Empirical Block Taxonomies Overview

• Empirical block taxonomies are frameworks that decompose complex systems into modular blocks for hierarchical, data-driven classification and comparison.
• They apply to diverse domains including network models, low-rank matrix approximations, and blockchain architectures using measurable features like node degrees and CUR error bounds.
• The hierarchical organization in these taxonomies facilitates efficient model selection, parameter estimation, and robust empirical inference with concrete statistical guarantees.

Empirical block taxonomies comprise a family of frameworks and methodologies for the empirical identification, classification, and comparative analysis of modular, block-structured systems, including network models, low-rank matrix approximations, blockchain architectures, and networks in natural or engineered systems. Central to these approaches are the decomposition of complex objects into atomic or block subunits, the hierarchical classification of such substructures, and the use of empirical (often data-driven) features—such as node degrees, block assignments, or system response curves—for efficient inference, modeling, and comparison across diverse instances or families.

1. Principles of Block Identification and Empirical Taxonomy

Empirical block taxonomies are built on the decomposition of complex objects—such as networks, matrices, or blockchain systems—into atomic or mesoscopic components that can be empirically observed, measured, and classified. The process is characterized by a bottom-up methodology that consolidates vocabularies and lifts modular building blocks through comparative analysis of multiple instances. A hierarchical taxonomy is then constructed: atomic units are grouped into increasingly coarse components, typically forming a tree whose branches correspond to main system features and subcomponents. For empirical comparison and model selection, operational layouts (or "design patterns") for each subcomponent are enumerated, and quantitative classification and distance functions are defined over the vector of observed component features (Tasca et al., 2017).

2. Block Models in Network Data: Definitions and Hierarchies

In network modeling, block models define a latent partition of the nodes; edge probabilities and/or other parameters are tied to these block assignments. The canonical Stochastic Block Model (SBM) assigns each node a latent block and draws edges independently with block-dependent probabilities. Several important hierarchical block-model families have been formalized (Noroozi et al., 2020):

Model (Abbr.)	Edge Probability Parameterization	Free Parameters
SBM	$P_{ij} = B_{z(i), z(j)}$	$K(K+1)/2 - 1$
DCBM	$P_{ij} = h_i B_{z(i),z(j)} h_j$	$n + K(K+1)/2 - 1$
PABM	$$P_{ij} = \Lambda^{(z(i),z(j))}_i \Lambda^{(z(j),z(i))}_j$$	$n K$
NBM $(K,L)$	Two-level $(K, L)$, $$P^{(k_1,k_2)} = B_{k_1,k_2} h^{(k_1,c(k_2))}(h^{(k_2,c(k_1))})^T$$	$K^2 + nL - KL$

The "Nested Block Model" (NBM) nests SBM, DCBM, and PABM as special cases, interpolating the complexity/expressivity trade-off by varying meta-community count $K(K+1)/2 - 1$0 and structure of nodewise parameters. Empirical block taxonomy in this context refers to selecting among, or parameterizing, these hierarchies by observed statistics such as degree variation, blockwise inhomogeneity, and mixing patterns (Noroozi et al., 2020).

3. Degree-driven Empirical Classification in SBMs

Empirical degree distributions are a sufficient statistic for robust recovery of block structure in SBMs, as shown by Channarond, Daudin, and Robin (Channarond et al., 2011). The empirical normalized degrees $K(K+1)/2 - 1$1, where $K(K+1)/2 - 1$2 is node $K(K+1)/2 - 1$3's degree, cluster exponentially tightly around their block-specific means $K(K+1)/2 - 1$4. If the separation $K(K+1)/2 - 1$5 is nonzero, an efficient "Largest Gaps" (LG) algorithm partitions the network into $K(K+1)/2 - 1$6 blocks by locating the $K(K+1)/2 - 1$7 largest gaps in the sorted $K(K+1)/2 - 1$8 sequence. This permits:

• Consistent block assignment with misclassification probability decaying exponentially in $K(K+1)/2 - 1$9.
• Plug-in estimation of block proportions and connection probabilities, with concentration to true parameters dominated by partition error.
• Model selection (block number estimation) by penalized gap statistics on the $P_{ij} = h_i B_{z(i),z(j)} h_j$0 sequence.

All steps operate in $P_{ij} = h_i B_{z(i),z(j)} h_j$1 time for a graph with $P_{ij} = h_i B_{z(i),z(j)} h_j$2 edges and $P_{ij} = h_i B_{z(i),z(j)} h_j$3 nodes, requiring only the degree vector, and deliver finite-sample error bounds (Channarond et al., 2011).

