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Simplicial Models and Topological Inference in Biological Systems
[chapter]

Vidit Nanda, Radmila Sazdanović

2013
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Natural Computing Series
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This article is a user's guide to algebraic topological methods for data analysis with a particular focus on applications to datasets arising in experimental biology. We begin with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets. From these structures, one computes topological statistics of the original data via the algebraic theory of (persistent) homology. These statistics are
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... hese statistics are shown to be computable and robust measures of the shape underlying a dataset. Finally, we showcase some appealing instances of topologydriven inference in biological settings -from the detection of a new type of breast cancer to the analysis of various neural structures. Simplicial Complexes. Start with a finite set V whose elements we call vertices. A simplicial complex with vertex set V is a collection K of subsets of V which is closed under inclusion. More precisely, we require that the following two conditions hold: • for each vertex v in V, the one-element set {v} lies in K, and • if τ is in K and σ ⊂ τ is a subset, then σ is also in K. Each element τ of K is called a simplex and its dimension (written dim τ) is defined to be #(τ) − 1 where # denotes cardinality (i.e., counts the number of vertices of τ). Any subset σ of τ is called a face of τ and this relationship is denoted by σ τ. We write K d to indicate the collection of d-dimensional simplices in K for each d 0. It is clear from the first property of simplicial complexes that the elements of V correspond in a one-to-one manner with those of K 0 , and it is therefore customary to speak of the two sets interchangeably. Consequently, one often encounters phrases resembling "let K be a simplicial complex" with no explicit mention of the underlying vertex set. Before proceeding any further, we will examine a small simplicial complex in some detail. Example 1.1. Given a vertex set V = {a, b, . . . , f, g}, we may construct a simplicial complex K in layers, one dimension at a time. Denote subsets of V by their elements in alphabetical order so that {a, b, c} is simply written abc. As we have already seen, K 0 is completely determined by V. Next, K 1 can contain any pair of distinct vertices in V and there is some freedom to choose such pairs. For instance, we select ac, ae, bc, bd, be, bg, cd, cg, dg, ef} . Fixing K 1 immediately constrains which simplices can lie in K 2 . For instance, abe is allowed in K 2 since all of its 1-dimensional faces ab, ae, be are in K 1 . However, acd is banned because ad ≺ acd but ad is not present in K 1 . We add the following (legal!) 2-dimensional simplices to K: K 2 = {abe, bcg, bcd, bdg, cdg} , and note that the only 3-dimensional simplex whose faces all exist in K 2 is bcdg. Let us include that simplex as well and get:

doi:10.1007/978-3-642-40193-0_6
fatcat:t6phmvf7lvgdlhsfifkfnfa5qy