The Reduced Neighborhood Topological Indices and Polynomial for the Treatment of COVID-19

A graphical index is a numeric value corresponding to a structurally invariant graph, and in molecular graph theory, these invariants are known as topological indices. In the field of Chemical and Medical Sciences, the topological indices are used to study the chemical, biological, medical, and pharmaceutical features of drugs. Concerning the previous deadly diseases, the COVID-19 pandemic has been considered the biggest life-threatening issue that modern medicines have ever tackled. COVID-19 is immedicable, and even the existing treatments are only helping a certain group of sufferers. Scientists have tested available antiviral agents and got a favorable impact on recovering from the pandemic. Some of these antiviral agents are remdesivir, chloroquine, hydroxychloroquine, theaflavin, and dexamethasone. Keeping given the importance of topological indices in the study of pharmaceutical and chemical drugs, in this paper, we calculate the reduced neighborhood topological indices and RNM-polynomial of some of the antiviral agents remdesivir, chloroquine, hydroxychloroquine, theaflavin, and dexamethasone. The results thus obtained may be useful for finding new medicine and vaccine for the treatment of COVID-19.

Heterogeneous Graph Matching Networks for Unknown Malware Detection

Information systems have widely been the target of malware attacks. Traditional signature-based malicious program detection algorithms can only detect known malware and are prone to evasion techniques such as binary obfuscation, while behavior-based approaches highly rely on the malware training samples and incur prohibitively high training cost. To address the limitations of existing techniques, we propose MatchGNet, a heterogeneous Graph Matching Network model to learn the graph representation and similarity metric simultaneously based on the invariant graph modeling of the program's execution behaviors. We conduct a systematic evaluation of our model and show that it is accurate in detecting malicious program behavior and can help detect malware attacks with less false positives. MatchGNet outperforms the state-of-the-art algorithms in malware detection by generating 50% less false positives while keeping zero false negatives.

GeniePath: Graph Neural Networks with Adaptive Receptive Paths

We present, GeniePath, a scalable approach for learning adaptive receptive fields of neural networks defined on permutation invariant graph data. In GeniePath, we propose an adaptive path layer consists of two complementary functions designed for breadth and depth exploration respectively, where the former learns the importance of different sized neighborhoods, while the latter extracts and filters signals aggregated from neighbors of different hops away. Our method works in both transductive and inductive settings, and extensive experiments compared with competitive methods show that our approaches yield state-of-the-art results on large graphs.

Invariant Graph Partition Comparison Measures

Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.

The Cycle Polynomial of a Permutation Group

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles:\[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbital chromatic polynomial of $\Gamma$ and $G$, where $\Gamma$ is a $G$-invariant graph, introduced by the first author, Jackson and Rudd. We pose the general problem of finding all such reciprocal pairs, and give a number of examples and characterisations: the latter include the cases where $\Gamma$ is a complete or null graph or a tree. The paper concludes with some comments on other polynomials associated with a permutation group.

Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus

We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus$\mathbb{T}^{2}$at a fixed energy level${\mathcal{E}}\subset T\mathbb{T}^{2}$strictly above Mañé’s strict critical value. We prove that, if for some rational direction${\it\zeta}\in S^{1}$there is no invariant graph${\mathcal{T}}\subset {\mathcal{E}}$over$\mathbb{T}^{2}$for the Euler–Lagrange flow with the property that all orbits on${\mathcal{T}}$have an asymptotic direction equal to${\it\zeta}$, then there are chaotic dynamics in${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in${\mathcal{E}}$vanishes, then in${\mathcal{E}}$there are invariant graphs for all asymptotic directions${\it\zeta}\in S^{1}$and integrable-like behavior on a large scale.