Equivalent Graph Research Articles

Intersection graph and writhe polynomial

We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.

PolyJuice: Detecting Mis-compilation Bugs in Tensor Compilers with Equality Saturation Based Rewriting

Tensor compilers are essential for deploying deep learning applications across various hardware platforms. While powerful, they are inherently complex and present significant challenges in ensuring correctness. This paper introduces PolyJuice, an automatic detection tool for identifying mis-compilation bugs in tensor compilers. Its basic idea is to construct semantically-equivalent computation graphs to validate the correctness of tensor compilers. The main challenge is to construct equivalent graphs capable of efficiently exploring the diverse optimization logic during compilation. We approach it from two dimensions. First, we propose arithmetic and structural equivalent rewrite rules to modify the dataflow of a tensor program. Second, we design an efficient equality saturation based rewriting framework to identify the most simplified and the most complex equivalent computation graphs for an input graph. After that, the outcome computation graphs have different dataflow and will likely experience different optimization processes during compilation. We applied it to five well-tested industrial tensor compilers, namely PyTorch Inductor, OnnxRuntime, TVM, TensorRT, and XLA, as well as two well-maintained academic tensor compilers, EinNet and Hidet. In total, PolyJuice detected 84 non-crash mis-compilation bugs, out of which 49 were confirmed with 20 fixed.

FlowCert: Translation Validation for Asynchronous Dataflow via Dynamic Fractional Permissions

Coarse-grained reconfigurable arrays (CGRAs) have gained attention in recent years due to their promising power efficiency compared to traditional von Neumann architectures. To program these architectures using ordinary languages such as C, a dataflow compiler must transform the original sequential, imperative program into an equivalent dataflow graph, composed of dataflow operators running in parallel. This transformation is challenging since the asynchronous nature of dataflow graphs allows out-of-order execution of operators, leading to behaviors not present in the original imperative programs. We address this challenge by developing a translation validation technique for dataflow compilers to ensure that the dataflow program has the same behavior as the original imperative program on all possible inputs and schedules of execution. We apply this method to a state-of-the-art dataflow compiler targeting the RipTide CGRA architecture. Our tool uncovers 8 compiler bugs where the compiler outputs incorrect dataflow graphs, including a data race that is otherwise hard to discover via testing. After repairing these bugs, our tool verifies the correct compilation of all programs in the RipTide benchmark suite.

A Low-Level Look at A-Normal Form

A-normal form (ANF) is a widely studied intermediate form in which local control and data flow is made explicit in syntax, and a normal form in which many programs with equivalent control-flow graphs have a single normal syntactic representation. However, ANF is difficult to implement effectively and, as we formalize, difficult to extend with new lexically scoped constructs such as scoped region-based allocation. The problem, as has often been observed, is that normalization of commuting conversions is hard. This traditional view of ANF that normalizing commuting conversions is hard, found in formal models and informed by high-level calculi, is wrong. By studying the low-level intensional aspects of ANF, we can derive a normal form in which normalizing commuting conversion is easy, does not require join points, or code duplication, or renormalization after inlining, and is easily extended with new lexically scoped effects. We formalize the connection between ANF and monadic form and their intensional properties, derive an imperative ANF, and design a compiler pipeline from an untyped -calculus with scoped regions, to monadic form, to a low-level imperative monadic form in which A-normalization is trivial and safe for regions. We prove that any such compiler preserves, or optimizes, stack and memory behaviour compared to ANF. Our formalization reconstructs and systematizes pragmatic choices found in practice, including current production-ready compilers. The main take-away from this work is that, in general, monadic form should be preferred over ANF, and A-normalization should only be done in a low-level imperative intermediate form. This maximizes the advantages of each form, and avoids all the standard problems with ANF.

The ultrametric backbone is the union of all minimum spanning forests

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all shortest paths and De Morgan's law consistency.

