Efficient Identification of Equivalences in Dynamic Graphs and Pedigree Structures

TL;DR

This paper introduces a new framework for efficiently identifying equivalences in dynamic graph structures like pedigree IBD graphs, enabling effective testing and analysis across varying parameters with minimal operations.

Contribution

The paper presents a novel, general framework for testing equivalences in complex, dynamic structures such as pedigree graphs, with proven theoretical and algorithmic properties.

Findings

Framework effectively identifies equivalences in dynamic graphs.

Algorithms demonstrate high accuracy in simulations.

The approach reduces computational complexity for graph analysis.

Abstract

We propose a new framework for designing test and query functions for complex structures that vary across a given parameter such as genetic marker position. The operations we are interested in include equality testing, set operations, isolating unique states, duplication counting, or finding equivalence classes under identifiability constraints. A motivating application is locating equivalence classes in identity-by-descent (IBD) graphs, graph structures in pedigree analysis that change over genetic marker location. The nodes of these graphs are unlabeled and identified only by their connecting edges, a constraint easily handled by our approach. The general framework introduced is powerful enough to build a range of testing functions for IBD graphs, dynamic populations, and other structures using a minimal set of operations. The theoretical and algorithmic properties of our approach are analyzed and proved. Computational results on several simulations demonstrate the effectiveness of our approach.