Foundations of a Connectivity Theory for Simplicial Complexes

Abstract

This paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert-Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids.

Structure and Interpretation of Computer Programs

“…Finally, graphs are extremely useful objects that are analysed in a variety of ways, each exposing relevant features; of these variants, the authors find two fields very promising: topological and algebraic graph theories. In particular, studying call graphs using a variant of Atkin's A-Homotopy theory is likely to yield interesting results [3]. Also, spectral methods applied to call graphs is an area that we think is worth investigating.…”

“…In scale-rich networks, as hubs tend to connect to lesser nodes, the rate of convergence is less rapid. Second, S(g) appears to be language independent 3 . Both near zero and higher S(g)s appear in all languages.…”

Simplicial Complexes of Networks and Their Statistical Properties

“…Consequently, the invariants of simplicial complexes may be defined based on their different aspects and each aspect provides completely different measures of the complex and, by extension, of the graph from which the complex was constructed. In the first case various algebraic topological measures may be associated such as homotopy and homology groups [6]. In the second case several invariants may be defined and numerically evaluated.…”

Persistent Path Homology of Directed Networks

“…In a series of papers released between 2012 and 2014, A. Grigor'yan, Y. Lin, Y. Muranov, and S.T. Yau formalized a notion of homology on digraphs called path homology [GLMY12], as well as a homotopy theory for digraphs that is compatible with path homology [GLMY14] and the homotopy theory for undirected graphs proposed in [BKLW01,BBLL06]. This notion of path homology is the central object of study in our work, and our contribution is to study the notion of persistent homology that arises from this construction.…”