Graph layout

The escalating complexity of electrical distribution power grids presents a significant challenge for system operators and engineers in assessing the grid's state intuitively and rapidly. Conventional visualisation methods, which often rely on dynamic and network-topology-based layouts, can complicate the interpretation of large datasets. This study introduces an innovative approach to the deterministic visualisation of distribution grids by integrating a power grid model with the Keçeci Layout algorithm, which is distinguished by its sequential and repeatable structure. The Keçeci Layout positions nodes sequentially along a primary axis and in a zigzag pattern on a secondary axis according to a predefined order, thereby establish an orderly and predictable visual representation of the grid's topological structure. This integration facilitates the visualisation of grid components such as buses, transformers, and distribution lines based on a specific logical sequence (e.g., line route or asset ID). Unlike traditional force-directed or geographic layouts, this deterministic approach ensures consistency and comparability by always producing the same visual output for the same grid model. This feature offers a critical advantage, particularly in detecting topological changes or anomalies within the grid. For instance, a post-fault topological alteration becomes immediately apparent in a visualisation generated by the Keçeci Layout, as the position of the nodes is directly linked to the sequential data structure. In this work, critical grid data, including power flow, voltage levels, and line loadings, are presented as visual layers overlaid on the Keçeci Layout. The use of colour-coding, arrow scaling, and other visual cues assists operators in developing a quick and clear understanding of the grid's instantaneous condition.[3][4] This methodology supports decision-making processes by enabling a more straightforward comprehension and interpretation of complex grid behaviours, especially in training, planning, and post-fault analysis scenarios. In conclusion, the integration of the power grid model with the Keçeci Layout has the potential to offer a standardised, intuitive, and powerful visualisation tool for the analysis and management of distribution power grids.

This paper provides a survey of results on the exact bandwidth, edgesum, and pro le of graphs. A bibliography o f work in these areas is provided. The emphasis is on composite graphs. This may be regarded as an update of the original survey of solved bandwidth problems by Chinn, Chv atalov a, Dewdney, and Gibbs 10] in 1982. Also several of the application areas involving these graph parameters are described.

Graph visualisation is a critical tool in scientific inquiry, yet standard algorithms like force-directed layouts often obscure inherent sequential information by prioritising the display of topological clusters. This limitation hinders the analysis of systems in fields like chemistry and environmental science, where processes are frequently defined by a linear order. This paper introduces and evaluates the Keçeci Layout, a deterministic, order-preserving algorithm designed specifically to visualise such systems. Operating on a predictable zigzag principle, the Keçeci Layout systematically arranges nodes according to their predefined sequence. This represents a deliberate trade-off, favouring absolute structural clarity and reproducibility over the discovery of emergent network topology. We demonstrate its utility through applications in visualising polymer chains and reaction pathways in chemistry, and mapping contaminant transport and life cycle stages in environmental science. A comparative analysis highlights its principal advantage in providing an unambiguous, stable representation of linear processes, and its main limitation, the intentional concealment of community structures. The research concludes that the Keçeci Layout is not a universal replacement for existing algorithms but an essential, specialised tool. It provides a vital framework for structural thinking when the primary insight to be conveyed is the sequence itself. The findings underscore the importance of selecting a visualisation algorithm that aligns with the specific research question, positing the Keçeci Layout as the optimal choice when progression and order are the central narrative.

This study presents a comparative analysis of a traditional, pure Python-based approach and a modern, JAX library-based approach for calculating the harmonic series, focusing on performance, scalability, and efficiency. The harmonic series is a divergent series of significant importance in mathematical analysis. In computational mathematics, the effective evaluation of such series is critical, particularly in applications requiring large-scale data and high precision. The traditional pure Python approach utilises the fractions.Fraction class to provide exact arithmetic, enhancing the code's readability and accuracy. However, this method exhibits linear time complexity for large values of n, limiting its performance in the context of repetitive calculations. Furthermore, memory usage and processing time increase linearly with n. In contrast, the JAX-based approach (the Oresmej module) leverages the advanced features of JAX, a library designed for high-performance scientific computing, including vectorisation (vmap), just-in-time (JIT) compilation, and automatic differentiation. This approach yields a significant speed advantage, especially in large-scale and repetitive operations. Additionally, caching is supported via @lru_cache, which prevents re-computation for identical inputs. The return value is defined as a tuple, thereby enhancing data integrity and cache compatibility. The comparative analysis was conducted using micro-benchmarking techniques. Following a warm-up period, the JAX-based approach was found to be several hundred times faster than pure Python, with cached functions producing results almost instantaneously. Therefore, for scientific research demanding high performance, scalability, and sustainable computation, approaches based on Cython (Cythonize), Numba, JAX, NumPy, and parallel processing (multiprocessing, joblib) are strongly recommended.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑_(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η_(n-1)+ 1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑_(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η_(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behaviour. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modelling.

