Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties

This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.

Connectivity modulations induced by reach&grasp movements: a multidimensional approach

Reach&grasp requires highly coordinated activation of different brain areas. We investigated whether reach&grasp kinematics is associated to EEG-based networks changes. We enrolled 10 healthy subjects. We analyzed the reach&grasp kinematics of 15 reach&grasp movements performed with each upper limb. Simultaneously, we obtained a 64-channel EEG, synchronized with the reach&grasp movement time points. We elaborated EEG signals with EEGLAB 12 in order to obtain event related synchronization/desynchronization (ERS/ERD) and lagged linear coherence between Brodmann areas. Finally, we evaluated network topology via sLORETA software, measuring network local and global efficiency (clustering and path length) and the overall balance (small-worldness). We observed a widespread ERD in α and β bands during reach&grasp, especially in the centro-parietal regions of the hemisphere contralateral to the movement. Regarding functional connectivity, we observed an α lagged linear coherence reduction among Brodmann areas contralateral to the arm involved in the reach&grasp movement. Interestingly, left arm movement determined widespread changes of α lagged linear coherence, specifically among right occipital regions, insular cortex and somatosensory cortex, while the right arm movement exerted a restricted contralateral sensory-motor cortex modulation. Finally, no change between rest and movement was found for clustering, path length and small-worldness. Through a synchronized acquisition, we explored the cortical correlates of the reach&grasp movement. Despite EEG perturbations, suggesting that the non-dominant reach&grasp network has a complex architecture probably linked to the necessity of a higher visual control, the pivotal topological measures of network local and global efficiency remained unaffected.

Using deep learning to classify pediatric posttraumatic stress disorder at the individual level

Background Children exposed to natural disasters are vulnerable to developing posttraumatic stress disorder (PTSD). Previous studies using resting-state functional neuroimaging have revealed alterations in graph-based brain topological network metrics in pediatric PTSD patients relative to healthy controls (HC). Here we aimed to apply deep learning (DL) models to neuroimaging markers of classification which may be of assistance in diagnosis of pediatric PTSD. Methods We studied 33 pediatric PTSD and 53 matched HC. Functional connectivity between 90 brain regions from the automated anatomical labeling atlas was established using partial correlation coefficients, and the whole-brain functional connectome was constructed by applying a threshold to the resultant 90 * 90 partial correlation matrix. Graph theory analysis was used to examine the topological properties of the functional connectome. A DL algorithm then used this measure to classify pediatric PTSD vs HC. Results Graphic topological measures using DL provide a potentially clinically useful classifier for differentiating pediatric PTSD and HC (overall accuracy 71.2%). Frontoparietal areas (central executive network), cingulate cortex, and amygdala contributed the most to the DL model’s performance. Conclusions Graphic topological measures based on fMRI data could contribute to imaging models of clinical utility in distinguishing pediatric PTSD from HC. DL model may be a useful tool in the identification of brain mechanisms PTSD participants.

Stop Bickering! Reconciling Signaling Pathway Databases with Network Topologies

A major goal of molecular systems biology is to understand the coordinated function of genes or proteins in response to cellular signals and to understand these dynamics in the context of disease. Signaling pathway databases such as KEGG, NetPath, NCI-PID, and Panther describe the molecular interactions involved in different cellular responses. While the same pathway may be present in different databases, prior work has shown that the particular proteins and interactions differ across database annotations. However, to our knowledge no one has attempted to quantify their structural differences. It is important to characterize artifacts or other biases within pathway databases, which can provide a more informed interpretation for downstream analyses. In this work, we consider signaling pathways as graphs and we use topological measures to study their structure. We find that topological characterization using graphlets (small, connected subgraphs) distinguishes signaling pathways from appropriate null models of interaction networks. Next, we quantify topological similarity across pathway databases. Our analysis reveals that the pathways harbor database-specific characteristics implying that even though these databases describe the same pathways, they tend to be systematically different from one another. We show that pathway-specific topology can be uncovered after accounting for database-specific structure. This work present the first step towards elucidating common pathway structure beyond their specific database annotations.

Topological measures for identifying and predicting the spread of complex contagions

The standard measure of distance in social networks – average shortest path length – assumes a model of “simple” contagion, in which people only need exposure to influence from one peer to adopt the contagion. However, many social phenomena are “complex” contagions, for which people need exposure to multiple peers before they adopt. Here, we show that the classical measure of path length fails to define network connectedness and node centrality for complex contagions. Centrality measures and seeding strategies based on the classical definition of path length frequently misidentify the network features that are most effective for spreading complex contagions. To address these issues, we derive measures of complex path length and complex centrality, which significantly improve the capacity to identify the network structures and central individuals best suited for spreading complex contagions. We validate our theory using empirical data on the spread of a microfinance program in 43 rural Indian villages.

Supplemental Material: Vein topology, structures, and distribution during the prograde formation of an Archean gold stockwork

Structural features investigated on the Cheechoo tonalite/granodiorite including the topological measures carried out on the outcrop and on drill cores, along with the gold grades and facies variability, and the structural measures of foliation and veins from the main stripped area.

Supplemental Material: Vein topology, structures, and distribution during the prograde formation of an Archean gold stockwork

Structural features investigated on the Cheechoo tonalite/granodiorite including the topological measures carried out on the outcrop and on drill cores, along with the gold grades and facies variability, and the structural measures of foliation and veins from the main stripped area.

Weak Convergence of Topological Measures

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.