Topology in cyber research

  title={Topology in cyber research},
  author={Steve Huntsman and Jimmy Paladino and Michael Robinson},
We give an idiosyncratic overview of applications of topology to cyber research, spanning the analysis of variables/assignments and control flow in computer programs, a brief sketch of topological data analysis in one dimension, and the use of sheaves to analyze wireless networks. The text is from a chapter in the forthcoming book Mathematics in Cyber Research, to be published by Taylor and Francis. 



Mathematical structure in human affairs

Elementary Applied Topology

Generalizing cyclomatic complexity via path homology

This work considers the application of path homology as a more powerful analogue of cyclomatic complexity and exhibits several classes of examples in this vein while experimentally demonstrating that path Homology gives identicial results tocyclomatic complexity for at least one detailed notion of structured control flow.

Topological Differential Testing

Topological differential testing is introduced, an approach to extracting the consensus behavior of a set of programs on a corpus of inputs using the topological notion of a simplicial complex to determine inputs that cause inconsistent behavior and in turn reveal input specifications.

Path Homologies of Deep Feedforward Networks

This work provides a characterization of two types of directed homology for fully-connected, feedforward neural network architectures and shows that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers within the network.

New ways to multiply 3 x 3-matrices

Path homology theory of multigraphs and quivers

We construct a new homology theory for the categories of quivers and multigraphs and describe the basic properties of introduced homology groups. We introduce a conception of homotopy in the category

On the path homology theory of digraphs and Eilenberg–Steenrod axioms

In the paper we continue the investigation of the path homology theory of digraphs that was constructed in our previous papers. We prove basic theorems that are similar to the theorems of classical

Two Bilinear (3 × 3)-Matrix Multiplication Algorithms of Complexity 25

This paper presents two algorithms of complexity 25 possessing the following two properties (symmetries): the matrices A1,B1, and C1 are identity, and if the algorithm involves a tripleA, B, C, then it also involves the triples B,C, A and C, A, B.


Any relation between the elements of a set X and the elements of a set Y is associated with two simplicial complexes K and L. A simplex of K is a finite set of elements of X related to a common