Nodal-line semimetals constitute a fascinating subclass within the family of topological materials in condensed matter physics. These unique materials are characterized by the intersection of their electronic energy bands along one or more closed or open lines within the Brillouin Zone, rather than at isolated points. These lines of intersection are termed "nodal lines," and along these lines, electrons can behave as if they were massless, leading to extraordinary electronic properties. The significance of nodal-line semimetals stems from the fundamental physical phenomena offered by these unique linear crossings, as well as the promising potential they hold for future technologies, particularly quantum computers. One of the most striking features of these materials is that their nodal lines are often topologically protected by specific crystal symmetries (e.g., mirror symmetry). This protection renders the nodal lines and their associated electronic states highly resilient to minor imperfections and disorder within the material. Considering that decoherence (the loss of quantum states by qubits) is the primary adversary of quantum computers, such topological stability is of vital importance. Nodal-line semimetals, through this inherent protection mechanism, could offer new pathways for developing more long-lived and fault-tolerant qubits. Another distinguishing characteristic of nodal-line semimetals is their ability to host exotic surface states with nearly flat energy bands, known as "drumhead surface states." These surface states are localized within the region bounded by the projection of the nodal line onto the surface and can possess a high density of states. The manipulation of these special surface states might enable novel quantum information processing schemes or topologically protected information carriers. Furthermore, the geometry and number of nodal lines can significantly influence the material's response to electromagnetic fields or mechanical strain. This makes nodal-line semimetals potential candidates for sensitive sensors or tuneable electronic/optical components. When a magnetic field is applied or symmetries are broken, a nodal line can split into Weyl points or a band gap can open, offering the possibility to dynamically control the material's properties. Such control mechanisms could be exploited for designing quantum switches or complex quantum circuit elements. In conclusion, nodal-line semimetals, with the rich physics offered by their linear band crossings and the advantages conferred by topological protection, carry immense potential for applications in quantum computing and beyond. Their drumhead surface states and tuneable electronic structures make them exciting systems for fundamental scientific research and prime candidates to become building blocks for future quantum technologies. Ongoing research in this area is anticipated to pave the way for more robust and powerful quantum computers.

Nodal-line semimetals constitute a fascinating subclass within the family of topological materials in condensed matter physics. These unique materials are characterized by the intersection of their electronic energy bands along one or more closed or open lines within the Brillouin Zone, rather than at isolated points. These lines of intersection are termed "nodal lines," and along these lines, electrons can behave as if they were massless, leading to extraordinary electronic properties. The significance of nodal-line semimetals stems from the fundamental physical phenomena offered by these unique linear crossings, as well as the promising potential they hold for future technologies, particularly quantum computers. One of the most striking features of these materials is that their nodal lines are often topologically protected by specific crystal symmetries (e.g., mirror symmetry). This protection renders the nodal lines and their associated electronic states highly resilient to minor imperfections and disorder within the material. Considering that decoherence (the loss of quantum states by qubits) is the primary adversary of quantum computers, such topological stability is of vital importance. Nodal-line semimetals, through this inherent protection mechanism, could offer new pathways for developing more long-lived and fault-tolerant qubits. Another distinguishing characteristic of nodal-line semimetals is their ability to host exotic surface states with nearly flat energy bands, known as "drumhead surface states." These surface states are localized within the region bounded by the projection of the nodal line onto the surface and can possess a high density of states. The manipulation of these special surface states might enable novel quantum information processing schemes or topologically protected information carriers. Furthermore, the geometry and number of nodal lines can significantly influence the material's response to electromagnetic fields or mechanical strain. This makes nodal-line semimetals potential candidates for sensitive sensors or tuneable electronic/optical components. When a magnetic field is applied or symmetries are broken, a nodal line can split into Weyl points or a band gap can open, offering the possibility to dynamically control the material's properties. Such control mechanisms could be exploited for designing quantum switches or complex quantum circuit elements. In conclusion, nodal-line semimetals, with the rich physics offered by their linear band crossings and the advantages conferred by topological protection, carry immense potential for applications in quantum computing and beyond. Their drumhead surface states and tuneable electronic structures make them exciting systems for fundamental scientific research and prime candidates to become building blocks for future quantum technologies. Ongoing research in this area is anticipated to pave the way for more robust and powerful quantum computers.