Dominant eigenvalue and universal winners of digraphs

Let A be a nonnegative irreducible matrix of order n and Ai(t) be the matrix obtained by increasing its ith diagonal entry by a positive number t. An index i∈[n]={1,…,n} is called a universal winner for A, if, letting ρ(⋅) denote the spectral radius, ρ(Ai(t))≥ρ(Aj(t)) for j∈[n] and for t∈(0,∞).Let G be a strongly connected digraph with vertex set [n]. Let WG be the set of all nonnegative matrices whose underlying graph structure is G. We say a vertex u structurally dominates another vertex v in G, if ρ(Au(t))≥ρ(Av(t)) for all t∈(0,∞) and for all A∈WG. We characterize the class of digraphs G that do not have a vertex that structurally dominates all other vertices. We say two vertices u and v structurally tie if they structurally dominate each other in G. We supply an equivalent graph theoretic condition for the structural tying of two vertices in G. Let S⊆[n] be nonempty. We characterize the class of strongly connected digraphs G with vertex set [n] such that S is the set of universal winners for each A∈WG. We also characterize the strongly connected digraphs whose vertex set can be partitioned into subsets P1,P2,…,Pk such that vertices inside a part Pi structurally tie with each other and vertices of Pi structurally dominate vertices of Pj strictly (without tying) for i<j.

Foundation of a new technique for geometric and non-geometric multi-shapes similarities degrees using boundary unfolding transformation with applications

This paper presents a novel approach for measuring multi-shapes degrees of similarities. A new shape transformation concept is suggested through mapping the closed boundary of the shape using unfolded process into one-to-one equivalent graph (or signal) where ample number of approaches for similarity testing can be applied. The degree of similarity is then measured by calculating the discrepancies between each of the two transformed unfolded-graphs autocorrelation functions. The proposed method is also effective in applying clustering technique for clustering groups of shapes into three hierarchical similarity levels depending on their mutual degree of similarity. The suggested similarity method handles two categories of shapes: geometric and non-geometric. Unlike other well-known techniques in the subject of machine learning and deep learning, the presented method represents a powerful analytical-based technique for similarity analysis of both regular and irregular shapes regardless of size or orientation, and in effectively handling compound or multi-shapes degrees of similarities. The results of testing the approach on selected datasets have successfully demonstrated the great effectiveness of the developed technique in yielding output multi-shape degrees of similarities in new graphical and comprehensive forms. Finally, it is pointed out that the new approach will have many potential real-life applications such as in industry, medicine, biology, security, and authentications.

Compilation of algorithm-specific graph states for quantum circuits

We present a quantum circuit compiler that prepares an algorithm-specific graph state from quantum circuits described in high level languages, such as Cirq and Q#. The computation can then be implemented using a series of non-Pauli measurements on this graph state. By compiling the graph state directly instead of starting with a standard lattice cluster state and preparing it over the course of the computation, we are able to better understand the resource costs involved and eliminate wasteful Pauli measurements on the actual quantum device. Access to this algorithm-specific graph state also allows for optimisation over locally equivalent graph states to implement the same quantum circuit. The compiler presented here finds ready application in measurement based quantum computing, NISQ devices and logical level compilation for fault tolerant implementations.

When Science Meets Art on Instagram: Examining the Effects of Visual Art on Emotions, Interest, and Social Media Engagement

While artistic representations of scientific subjects, phenomena, and data have gained traction on social media, their effects on audience engagement remain understudied. Using an experiment with Instagram users in the United States ( N = 655), this study found that exposure to artistic representations of COVID-19 information on Instagram significantly increased individuals’ interest and positive emotions compared with exposure to informationally equivalent data graphs. Positive emotions also mediated the relationship of art exposure on one’s liking and sharing of Instagram posts. We discuss the potential roles of visual art and social media in shaping public engagement with science.

