ such that $(e_j,r,e_{j+1})$ holds for $j=1,\ldots,m-1$, and its transitive closure is the set of all ordered pairs $(e_j,r,e_k)$ whenever $j<k$. Any linear projection $P$ satisfying $P^2=P$ can represent such a chain by assigning embeddings with
$$
P e_1 = P e_2 = \cdots = P e_m
$$
and choosing $R_r$ so that $R_r(P e_j)=P e_{j+1})$. Because $P e_j$ is constant along the chain, all derived edges $(e_j,r,e_k)$ with $j<k$ satisfy the same rule.

For transitive relations in Rot-Pro, the paper’s proof sketch sets for each coordinate $k$
$$
a_r(k)=1,\qquad b_r(k)=0,
$$
so that $P_r(k)$ projects onto a single $1$-D subspace, and forces $\theta_r(k)=2n\pi$ so that $R_r(k)$ is the identity on the projected line. Then for any chain $e_1\to e_2\to e_3$ under $r$,
$$
P_r e_2 = R_r(P_r e_1)=P_r e_1,\qquad
P_r e_3 = R_r(P_r e_2)=P_r e_2=P_r e_1,
$$
hence $R_r(P_r e_1)=P_r e_3$ and $(e_1,r,e_3)$ is scored perfectly, with zero distance. Concretely, all projected points lie on the same line given by
$$
\cos \theta_p(k)\cdot \mathrm{Re}(e_j(k))-\sin \theta_p(k)\cdot \mathrm{Im}(e_j(k))=\text{constant}.
$$

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
$$
\text{Florida\_State\_University} \to \text{United\_States} \to \text{North\_America} \to \text{Americas}.
$$
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 $c_k$, and $\theta_r(k)=0$ so that $R_r$ acts as the identity. Thus $(e_1,r,e_4)$ is recovered even though it was not seen at training time.

Empirically, on the synthetic Countries tasks $S_1,S_2,S_3$, each requiring one-, two- and three-hop transitive inference, Rot-Pro achieves AUC-PR of $1.00/1.00/0.998$ respectively, whereas RotatE drops to $0.95$ on the three-hop task. On YAGO3-10’s specialized transitivity test sets $(S_1,S_2,S_3)$, Rot-Pro outperforms RotatE by over $+20$ MRR points on $S_3$. Visualizations of the learned relational rotation phases show that non-trivial projection dimensions settle at $\theta_r\approx 0$ or $2\pi$, confirming that $R_r$ 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
$$
H=(V,E),
$$
where $V$ is a set of vertices and $E$ is a set of edges, each edge $e\in E$ being a pair
$$
e=(t(e),h(e)),
$$
with $t(e)\subseteq V$ its tail (invertex) and $h(e)\subseteq V$ its head (outvertex). A metapath from $B$ to $C$ is any set of edges $M\subseteq E$ satisfying three conditions: the source inputs are all covered, the target outputs are all produced, and each edge in $M$ lies on some simple-path connecting some $b\in B$ to some $c\in C$ [2509.04543].

The dominantness conditions are central. A metapath $M(B,C)$ is edge-dominant if no proper subset $M'\subsetneq M$ is a metapath from $B$ to $C$. It is input-dominant if there is no metapath $M'(B',C)$ with $B'\subsetneq B$. It is dominant if it is both edge-dominant and input-dominant. An irreducible metapath over $X'$ is a dominant metapath $M(A,\{x\})$ with $A\subseteq X'\setminus\{x\}$ and $x\in X'$ that cannot be factored within $X'$:
$$
M(A,\{x\}) = M(A,Z)\cdot M(Z,\{x\})
$$
with both factors nonempty metapaths satisfying the dominance conditions.

The TPP problem is: given $H=(V,E)$ and a designated subset $X'\subseteq V$, construct a hypergraph
$$
H'=(X',E')
$$
such that: $(i)$ $H'$ contains only edges whose source and target lie in $X'$, $(ii)$ $H'$ preserves all dominant transitive relationships among $X'$, and $(iii)$ $H'$ 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 $H'=(X',E')$ is the TPP of $H$ over $X'$ if: for every edge $e'=(A,W)\in E'$ and every $x\in W$ there exists in $H$ a dominant metapath $M(A,\{x\})$ which is irreducible over $X'$; if in $H$ there is any dominant irreducible metapath $M(A,\{x\})$ with $A\subseteq X'$ and $x\in X'$, then $H'$ contains an edge $e'=(A,W)$ with $x\in W$; and no two distinct edges in $E'$ share the same tail set.