Nodal-line semimetals constitute a fascinating subclass within the family of topological materials in condensed matter physics. These unique materials are characterized by the intersection of their electronic energy bands along one or more closed or open lines within the Brillouin Zone, rather than at isolated points. These lines of intersection are termed "nodal lines," and along these lines, electrons can behave as if they were massless, leading to extraordinary electronic properties. The significance of nodal-line semimetals stems from the fundamental physical phenomena offered by these unique linear crossings, as well as the promising potential they hold for future technologies, particularly quantum computers. One of the most striking features of these materials is that their nodal lines are often topologically protected by specific crystal symmetries (e.g., mirror symmetry). This protection renders the nodal lines and their associated electronic states highly resilient to minor imperfections and disorder within the material. Considering that decoherence (the loss of quantum states by qubits) is the primary adversary of quantum computers, such topological stability is of vital importance. Nodal-line semimetals, through this inherent protection mechanism, could offer new pathways for developing more long-lived and fault-tolerant qubits. Another distinguishing characteristic of nodal-line semimetals is their ability to host exotic surface states with nearly flat energy bands, known as "drumhead surface states." These surface states are localized within the region bounded by the projection of the nodal line onto the surface and can possess a high density of states. The manipulation of these special surface states might enable novel quantum information processing schemes or topologically protected information carriers. Furthermore, the geometry and number of nodal lines can significantly influence the material's response to electromagnetic fields or mechanical strain. This makes nodal-line semimetals potential candidates for sensitive sensors or tuneable electronic/optical components. When a magnetic field is applied or symmetries are broken, a nodal line can split into Weyl points or a band gap can open, offering the possibility to dynamically control the material's properties. Such control mechanisms could be exploited for designing quantum switches or complex quantum circuit elements. In conclusion, nodal-line semimetals, with the rich physics offered by their linear band crossings and the advantages conferred by topological protection, carry immense potential for applications in quantum computing and beyond. Their drumhead surface states and tuneable electronic structures make them exciting systems for fundamental scientific research and prime candidates to become building blocks for future quantum technologies. Ongoing research in this area is anticipated to pave the way for more robust and powerful quantum computers.

The visualization of scientific data is a fundamental step in uncovering hidden patterns and relationships within complex systems. While conventional force-directed graph layout algorithms (e.g., spring layout) are effective at displaying the general topology and clustering tendencies of systems, they possess a critical shortcoming: they are inherently non-deterministic and obscure the natural, sequential, or index-based identities of nodes. This poses a significant limitation when analysing systems that contain an intrinsic sequential logic, such as quantum circuits, metabolic pathways, crystal structures, or chronological data structures. This paper introduces the "Keçeci Layout," a deterministic and order-preserving algorithm designed to fill this analytical gap. The algorithm operates by arranging nodes sequentially along a primary axis according to their IDs, employing a predictable zigzag pattern on a secondary axis to prevent node overlaps. This approach establishes a direct and intuitive bridge between a system's abstract data structure (e.g., qubit indices, reaction steps) and its visual representation. To demonstrate the versatility and efficacy of the Keçeci Layout, four case studies from distinct scientific disciplines are presented: Chemistry, Physics, Biology, Quantum Computing, Materials Science, Molecular Biology, Computer Science, etc. In conclusion, the Keçeci Layout is proven to be a powerful scientific tool that offers analytical depth by remaining faithful to the underlying data structure, moving beyond purely aesthetic presentation. The algorithm provides a significant advantage over conventional methods, particularly for the visual analysis of ordered, hierarchical, or index-dependent systems.

The visualization of scientific data is a fundamental step in uncovering hidden patterns and relationships within complex systems. While conventional force-directed graph layout algorithms (e.g., spring layout) are effective at displaying the general topology and clustering tendencies of systems, they possess a critical shortcoming: they are inherently non-deterministic and obscure the natural, sequential, or index-based identities of nodes. This poses a significant limitation when analysing systems that contain an intrinsic sequential logic, such as quantum circuits, metabolic pathways, crystal structures, or chronological data structures. This paper introduces the "Keçeci Layout," a deterministic and order-preserving algorithm designed to fill this analytical gap. The algorithm operates by arranging nodes sequentially along a primary axis according to their IDs, employing a predictable zigzag pattern on a secondary axis to prevent node overlaps. This approach establishes a direct and intuitive bridge between a system's abstract data structure (e.g., qubit indices, reaction steps) and its visual representation. To demonstrate the versatility and efficacy of the Keçeci Layout, four case studies from distinct scientific disciplines are presented: Chemistry, Physics, Biology, Quantum Computing, Materials Science, Molecular Biology, Computer Science, etc. In conclusion, the Keçeci Layout is proven to be a powerful scientific tool that offers analytical depth by remaining faithful to the underlying data structure, moving beyond purely aesthetic presentation. The algorithm provides a significant advantage over conventional methods, particularly for the visual analysis of ordered, hierarchical, or index-dependent systems.

