H1 Pan-Graph-Matrix: Allele-Centric Pangenome Analysis

• H1 Pan-Graph-Matrix is an allele-centric representation that encodes genomic variants via a binary incidence matrix for precise haplotype mapping.
• The framework uses adaptive per-allele compression, choosing between dense bitmaps and sparse lists to optimize storage based on carrier sparsity.
• H1 is information-equivalent to its path-centric dual H2, enabling efficient carrier enumeration, rare variant analysis, and scalable cohort stratification.

The H1 pan-graph-matrix is an allele-centric representation for population-scale pangenome analysis, fundamentally designed to encode exact haplotype membership using adaptive, per-allele compression. In the H1 formulation, alleles—defined as concrete genomic variants including single-nucleotide changes (SNV), indels, and structural variants—are treated as first-class objects and mapped to haplotype carriers through a binary incidence matrix. This direct allele-to-haplotype mapping allows efficient carrier enumeration, frequency calculation, and intersection queries, exploiting carrier sparsity for scalable storage and rapid retrieval. The H1 framework provides a unified, population-aware foundation suitable for diverse downstream analyses and is strictly information-equivalent to its path-centric dual, H2 (garrone, 24 Dec 2025).

1. Formal Definition and Core Structure

Let $H$ be the total number of haplotypes in a cohort (e.g., $H=400$ for 200 diploid individuals), and $m$ the number of distinct alleles (SNV/INDEL and structural) observed. The H1 pan-graph-matrix represents variation by an $m \times H$ binary matrix

$A = [a_{ij}] \in \{0,1\}^{m\times H}$

where row $i$ corresponds to the $i$-th allele (e.g., "G→T at position 1,234,567" or "insertion of 5 kb at chr1:5,000,000") and column $j$ to haplotype $j$. The entry $a_{ij} = 1$ if and only if haplotype $H=400$0 carries allele $H=400$1, and $H=400$2 otherwise. This structure exposes alleles as direct, queryable entities and enables linear-time carrier enumeration proportional to per-allele carrier count $H=400$3 or $H=400$4 for dense representation.

2. Mathematical Characterization and Sparsity

The matrix $H=400$5 encodes the binary allele–haplotype incidence relation, with notation:

• $H=400$6: number of haplotypes
• $H=400$7: number of alleles
• $H=400$8 if haplotype $H=400$9 carries allele $m$0
• $m$1: carrier count per allele, $m$2

Sparsity patterns in $m$3 arise naturally:

• Rare alleles exhibit $m$4 and result in highly sparse rows.
• Common alleles yield $m$5 and dense rows.

The matrix can be visualized as:

$m$6

This formulation directly supports efficient population stratification, carrier set intersection, and cohort-level frequency calculations.

3. Adaptive Per-Allele Compression

H1 employs a per-row adaptive compression scheme, exploiting the carrier distribution for each allele. There are two canonical encoding choices:

• Dense bitmap: Store an $m$7-bit vector per row ($m$8 bits).
• Sparse list: Store $m$9 integer carrier indices ($m \times H$0 bits).

The break-even carrier count $m \times H$1, at which sparse and dense representations are equally costly:

$m \times H$2

For $m \times H$3, $m \times H$4. H1 chooses, for each allele $m \times H$5, the encoding with lower bit-cost, yielding storage efficiency near the theoretical minimum across all possible carrier cardinalities. A small per-row header records the encoding type and $m \times H$6.

4. Comparative Analysis with Existing Formats

H1’s allele-centric paradigm contrasts with established formats by decoupling sequence orientation and path structure from carrier incidence. Salient distinctions are summarized below:

Representation	Primary Unit	Carrier Query Mode	Compression Driver
VCF/BCF	Genomic site	Indirect (genotypes)	File-level codecs
PBWT (BGT, GTC)	Haplotype string	Weak for allele query	Ordering in transform
Pangenome graphs	Graph node/edge	Implicit via path idxs	Sequence redundancy
H1	Allele incidence row	Direct by allele	Carrier sparsity
H2	Haplotype path	Inverted index	Topological ordering

Quantitative evaluations (2 Mb window, chr 1, $m \times H$7):

Variant Class	Sites	Bitmap-Only (bits)	H1 Hybrid (bits)	Hybrid/Bitmap Ratio
SNV/INDEL	24,921	$m \times H$8	$m \times H$9	0.31
Structural variants	45	$A = [a_{ij}] \in \{0,1\}^{m\times H}$0	$A = [a_{ij}] \in \{0,1\}^{m\times H}$1	0.22

Bitmap-only stores every row as a bitvector; H1 hybrid adaptively encodes each allele.

