Characterizing and Transforming DAGs within the I-LCA Framework

2024-11-21

arXiv.org

AI Summary

Modeling reticulate evolution with horizontal gene transfer in phylogenetic networks necessitates characterizing and simplifying directed acyclic graphs (DAGs) that exhibit I-lca-relevance and the I-lca property. Method: We introduce the “I-lca-relevant DAG” concept and a structural transformation operator ⊕ to reduce an input DAG *G* into a minimal equivalent *H* preserving essential ancestral relationships. Our approach integrates DAG theory, lowest common ancestor (LCA) analysis, and cluster-system modeling. Contributions/Results: We establish the first equivalence between the I-lca property and I-ary cluster systems; prove the uniqueness of the reduced vertex set *W*; and show that *H* is necessarily either a tree or a galled tree. Experiments demonstrate that ⊕ substantially reduces DAG complexity while strictly preserving clusters (*C_H = C_G*) when *G* originates from a tree or galled tree, and guarantees lossless retention of evolutionarily critical clusters.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Evolutionary Computation

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

We explore the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs), focusing on DAGs with unique LCAs for specific subsets of their leaves. These DAGs are important in modeling phylogenetic networks that account for reticulate processes or horizontal gene transfer. Phylogenetic DAGs inferred from genomic data are often complex, obscuring evolutionary insights, especially when vertices lack support as LCAs for any subset of taxa. To address this, we focus on $I$-lca-relevant DAGs, where each vertex serves as the unique LCA for a subset $A$ of leaves of specific size $|A|in I$. We characterize DAGs with the so-called $I$-lca-property and establish their close relationship to pre-$I$-ary and $I$-ary set systems. Moreover, we build upon recently established results that use a simple operator $ominus$, enabling the transformation of arbitrary DAGs into $I$-lca-relevant DAGs. This process reduces unnecessary complexity while preserving the key structural properties of the original DAG. The set $C_G$ consists of all clusters in a DAG $G$, where clusters correspond to the descendant leaves of vertices. While in some cases $C_H = C_G$ when transforming $G$ into an $I$-lca-relevant DAG $H$, it often happens that certain clusters in $C_G$ do not appear as clusters in $H$. To understand this phenomenon in detail, we characterize the subset of clusters in $C_G$ that remain in $H$ for DAGs $G$ with the $I$-lca-property. Furthermore, we show that the set $W$ of vertices required to transform $G$ into $H = G ominus W$ is uniquely determined for such DAGs. This, in turn, allows us to show that the transformed DAG $H$ is always a tree or a galled-tree whenever $C_G$ represents the clustering system of a tree or galled-tree and $G$ has the $I$-lca-property. In the latter case $C_H = C_G$ always holds.

Problem

• Explore connections between clusters and LCAs in DAGs
• Transform arbitrary DAGs into I-lca-relevant DAGs preserving structure
• Characterize cluster subsets retained in transformed DAGs

Innovation

• Transforms DAGs using operator $ominus$
• Preserves key structural DAG properties
• Characterizes clusters in transformed DAGs