The visualization of scientific data is a fundamental step in uncovering hidden patterns and relationships within complex systems. While conventional force-directed graph layout algorithms (e.g., spring layout) are effective at displaying the general topology and clustering tendencies of systems, they possess a critical shortcoming: they are inherently non-deterministic and obscure the natural, sequential, or index-based identities of nodes. This poses a significant limitation when analysing systems that contain an intrinsic sequential logic, such as quantum circuits, metabolic pathways, crystal structures, or chronological data structures. This paper introduces the "Keçeci Layout," a deterministic and order-preserving algorithm designed to fill this analytical gap. The algorithm operates by arranging nodes sequentially along a primary axis according to their IDs, employing a predictable zigzag pattern on a secondary axis to prevent node overlaps. This approach establishes a direct and intuitive bridge between a system's abstract data structure (e.g., qubit indices, reaction steps) and its visual representation. To demonstrate the versatility and efficacy of the Keçeci Layout, four case studies from distinct scientific disciplines are presented: Chemistry, Physics, Biology, Quantum Computing, Materials Science, Molecular Biology, Computer Science, etc. In conclusion, the Keçeci Layout is proven to be a powerful scientific tool that offers analytical depth by remaining faithful to the underlying data structure, moving beyond purely aesthetic presentation. The algorithm provides a significant advantage over conventional methods, particularly for the visual analysis of ordered, hierarchical, or index-dependent systems.

The visualization of scientific data is a fundamental step in uncovering hidden patterns and relationships within complex systems. While conventional force-directed graph layout algorithms (e.g., spring layout) are effective at displaying the general topology and clustering tendencies of systems, they possess a critical shortcoming: they are inherently non-deterministic and obscure the natural, sequential, or index-based identities of nodes. This poses a significant limitation when analysing systems that contain an intrinsic sequential logic, such as quantum circuits, metabolic pathways, crystal structures, or chronological data structures. This paper introduces the "Keçeci Layout," a deterministic and order-preserving algorithm designed to fill this analytical gap. The algorithm operates by arranging nodes sequentially along a primary axis according to their IDs, employing a predictable zigzag pattern on a secondary axis to prevent node overlaps. This approach establishes a direct and intuitive bridge between a system's abstract data structure (e.g., qubit indices, reaction steps) and its visual representation. To demonstrate the versatility and efficacy of the Keçeci Layout, four case studies from distinct scientific disciplines are presented: Chemistry, Physics, Biology, Quantum Computing, Materials Science, Molecular Biology, Computer Science, etc. In conclusion, the Keçeci Layout is proven to be a powerful scientific tool that offers analytical depth by remaining faithful to the underlying data structure, moving beyond purely aesthetic presentation. The algorithm provides a significant advantage over conventional methods, particularly for the visual analysis of ordered, hierarchical, or index-dependent systems.

The study of evolution of networks has received increased interest with the recent discovery that many real-world networks possess many things in common, in particular the manner of evolution of such networks. By adding a dimension of time to graph analysis, evolving graphs present opportunities and challenges to extract valuable information. This paper introduces the Evolving Graph Markup Language (EGML), an XML application for representing evolving graphs and related results. Along with EGML, a software tool is provided for the study of evolving graphs. New evolving graph drawing techniques based on the force-directed graph layout algorithm are also explored. Our evolving graph techniques reduce vertex movements between graph instances, so that an evolving graph can be viewed with smooth transitions.

When exploiting the power of node-link diagrams to represent real-world data such as web structures, airline routes, electrical, telecommunication and social networks, link congestion frequently arises. Such areas in the diagram-with dense, overlapping links-are not readable: connectivity, node shapes, labels, and contextual information are obscured. In response, graph-layout research has begun to consider the modification of link shapes with techniques such as link routing and bundling. In this paper, we delve into the interactive techniques afforded by variant use of link curvature, delineating a six-dimensional design space that is populated by four families of interactive techniques: bundling, fanning, magnets, and legends. Our taxonomy encompasses existing techniques and reveals several novel link interactions. We describe the implementation of these techniques and illustrate their potential for exploring dense graphs with multiple types of links.

Background: The advent of "omics" science has brought new perspectives in contemporary biology through the high-throughput analyses of molecular interactions, providing new clues in protein/gene function and in the organization of biological pathways. Biomolecular interaction networks, or graphs, are simple abstract representations where the components of a cell (e.g. proteins, metabolites etc.) are represented by nodes and their interactions are represented by edges. An appropriate visualization of data is crucial for understanding such networks, since pathways are related to functions that occur in specific regions of the cell. The force-directed layout is an important and widely used technique to draw networks according to their topologies. Placing the networks into cellular compartments helps to quickly identify where network elements are located and, more specifically, concentrated. Currently, only a few tools provide the capability of visually organizing networks by cellular compartments. Most of them cannot handle large and dense networks. Even for small networks with hundreds of nodes the available tools are not able to reposition the network while the user is interacting, limiting the visual exploration capability. Results: Here we propose CellNetVis, a web tool to easily display biological networks in a cell diagram employing a constrained force-directed layout algorithm. The tool is freely available and open-source. It was originally designed for networks generated by the Integrated Interactome System and can be used with networks from others databases, like InnateDB. Conclusions: CellNetVis has demonstrated to be applicable for dynamic investigation of complex networks over a consistent representation of a cell on the Web, with capabilities not matched elsewhere.

