Symmetric Adjacency Research Articles

THE GENERAL SYMMETRIC DEGREE AND GENERAL SYMMETRIC DEGREE ADJACENCY POLYNOMIAL OF A GRAPH AND ITS ENERGY

The spectral graph theory is concerned with the relationship between the spectra of specific matrices associated with a graph and the structural properties of that graph. This paper introduces a general symmetric degree adjacency of a graph G, general symmetric degree of a graph G, and complete weighted graph of a graph. Here we explore its characteristic polynomial using Sachs theorem. We also investigate the bounds for index and energy of general symmetric degree adjacency of a graph. We give two algorithms, first one to find general symmetric degree adjacency of a graph G and second one to find general symmetric degree of a graph G.

GBVSSL: Contrastive Semi-Supervised Learning Based on Generalized Bias-Variance Decomposition

Mainstream semi-supervised learning (SSL) techniques, such as pseudo-labeling and contrastive learning, exhibit strong generalization abilities but lack theoretical understanding. Furthermore, pseudo-labeling lacks the label enhancement from high-quality neighbors, while contrastive learning ignores the supervisory guidance provided by genuine labels. To this end, we first introduce a generalized bias-variance decomposition framework to investigate them. Then, this research inspires us to propose two new techniques to refine them: neighbor-enhanced pseudo-labeling, which enhances confidence-based pseudo-labels by incorporating aggregated predictions from high-quality neighbors; label-enhanced contrastive learning, which enhances feature representation by combining enhanced pseudo-labels and ground-truth labels to construct a reliable and complete symmetric adjacency graph. Finally, we combine these two new techniques to develop an excellent SSL method called GBVSSL. GBVSSL significantly surpasses previous state-of-the-art SSL approaches in standard benchmarks, such as CIFAR-10/100, SVHN, and STL-10. On CIFAR-100 with 400, 2500, and 10,000 labeled samples, GBVSSL outperforms FlexMatch by 3.46%, 2.72%, and 2.89%, respectively. On the real-world dataset Semi-iNat 2021, GBVSSL improves the Top-1 accuracy over CCSSL by 4.38%. Moreover, GBVSSL exhibits faster convergence and enhances unbalanced SSL. Extensive ablation and qualitative studies demonstrate the effectiveness and impact of each component of GBVSSL.

Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices

This paper introduces and explores the concept of secondary k-Range Symmetric (RS) Neutrosophic Fuzzy Matrices (NFM) and establishes its properties and relationships with other symmetric and secondary symmetric NFMs. The study defines secondary k-RS NFMs and provides insightful numerical examples to illustrate their characteristics. The paper investigates the interconnections among s-k-RS, s-RS, k-RS, and RS NFMs, discuss on their mutual relations. Additionally, the necessary and sufficient conditions for a given NFM to qualify as a s-k-RS NFM are identified. The research demonstrates that k-symmetry implies k-RS, and vice versa, contributing to a comprehensive understanding between different types of symmetries in NFMs. Graphical representations of RS, column symmetric, and kernel symmetric adjacency and incidence NFMs are presented, unveiling intriguing patterns and relationships. While every adjacency NFM is symmetric, range symmetric, column symmetric, and kernel symmetric, the incidence matrix satisfies only kernel symmetric conditions. The study further establishes that every range symmetric adjacency NFM is a kernel symmetric adjacency NFM, though the converse does not hold in general. The existence of multiple generalized inverses of NFMs in Fn is explored, with additional equivalent conditions for certain g-inverses of s-κ-RS NFMs to retain the s-κ-RS property. We conclude by characterizing the generalized inverses belonging to specific sets {1, 2}, {1, 2, 3}, and {1, 2, 4} of s-k-RS NFMs, providing a comprehensive framework for understanding the structure and properties of secondary k-Range Symmetric Neutrosophic Fuzzy Matrices. This research contributes to the mathematical literature by introducing a novel class of NFMs and establishing their fundamental properties and relationships, presenting new perspectives on matrix theory in the context of neutrosophic fuzzy logic.

