New Betweenness Centrality Node Attack Strategies for Real-World Complex Weighted Networks

In this work, we introduce a new node attack strategy removing nodes with the highest conditional weighted betweenness centrality (CondWBet), which combines the weighted structure of the network and the node’s conditional betweenness. We compare its efficacy with well-known attack strategies from literature over five real-world complex weighted networks. We use the network weighted efficiency (WEFF) like a measure encompassing the weighted structure of the network, in addition to the commonly used binary-topological measure, i.e., the largest connected cluster (LCC). We find that if the measure is WEFF, the CondWBet strategy is the best to decrease WEFF in 3 out of 5 cases. Further, CondWBet is the most effective strategy to reduce WEFF at the beginning of the removal process, whereas the Strength that removes nodes with the highest sum of the link weights first shows the highest efficacy in the final phase of the removal process when the network is broken into many small clusters. These last outcomes would suggest that a better attacking in weighted networks strategy could be a combination of the CondWBet and Strength strategies.

Topology identifies emerging adaptive mutations in SARS-CoV-2

The COVID-19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARS-CoV-2. As the virus spreads and evolves it acquires new mutations that could have important public health consequences, including higher transmissibility, morbidity, mortality, and immune evasion, among others. Ideally, we would like to quickly identify new mutations that could confer adaptive advantages to the evolving virus by leveraging the large number of SARS-CoV-2 genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. The large number of currently available genomes, more than a million at this moment, however precludes the efficient use of phylogeny-based techniques. Here, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations of the coronavirus SARS-CoV-2 in the ongoing COVID-19 pandemic. Our method relies on a novel topological tool for the analysis of viral genome datasets based on persistent homology. It systematically identifies convergent events in viral evolution merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the complete GISAID dataset as of February 2021, comprising 303,651 high-quality SARS-CoV-2 isolates taken from patients all over the world since the beginning of the pandemic. A complete list of mutations showing topological signals of convergence is compiled. We find that topologically salient mutations on the receptor-binding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape. Moreover, for many adaptive mutations the topological signal precedes an increase in prevalence. We demonstrate the capability of our method to effectively identify emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we are able to extract geo-temporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants.

Integration with respect to deficient topological measures on locally compact spaces

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that $\begin{array}{} \int\limits_X \end{array}$f dμ = 0, but $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for $\begin{array}{} \int\limits_X \end{array}$f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem.

A version of Hake’s theorem for Kurzweil–Henstock integral in terms of variational measure

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake’s theorem in terms of a variational measure.

DEGREE-BASED GINI INDEX FOR GRAPHS

In Balaji and Mahmoud [1], the authors introduced a distance-based Gini index for rooted trees. In this paper, we introduce a degree-based Gini index (or just simply degree Gini index) for graphs. The latter index is a topological measure on a graph capturing the proximity to regular graphs. When applied across the random members of a class of graphs, we can identify an average measure of regularity for the class. Whence, we can compare the classes of graphs from the vantage point of closeness to regularity.We develop a simplified computational formula for the degree Gini index and study its extreme values. We show that the degree Gini index falls in the interval [0, 1). The main focus in our study is the degree Gini index for the class of binary trees. Via a left-packing transformation, we show that, for an arbitrary sequence of binary trees, the Gini index has inferior and superior limits in the interval [0, 1/4]. We also show, via the degree Gini index, that uniform rooted binary trees are more regular than binary search trees grown from random permutations.

Relative Helicity and Jog Densities in Continuum Descriptions of Dislocations

ABSTRACTDislocations are line like crystal defects mediating plasticity in single crystals. In the current contribution we review classical continuum concepts of dislocation theory from a topological view point. Subsequently, we introduce a new measure for the density of jogs mutually impaired on each other by dislocations on different slip systems. This jog density is closely related to a topological measure of the interlinkage of the dislocations on the involved slip systems, known as relative helicity in other branches of physics.

A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting

A Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.