4. Empirical Block Methods in Low-rank Matrix Approximation

Block-based empirical taxonomies arise prominently in matrix decomposition. Block Discrete Empirical Interpolation Methods (Block-DEIM) generalize scalar DEIM by selecting index blocks rather than single indices, facilitating efficient column/row sampling for CUR factorizations (Gidisu et al., 2022):

• In each step, blocks of size $P_{ij} = h_i B_{z(i),z(j)} h_j$4 (rather than a single index) are jointly chosen to maximize the "volume" (determinant) of a submatrix, using either the MaxVol heuristic or a strong Rank-Revealing QR (RRQR).
• Adaptive block size is supported by inspecting near-equal entries in residuals.
• Complexity is reduced compared to scalar DEIM: for large matrices ($P_{ij} = h_i B_{z(i),z(j)} h_j$5), Block-DEIM achieves $P_{ij} = h_i B_{z(i),z(j)} h_j$6–$P_{ij} = h_i B_{z(i),z(j)} h_j$7 speedups while maintaining CUR error bounds within the same theoretical factor, and typically reducing error-constants by $P_{ij} = h_i B_{z(i),z(j)} h_j$8.

The empirical block taxonomy for these methods classifies algorithms along axes: scalar vs. block, fixed vs. adaptive block size, selection rule (MaxVol or RRQR), and hierarchy (DEIM $P_{ij} = h_i B_{z(i),z(j)} h_j$9 B-DEIM-RRQR/MaxVol $n + K(K+1)/2 - 1$0 AdapBlock-DEIM $n + K(K+1)/2 - 1$1 QDEIM). Specific regimes of $n + K(K+1)/2 - 1$2 guide selection of the most appropriate method (Gidisu et al., 2022).

5. Taxonomies of Networks via Empirical Mesoscopic Structure

A distinctive empirical block taxonomy arises in the study of networks via their community structure. The framework of mesoscopic response functions (MRFs) quantifies how the partitioning of a network into blocks (communities) evolves across scales, parametrized by a resolution parameter $n + K(K+1)/2 - 1$3 (Onnela et al., 2010):

• Each network is characterized by three MRFs $n + K(K+1)/2 - 1$4 as functions of the antiferromagnetic fraction $n + K(K+1)/2 - 1$5 of node pairs.
• Distances between networks are computed as $n + K(K+1)/2 - 1$6 differences in these curves, aggregated (via PCA) into a single distance metric $n + K(K+1)/2 - 1$7.
• Agglomerative hierarchical clustering with these distances yields a comprehensive dendrogram ("taxonomy") on large corpora (e.g., 746 networks across 14 categories).
• Resulting taxonomies reveal non-trivial structure, clustering networks with similar functional, temporal, or biological properties even across disciplinary boundaries.

A salient implication is that mesoscopic community structure constitutes a "fingerprint" for network taxonomies, more discriminative than global or local graph statistics (Onnela et al., 2010).

6. Block Taxonomies in Blockchain Systems

A canonical application of empirical block taxonomies in software architectures is the classification of blockchain systems into a tree of atomic, empirically observed components (Tasca et al., 2017):

• Component decomposition yields seven major branches: Consensus, Transaction Capabilities, Native Currency & Tokenization, Extensibility, Security & Privacy, Codebase, Identity Management, and Charging & Rewarding.
• Each branch is subdivided into subcomponents, further classified by empirical layouts (e.g., consensus: PoW/PoS/PoA; transaction model: UTXO vs. account-balance; data structure: Merkle tree variants; scripting language: Turing-complete or not).
• Blockchain instances are encoded as discrete feature vectors; distances are computed (e.g., weighted Hamming) to compare architectures systematically.
• The taxonomy tree provides a flexible and extensible backbone for mapping, comparing, and evolving blockchain technologies as new empirical layouts emerge.

This approach enables navigability of architectural design space and principled comparison across heterogeneous blockchain systems based on empirical feature measurement (Tasca et al., 2017).

7. Methodological Themes and Implications

Across these domains, empirical block taxonomies share methodological principles:

• Decomposition into empirical structural or functional units (blocks, components, communities).
• Extraction of sufficient statistics or low-dimensional summaries (e.g., degrees, MRFs, block matrix entries) for classification and inference.
• Empirical, data-driven selection or estimation of models within nested block-family hierarchies.
• Hierarchical organization allows adaptation to increasing complexity and new layouts while maximizing parsimony and interpretability.

A plausible implication is that generalized empirical block taxonomies offer a scalable, modular, and comparably universal methodology for analyzing and classifying complex systems across network science, matrix computations, and digital architectures, with each instance supporting precise statistical guarantees and efficient computational procedures (Channarond et al., 2011, Noroozi et al., 2020, Gidisu et al., 2022, Onnela et al., 2010, Tasca et al., 2017).