In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.

In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.

Keçeci Layout is a deterministic node layout algorithm designed for graph visualization in Python. Its primary purpose is to position the nodes of a graph in a predefined, sequential, and repeatable manner. The algorithm processes nodes sequentially, placing them along a user-defined primary axis (e.g., top-down or left-to-right) while applying an offset on the secondary axis in a zigzag pattern. This zigzag pattern helps prevent node overlaps while maintaining an orderly structure. The function is designed to be compatible with popular Python graph libraries, including NetworkX, Rustworkx, igraph, Networkit, and Graphillion (via GraphSet objects). It takes a graph object from one of these libraries as input, processes the nodes (usually by sorting their IDs), and returns a Python dictionary mapping each node's identifier (in the library-specific format) to its calculated (x, y) coordinates. Users can customize the node spacing along the primary and secondary axes (`primary_spacing`, `secondary_spacing`), the main direction of the layout (`primary_direction`), and the starting side of the zigzag pattern (`secondary_start`) through parameters. This regular and predictable structure is useful, particularly when the order of nodes is significant or when a simple, aesthetically pleasing, and easily traceable graph visualization is desired. Keçeci Layout is a deterministic node layout algorithm designed for graph visualization in Python. Its primary purpose is to position the nodes of a graph in a predefined, sequential, and repeatable manner. The algorithm processes nodes sequentially, placing them along a user-defined primary axis (e.g., top-down or left-to-right) while applying an offset on the secondary axis in a zigzag pattern. This zigzag pattern helps prevent node overlaps while maintaining an orderly structure. The function is designed to be compatible with popular Python graph libraries, including NetworkX, Rustworkx, igraph, Networkit, and Graphillion (via GraphSet objects). It takes a graph object from one of these libraries as input, processes the nodes (usually by sorting their IDs), and returns a Python dictionary mapping each node's identifier (in the library-specific format) to its calculated (x, y) coordinates. Users can customize the node spacing along the primary and secondary axes (`primary_spacing`, `secondary_spacing`), the main direction of the layout (`primary_direction`), and the starting side of the zigzag pattern (`secondary_start`) through parameters. This regular and predictable structure is useful, particularly when the order of nodes is significant or when a simple, aesthetically pleasing, and easily traceable graph visualization is desired.

The page number problem is to determine the minimum number of pages in a book in which a graph G can be embedded with the vertices placed in a sequence along the spine and the edges on the pages of the book such that no two edges cross each other in any drawing. In this paper we have (a) statistically evaluated five heuristics for ordering vertices on the spine for minimum number of edge crossings with all the edges placed in a single page, (b) statistically evaluated four heuristics for distributing edges on a minimum number of pages with no crossings for a fixed ordering of vertices on the spine and (c) implemented and experimentally evaluated a hybrid evolutionary algorithm (HEA) for solving the pagenumber problem. In accordance with the results of (a) and (b) above, in HEA, placement of vertices on the spine is decided using a random depth first search of the graph and an edge embedding heuristic adapted from Chung et al. is used to distribute the edges on a minimal number of pages. The results of experiments with HEA on selected standard and random graphs show that the algorithm achieves the optimal pagenumber for the standard graphs. HEA performance is also compared with the Genetic Algorithm described by Kapoor et al. It is observed that HEA gives a better solution for most of the graph instances.

We present a new elegant algorithm for layout of biological signaling pathways. It uses a force-directed layout scheme, taking into account directional and regional constraints enforced by different molecular interaction types and subcellular locations in a cell. The algorithm has been successfully implemented as part of a pathway integration and analysis toolkit named Patika, and results with respect to computational complexity and quality of the layout have been found satisfactory. The structure of pathway graphs highly depend on the type of the pathways and the ontology used to represent the biological phenomenon. We assume the basics

Visualization is crucial to the effective analysis of biological pathways. A poorly laid out pathway confuses the user, while a well laid out one improves the userÕs comprehension of the underlying biological phenomenon. We present a new, elegant algorithm for layout of biological signaling pathways. Our algorithm uses a force-directed layout scheme, taking into account directional and rectangular regional constraints enforced by different molecular interaction types and subcellular locations in a cell. The algorithm has been successfully implemented as part of a pathway visualization and analysis toolkit named PATIKA PATIKA, and results with respect to computational complexity and quality of the layout have been found satisfactory. The algorithm may be easily adapted to be used in other applications with similar conventions and constraints as well. PATIKA PATIKA version 1.0 beta is available upon request at .