Hamiltonian-path based constraint reduction for deployable polyhedral mechanisms

Most of the deployable polyhedral mechanisms (DPMs) are multi-loop overconstrained mechanisms that causes barriers for their applications due to the issues in assembly, operation and control. Yet, constraint reduction for these multi-loop overconstrained mechanisms is extremely challenging in kinematics. In this paper, by introducing the Hamiltonian path to investigate the 3D topological graphs of a group of Sarrus-inspired DPMs, we propose a systematic method for constraint reduction of multi-loop overconstrained DPMs. We demonstrate that through the removal of redundant joints with the assistant of tetrahedral Hamiltonian path, one equivalent simplest topological graph of tetrahedral mechanism is identified as a reduction basic unit. Subsequently, one simplest form of Sarrus-inspired cubic mechanism is obtained by investigating two Hamiltonian paths of its dual octahedron and sequentially arranging basic units. Furthermore, a total of nineteen simplest forms of Sarrus-inspired dodecahedral mechanisms are identified from seventeen Hamiltonian paths of its dual icosahedron. The overconstraints in each simplest Sarrus-inspired DPM are greatly reduced while preserving the original one-degree-of-freedom (DOF) motion behavior. The method proposed in this paper not only lays the groundwork for further research in wider deployable polyhedrons, but also inspires the reduction of other multi-loop overconstrained mechanisms, with potential applications in the fields of manufacturing, architecture and space exploration.

Hypergraph reconstruction from uncertain pairwise observations

The network reconstruction task aims to estimate a complex system’s structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities—the pairwise case. Here, using Bayesian inference, we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.

A probabilistic model for networks generated by actors’ characteristics

This work presents the affinity network model for random graphs, consisting of a broad family of random graph models depending on some parameters. In this model, we suppose that each individual randomly chooses a set of characteristics that represent him according to a certain probability measure. The connections between two individuals depend on their shared characteristics and are valued according to a function that measures what we call affinity in the network. According to the choice of this function, the network’s density can vary from sparse to complete graph, causing the model to be very flexible, which makes it suitable to fit with real networks. To illustrate the behavior of the affinity network model, we present a Monte Carlo simulation study. We tune the model’s generating parameters, analyze its topological measurements, and compare them with equivalent graphs with edges occurring independently, disregarding actors’ characteristics.

Dessins D’Enfants, Brauer Graph Algebras and Galois Invariants

In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d’enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d’enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced action of the absolute Galois group.

Zagreb Connection Indices for Metal Trihalides

Abstract The progression of the 2D novel, metal trihalides MX 3 has been learned in Dirac half-metallicity topological spintronic possessions because of their interest that are inherited magnetic processes reflected by 2D metal trihalides. Additionally, the effects of the orbital pair create a big relative spin. Furthermore, because of spin-orbital coupling joined with force and high bend temperature, 2D materials like Bil 3 discover significant applications in room temperature radiation indicators bringing about an unfamiliar band gap. The Zagreb indices are widely considered T-indices. We study the second-degree-based Zagreb connection indices that are equivalent graph invariants. Our purpose is to calculate the connection indices of metal trihalides. Ultimately, we make a comparison of these indices for their expected values and give a graphical representation of metal trihalides using numerical tables and 3D plots.

Artistic representations of data can help bridge the US political divide over climate change

Visual art has been used to revamp the portrayal of climate change with the aims of engaging emotions and expanding nonexperts’ psychological capacity to perceive its relevance. However, empirical evidence supporting the effectiveness of artistic representation of data as a tool for public communication is lacking. Using controlled experiments with two national samples of U.S. adults (total N = 671), here we found that artistic visualizations elicited stronger positive emotions than informationally equivalent data graphs but did not differ in their perceived credibility or effectiveness as visual aids for learning. When used to prompt individual reflection, artistic visualizations appeared to mitigate the political division in viewers’ perceived relevance of climate change that could otherwise be exacerbated by exposure to data graphs.

Design and Modeling Framework for DexTeR: Dexterous Continuum Tensegrity Manipulator

AbstractThe field of tensegrity faces challenges in design to facilitate efficient fabrication, and modeling due to the antagonistic nature of tension and compression elements. The research presents design methodology, and modeling framework for a human-spine inspired Dexterous continuum Tensegrity manipulatoR (DexTeR). DexTeR is a continuum manipulator that comprises of an assembly of “vertebra” modules fabricated using two curved links and 12 strings, and actuated using motor-tendon actuators. The fabrication methodology involves the construction of the equivalent graph of the module and finding the Euler path that traverses every edge of the graph exactly once. The vertices and edges of the graph correspond to the holes and strings or links of the mechanism. Unlike traditional rigid manipulators, the design results in centralization of the majority of the weight of the actuators at the base with negligible effect on the manipulator dynamics. For the first time in literature, we fabricate a tensegrity manipulator that is assembled using ten modules to conceptually validate the time and cost efficiency of the approach. A dynamic model of a vertebra module is presented using the Euler–Newton approach with screw theory representation. Each rigid link is represented using a screw, a six-dimensional vector with components of angular rotation, and linear translation. The nonlinearity in the system arises from the discontinuous behavior of the strings and the “closed-chain” nature of the mechanism. The behavior of the strings is piece-wise continuous to model their slack, compliant, or tension states.

