Graph Lineages and Skeletal Graph Products
Eric Mjolsness, Cory B. Scott

TL;DR
This paper introduces a hierarchical algebraic framework for structured graph lineages, enabling efficient modeling of complex systems and applications in neural networks and multigrid methods.
Contribution
It defines graph lineages with exponential growth, skeletal operators, and a category of graded graphs, forming an algebraic type theory for hierarchical graph structures.
Findings
Developed skeletal algebraic operators for graded graphs
Applied framework to deep neural networks and multigrid methods
Established approach for approaching continuum limit objects
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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGraph Theory and Algorithms · Advanced Graph Neural Networks · Data Visualization and Analytics