A typical data-driven visualization of electroencephalography (EEG) coherence is a graph layout, with vertices representing electrodes and edges representing significant coherences between electrode signals. A drawback of this layout is its visual clutter for multichannel EEG. To reduce clutter, we define a functional unit (FU) as a data-driven region of interest (ROI). An FU is a spatially connected set of electrodes recording pairwise significantly coherent signals, represented in the coherence graph by a spatially connected clique. Earlier we presented two methods to detect FUs, a maximal clique based (MCB) method (time complexity O(3 n/3 ), with n the number of vertices) and a more efficient watershed based (WB) method (time complexity O(n 2 log n) ). To reduce the potential over-segmentation of the WB method, we introduce an improved watershed based (IWB) method (time complexity O(n 2 log n)). The IWB method merges basins representing FUs during the segmentation if they are spatially connected and if their union is a clique. The WB and IWB method both are up to a factor of 100,000 faster than the MCB method for a typical multichannel setting with 128 EEG channels, thus making interactive visualization of multichannel EEG coherence possible. Results show that, considering the MCB method as the gold standard, the difference between IWB and MCB FU maps is smaller than between WB and MCB FU maps. We also introduce two novel group maps for data-driven group analysis as extensions of the IWB method. First, the group mean coherence map preserves dominant features from a collection of individual FU maps. Second, the group FU size map visualizes the average FU size per electrode across a collection of individual FU maps. Finally, we employ an extensive case study to evaluate the IWB FU map and the two new group maps for data-driven group analysis. Results, in accordance with conventional findings, indicate differences in EEG coherence between younger and older adults. However, they also suggest that an initial selection of hypothesis-driven ROIs could be extended with additional datadriven ROIs.

Architectural design is a highly complex practice that involves a wide diversity of disciplines, technologies, proprietary design software, expertise, and an almost infinite number of constraints, across a vast array of design tasks. Enabling intuitive, accessible, and scalable design processes is an important step towards performance-driven and sustainable design for all. To that end, we introduce Architext, a novel semantic generation assistive tool. Architext enables design generation with only natural language prompts, given to large-scale Language Models, as input. We conduct a thorough quantitative evaluation of Architext's downstream task performance, focusing on semantic accuracy and diversity for a number of pre-trained language models ranging from 120 million to 6 billion parameters. Architext models are able to learn the specific design task, generating valid residential layouts at a near 100% rate. Accuracy shows great improvement when scaling the models, with the largest model (GPT-J) yielding impressive accuracy ranging between 25% to over 80% for different prompt categories. We open source the finetuned Architext models and our synthetic dataset, hoping to inspire experimentation in this exciting area of design research.

In this paper, we present GPLAN, software aimed at constructing dimensioned floorplan layouts based on graph-theoretical and optimization techniques. GPLAN takes user requirements as input in the following two forms: i. Adjacency graph: It allows user to draw an adjacency graph on a GUI(graphical user interface) corresponding to which GPLAN produces a set of dimensioned floorplans with a rectangular boundary, where each floorplan is topologically distinct from others. ii. Dimensionless layout: Here, user can draw any layout with rectangular or non-rectangular boundary on a GUI and GPLAN transforms it into a dimensioned floorplan while preserving adjacencies, positions, shapes of the rooms. The above approaches represent different ways of inserting adjacencies and GPLAN generate dimensioned floorplans corresponding to the given adjacencies. The larger aim is to provide alternative platforms to user for producing dimensioned floorplans for all given (architectural) constraints, which ...

Social network analysis software such as NodeXL has been used to describe participation and interaction in numerous social networks, but it has not yet been widely used to examine dynamics in online classes, where participation is frequently required ...

Innovative improvements in the area of human-computer interaction and user interfaces have enabled intuitive and effective applications for a variety of problems. On the other hand, there has also been the realization that several real-world optimization problems still cannot be totally automated. Very often, user interaction is necessary for refining the optimization problem, managing the computational resources available, or validating or adjusting a computer-generated solution. This paper presents an interactive framework called user hints for having humans help optimization methods to solve difficult problems. In the framework users play a dynamic and important role by providing hints. Hints are actions that help to insert domain knowledge, to escape from local minima, to reduce the space of solutions to be explored, or to avoid ambiguity when there is more than one optimal solution. User hints are given in an intuitive way through a graphical interface. Visualization tools are also included in order to inform the user about the state of the optimization process. We discuss applications of the user hints framework to the graph drawing and the map labeling problems. An evaluation of some user hints systems indicates that optimization processes can benefit from human interaction.

Multimodal models that combine different media like text,  image, audio, and graph have revolutionised the architectural design process, which could provide automated solutions to assist the architects during the early design stages. Recent studies use Graph Neural Networks (GNNs) to learn topological information and Convolution Neural Networks (CNNs) to learn spatial information from floorplans. This paper proposes a deep learning multimodal model incorporating GNNs and the Stable Diffusion model to learn the floorplan's topological and spatial information. The authors trained a Stable Diffusion model on samples from the RPLAN dataset. They used graph embedding for conditional generation and experimented with three approaches to whole-graph embedding techniques. The proposed Stable Diffusion model maps the user input, a graph representing the room types and their relationships, to the output, the predicted floorplans, as a raster image. The Graph2Vec and contrastive learning methods generate superior representational capabilities and yield good and comparable results in both computationally derived scores and evaluations conducted by human assessors, compared to the Graph Encoder-CNN Decoder.