Some generalized centralities in higher-order networks represented by simplicial complexes

Abstract Higher-order interactions, that is, interactions among the units of group size greater than two, are a fundamental structural feature of a variety of complex systems across the scale. Simplicial complexes are combinatorial objects that can capture and model the higher-order interactions present in a given complex system and thus represent the complex system as a higher-order network comprising simplices. In this work, a given simplicial complex is viewed as a finite union of d-exclusive simplicial complexes. Thus, to represent a complex system as a higher-order network given by a simplicial complex that captures all orders of interactions present in the system, a family of symmetric adjacency tensors A(d) of dimension d + 1 and appropriate order has been used. Each adjacency tensor A(d) represents a d-exclusive simplicial complex and for d≥2 it represents exclusively higher-order interactions of the system. For characterizing the structure of d-exclusive simplicial complexes, the notion of generalized structural centrality indices namely, generalized betweenness centrality and generalized closeness centrality has been established by developing the concepts of generalized walk and generalized distance in the simplicial complex. Generalized centrality indices quantify the contribution of δ-simplices in any d-exclusive simplicial complex Δ, where δ<d and if d≥2, it describes the contribution of δ-faces to the higher-order interactions of Δ. These generalized centrality indices provide local structural descriptions, which lead to mesoscale insights into the simplicial complex that comprises the higher-order network. An important theorem providing a general technique for the characterization of connectedness in d-exclusive simplicial complexes in terms of irreducibility of its adjacency tensor has been established. The concepts developed in this work together with concepts of generalized simplex deletion in d-exclusive simplicial complexes have been illustrated using examples. The effect of deletions on the generalized centralities of the complexes in the examples has been discussed.

On synchronization of the double sphere Kuramoto model with connected undirected graphs

M.A. Lohe first proposed the double sphere Kuramoto model (On the double sphere model of synchronization, Physica D: Nonlinear Phenomena, 412:132642, 2020), which is a more general model than the high-dimensional Kuramoto model defined on a sphere. For the double sphere Kuramoto model with a complete graph, some theoretic results on synchronization were obtained. But for the case of general graphs, there is no report yet. In this paper, for the double sphere Kuramoto model composed of identical oscillators interacting through a connected undirected graph with a symmetric nonnegative adjacency matrix, the synchronization is proved by using LaSalle’s invariance principle. Moreover, the exponential synchronization is achieved, and the supremum of exponential synchronization rates is exactly obtained.

The spectra of digraphs with Morita equivalent C⁎-algebras

Eilers et al. have recently completed the geometric classification of unital digraph C⁎-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the nonzero elements of the spectrum of a finite digraph, which in this paper is allowed to have loops and parallel edges. We consider several different digraph spectra that have been studied in the literature, answering this question for the Laplace and adjacency spectra, their skew counterparts, the symmetric adjacency spectrum, the adjacency spectrum of the line digraph, the Hermitian adjacency spectrum, and the normalized Laplacian, considering in most cases two ways that these spectra can be defined in the presence of parallel edges. We show that the adjacency spectra of the digraph and line digraph are preserved by a subset of the moves, and the skew adjacency and Laplace spectra are preserved by the Cuntz splice. We give counterexamples to show that the other spectra are not preserved by the remaining moves. The same results hold if one restricts to the class of strongly connected digraphs.

Computational and Theoretical Challenges for Computing the Minimum Rank of a Graph

The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs. History: This “Challenge” paper was invited by the Editor in Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida. Funding: This work was supported by the National Science Foundation [Grant DMS-1720225]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1219 .

Computational Robotics: An Alternative Approach for Predicting Terrorist Networks

Increasing terrorist activities globally have attracted the attention of many researchers, policy makers and security agencies towards counterterrorism. The clandestine nature of terrorist networks have made them difficult for detection. Existing works have failed to explore computational characterization to design an efficient threat-mining surveillance system. In this paper, a computationally-aware surveillance robot that auto-generates threat information, and transmit same to the cloud-analytics engine is developed. The system offers hidden intelligence to security agencies without any form of interception by terrorist elements. A miniaturized surveillance robot with Hidden Markov Model (MSRHMM) for terrorist computational dissection is then derived. Also, the computational framework for MERHMM is discussed while showing the adjacency matrix of terrorist network as a determinant factor for its operation. The model indicates that the terrorist network have a property of symmetric adjacency matrix while the social network have both asymmetric and symmetric adjacency matrix. Similarly, the characteristic determinant of adjacency matrix as an important operator for terrorist network is computed to be -1 while that of a symmetric and an asymmetric in social network is 0 and 1 respectively. In conclusion, it was observed that the unique properties of terrorist networks such as symmetric and idempotent property conferred a special protection for the terrorist network resilience. Computational robotics is shown to have the capability of utilizing the hidden intelligence in attack prediction of terrorist elements. This concept is expected to contribute in national security challenges, defense and military intelligence.

