arXiv subjects

Daniel Scofield

Publications and source records attributed to Daniel Scofield.

6 recordsLinked to original sources

arXiv2024

Structure of the chromatic polynomial

Recently, big data techniques such as machine learning and topological data analysis have made their way to theoretical mathematics. Motivated by the recent work with polynomial invariants for knots, we use manifold learning and topological data analysis techniques to explore the structure and properties of the point cloud consisting of the chromatic polynomials of graphs up to 10 crossings. Although chromatic, as well as the Tutte polynomial fail to distinguish graphs, according to a conjecture by Bollobas, Pebody and Riordan they approximate the space of random graphs. In this work we compare structures in the chromatic data revealed using filtered PCA and Ball Mapper techniques, and relate them with a range of numerical invariants for graphs.

math.AT↗

arXiv2022

Beyond the Hype: A Real-World Evaluation of the Impact and Cost of Machine Learning-Based Malware Detection

In this paper, we present a scientific evaluation of four prominent malware detection tools to assist an organization with two primary questions: To what extent do ML-based tools accurately classify previously- and never-before-seen files? Is it worth purchasing a network-level malware detector? To identify weaknesses, we tested each tool against 3,536 total files (2,554 or 72\% malicious, 982 or 28\% benign) of a variety of file types, including hundreds of malicious zero-days, polyglots, and APT-style files, delivered on multiple protocols. We present statistical results on detection time and accuracy, consider complementary analysis (using multiple tools together), and provide two novel applications of the recent cost-benefit evaluation procedure of Iannacone \& Bridges. While the ML-based tools are more effective at detecting zero-day files and executables, the signature-based tool may still be an overall better option. Both network-based tools provide substantial (simulated) savings when paired with either host tool, yet both show poor detection rates on protocols other than HTTP or SMTP. Our results show that all four tools have near-perfect precision but alarmingly low recall, especially on file types other than executables and office files -- 37% of malware tested, including all polyglot files, were undetected. Priorities for researchers and takeaways for end users are given.

cs.CR↗

arXiv2020

Extremal Khovanov homology and the girth of a knot

We utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper we also define the girth of a link, discuss relations to other knot invariants, and the possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to $\ell>2$, than the girth of the link is equal to $\ell.$

math.GT↗

arXiv2018

Patterns in Khovanov link and chromatic graph homology

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

math.GT↗

arXiv2014

Classification of Complex Cyclic Leibniz Algebras

Since Leibniz algebras were introduced by Loday as a generalization of Lie algebras, there has been a lot of interest in which results of the latter extend to the former. Cyclic algebras, those generated by one element, are a useful tool for studying these concepts. In fact, they have no Lie algebra counterpart. Their simple structure lends itself to elegant counterexamples to the extension of several important results from Lie algebras to Leibniz algebras. In this paper, we give a classification of cyclic Leibniz algebras.

math.RA↗

arXiv2014

Cyclic Leibniz Algebras

Leibniz algebras generated by one element, called cyclic, provide simple and illuminating examples of many basic concepts. It is the purpose of this paper to illustrate this fact.

math.RA↗