We describe a package of practical tools and libraries for manipulating graphs and their drawings. Our design, which aimed at facilitating the combination of the package components with other tools, includes stream and event interfaces for graph operations, high-quality static and dynamic layout algorithms, and the ability to handle sizable graphs. We conclude with a description of the applications of this package to a variety of software engineering tools.

Social network analysis uses techniques from graph theory to analyze the structure of relationships among social actors such as individuals or groups. We investigate the effect of the layout of a social network on the inferences drawn by observers about the number of social groupings evident and the centrality of various actors in the network. We conducted an experiment in which eighty subjects provided answers about three drawings. The subjects were not told that the drawings were chosen from five different layouts of the same graph. We found that the layout has a significant effect on their inferences and present some initial results about the way certain Euclidean features will affect perceptions of structural features of the network. There is no "best" layout for a social network; when layouts are designed one must take into account the most important features of the network to be presented as well as the network itself.

This paper presents strategies for speeding up calculation of graph metrics and layout by exploiting the parallel architecture of modern day Graphics Processing Units (GPU), specifically Compute Unified Device Architecture (CUDA) by Nvidia. Graph centrality metrics like Eigenvector, Betweenness, Page Rank and layout algorithms like Fruchterman-Rheingold are essential components of Social Network Analysis (SNA). With the growth in adoption of SNA in different domains and increasing availability of huge networked datasets for analysis, social network analysts require faster tools that are also scalable. Our results, using NodeXL, show up to 802 times speedup for a Fruchterman-Rheingold graph layout and up to 17,972 times speedup for Eigenvector centrality metric calculations on a 240 core CUDA-capable GPU.

The visualization of complex conceptual structures is a key component of support tools for many applications in science and engineering. A graph is an abstract data structure that is used to model information. Hence, many information visualization systems require graphs to be drawn so that they are easy to read and understand. Also many practical applications like VLSI design need the drawing of a graph in some specific way. In this paper, we address the problem of drawing and visualization of graphs using various algorithms by presenting an integrated software environment for both the design and the visualization of graphs.

The accuracy of mathematical models and numerical methods are of significant importance to obtain reliable simulation results. Therefore, employing a mesh for a partial differential equation (PDE) simulation which results in less analysis error is desirable for such modeling. Quadrilateral meshes often contain fewer elements than triangular meshes, and thus result in more computationally efficient PDE simulations. Similarly, their associated PDE analyses are typically more accurate [1]. Furthermore, quadrilateral meshes are preferred in dynamic simulations such as car crashes or dynamic fracture studies. The motion of the mesh elements in such computational studies results in tangled meshes which makes mesh untangling and quality improvement methods important topics of study. Mesh tangling occurs when mesh elements overlap. In order to address this issue, various untangling methods have been introduced in the literature. Following the untangling phase, mesh quality improvement is typically applied to the resulting mesh in order to improve its quality. A common theme for the previous methods is the application of these two processes as separate steps rather than applying both in one step. This usually results in greater computational expense.

Particle advection is the approach for extraction of integral curves from vector fields. Efficient parallelization of particle advection is a challenging task due to the problem of load imbalance, in which processes are assigned unequal workloads, causing some of them to idle as the others are performing compute. Various approaches to load balancing exist, yet they all involve trade-offs such as increased inter-process communication, or the need for central control structures. In this work, we present two local load balancing methods for particle advection based on the family of diffusive load balancing. Each process has access to the blocks of its neighboring processes, which enables dynamic sharing of the particles based on a metric defined by the workload of the neighborhood. The approaches are assessed in terms of strong and weak scaling as well as load imbalance. We show that the methods reduce the total run-time of advection and are promising with regard to scaling as they operate locally on isolated process neighborhoods.

McDiarmid, C. and Z. Miller, Lattice bandwidth of random graphs, Discrete Applied Mathematics 30 (1991) 221-227. The bandwidth of a random graph has been well studied. A natural generalization of bandwidth involves replacing the path as host graph by a multi-dimensional lattice. In this paper we investigate the corresponding behavior for random graphs.

Graph drawing is an information visualization technology for illustrating relations between objects. Interactive graph drawing is often important since it is difficult to statically lay out complex graphs. For the interactive drawing of general undirected graphs, we have proposed the high-dimensional approach, which uses static graph layouts in high-dimensional spaces to dynamically find twodimensional layouts according to user interaction. Although the resulting interactive graph drawing method was found to be fast, other properties of it are not yet clear. In this paper, we analyze the high-dimensional approach to further explore its properties. Specifically, we perform the following two kinds of its analysis: (1) sensitivity analysis for investigating how the high-dimensional approach places graph nodes on the two-dimensional plane; (2) empirical analysis for examining the appropriateness of underlying graph layout methods. The results show that, as an underlying graph layout method, Kruskal and Seery's method based on Torgerson's multidimensional scaling method is more appropriate for the high-dimensional approach than other methods for computing graph layouts in high-dimensional spaces.

