Complex-Index Matrices in Graph Provenance

• Complex-Index Matrices are a semiring-based construction that embeds structural provenance directly into matrix entries.
• They enable standard matrix multiplication to compute aggregated values while tracking the vertices, edges, and paths responsible for those values.
• They preserve semiring legality in sparse linear algebra, supporting applications such as shortest-path witness extraction in graph algorithms.

Complex-index matrices are a semiring-based construction in which some information that would ordinarily remain in the row or column indices of a matrix is moved into the matrix entries themselves, so that GraphBLAS-style linear algebra can compute not only aggregated values but also the vertices, edges, paths, or intermediate indices responsible for those values. In the formulation developed in “GraphBLAS Mathematical Opportunities: Parallel Hypersparse, Matrix Based Graph Streaming, and Complex-Index Matrices” (Jananthan et al., 23 Sep 2025), a complex-index matrix is not an ordinary complex matrix over $\mathbb C$; rather, it is a matrix over a complex-like semiring on $\mathbb S\times \mathbb S$ with a nilpotent-style rule $i\otimes i=0^\mathbb S$. The construction is designed to preserve semiring legality for matrix multiplication while enriching entries with provenance data (Jananthan et al., 23 Sep 2025).

1. Algebraic definition

The construction starts from an arbitrary semiring

$$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$

A new semiring is then defined on the Cartesian product $\mathbb S\times \mathbb S$ by

$$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$

and

$$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$

Its additive and multiplicative identities are

$$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$

The paper states that

$$(\mathbb S\times \mathbb S,\oplus,\otimes,(0^\mathbb S,0^\mathbb S),(1^\mathbb S,0^\mathbb S))$$

is a semiring (Jananthan et al., 23 Sep 2025).

The notation is intentionally complex-like. One sets

$i := (0^\mathbb S,1^\mathbb S),$

and writes

$\mathbb S\times \mathbb S$0

With this identification,

$\mathbb S\times \mathbb S$1

This is the defining algebraic distinction from ordinary complex arithmetic. The original semiring embeds via

$\mathbb S\times \mathbb S$2

so ordinary $\mathbb S\times \mathbb S$3-valued matrices appear as the special case with zero imaginary part. The paper also defines

$\mathbb S\times \mathbb S$4

and applies these pointwise to matrices (Jananthan et al., 23 Sep 2025).

2. Matrix structure and propagation law

A complex-index matrix is an associative array or matrix

$\mathbb S\times \mathbb S$5

typically written entrywise as

$\mathbb S\times \mathbb S$6

For graph problems, the key case is

$\mathbb S\times \mathbb S$7

The motivating heuristic given in the paper is to redefine an adjacency array by setting

$\mathbb S\times \mathbb S$8

if there is an edge from $\mathbb S\times \mathbb S$9 to $i\otimes i=0^\mathbb S$0, and $i\otimes i=0^\mathbb S$1 otherwise. In that encoding, the real part can store the source key and the imaginary part can store the destination key (Jananthan et al., 23 Sep 2025).

Matrix multiplication is the ordinary array product induced by the new semiring. If

$i\otimes i=0^\mathbb S$2

then

$i\otimes i=0^\mathbb S$3

Entrywise, if

$i\otimes i=0^\mathbb S$4

then

$i\otimes i=0^\mathbb S$5

This propagation law is the core device: it keeps the base-value product in the real part and mixes left and right metadata in the imaginary part (Jananthan et al., 23 Sep 2025).

The motivating problem is standard in semiring graph algorithms. Over the min-plus semiring,

$i\otimes i=0^\mathbb S$6

the entry $i\otimes i=0^\mathbb S$7 gives the least weight of an $i\otimes i=0^\mathbb S$8-hop path from $i\otimes i=0^\mathbb S$9 to $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$0, but it does not directly return the path. Complex-index matrices are introduced precisely to carry that witness information through ordinary matrix multiplication (Jananthan et al., 23 Sep 2025).

3. Provenance constructions and graph algorithms

The main worked construction in the paper enriches shortest-path computation so that matrix multiplication returns both optimal weights and optimal witnesses. The enriched semiring is

$$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$1

where $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$2 is the set of finite vertex strings and $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$3 is its power set. The operations are defined so that the first component stores weight and the second component stores a set of witness paths: $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$4 and

$$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$5

The identities are

$$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$6

The corresponding complex-index adjacency array is defined by

$$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$7

The real part stores the full one-edge path $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$8, while the imaginary part stores the suffix $$(\mathbb{S}, \oplus, \otimes, 0^\mathbb{S}, 1^\mathbb{S}).$$9 (Jananthan et al., 23 Sep 2025).

With

$\mathbb S\times \mathbb S$0

the paper defines a recurrence in which $\mathbb S\times \mathbb S$1 is obtained from $\mathbb S\times \mathbb S$2 and $\mathbb S\times \mathbb S$3 by matrix multiplication over $\mathbb S\times \mathbb S$4. Its main proposition states that if

$\mathbb S\times \mathbb S$5

then $\mathbb S\times \mathbb S$6 is the least weight among all $\mathbb S\times \mathbb S$7-hop paths from $\mathbb S\times \mathbb S$8 to $\mathbb S\times \mathbb S$9, and $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$0 is the set of all $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$1-hop paths from $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$2 to $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$3 with weight $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$4 (Jananthan et al., 23 Sep 2025). The construction is thus dynamic programming expressed as semiring matrix multiplication.

