Graph-theoretic Framework Research Articles

A new graph theory to unravel the bulk-boundary correspondence of graphene nanoribbons

We developed a new graph theory rooted in Clar's sextet rule to unravel the bulk-boundary correspondence of graphene. This methodology is specifically focused on the topological invariant and the chiral winding number, which enables the chemical rationalization of edge and boundary states in one-dimensional graphene nanoribbon (GNR) materials. The Clar structure derived from facile Lewis structures facilitates direct prediction of free radical distribution along edges and boundaries of GNRs across various geometric configurations. We then extend this graph theoretical framework to include metallic GNRs, demonstrating its power in several paradigms where conventional topological theories show limitations. Upon reducing the topological parameters and hence the complexity, the new approach provides a visual comprehension for the electronic topology and hence conductivity of GNR, greatly simplifying the formulation of design principles for future application of graphene interconnects.

Graph theoretical descriptors differentiate d-Mannose isomers in the principal component proposed feature space: A computational approach

The intricate nature of carbohydrates, particularly monosaccharides, stems from the existence of several chiral centers within their tertiary structures. Predicting and characterizing the molecular geometries and electrostatic landscapes of these substances is difficult due to their complex electrical properties. Moreover, these structures can display a substantial degree of conformational flexibility due to the presence of many rotatable bonds. Moreover, identifying and distinguishing between D and L enantiomers of monosaccharides presents a significant analytical obstacle since there is a need for empirically measurable properties that can distinguish them. This work uses Principal Component Analysis (PCA) to explore the chemical information included in 3D descriptors in order to comprehend the conformational space of d-Mannose stereoisomers. The isomers may be discriminated by utilizing 3D matrix-based indices, geometrical descriptors, and RDF descriptors. The isomers can be distinguished by descriptors, such as the Harary-like index from the reciprocal squared geometrical matrix (H_RG), Harary-like index from Coulomb matrix (H_Coulomb), Wiener-like index from Coulomb matrix (Wi_Coulomb), Wiener-like index from geometrical matrix (Wi_G), Graph energy from Coulomb matrix (SpAbs_Coulomb), Spectral absolute deviation from Coulomb matrix (SpAD_Coulomb), and Spectral positive sum from Coulomb matrix (SpPos_Coulomb). Among these descriptors, the first two, H_RG and H_Coulomb, perform the best in differentiation among the 3D-Matrix-Based Descriptors (3D-MBD) class. The results obtained from this study provide a significant chemical insight into the structural characteristics of the compounds inside the graph theoretical framework. These findings are likely to serve as the basis for developing new methods for analytical experiments.

The meso-connectomes of mouse, marmoset, and macaque: network organization and the emergence of higher cognition.

The recent publications of the inter-areal connectomes for mouse, marmoset, and macaque cortex have allowed deeper comparisons across rodent vs. primate cortical organization. In general, these show that the mouse has very widespread, "all-to-all" inter-areal connectivity (i.e. a "highly dense" connectome in a graph theoretical framework), while primates have a more modular organization. In this review, we highlight the relevance of these differences to function, including the example of primary visual cortex (V1) which, in the mouse, is interconnected with all other areas, therefore including other primary sensory and frontal areas. We argue that this dense inter-areal connectivity benefits multimodal associations, at the cost of reduced functional segregation. Conversely, primates have expanded cortices with a modular connectivity structure, where V1 is almost exclusively interconnected with other visual cortices, themselves organized in relatively segregated streams, and hierarchically higher cortical areas such as prefrontal cortex provide top-down regulation for specifying precise information for working memory storage and manipulation. Increased complexity in cytoarchitecture, connectivity, dendritic spine density, and receptor expression additionally reveal a sharper hierarchical organization in primate cortex. Together, we argue that these primate specializations permit separable deconstruction and selective reconstruction of representations, which is essential to higher cognition.