Fragment-based deep molecular generation using hierarchical chemical graph representation and multi-resolution graph variational autoencoder.

Graph generative models have recently emerged as an interesting approach to construct molecular structures atom-by-atom or fragment-by-fragment. In this study, we adopt the fragment-based strategy and decompose each input molecule into a set of small chemical fragments. In drug discovery, a few drug molecules are designed by replacing certain chemical substituents with their bioisosteres or alternative chemical moieties. This inspires us to group decomposed fragments into different fragment clusters according to their local structural environment around bond-breaking positions. In this way, an input structure can be transformed into an equivalent three-layer graph, in which individual atoms, decomposed fragments, or obtained fragment clusters act as graph nodes at each corresponding layer. We further implement a prototype model, named multi-resolution graph variational autoencoder (MRGVAE), to learn embeddings of constituted nodes at each layer in a fine-to-coarse order. Our decoder adopts a similar but conversely hierarchical structure. It first predicts the next possible fragment cluster, then samples an exact fragment structure out of the determined fragment cluster, and sequentially attaches it to the preceding chemical moiety. Our proposed approach demonstrates comparatively good performance in molecular evaluation metrics compared with several other graph-based molecular generative models. The introduction of the additional fragment cluster graph layer will hopefully increase the odds of assembling new chemical moieties absent in the original training set and enhance their structural diversity. We hope that our prototyping work will inspire more creative research to explore the possibility of incorporating different kinds of chemical domain knowledge into a similar multi-resolution neural network architecture.

A novel design methodology for fixed-shaft hybrid powertrain with a single-motor

Hybrid power configuration has two kinds of power sources, how to design the configuration to couple the energy of the two power sources is an important part of the development of hybrid power technology. In the systematic design scheme of hybrid power, most scholars focus on the design of planetary gear hybrid power configuration, but lack of systematic design scheme of fixed-shaft hybrid power configuration. In this paper, based on the equivalent tree graph method, the fixed-shaft hybrid configuration is transformed into the corresponding tree graph representation, and all possible configurations are generated by the combination of the trunk, branches and leaves. Add the corresponding design constraints, use Matlab to realize automatic generation and screening, and finally obtain the required configuration after specification. The configuration is verified in the Simulink &amp; Amesim co-simulation environment. The results show that the proposed single-motor fixed-shaft hybrid power configuration has certain advantages in economy compared with the traditional P2 and P3 configurations, and its dynamic performance is the same as that of P2 configurations and superior to P3 configurations.

Secure Networking with Software-Defined Reconfigurable Intelligent Surfaces

Reconfigurable intelligent surfaces (RIS) are considered of paramount importance to improve air-ground and THz communications performance for 6G systems. Recently, RISs were proposed in Physical Layer Security (PLS), as they can (i) improve the secrecy capacity due to the controlled directional reflections' capability of RIS elements and (ii) avoid potential eavesdroppers, redirecting data streams towards the intended users. This paper proposes the integration of a multi-RISs system within a Software Defined Networking (SDN) architecture to provide a specific control layer for secure data flows forwarding. The optimisation problem is properly characterised in terms of an objective function and an equivalent graph theory model is considered to address the optimal solution. Moreover, different heuristics are proposed, trading off complexity and PLS performance, to evaluate the more suitable multi-beam routing strategy. Numerical results are also provided, focusing on a worst case scenario which points out the improvement of the secrecy rate from the increase in the number of eavesdroppers. Furthermore, the security performance is investigated for a specific user mobility pattern in a pedestrian scenario.

Queer dual equivalence graphs

We introduce a new paradigm for proving the Schur $P$-positivity. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur $P$-positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalization with a new application to the product of Schur $P$-functions.