Joint Embedding of Graphs.

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the joint embedding method identifies a linear subspace spanned by rank one symmetric matrices and projects adjacency matrices of graphs into this subspace. The projection coefficients can be treated as features of the graphs, while the embedding components can represent vertex features. We also propose a random graph model for multiple graphs that generalizes other classical models for graphs. We show through theory and numerical experiments that under the model, the joint embedding method produces estimates of parameters with small errors. Via simulation experiments, we demonstrate that the joint embedding method produces features which lead to state of the art performance in classifying graphs. Applying the joint embedding method to human brain graphs, we find it extracts interpretable features with good prediction accuracy in different tasks.

Bayesian Regression With Undirected Network Predictors With an Application to Brain Connectome Data

This article focuses on the relationship between a measure of creativity and the human brain network for subjects in a brain connectome dataset obtained using a diffusion weighted magnetic resonance imaging procedure. We identify brain regions and interconnections that have a significant effect on creativity. Brain networks are often expressed in terms of symmetric adjacency matrices, with row and column indices of the matrix representing the regions of interest (ROI), and a cell entry signifying the estimated number of fiber bundles connecting the corresponding row and column ROIs. Current statistical practices for regression analysis with the brain network as the predictor and the measure of creativity as the response typically vectorize the network predictor matrices prior to any analysis, thus failing to account for the important structural information in the network. This results in poor inferential and predictive performance in presence of small sample sizes. To answer the scientific questions discussed above, we develop a flexible Bayesian framework that avoids reshaping the network predictor matrix, draws inference on brain ROIs and interconnections significantly related to creativity, and enables accurate prediction of creativity from a brain network. A novel class of network shrinkage priors for the coefficient corresponding to the network predictor is proposed to achieve these goals simultaneously. The Bayesian framework allows characterization of uncertainty in the findings. Empirical results in simulation studies illustrate substantial inferential and predictive gains of the proposed framework in comparison with the ordinary high-dimensional Bayesian shrinkage priors and penalized optimization schemes. Our framework yields new insights into the relationship of brain regions with creativity, also providing the uncertainty associated with the scientific findings. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

A Fast and Exact Greedy Algorithm for the Core–Periphery Problem

The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce a well-connected subgraph and share connections with peripheral nodes, while the peripheral nodes are loosely connected to the core nodes and other peripheral nodes. We propose a polynomial-time algorithm to detect core–periphery structures in networks having a symmetric adjacency matrix. The core set is defined as the solution of a combinatorial optimization problem, which has a pleasant symmetry with respect to graph complementation. We provide a complete description of the optimal solutions to that problem and an exact and efficient algorithm to compute them. The proposed approach is extended to networks with loops and oriented edges. Numerical simulations are carried out on both synthetic and real-world networks to demonstrate the effectiveness and practicability of the proposed algorithm.

Semiparametric model for covariance regression analysis

Estimating covariance matrices is an important research topic in statistics and finance. A semiparametric model for covariance matrix estimation is proposed. Specifically, the covariance matrix is modeled as a polynomial function of the symmetric adjacency matrix with time varying parameters. The asymptotic properties for the time varying coefficient and the associated semiparametric covariance estimators are established. A Bayesian information criterion to select the order of the polynomial function is also investigated. Simulation studies and an empirical example are presented to illustrate the usefulness of the proposed method.

Franck-Condon factors by counting perfect matchings of graphs with loops.

We show that the Franck-Condon Factor (FCF) associated with a transition between initial and final vibrational states in two different potential energy surfaces, having N and M vibrational quanta, respectively, is equivalent to calculating the number of perfect matchings of a weighted graph with loops that has P = N + M vertices. This last quantity is the loop hafnian of the (symmetric) adjacency matrix of the graph which can be calculated in O(P32P/2) steps. In the limit of small numbers of vibrational quanta per normal mode, our loop hafnian formula significantly improves the speed at which FCFs can be calculated. Our results more generally apply to the calculation of the matrix elements of a bosonic Gaussian unitary between two multimode Fock states having N and M photons in total and provide a useful link between certain calculations of quantum chemistry, quantum optics, and graph theory.