Despite extensive research, it is still difficult to produce effective interactive layouts for large graphs. Dense layout and occlusion make food webs, ontologies, and social networks difficult to understand and interact with. We propose a new interactive Visual Analytics component called TreePlus that is based on a tree-style layout. TreePlus reveals the missing graph structure with visualization and interaction while maintaining good readability. To support exploration of the local structure of the graph and gathering of information from the extensive reading of labels, we use a guiding metaphor of "Plant a seed and watch it grow." It allows users to start with a node and expand the graph as needed, which complements the classic overview techniques that can be effective at (but often limited to) revealing clusters. We describe our design goals, describe the interface, and report on a controlled user study with 28 participants comparing TreePlus with a traditional graph interface for six tasks. In general, the advantage of TreePlus over the traditional interface increased as the density of the displayed data increased. Participants also reported higher levels of confidence in their answers with TreePlus and most of them preferred TreePlus.

In the information era, people’s lives are deeply impacted by IT via exposure to social media, emails, RSS feed, chats, web pages, etc. Such data is considered very valuable nowadays since it may help companies to better their strategies. For example, companies can analyse their customers’ trends or their competitors marketing interventions and adjust their strategies accordingly. Several decisional tools have been developed but most of them rely on relational databases. This makes it difficult for decision makers to take advantage of unstructured data which today represents more than 85% of the available data. Thus, there is a rising need for a suitable management process of unstructured data through collecting, managing, transferring and transforming it into a meaningful informed data. This paper will introduce a new tool for Big Unstructured Data for the Competitive Intelligence named Xplor EveryWhere (XEW). It will also describe the enhancement brought to its newest feature XEWG...

We demonstrate the Temporal Graph Explorer, a distributed opensource framework that enables time-dependent graph exploration and analysis on large real-world networks using a rich temporal property graph model and dynamic graph operators. In this demonstration we show how the evolution of a graph plays an essential role in many analytical questions. Besides retrieving a snapshot from a past graph state or calculating the difference between two graph snapshots, users can use our application to summarize the graph to reduce its complexity and to obtain deeper insights into its evolution. Visitors of the demo can visually experience these advanced temporal operators and their results or build and manipulate analytical programs in a declarative way without extended programming skills and run them on a distributed local or remote environment. We provide realworld temporal graph data from bicycle rentals of New York City with millions of rentals per month, also demonstrating horizontal sc...

The lab has been influential with contributions in algorithms, interaction techniques and experimental results in Network Visualisation, Interactive Optimisation and Geographic and Cartographic visualisation. It has also been a leader in the emerging topic of Immersive Analytics, which explores natural interactions and immersive display technologies in support of data analytics. We reflect on advances in these areas but also sketch our vision for future research and developments in data visualisation more broadly.

The lab has been influential with contributions in algorithms, interaction techniques and experimental results in Network Visualisation, Interactive Optimisation and Geographic and Cartographic visualisation. It has also been a leader in the emerging topic of Immersive Analytics, which explores natural interactions and immersive display technologies in support of data analytics. We reflect on advances in these areas but also sketch our vision for future research and developments in data visualisation more broadly.

Constellation is a visualization system for the results of queries from the MindNet natural language semantic network. Constellation is targeted at helping MindNet's creators and users refine their algorithms, as opposed to understanding the structure of language. We designed a special-purpose graph layout algorithm which exploits higher-level structure in addition to the basic node and edge connectivity. Our layout prioritizes the creation of a semantic space to encode plausibility instead of traditional graph drawing metrics like minimizing edge crossings. We make careful use of several perceptual channels both to minimize the visual impact of edge crossings and to emphasize highlighted constellations of nodes and edges.

One of the main issues in colour image processing is changing objects' colour due to colour of illumination source. Colour constancy methods tend to modify overall image colour as if it was captured under natural light illumination. Without colour constancy, the colour would be an unreliable cue to object identity. Till now, many methods in colour constancy domain are presented. They are in two categories; statistical methods and learning-based methods. This paper presents a new statistical weighted algorithm for illuminant estimation. Weights are adjusted to highlight two key factors in the image for illuminant estimation, that is contrast and brightness. The focus was on the convex part of the contrast stretching function to create the weights. Moreover, a novel partitioning mechanism in the colour domain that leads to improvement in efficiency is proposed. The proposed algorithm is evaluated on two benchmark linear image databases according to two evaluation metrics. The experimental results showed that it is competitive to the statistical state of the art methods. In addition to its low computational cost, it has the advantage of improving the efficiency of statistics-based algorithms for dark images and images with low brightness contrast. Moreover, it is robust to camera change types. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.