5. Construction Workflow from Raw Pangenome Data

H1 construction proceeds via the following:

1. Input: Phased variant calls (SNV/INDEL, SV) for $A = [a_{ij}] \in \{0,1\}^{m\times H}$2 samples ($A = [a_{ij}] \in \{0,1\}^{m\times H}$3 haplotypes).
2. Allele enumeration: Compile all $A = [a_{ij}] \in \{0,1\}^{m\times H}$4 distinct alleles.
3. Row encoding:
   • For each allele $A = [a_{ij}] \in \{0,1\}^{m\times H}$5:
     • Collect carrier haplotype IDs from callset ($A = [a_{ij}] \in \{0,1\}^{m\times H}$6).
     • Compute $A = [a_{ij}] \in \{0,1\}^{m\times H}$7, $A = [a_{ij}] \in \{0,1\}^{m\times H}$8.
     • Store as sorted integer list if sparse cost < dense, else as bitvector.
4. Metadata: Store per-row header (encoding type, $A = [a_{ij}] \in \{0,1\}^{m\times H}$9).
5. Optional: Associate sequence payloads or functional annotations externally for regulated sharing or analysis.

6. Duality and Information Equivalence to H2

is the path-centric dual representation. In H2:

• Each haplotype $i$0 is an ordered list of abstract graph edges (reference segments + variant branches).
• Edge-to-haplotype membership is given by a transposed index (i.e., $i$1), preserving complete information equivalence.

Conversion guarantees:

• H1 → H2: For haplotype $i$2, walk graph by column $i$3 of $i$4 in genomic order.
• H2 → H1: For each allele/edge, assemble haplotype list for row $i$5 of $i$6.

No information is lost; the two encodings enable allele-centric or path-centric queries, projected from the same incidence relation.

7. Evaluation in Real Cohorts

Empirical evaluation utilized the 1000 Genomes Project 30× high-coverage 2020 cohort (GRCh38, 200 individuals, $i$7) across a 2 Mb region of chromosome 1:

• Compression: 69% reduction for SNV/INDEL (0.31×), 78% for structural variants (0.22×), relative to bitmap-only encoding.
• Scalability: Build time $i$8; dominated by rare variant rows in sparse regime.
• Query time: Carrier enumeration for rare allele ($i$9): $i$0. Common allele ($i$1200 carriers): $i$2 for sparse, $i$3 for bitmap word-scans.

8. Downstream Applications and Integration Practices

H1’s structure directly enables the following applications:

• Rare-variant interpretation: Efficient carrier subset extraction and cohort stratification via intersection of sparse lists.
• Structural variant analyses: Leverages extreme sparsity for minimal storage and rapid allele-carrier queries.
• Pharmacogenomics and drug-target analyses: Enables cohort-wide filtering by allele incidence without genome-wide scan overhead.
• Privacy-aware sharing: Incidence rows can be distributed absent sequence payloads; sequence attached only for approved users.
• Integration guidance:
  • Construct H1 for the full cohort; annotate externally (e.g., functional consequence, gene context).
  • Derive H2 for localized reconstruction (e.g., region-specific haplotypes).
  • Employ hybrid encoding thresholds matched to the cohort’s $i$4; per-chromosome tuning unnecessary.

This suggests that allele-centric, adaptive, sparsity-driven matrix representation such as H1 can be a foundational tool in scalable pangenome analysis, offering near-optimal storage and direct analytical access while maintaining full information equivalence with dual path-centric representations (garrone, 24 Dec 2025).