The theoretical guarantees are stated as theorems. Theorem 2.2 gives uniqueness and idempotence: the TPP over $X'$ is unique, and applying TPP twice yields the same result. Theorem 2.3 gives minimality: no proper subset of $E'$ preserves all dominant irreducible metapaths on $X'$. Theorem 2.4 gives completeness: every dominant metapath in $H$ whose source and target lie in $X'$ can be decomposed into a sequence of irreducible metapaths, each realized by an edge in $E'$.

## 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 $2^{|X'|}$ subsets $V'\subseteq X'\setminus\{x\}$, TPP only needs, for each $x\in X'$, one call to enumerate all edge-dominant metapaths from $X'\setminus\{x\}$ to $\{x\}$ and then applies a polynomial-time filter via a set-trie to retain only those that are irreducible [2509.04543].

A set-trie stores key-to-value associations where each key is a subset of a ground universe $U$. It supports two critical queries: given $S\subseteq U$, retrieve all values whose key $K$ satisfies $K\subseteq S$; and given $S\subseteq U$, retrieve all values whose key $K$ satisfies $K\supseteq S$. The main algorithm initializes $E'\leftarrow\emptyset$, builds a set-trie $S$ of inverter-to-edge-index over all $E$, and for each $x\in X'$ lets $S_x\leftarrow X'\setminus\{x\}$. It then calls
$$
\mathrm{AllPaths}(E,S_x,\{x\})
$$
to enumerate all edge-dominant metapaths from $S_x$ to $\{x\}$. The result is filtered to keep only those metapaths whose inverting set $t(M)$ is minimal among all enumerated paths, that is, irreducible; this filter inserts each $t(M)$ into a set-trie $T$ and rejects any $t(M)$ that has a strict subset already in $T$. For each surviving metapath, the algorithm adds an edge $e'=(A,\{x\})$ to $E'$, 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 $S$, and prunes any partial path whose tail-set is a superset of an already-found path. This process calls a subset-query on $S$ exactly once per $x\in X'$, not $2^{|X'|}$ times. The irreducibility filter in step 3(c) runs in
$$
O(p\cdot k),
$$
where $p=|\text{enumerated paths}|$ and $k$ is average invertex-set size, both polynomial in $|E|$.

The complexity discussion is explicit. Let $n=|V|$, $m=|E|$, and $r=|X'|$. Basu and Blanning must, for each $x\in X'$, consider every $V'\subseteq X'\setminus\{x\}$, that is, $2^{r-1}$ subsets, and for each run an NP-hard dominant-metapath enumeration, yielding worst-case time
$$
O(2^r\cdot T_{\mathrm{dominant}}).
$$
TPP runs AllPaths exactly $r$ 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 $O(p\cdot r)$, and the final edge-merge is $O(m)$. Empirically this runs in near-linear time in $m$ on real data.

The reported case studies are concrete. In the aircraft landing-gear supply chain, with $|V|\approx 30$ and $|E|\approx 60$, the BBP path-finder exhausted $16$ GB RAM in $1$ h and produced no projection in $24$ h, whereas TPP enumerated all $5\ 248$ edge-dominant paths in $<120$ s and completed projection over $X'$ in $<0.2$ s. In the university network security metagraph, with $|V|\approx 1\ 000$ and $|E|\approx 1\ 200$, BBP ran out of memory and did not terminate in $24$ h, whereas TPP completed projection over the relevant zone-subset in $\sim 0.1$ 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 $\le$ 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 $P_r$ with $P_r^2=P_r$, and transitivity depends on the additional requirement that $R_r$ act trivially on $\mathrm{Range}(P_r)$, implemented by $\theta_r=0 \bmod 2\pi$ on the projection subspace [2110.14450]. In the directed-hypergraph formulation, TPP is a projection from $H=(V,E)$ to $H'=(X',E')$ whose defining guarantees are existence, completeness, uniqueness of inverter, uniqueness, minimality, and idempotence [2509.04543].

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.

Source: https://www.emergentmind.com/topics/transitivity-preserving-projection-tpp