Out‐of‐band transaction pool sync for large dynamic blockchain networks

AbstractSynchronization of transaction pools (mempools) has shown potential for improving the performance and block propagation delay of state‐of‐the‐art blockchains. Indeed, various heuristics have been proposed in the literature to incorporate early exchanges of unconfirmed transactions into the block propagation protocol. In this work, we take a different approach, maintaining transaction synchronization externally (and independently) of the block propagation channel. In the process, we formalize the synchronization problem within a graph theoretic framework and introduce a novel algorithm (SREP—set reconciliation‐enhanced propagation) with quantifiable guarantees. We analyze the algorithm's performance for various realistic network topologies and show that it converges on static connected graphs in a time bounded by the diameter of the graph. In graphs with dynamic edges, SREP converges in an expected time that is linear in the number of nodes. We confirm our analytical findings through extensive simulations that include comparisons with MempoolSync, a recent approach from the literature. Our simulations show that SREP incurs reasonable bandwidth overhead and scales gracefully with the size of the network (unlike MempoolSync).

Reduced white matter maturation in the central auditory system of children living with HIV.

School-aged children experience crucial developmental changes in white matter (WM) in adolescence. The human immunodeficiency virus (HIV) affects neurodevelopment. Children living with perinatally acquired HIV (CPHIVs) demonstrate hearing and neurocognitive impairments when compared to their uninfected peers (CHUUs), but investigations into the central auditory system (CAS) WM integrity are lacking. The integration of the CAS and other brain areas is facilitated by WM fibers whose integrity may be affected in the presence of HIV, contributing to neurocognitive impairments. We used diffusion tensor imaging (DTI) tractography to map the microstructural integrity of WM between CAS regions, including the lateral lemniscus and acoustic radiation, as well as between CAS regions and non-auditory regions of 11-year-old CPHIVs. We further employed a DTI-based graph theoretical framework to investigate the nodal strength and efficiency of the CAS and other brain regions in the structural brain network of the same population. Finally, we investigated associations between WM microstructural integrity outcomes and neurocognitive outcomes related to auditory and language processing. We hypothesized that compared to the CHUU group, the CPHIV group would have lower microstructural in the CAS and related regions. Our analyses showed higher mean diffusivity (MD), a marker of axonal maturation, in the lateral lemniscus and acoustic radiations, as well as WM between the CAS and non-auditory regions predominantly in frontotemporal areas. Most affected WM connections also showed higher axial and radial diffusivity (AD and RD, respectively). There were no differences in the nodal properties of the CAS regions between groups. The MD of frontotemporal and subcortical WM-connected CAS regions, including the inferior longitudinal fasciculus, inferior fronto-occipital fasciculus, and internal capsule showed negative associations with sequential processing in the CPHIV group but not in the CHUU group. The current results point to reduced axonal maturation in WM, marked by higher MD, AD, and RD, within and from the CAS. Furthermore, alterations in WM integrity were associated with sequential processing, a neurocognitive marker of auditory working memory. Our results provide insights into the microstructural integrity of the CAS and related WM in the presence of HIV and link these alterations to auditory working memory.

Taxonomy-Driven Graph-Theoretic Framework for Manufacturing Cybersecurity Risk Modeling and Assessment

Abstract Identifying, analyzing, and evaluating cybersecurity risks are essential to devise effective decision-making strategies to secure critical manufacturing against potential cyberattacks. However, a manufacturing-specific quantitative approach is lacking to effectively model threat events and evaluate the unique cybersecurity risks in discrete manufacturing systems. In response, this paper introduces the first taxonomy-driven graph-theoretic model and framework to formally represent this unique cybersecurity threat landscape and identify vulnerable manufacturing assets requiring prioritized control. First, the proposed framework characterizes threat actors’ techniques, tactics, and procedures using taxonomical classifications of manufacturing-specific threat attributes and integrates these attributes into cybersecurity risk modeling. This facilitates the systematic generation of comprehensive and generalizable cyber-physical attack graphs for discrete manufacturing systems. Second, using the attack graph formalism, the proposed framework enables concurrent modeling and analysis of a wide variety of cybersecurity threats comprising varying attack vectors, locations, vulnerabilities, and consequences. The risk model captures the cascading attack impact of varying attack methods through different cyber and physical entities in manufacturing systems, leading to specific consequences. Then, the constructed cyber-physical attack graphs are analyzed to comprehend threat propagation through the discrete manufacturing value chain and identify potential attack paths. Third, a quantitative risk assessment approach is presented to evaluate the cybersecurity risk associated with potential attack paths. It also identifies the attack path with the maximum likelihood of success, pointing out critical manufacturing assets requiring prioritized control. Finally, the proposed risk modeling and assessment framework is demonstrated using an illustrative example.

