--- title: 'H1 Pan-Graph-Matrix: Allele-Centric Pangenome Analysis' url: https://www.emergentmind.com/topics/h1-pan-graph-matrix type: topic --- # H1 Pan-Graph-Matrix: Allele-Centric Pangenome Analysis 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 [2512.21320]. ## 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 $j$ carries allele $i$, and $0$ otherwise. This structure exposes alleles as direct, queryable entities and enables linear-time carrier enumeration proportional to per-allele carrier count $k_i = \sum_{j=1}^H A_{ij}$ or $H/\mathrm{word\_size}$ for dense representation. ## 2. Mathematical Characterization and Sparsity The matrix $A$ encodes the binary allele–haplotype incidence relation, with notation: - $H$: number of haplotypes - $m$: number of alleles - $A_{ik} = 1$ if haplotype $k$ carries allele $i$ - $k_i$: carrier count per allele, $k_i = \sum_{j=1}^H A_{ij}$ Sparsity patterns in $A$ arise naturally: - **Rare alleles** exhibit $k_i \ll H$ and result in highly sparse rows. - **Common alleles** yield $k_i \approx H$ and dense rows. The matrix can be visualized as: \[ A = \begin{pmatrix} 1 & 0 & 1 & \cdots & 0 \\ 0 & 1 & 1 & \cdots & 1 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & 1 & \cdots & 0 \end{pmatrix} \quad A \in \{0,1\}^{m\times H}, \quad k_i = \sum_{j=1}^H A_{ij} \] 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 $H$-bit vector per row ($\mathrm{cost} = H$ bits). - **Sparse list:** Store $k_i$ integer carrier indices ($\mathrm{cost} = k_i \cdot \lceil \log_2 H \rceil$ bits). The break-even carrier count $k^*$, at which sparse and dense representations are equally costly: \[ H \approx k^* \log_2 H \implies k^* \approx \frac{H}{\log_2 H} \] For $H = 400$, $k^* \approx 46$. H1 chooses, for each allele $i$, 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 $k_i$. ## 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, $H=400$): | Variant Class | Sites | Bitmap-Only (bits) | H1 Hybrid (bits) | Hybrid/Bitmap Ratio | |----------------------|---------|--------------------|------------------|---------------------| | SNV/INDEL | 24,921 | $9.97 \times 10^6$ | $3.11 \times 10^6$ | 0.31 | | Structural variants | 45 | $1.80 \times 10^4$ | $3.98 \times 10^3$ | 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 $N$ samples ($2N$ haplotypes). 2. **Allele enumeration:** Compile all $m$ distinct alleles. 3. **Row encoding:** - For each allele $i$: - Collect carrier haplotype IDs from callset ($k_i$). - Compute $\mathrm{dense\_cost} = H$, $\mathrm{sparse\_cost} = k_i \cdot \lceil \log_2 H \rceil$. - Store as sorted integer list if sparse cost < dense, else as bitvector. 4. **Metadata:** Store per-row header (encoding type, $k_i$). 5. **Optional:** Associate sequence payloads or functional annotations externally for regulated sharing or analysis. ## 6. Duality and Information Equivalence to H2 H2 is the path-centric dual representation. In H2: - Each haplotype $j$ is an ordered list of abstract graph edges (reference segments + variant branches). - Edge-to-haplotype membership is given by a transposed index (i.e., $A^T$), preserving complete information equivalence. Conversion guarantees: - **H1 → H2:** For haplotype $j$, walk graph by column $j$ of $A$ in genomic order. - **H2 → H1:** For each allele/edge, assemble haplotype list for row $i$ of $A$. 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, $H=400$) 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 $O(m \cdot \bar{k} + m \cdot \log H)$; dominated by rare variant rows in sparse regime. - **Query time:** Carrier enumeration for rare allele ($k_i \approx 2$): $O(2)$. Common allele ($\sim$200 carriers): $O(200)$ for sparse, $O(H/\mathrm{word}) \sim O(6)$ 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 $H$; 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 [2512.21320]. Source: https://www.emergentmind.com/topics/h1-pan-graph-matrix