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.

Covariance Matrix Estimation Via Network Structure

In this article, we employ a regression formulation to estimate the high dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.

Non-Singular graphs with a Singular Deck

The n-vertex graph G(=Γ(G)) with a non-singular real symmetric adjacency matrix G, having a zero diagonal and singular (n−1)×(n−1) principal submatrices is termed a NSSD, a Non-Singular graph with a Singular Deck. NSSDs arose in the study of the polynomial reconstruction problem and were later found to characterise non-singular molecular graphs that are distinct omni-conductors and ipso omni-insulators. Since both matrices G and G−1 represent NSSDs Γ(G) and Γ(G−1), the value of the nullity of a one-, two- and three-vertex deleted subgraph of G is shown to be determined by the corresponding subgraph in Γ(G−1). Constructions of infinite subfamilies of non-NSSDs are presented. NSSDs with all two-vertex deleted subgraphs having a common value of the nullity are referred to as G-nutful graphs. We show that their minimum vertex degree is at least 4.

Adjacency in hospital planning

Adjacencies stand at the beginning of a multitude of planning tasks. Especially in hospital planning they are essential for describing relationships between different organizational units - e.g. ‘close’, ‘distant’ or ‘neutral’. Mathematically, these terms map to relative weights between each pair of units in the range [-1, 1] which are put into a (symmetric) adjacency matrix. This matrix subsequently determines relative locations of individual spaces (preliminary space layout). The paper deals with the effective definition of this adjacency matrix in the context of early-stage architectural planning. In contrast to current planning practice, which looks at each adjacency relation in isolation, our approach uses a Newtonian gravitation model to propagate changes to a single relationship immediately to the whole space layout. As a result, we are able to supply architects with a design tool that accelerates the definition of adjacencies and lets them preview the preliminary space layout at the same time.

Directed Connected Operators: Asymmetric Hierarchies for Image Filtering and Segmentation

Connected operators provide well-established solutions for digital image processing, typically in conjunction with hierarchical schemes. In graph-based frameworks, such operators basically rely on symmetric adjacency relations between pixels. In this article, we introduce a notion of directed connected operators for hierarchical image processing, by also considering non-symmetric adjacency relations. The induced image representation models are no longer partition hierarchies (i.e., trees), but directed acyclic graphs that generalize standard morphological tree structures such as component trees, binary partition trees or hierarchical watersheds. We describe how to efficiently build and handle these richer data structures, and we illustrate the versatility of the proposed framework in image filtering and image segmentation.

Universal state transfer on graphs

A continuous-time quantum walk on a graph G is given by the unitary matrix U(t)=exp(−itA), where A is the adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any ϵ>0, there is a time t, where the (a,b)-entry of U(t) satisfies |U(t)a,b|≥1−ϵ. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for ϵ=0. In this work, we study a natural extension of this notion called universal state transfer wherein state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (i) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (each eigenvector has entries which are equal in magnitude). (ii) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, the switching automorphism group is cyclic. (iii) There is a family of complex oriented prime-length cycles which has universal pretty good state transfer. This provides a concrete example of a family of graphs with this universal property. (iv) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. Finally, we provide a spectral characterization of universal perfect state transfer for graphs switching equivalent to circulants.

Toward effective group formation in computer-supported collaborative learning

Group formation task as a starting point for computer-supported collaborative learning plays a key role in achieving pedagogical goals. Various approaches have been reported in the literature to address this problem, but none have offered an optimal solution. In this research, an online learning environment was modeled as a weighted undirected complete graph in which each learner was implied as a node and the relationship between them was denoted as a weighted arc. The weight of each link indicated the similarity degree between the corresponding individuals. The similarity between two students was measured as the mean of their absolute interest levels. The graph was also represented through a symmetric adjacency matrix. Then, a novel binary integer programming formulation was proposed to model the group formation problem and optimally assign each learner to the most appropriate group. The method was utilized to divide an online class of 32 learners into 8 groups of size 4. Findings indicated that the suggested model was successful in optimally solving the problem in 20.53 seconds, on average. The performance of the method was also compared with a modified version of K-means clustering algorithm. Although, the running time of the suggested technique was not as good as the clustering algorithm, it generated better outcomes in theory and in practice.