Graph Connectivity in Log Steps Using Label Propagation

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

Natural Language Processing markers in first episode psychosis and people at clinical high-risk

Recent work has suggested that disorganised speech might be a powerful predictor of later psychotic illness in clinical high risk subjects. To that end, several automated measures to quantify disorganisation of transcribed speech have been proposed. However, it remains unclear which measures are most strongly associated with psychosis, how different measures are related to each other and what the best strategies are to collect speech data from participants. Here, we assessed whether twelve automated Natural Language Processing markers could differentiate transcribed speech excerpts from subjects at clinical high risk for psychosis, first episode psychosis patients and healthy control subjects (total N = 54). In-line with previous work, several measures showed significant differences between groups, including semantic coherence, speech graph connectivity and a measure of whether speech was on-topic, the latter of which outperformed the related measure of tangentiality. Most NLP measures examined were only weakly related to each other, suggesting they provide complementary information. Finally, we compared the ability of transcribed speech generated using different tasks to differentiate the groups. Speech generated from picture descriptions of the Thematic Apperception Test and a story re-telling task outperformed free speech, suggesting that choice of speech generation method may be an important consideration. Overall, quantitative speech markers represent a promising direction for future clinical applications.

Distant residues modulate conformational opening in SARS-CoV-2 spike protein

Infection by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) involves the attachment of the receptor-binding domain (RBD) of its spike proteins to the ACE2 receptors on the peripheral membrane of host cells. Binding is initiated by a down-to-up conformational change in the spike protein, the change that presents the RBD to the receptor. To date, computational and experimental studies that search for therapeutics have concentrated, for good reason, on the RBD. However, the RBD region is highly prone to mutations, and is therefore a hotspot for drug resistance. In contrast, we here focus on the correlations between the RBD and residues distant to it in the spike protein. This allows for a deeper understanding of the underlying molecular recognition events and prediction of the highest-effect key mutations in distant, allosteric sites, with implications for therapeutics. Also, these sites can appear in emerging mutants with possibly higher transmissibility and virulence, and preidentifying them can give clues for designing pan-coronavirus vaccines against future outbreaks. Our model, based on time-lagged independent component analysis (tICA) and protein graph connectivity network, is able to identify multiple residues that exhibit long-distance coupling with the RBD opening. Residues involved in the most ubiquitous D614G mutation and the A570D mutation of the highly contagious UK SARS-CoV-2 variant are predicted ab initio from our model. Conversely, broad-spectrum therapeutics like drugs and monoclonal antibodies can target these key distant-but-conserved regions of the spike protein.

Wave Transport and Localization in Prime Number Landscapes

In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.

BC tree-based spectral sampling for big complex network visualization

Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling $$BC\_SV$$ B C _ S V and spectral edge sampling $$BC\_SS$$ B C _ S S by computing effective resistance values of vertices and edges for each connected component. Furthermore, we present $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D , graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-based graph decomposition and distributed computing. Experimental results demonstrate that $$BC\_SV$$ B C _ S V and $$BC\_SS$$ B C _ S S are significantly faster than previous spectral graph sampling methods while preserving the same sampling quality. $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D obtain further significant runtime improvement over sequential approaches, and $$DBC\_GD$$ D B C _ G D further achieves significant improvements in quality metrics over sequential graph drawing layouts.

Mutations in SARS-CoV-2 Variants Modulate the Microscopic Dynamics of Neutralizing Antibodies

Monoclonal antibodies have emerged as viable treatment for the COVID-19 disease caused by the SARS-CoV-2 virus. But the new viral variants can reduce the efficacy of the currently available antibodies, as well as diminish the vaccine induced immunity. Here, we demonstrate how the microscopic dynamics of the SARS-CoV-2 neutralizing monoclonal antibodies, can be modulated by the mutations present in the spike proteins of the variants currently circulating in the world population. We show that the dynamical perturbation in the antibody structure can be diverse, depending both on the nature of the antibody and on the location of the mutation. The correlated motion between the antibody and the receptor binding domain (RBD) can also be changed, altering the binding affinity. By constructing a protein graph connectivity network, we could delineate the mutant induced modifications in the allosteric information flow pathway through the antibody, and observed the presence of both localized and long distance effects. We identified a loop consisting of residues 470-490 in the RBD which works like an anchor preventing the detachment of the antibodies, and individual mutations in that region can significantly affect the antibody binding propensity. Our study provides fundamental and atomistically detailed insight on how virus neutralization by monoclonal antibody can be impacted by the mutations in the epitope, and can potentially facilitate the rational design of monoclonal antibodies, effective against the new variants of the novel coronavirus.

Learning Attributed Graph Representation with Communicative Message Passing Transformer

Constructing appropriate representations of molecules lies at the core of numerous tasks such as material science, chemistry, and drug designs. Recent researches abstract molecules as attributed graphs and employ graph neural networks (GNN) for molecular representation learning, which have made remarkable achievements in molecular graph modeling. Albeit powerful, current models either are based on local aggregation operations and thus miss higher-order graph properties or focus on only node information without fully using the edge information. For this sake, we propose a Communicative Message Passing Transformer (CoMPT) neural network to improve the molecular graph representation by reinforcing message interactions between nodes and edges based on the Transformer architecture. Unlike the previous transformer-style GNNs that treat molecule as a fully connected graph, we introduce a message diffusion mechanism to leverage the graph connectivity inductive bias and reduce the message enrichment explosion. Extensive experiments demonstrated that the proposed model obtained superior performances (around 4% on average) against state-of-the-art baselines on seven chemical property datasets (graph-level tasks) and two chemical shift datasets (node-level tasks). Further visualization studies also indicated a better representation capacity achieved by our model.

Connectivity of Semiring Valued Graphs

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of semiring valued graphs, also known as S-valued symmetry graphs, in 2015. Since then, works on S-valued symmetry graphs such as vertex dominating set, edge dominating set, regularity, etc. have been done. However, the connectivity of S-valued graphs has not been studied. Motivated by this, in this paper, the concept of connectivity in S-valued graphs has been studied. We have introduced the term vertex S-connectivity and edge S-connectivity and arrived some results for connectivity of a complete S-valued symmetry graph, S-path and S-star. Unlike the graph theory, we have observed that the inequality for connectivity κ(G)≤κ′(G)≤δ(G) holds in the case of S-valued graphs only when there is a symmetry of the graph as seen in Examples 3–5.