Single-shot global localization via graph-theoretic correspondence matching

ABSTRACT This paper describes a method of single-shot global localization based on graph-theoretic matching of instances between a query and a prior map. The proposed framework employs correspondence matching based on the maximum clique problem (MCP). The framework is potentially applicable to other map and/or query modalities thanks to the graph-based abstraction of the problem, while many existing global localization methods rely on a query and the dataset in the same modality. We implement it with a semantically labeled 3D point cloud map, and a semantic segmentation image as a query. Leveraging the graph-theoretic framework, the proposed method realizes global localization exploiting only the map and the query. The method shows promising results on multiple large-scale simulated maps of urban scenes.

Framework for cyber risk loss distribution of hospital infrastructure: Bond percolation on mixed random graphs approach.

Networks like those of healthcare infrastructure have been a primary target of cyberattacks for over a decade. From just a single cyberattack, a healthcare facility would expect to see millions of dollars in losses from legal fines, business interruption, and loss of revenue. As more medical devices become interconnected, more cyber vulnerabilities emerge, resulting in more potential exploitation that may disrupt patient care and give rise to catastrophic financial losses. In this paper, we propose a structural model of an aggregate loss distribution across multiple cyberattacks on a prototypical hospital network. Modeled as a mixed random graph, the hospital network consists of various patient-monitoring devices and medical imaging equipment as random nodes to account for the variable occupancy of patient rooms and availability of imaging equipment that are connected by bidirectional edges to fixed hospital and radiological information systems. Our framework accounts for the documented cyber vulnerabilities of a hospital's trusted internal network of its major medical assets. To our knowledge, there exist no other models of an aggregate loss distribution for cyber risk in this setting. We contextualize the problem in the probabilistic graph-theoretical framework using a percolation model and combinatorial techniques to compute the mean and variance of the loss distribution for a mixed random network with associated random costs that can be useful for healthcare administrators and cybersecurity professionals to improve cybersecurity management strategies. By characterizing this distribution, we allow for the further utility of pricing cyberrisk.

Classifying schizophrenic and controls from fMRI data using graph theoretic framework and community detection

Classifying schizophrenic and controls from fMRI data using graph theoretic framework and community detection

LLT Cumulants and Graph Coloring

The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the Macdonald cumulant has an explicit positive expansion in terms of LLT cumulants of ribbon shapes, generalizing the classical decomposition of Macdonald polynomials. We also find a natural combinatorial interpretation of the LLT cumulant of a given directed graph as a weighted generating function of colorings of its subgraphs.We use this graph theoretical framework to prove various positivity results. This includes monomial positivity, positivity in fundamental quasisymmetric functions and related positivity of the coefficients of Schur polynomials indexed by hook shapes. We also prove $e$-positivity for vertical-shape LLT cumulants, after the shift of variable $q \to q+1$, which refines a recent result of Alexandersson and Sulzgruber. All these results give evidence towards Schur-positivity of LLT cumulants, which we conjecture here. We prove that this conjecture implies Schur-positivity of Macdonald cumulants, and we give more evidence by proving the conjecture for LLT cumulants of melting lollipops that refines a recent result of Huh, Nam and Yoo.