A second construction tracks contributing values and intermediate indices in a general matrix product. The paper defines an auxiliary provenance semiring $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$5, where

$$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$6

together with operations $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$7 and a multiplicative rule that combines tuples sharing the same intermediate key. An input matrix is encoded as a complex-index matrix $$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$8 whose real part carries left-value and left-key metadata and whose imaginary part carries right-value and right-key metadata. If

$$(x_1,x_2)\oplus (y_1,y_2) = (x_1\oplus y_1,\; x_2\oplus y_2),$$9

over this semiring, then the paper states that, for any $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$0,

$$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$1

and

$$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$2

are obtained as projections of $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$3, and that the ordinary product value is recovered from the same enriched data (Jananthan et al., 23 Sep 2025). In this form, complex-index multiplication functions as an algebraic provenance-tracking mechanism.

4. Relation to GraphBLAS, sparsity, and ordinary complex matrices

The construction is presented as a value-domain lift for GraphBLAS and associative arrays. A complex-index matrix remains a sparse or hypersparse matrix in the GraphBLAS sense when only few entries are nonzero; the change is not in the storage pattern but in the entry type. The paper explicitly distinguishes:

• ordinary sparse matrix: sparse structure, scalar values;
• complex-index sparse matrix: sparse structure, richer scalar payloads containing value and index or provenance information (Jananthan et al., 23 Sep 2025).

It is also closely tied to associative arrays. Mathematically, a complex-index matrix is an associative array over $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$4, or over a further enriched semiring such as $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$5 or $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$6. The row and column dimensions are not expanded. This is not a tensor construction and not a block-matrix reindexing; instead, additional logical dimensions are encoded inside each entry (Jananthan et al., 23 Sep 2025).

The analogy to ordinary complex matrices is explicit but limited. The multiplication law

$$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$7

corresponds to enforcing

$$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$8

not $$(x_1,x_2)\otimes (y_1,y_2) = (x_1\otimes y_1,\; (x_1\otimes y_2)\oplus (x_2\otimes y_1)).$$9. The paper therefore states that the construction is closer to a semiring of dual numbers or a nilpotent extension than to true complex numbers (Jananthan et al., 23 Sep 2025).

This distinction is also useful terminologically. In other matrix literatures, the phrase “Complex-Index Matrices” is not standard terminology (Banica, 2017). In work on multiplicative nested sums, the matrix formalism naturally allows complex-valued entries, but the indices remain ordinary integers; that framework therefore supports complex-valued index matrices, not complex-valued summation indices (Jiu et al., 2016).

5. Computational implications

The principal computational consequence is that provenance-sensitive operations can be expressed inside ordinary semiring sparse matrix multiplication. The paper identifies shortest-path witness extraction, contributor tracking, and intermediate-key recovery as central use cases. Because the construction remains a semiring, associativity and distributivity are preserved, so GraphBLAS execution models remain valid: partial products can be formed independently, reductions are legal, and sparse kernels remain algebraically composable (Jananthan et al., 23 Sep 2025).

The implementation does not require literal storage of ordinary complex values. The paper states that “underlying implementations can be achieve[d] with a variety of efficient means that may not actually require storing complex values per se.” It suggests that entries could be represented by a pair of payloads, a struct, parallel matrices for real and imaginary parts, or a custom semiring object (Jananthan et al., 23 Sep 2025).

The limitations are equally explicit. Complex-index entries are larger than simple scalars, and in the path-recovery examples they may contain sets of strings or sets of tuples. Witness sets can grow combinatorially when ties occur. The paper notes that no benchmarks are provided for complex-index matrices, and it identifies several likely implementation issues: memory growth, higher constant factors, and potentially explosive intermediate states. It also notes that practical implementations would likely need pruning, bounded witness counts, compression, canonicalization of stored metadata, or specialized kernels for common cases (Jananthan et al., 23 Sep 2025).

6. Scope, misconceptions, and adjacent usages

The formal notion of a complex-index matrix is the one introduced in the GraphBLAS setting: a matrix over a semiring on $$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$0 whose two channels are used to encode both values and structural provenance (Jananthan et al., 23 Sep 2025). A common misconception is to read the term as referring to ordinary complex matrices over $$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$1. The formalism does not do that: the symbol $$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$2 is a bookkeeping device with

$$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$3

and the two components are meant to carry index information, not to instantiate standard complex arithmetic (Jananthan et al., 23 Sep 2025).

Another misconception is that the construction replaces row and column keys by value payloads. It does not. The row and column indices remain $$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$4; selected key information is injected into entries so that matrix multiplication can propagate witnesses and intermediate indices (Jananthan et al., 23 Sep 2025).

Outside this GraphBLAS usage, the phrase is best treated cautiously. In Hadamard-matrix and operator-algebraic contexts, the phrase “Complex-Index Matrices” is explicitly described as not standard terminology (Banica, 2017). In matrix representations of multiplicative nested sums, the relevant extension is to arbitrary, including complex, weight functions $$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$5, while the summation indices themselves remain integer variables in finite chains (Jiu et al., 2016). A plausible implication is that the GraphBLAS notion is distinctive because it formalizes “index” as payload-bearing semiring data rather than as a conventional matrix label set.

In that precise sense, complex-index matrices are a semiring-theoretic method for fusing structural graph provenance with sparse linear algebra. Their defining contribution is that ordinary matrix multiplication, once lifted to the semiring

$$(0^\mathbb S,0^\mathbb S),\qquad (1^\mathbb S,0^\mathbb S).$$6

can compute both values and the indices responsible for those values within the same algebraic kernel (Jananthan et al., 23 Sep 2025).