Grid Graphs Research Articles

Super-resolution of diffusion-weighted images using space-customized learning model.

Diffusion-weighted imaging (DWI) is a noninvasive method used for investigating the microstructural properties of the brain. However, a tradeoff exists between resolution and scanning time in clinical practice. Super-resolution has been employed to enhance spatial resolution in natural images, but its application on high-dimensional and non-Euclidean DWI remains challenging. This study aimed to develop an end-to-end deep learning network for enhancing the spatial resolution of DWI through post-processing. We proposed a space-customized deep learning approach that leveraged convolutional neural networks (CNNs) for the grid structural domain (x-space) and graph CNNs (GCNNs) for the diffusion gradient domain (q-space). Moreover, we represented the output of CNN as a graph using correlations defined by a Gaussian kernel in q-space to bridge the gap between CNN and GCNN feature formats. Our model was evaluated on the Human Connectome Project, demonstrating the effective improvement of DWI quality using our proposed method. Extended experiments also highlighted its advantages in downstream tasks. The hybrid convolutional neural network exhibited distinct advantages in enhancing the spatial resolution of DWI scans for the feature learning of heterogeneous spatial data.

Equitable Irregular Edge-Weighting of Polar Grid Graph and Mongolian Tent Graph

An k-edge-weighting of a graph G=(V,E) is a map φ: E(G) → {1, 2, 3, ..., k} where k ≥ 1 is an integer. Denote Sφ(v) is the sum of edge-weights appearing on the edges incident at the vertex v under φ. An k-edge-weighting of G is equitable irregular if |Sφ (u)−Sφ (v)| ≤ 1, for every pair of adjacent vertices u and v in G. Theequitable irregular strength of an equitable irregular graph G is the smallest positive integer k such that there is a k-edge weighting of G and is denoted by Se (G). In this paper, we discuss the equitable irregularity of Polar grid and Mongolian tent graphs

Inclusive Local Irregularity Vertex Coloring In Grid Graph Family

Let is a simple graph and connected where is vertex set and is edge set. A maping as vertex k- labeling and function : is inclusive local irregularity vertex coloring, with . The minimum number of colors produced from inclusive local irregularity vertex coloring of graph is called inclusive chromatic number local irregularity, denoted by . On this paper, we learn about the inclusive local irregularity vertex coloring and determine the chromatic number on grid graph family.

Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications.

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks.

Resistance values under transformations in regular triangular grids

In Evans and Francis (2022) and Hendel (2021) the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced s times provided the original grid has at least 4s rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously.

Graph-based decomposition for electrical wiring design in interior finishing

Electrical wiring design within interior finishing is subject to various practical criteria, thereby resulting in time-intensive manual implementation. Despite this, most applications offer only basic collision detection tools, limiting design capabilities. In response, a graph-based decomposition approach is introduced, framing electrical wiring design as an optimization problem on a non-uniform grid graph derived from building information. This method breaks down the problem into two distinct subproblems, which are sequentially solved to achieve an optimal overall solution. Through experimentation across synthetic and real-world instances, performance improvements are demonstrated compared to two conventional integer programming solvers, yielding cost reductions averaging 12.3% when compared to a leading commercial application for interior design. These findings demonstrate the potential integration of this approach into building information modeling platforms, offering an efficient toolset for electrical wiring design in interior finishing.

Complexity results on k-independence in some graph products

For a positive integer k, a subset S of vertices of a graph G is k-independent if each vertex in S has at most k − 1 neighbors in S. We consider k-independent sets in two graph products: Cartesian and complementary prism. We show that the problem of determining k-independence remains NP-complete even for Cartesian products and complementary prisms. Furthermore, we present results on k-independence in the Cartesian product of two paths, known as grid graphs.

Learn to solve dominating set problem with GNN and reinforcement learning

The Dominating Set Problem has a wide range of applications in many industrial areas and the problem has been proven to be NP-hard. The idea using neural networks to solve combinatorial optimization problems has been shown to be effective and time-saving in recent years. Inspired by these studies, to solve the Dominating Set Problem, we train a neural network by Double Deep Q-Networks (DDQN). To better capture the features and structures of the graph, we use a message passing network for the graph representation. We validate our model on random graphs of different sizes, and even on several different lattice graphs, which show our model is effective.

Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field associated to one spin value. In this energy landscape there is one stable configuration and q−1 metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as β→∞ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. Finally, we identify all the minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration by giving a geometric characterization.

Poidge-convexity in triangular lattices

We adapt the notion of poidge-convexity to subgraphs of the triangular lattice, and investigate the poidge-convexity of graphs belonging to two families of such subgraphs: paths and grid graphs.