Graph-theoretic approach to detecting real-time intents within purchase conversion funnel using clickstream data

PurposePurchase conversion prediction aims to improve user experience and convert visitors into real buyers to drive sales of firms; however, the total conversion rate is low, especially for e-retailers. To date, little is known about how e-retailers can scientifically detect users' intents within a purchase conversion funnel during their ongoing sessions and strategically optimize real-time marketing tactics corresponding to dynamic intent states. This study mainly aims to detect a real-time state of the conversion funnel based on graph theory, which refers to a five-class classification problem in the overt real-time choice decisions (RTCDs)—click, tag-to-wishlist, add-to-cart, remove-from-cart and purchase—during an ongoing session.Design/methodology/approachThe authors propose a novel graph-theoretic framework to detect different states of the conversion funnel by identifying a user's unobserved mindset revealed from their navigation process graph, namely clickstream graph. First, the raw clickstream data are identified into individual sessions based on a 30-min time-out heuristic approach. Then, the authors convert each session into a sequence of temporal item-level clickstream graphs and conduct a temporal graph feature engineering according to the basic, single-, dyadic- and triadic-node and global characteristics. Furthermore, the synthetic minority oversampling technique is adopted to address with the problem of classifying imbalanced data. Finally, the authors train and test the proposed approach with several popular artificial intelligence algorithms.FindingsThe graph-theoretic approach validates that users' latent intent states within the conversion funnel can be interpreted as time-varying natures of their online graph footprints. In particular, the experimental results indicate that the graph-theoretic feature-oriented models achieve a substantial improvement of over 27% in line with the macro-average and micro-average area under the precision-recall curve, as compared to the conventional ones. In addition, the top five informative graph features for RTCDs are found to be Transitivity, Edge, Node, Degree and Reciprocity. In view of interpretability, the basic, single-, dyadic- and triadic-node and global characteristics of clickstream graphs have their specific advantages.Practical implicationsThe findings suggest that the temporal graph-theoretic approach can form an efficient and powerful AI-based real-time intent detecting decision-support system. Different levels of graph features have their specific interpretability on RTCDs from the perspectives of consumer behavior and psychology, which provides a theoretical basis for the design of computer information systems and the optimization of the ongoing session intervention or recommendation in e-commerce.Originality/valueTo the best of the authors' knowledge, this is the first study to apply clickstream graphs and real-time decision choices in conversion prediction and detection. Most studies have only meditated on a binary classification problem, while this study applies a graph-theoretic approach in a five-class classification problem. In addition, this study constructs temporal item-level graphs to represent the original structure of clickstream session data based on graph theory. The time-varying characteristics of the proposed approach enhance the performance of purchase conversion detection during an ongoing session.

Genetic risk for schizophrenia is associated with increased proportion of indirect connections in brain networks revealed by a semi-metric analysis: evidence from population sample stratified for polygenic risk.

Research studies based on tractography have revealed a prominent reduction of asymmetry in some key white-matter tracts in schizophrenia (SCZ). However, we know little about the influence of common genetic risk factors for SCZ on the efficiency of routing on structural brain networks (SBNs). Here, we use a novel recall-by-genotype approach, where we sample young adults from a population-based cohort (ALSPAC:N genotyped = 8,365) based on their burden of common SCZ risk alleles as defined by polygenic risk score (PRS). We compared 181 individuals at extremes of low (N = 91) or high (N = 90) SCZ-PRS under a robust diffusion MRI-based graph theoretical SBN framework. We applied a semi-metric analysis revealing higher SMR values for the high SCZ-PRS group compared with the low SCZ-PRS group in the left hemisphere. Furthermore, a hemispheric asymmetry index showed a higher leftward preponderance of indirect connections for the high SCZ-PRS group compared with the low SCZ-PRS group (PFDR < 0.05). These findings might indicate less efficient structural connectivity in the higher genetic risk group. This is the first study in a population-based sample that reveals differences in the efficiency of SBNs associated with common genetic risk variants for SCZ.

Stochastic control of ecological networks.

