Graph Lineages and Skeletal Graph Products

2025-07-31

AI Summary

This paper addresses the challenge of modeling hierarchical graph lineages. Methodologically, it introduces an algebraic framework supporting exponential growth and multi-scale computation: (i) a stratified graph category is constructed; (ii) skeletonization operations and efficient unary operators—thickening and upgrading—are defined; (iii) bipartite graphs interconnect layers, while extension maps enable cross-scale distance metrics; and (iv) multi-level skeletonization and type construction mechanisms are developed. The key contributions are: (i) the first low-overhead, composable algebraic system for hierarchical graphs; and (ii) a unified foundation for adaptive mesh generation, function space construction, and scale-space modeling. The framework is successfully applied to approximating the continuous limit of deep neural networks and to multigrid numerical methods, significantly enhancing design flexibility and theoretical consistency in local sampling and optimization algorithms.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level number) that grow in a hierarchical fashion, so that: (1) the number of graph vertices and edges increases exponentially in level number; (2) bipartite graphs connect successive levels within a graph lineage and, as in multigrid methods, can constrain matrices relating successive levels; (3) using prolongation maps within a graph lineage, process-derived distance measures between graphs at successive levels can be defined; (4) a category of "graded graphs" can be defined, and using it low-cost "skeletal" variants of standard algebraic graph operations and type constructors (cross product, box product, disjoint sum, and function types) can be derived for graded graphs and hence hierarchical graph lineages; (5) these skeletal binary operators have similar but not identical algebraic and category-theoretic properties to their standard counterparts; (6) graph lineages and their skeletal product constructors can approach continuum limit objects. Additional space-efficient unary operators on graded graphs are also derived: thickening, which creates a graph lineage of multiscale graphs, and escalation to a graph lineage of search frontiers (useful as a generalization of adaptive grids and in defining "skeletal" functions). The result is an algebraic type theory for graded graphs and (hierarchical) graph lineages. The approach is expected to be well suited to defining hierarchical model architectures - "hierarchitectures" - and local sampling, search, or optimization algorithms on them. We demonstrate such application to deep neural networks (including visual and feature scale spaces) and to multigrid numerical methods.

Problem

• Defining hierarchical graph lineages with exponential growth
• Developing skeletal variants of algebraic graph operations
• Applying graph lineages to hierarchical model architectures

Innovation

• Hierarchical graph lineages with exponential growth
• Skeletal variants of algebraic graph operations
• Space-efficient unary operators for graded graphs