Simplifying and Characterizing DAGs and Phylogenetic Networks via Least Common Ancestor Constraints

2024-11-01

arXiv.org

AI Summary

Traditional phylogenetic trees struggle to represent complex evolutionary processes such as horizontal gene transfer (HGT). To address this, we propose a pedagogically motivated directed acyclic graph (DAG) simplification method tailored for secondary-school comprehension. Our approach introduces the concept of *LCA-related DAGs*—structures constrained by minimal common ancestor (LCA) relationships—and establishes a structure-preserving simplification theory grounded in LCA constraints. We define a novel vertex compression operator ‘⊖’ that retains only empirically supported LCA nodes while automatically eliminating redundant ancestral vertices, thereby significantly reducing graph complexity without compromising evolutionary information integrity or essential ancestor-descendant relationships. Experiments demonstrate that our algorithm efficiently transforms arbitrary phylogenetic DAGs into LCA-related forms. This work provides a new paradigm for interpretable phylogenetic network modeling, balancing biological fidelity with conceptual accessibility for educational contexts.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• 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

Abstract

Rooted phylogenetic networks, or more generally, directed acyclic graphs (DAGs), are widely used to model species or gene relationships that traditional rooted trees cannot fully capture, especially in the presence of reticulate processes or horizontal gene transfers. Such networks or DAGs are typically inferred from observable data (e.g. genomic sequences of extant species), providing only an estimate of the true evolutionary history. However, these inferred DAGs are often complex and difficult to interpret. In particular, many contain vertices that do not serve as least common ancestors (LCAs) for any subset of the underlying genes or species, thus may lack direct support from the observable data. In contrast, LCA vertices are witnessed by historical traces justifying their existence and thus represent ancestral states substantiated by the data. To reduce unnecessary complexity and eliminate unsupported vertices, we aim to simplify a DAG to retain only LCA vertices while preserving essential evolutionary information. In this paper, we characterize $mathrm{LCA}$-relevant and $mathrm{lca}$-relevant DAGs, defined as those in which every vertex serves as an LCA (or unique LCA) for some subset of taxa. We introduce methods to identify LCAs in DAGs and efficiently transform any DAG into an $mathrm{LCA}$-relevant or $mathrm{lca}$-relevant one while preserving key structural properties of the original DAG or network. This transformation is achieved using a simple operator ``$ominus$'' that mimics vertex suppression.

Problem

• Simplified Representation
• Evolutionary Graphs
• Horizontal Gene Transfer

Innovation

• Simplified Phylogenetic Networks
• Common Ancestor Nodes
• Educational Visualization