The paper models the maintenance of ecological networks in forest environments, built from bioreserves, patches and corridors, when these grids are subject to random processes such as extreme natural events. It also outlines a management plan to support the optimized results. After presenting the random graph-theoretic framework, we apply the stochastic optimal control to the graph dynamics. Our results show that the preservation of the network architecture cannot be achieved, under stochastic control, over the entire duration. It can only be accomplished, at the cost of sacrificing the links between the patches, by increasing the usage of the control devices. This would have a negative effect on the species migration by causing congestion among the channels left at their disposal. The optimal scenario, in which the shadow price is at its lowest and all connections are well-preserved, occurs at half of the course, be it the only optimal stopping moment found on the stochastic optimal trajectories. In such a scenario, the optimal forestry management policy has to integrate agility, integrated response, and quicker response time.

Structural rigidity theory applied to the scaffold-free (dis)assembly of space frames using cooperative robotics

This paper presents a fabrication-informed design method for triangulated space frame structures that remain stable during all phases of their robotic assembly and disassembly without requiring external scaffolding. A graph theoretic framework, based on rigidity theory, is developed to allow the structure, its support conditions, and the impact of robotic support constraints to be simultaneously represented in a single topological framework. The structural system is sequentially designed with an assembly logic based on Henneberg graph-construction steps, which are executed with two robots through a cooperative rigidity-preserving sequence. Ensuring planarity of the resulting graph during these construction steps is shown to lead to intrinsic disassembly potentials within the system. A graph-based algorithm is presented to locate, isolate and remove locally rigid tetrahedral cells formed in the structural system. This algorithm is then utilized to compute a rigidity-preserving robotic disassembly sequence. The design method is demonstrated in the case study design of a wooden space frame arch structure that is robotically (dis)assembled.

Altered grey matter integrity and network vulnerability relate to epilepsy occurrence in patients with multiple sclerosis.

The aim of this study was to investigate the relevance of compartmentalized grey matter (GM) pathology and network reorganization in multiple sclerosis (MS) patients with concomitant epilepsy. From 3-T magnetic resonance imaging scans of 30 MS patients with epilepsy (MSE group; age 41 ± 15 years, 21 females, disease duration 8 ± 6 years, median Expanded Disability Status Scale [EDSS] score 3), 60 MS patients without epilepsy (MS group; age 41 ± 12 years, 35 females, disease duration 6 ± 4 years, EDSS score 2), and 60 healthy subjects (HS group; age 40 ± 13 years, 27 females) the regional volumes of GM lesions and of cortical, subcortical and hippocampal structures were quantified. Network topology and vulnerability were modelled within the graph theoretical framework. Receiver-operating characteristic (ROC) curve analysis was applied to assess the accuracy of GM pathology measures to discriminate between MSE and MS patients. Higher lesion volumes within the hippocampus, mesiotemporal cortex and amygdala were detected in the MSE compared to the MS group (all p &lt; 0.05). The MSE group had lower cortical volumes mainly in temporal and parietal areas compared to the MS and HS groups (all p &lt; 0.05). Lower hippocampal tail and presubiculum volumes were identified in both the MSE and MS groups compared to the HS group (all p &lt; 0.05). Network topology in the MSE group was characterized by higher transitivity and assortativity, and higher vulnerability compared to the MS and HS groups (all p &lt; 0.05). Hippocampal lesion volume yielded the highest accuracy (area under the ROC curve 0.80 [0.67-0.91]) in discriminating between MSE and MS patients. High lesion load, altered integrity of mesiotemporal GM structures, and network reorganization are associated with a greater propensity for epilepsy occurrence in people with MS.

Network representation and analysis of energy coupling mechanisms in cellular metabolism by a graph-theoretical approach.