Tipsy cop and tipsy robber: Collisions of biased random walks on graphs

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the tipsy cop and drunken robber is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including δ-regular trees and δ-regular trees rooted to a Δ-regular tree, where Δ≥δ. Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

The issue of counting independent sets of a graph, G, represented as i(G), is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, i(G) is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the ‘branch-and-bound’ technique and is applied to compute i(Gm,n) for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the ’basic cases’ linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of i(Gm,n) compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs.

Thinness and its variations on some graph families and coloring graphs of bounded thinness

Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs. We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs CRn. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of n × n grids GRn and m×n grids GRm,n, proving that n−1/3 ≤ thin(GRn) ≤ n+1/2. Regarding the precedence thinness, we prove that prec-thin(GRn,2) = n+1/2 and that n− 1 + 3/2 ≤ prec-thin(GRn) ≤ n− 1 2. As applications, we show that the k-coloring problem is NP-complete for precedence 2-thin graphs and for proper 2-thin graphs, when k is part of the input. On the positive side, it is polynomially solvable for precedence proper 2-thin graphs, given the order and partition.

Structure shapes the representation of a novel category.

Concepts contain rich structures that support flexible semantic cognition. These structures can be characterized by patterns of feature covariation: Certain features tend to cluster in the same items (e.g., feathers, wings, can fly). Existing computational models demonstrate how this kind of structure can be leveraged to slowly learn the distinctions between categories, on developmental timescales. However, it is not clear whether and how we leverage feature structure to quickly learn a novel category. We thus investigated how the internal structure of a new category is first extracted from experience, with the prediction that feature-based structure would have a rapid and broad influence on the learned category representation. Across three experiments, novel categories were designed with patterns of feature associations determined by carefully constructed graph structures, with Modular graphs-exhibiting strong clusters of feature covariation-compared against Random and Lattice graphs. In Experiment 1, a feature inference task using verbal stimuli revealed that Modular structure broadly facilitated category learning. Experiment 2 replicated this effect in visual categories. In Experiment 3, a statistical learning paradigm revealed that this Modular benefit relates to high-level structure rather than pairwise feature associations and persists even when category structure is incidental to the task. A neural network model was readily able to account for these effects, suggesting that correlational feature structure may be encoded within rapidly learned, distributed category representations. These findings constrain theories of category representation and link theories of category learning with structure learning more broadly. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Ordered scheduling in control-flow distributed transactional memory

Consider the control-flow model of transaction execution in a distributed system modeled as a communication graph where shared objects positioned at nodes of the graph are immobile but the transactions accessing the objects send requests to the nodes where objects are located to read/write those objects. The control-flow model offers benefits to applications in which the movement of shared objects is costly due to their sizes and security purposes. In this paper, we study the ordered scheduling problem of committing dependent transactions according to their predefined priorities in this model. The considered problem naturally arises in areas, such as loop parallelization and state-machine-based computing, where producing executions equivalent to a priority order is needed to satisfy certain properties. Specifically, we study ordered scheduling considering two performance metrics fundamental to any distributed system: (i) execution time - total time to commit all the transactions and (ii) communication cost - the total distance traversed in accessing required shared objects. We design scheduling algorithms that are individually or simultaneously efficient for both the metrics and rigorously evaluate them through several benchmarks on random and grid graphs, validating their efficiency. To our best knowledge, this is the first study of ordered scheduling in the control-flow model of distributed transaction execution.

The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs

The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

The chromatic polynomial of grid graph $$P_3 \square P_n$$

The chromatic polynomial of grid graph $$P_3 \square P_n$$

Controllability of networked multiagent systems based on linearized Turing’s model

Turing’s model has been widely used to explain how simple, uniform structures can give rise to complex, patterned structures during the development of organisms. However, it is very hard to establish rigorous theoretical results for the dynamic evolution behavior of Turing’s model since it is described by nonlinear partial differential equations. We focus on controllability of Turing’s model by linearization and spatial discretization. This linearized model is a networked system whose agents are second order linear systems and these agents interact with each other by Laplacian dynamics on a graph. A control signal can be added to agents of choice. Under mild conditions on the parameters of the linearized Turing’s model, we prove the equivalence between controllability of the linearized Turing’s model and controllability of a Laplace dynamic system with agents of first order dynamics. When the graph is a grid graph or a cylinder grid graph, we then give precisely the minimal number of control nodes and a corresponding control node set such that the Laplace dynamic systems on these graphs with agents of first order dynamics are controllable.

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.