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1. Distances Between Immersed Graphs: Metric Properties

   https://doi.org/10.1007/s44007-022-00037-8  

   Buchin, Maike; Chambers, Erin; Fang, Pan; Fasy, Brittany Terese; Gasparovic, Ellen; Munch, Elizabeth; Wenk, Carola (January 2023, La Matematica)

   Abstract Graphs in metric spaces appear in a wide range of data sets, and there is a large body of work focused on comparing, matching, or analyzing collections of graphs in different ambient spaces. In this survey, we provide an overview of a diverse collection of distance measures that can be defined on the set of finite graphs immersed (and in some cases, embedded) in a metric space. For each of the distance measures, we recall their definitions and investigate which of the properties of a metric they satisfy. Furthermore we compare the distance measures based on these properties and discuss their computational complexity. 

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2. Sketching Merge Trees for Scientific Visualization

   https://doi.org/10.1109/TopoInVis60193.2023.00013  

   Li, Mingzhe; Palande, Sourabh; Yan, Lin; Wang, Bei (October 2023, 2023 Topological Data Analysis and Visualization (TopoInVis))

   Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a set of merge trees using techniques from matrix sketching. That is, given a large set T of merge trees, we would like to find a much smaller set of basis trees S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be approximated via matrix sketching techniques such as principal component analysis and column subset selection. However, until now, there has not been any work on sketching a set of merge trees. We develop a framework for sketching a set of merge trees that combines matrix sketching with tools from optimal transport. In particular, we vectorize a set of merge trees into high-dimensional vectors while preserving their structures and structural relations. We demonstrate the applications of our framework in sketching merge trees that arise from time-varying scientific simulations. Specifically, our framework obtains a set of basis trees as representatives that capture the “modes” of physical phenomena for downstream analysis and visualization. 

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3. Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications

   https://doi.org/10.3390/mca25040067  

   Kirkpatrick, Anna; Patton, Kalen; Tetali, Prasad; Mitchell, Cassie (December 2020, Mathematical and Computational Applications)

   null (Ed.)

   Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult to know whether the resultant values fall within expected dispersion thresholds for a given energy function. The goal of this study was to construct a Markov chain capable of examining the dispersion of RNA secondary structures and branching properties obtained from NNTM energy function minimization independent of a specific nucleotide sequence. Plane trees are studied as a model for RNA secondary structure, with energy assigned to each tree based on the NNTM, and a corresponding Gibbs distribution is defined on the trees. Through a bijection between plane trees and 2-Motzkin paths, a Markov chain converging to the Gibbs distribution is constructed, and fast mixing time is established by estimating the spectral gap of the chain. The spectral gap estimate is obtained through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. The resulting algorithm can be used as a tool for exploring the branching structure of RNA, especially for long sequences, and to examine branching structure dependence on energy model parameters. Full exposition is provided for the mathematical techniques used with the expectation that these techniques will prove useful in bioinformatics, computational biology, and additional extended applications. 

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4. The effect of genome graph expressiveness on the discrepancy between genome graph distance and string set distance

   Qiu, Yutong; Kingsford, Carl (January 2022, ISCB ISMB 2022)

   Motivation: Intra-sample heterogeneity describes the phenomenon where a genomic sample contains a diverse set of genomic sequences. In practice, the true string sets in a sample are often unknown due to limitations in sequencing technology. In order to compare heterogeneous samples, genome graphs can be used to represent such sets of strings. However, a genome graph is generally able to represent a string set universe that contains multiple sets of strings in addition to the true string set. This difference between genome graphs and string sets is not well characterized. As a result, a distance metric between genome graphs may not match the distance between true string sets. Results: We extend a genome graph distance metric, Graph Traversal Edit Distance (GTED) proposed by Ebrahimpour Boroojeny et al., to FGTED to model the distance between heterogeneous string sets and show that GTED and FGTED always underestimate the Earth Mover’s Edit Distance (EMED) between string sets. We introduce the notion of string set universe diameter of a genome graph. Using the diameter, we are able to upper-bound the deviation of FGTED from EMED and to improve FGTED so that it reduces the average error in empirically estimating the similarity between true string sets. On simulated T-cell receptor sequences and actual Hepatitis B virus genomes, we show that the diameter-corrected FGTED reduces the average deviation of the estimated distance from the true string set distances by more than 250%. Availability and implementation: Data and source code for reproducing the experiments are available at: https:// github.com/Kingsford-Group/gtedemedtest/. 

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5. A Consensus Approach to Infer Tumor Evolutionary Histories

   https://doi.org/10.1145/3233547.3233584  

   Govek, Kiya; Sikes, Camden; Oesper, Layla (August 2018, BCB '18 Proceedings of the 2018 ACM International Conference on Bioinformatics, Computational Biology, and Health Informatics)

   Inspired by recent efforts to model cancer evolution with phylogenetic trees, we consider the problem of finding a consensus tumor evolution tree from a set of conflicting input trees. In contrast to traditional phylogenetic trees, the tumor trees we consider contain features such as mutation labels on internal vertices (in addition to the leaves) and allow multiple mutations to label a single vertex. We describe several distance measures between these tumor trees and present an algorithm to solve the consensus problem called GraPhyC. Our approach uses a weighted directed graph where vertices are sets of mutations and edges are weighted using a function that depends on the number of times a parental relationship is observed between their constituent mutations in the set of input trees. We find a minimum weight spanning arborescence in this graph and prove that the resulting tree minimizes the total distance to all input trees for one of our presented distance measures. We evaluate our GraPhyC method using both simulated and real data. On simulated data we show that our method outperforms a baseline method at finding an appropriate representative tree. Using a set of tumor trees derived from both whole-genome and deep sequencing data from a Chronic Lymphocytic Leukemia patient we find that our approach identifies a tree not included in the set of input trees, but that contains characteristics supported by other reported evolutionary reconstructions of this tumor. 

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