Mechanisms coupling the chemical reactions of oxidation and ATP synthesis in cellular metabolism by the fundamental biological process of oxidative phosphorylation (OX PHOS) in mitochondria provide > 90% of the energy requirements in living organisms. Mathematical graph theory methods have been extensively used to characterize various metabolic, regulatory, and disease networks in biology. However, networks of energy coupling mechanisms in OX PHOS have not been represented and analyzed previously by these approaches. Here, the problem of biological energy coupling is translated into a graph-theoretical framework, and all possible coupling schemes between oxidation and ATP synthesis are represented as graphs connecting these processes by various intermediates or states. The problem is shown to be transformed into the hard problem of finding a Hamiltonian tour in the networks of possible constituent mechanisms, given the constraints of a cyclical nature of operation of enzymes and biological molecular machines. Accessible mathematical proofs of three theorems that guarantee sufficient conditions for the existence of a Hamiltonian cycle in simple graphs are provided. The results of the general theorems are applied to the set of possible coupling mechanisms in OX PHOS and shown to (1) unequivocally differentiate between the major theories and mechanisms of energy coupling, (2) greatly reduce the possibilities for detailed consideration, and (3) deduce the biologically selected mechanism using additional constraints from the cumulative experimental record. Finally, an algorithm is constructed to implement the graph-theoretical procedure. In summary, the enormous power and generality of mathematical theorems and approaches in graph theory are shown to help solve a fundamental problem in biology.

A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸n

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type 𝔸n. The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type 𝔸n. With these tools, connected principal unit forms of Dynkin type 𝔸n are also classified up to strong congruence.

Generalizing the distribution of missing sums in sumsets

Given a finite set of integers A, its sumset is A+A:={ai+aj|ai,aj∈A}. We examine |A+A| as a random variable, where A⊂In=[0,n−1], the set of integers from 0 to n−1, so that each element of In is in A with a fixed probability p∈(0,1). Recently, Martin and O'Bryant studied the case in which p=1/2 and found a closed form for E[|A+A|]. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for Var(|A+A|) and bounds on the number of missing sums in A+A, mn;p(k):=P(2n−1−|A+A|=k). Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for E[|A+A|] and Var(|A+A|) for all p∈(0,1) and establish good bounds for E[|A+A|] and mn;p(k). We continue to investigate mn;p(k) by studying mp(k)=limn→∞⁡mn;p(k), proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for p=1/2, m1/2(6)>m1/2(7)<m1/2(8). This distribution is not unimodal, and is said to have a “divot” at 7. We report results investigating this divot as p varies, and through both theoretical and numerical analysis, prove that for p≥0.68 there is a divot at 1; that is, mp(0)>mp(1)<mp(2). Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets A+B where B is correlated to A by the probabilities P(i∈B|i∈A)=p1 and P(i∈B|i∉A)=p2. We provide some preliminary results using the extension of this framework.

Graph-theoretic estimation of reconfigurability in origami-based metamaterials

Origami-based mechanical metamaterials have recently received significant scientific interest due to their versatile and reconfigurable architectures. However, it is often challenging to account for all possible geometrical configurations of the origami assembly when each origami cell can take multiple phases. Here, we investigate the reconfigurability of a tessellation of origami-based cellular structures composed of bellows-like unit cells, specifically Tachi-Miura Polyhedron (TMP). One of the unique features of the TMP is that a single cell can take four different phases in a rigid foldable manner. Therefore, the TMP tessellation can achieve various shapes out of one original assembly. To assess the geometrical validity of the astronomical number of origami phase combinations, we build a graph-theoretic framework to describe the connectivity of unit cells and to analyze the reconfigurability of the tessellations. Our approach can pave the way to develop a systematic computational tool to design origami-based mechanical metamaterials with tailored properties.

PROBABILISTIC FRAMEWORK FOR LOSS DISTRIBUTION OF SMART CONTRACT RISK

Smart contract risk can be defined as a financial risk of loss due to cyber attacks on or contagious failures of smart contracts. Its quantification is of paramount importance to technology platform providers as well as companies and individuals when considering the deployment of this new technology. That is why, as our primary contribution, we propose a structural framework of aggregate loss distribution for smart contract risk under the assumption of a tree-stars graph topology representing the network of interactions among smart contracts and their users. To our knowledge, there exist no theoretical frameworks or models of an aggregate loss distribution for smart contracts in this setting. To achieve our goal, we contextualize the problem in the probabilistic graph-theoretical framework using bond percolation models. We assume that the smart contract network topology is represented by a random tree graph of finite size, and that each smart contract is the center of a random star graph whose leaves represent the users of the smart contract. We allow for heterogeneous loss topology superimposed on this smart contract and user topology and provide analytical results and instructive numerical examples.