Hypergraph Research Articles

Simplicial network analysis on EEG signals

It is well known that the epileptic signal analysis helps with automating the diagnosis and detection of a seizure, instead of relying on the traditional process of expert - visual inspection, which is both time-consuming and tedious. Recently, application of network analysis has grown as a surmounting approach for the interpretation of signals. In general, network-based signal analysis approach involves the conversion of time series procured at different physiological conditions into networks like hyper graphs, visibility graphs etc. followed by the derivation of network topological properties. In this paper, we make use of newly developed simplicial approach, where the cliques of visibility graphs are considered as simplices and are analyzed to obtain maximal cliques. Employing simplices would help in securing not only global and significant details but also localized and subtle details of dynamical behavior, for a given time series. The maximal cliques are mathematically evaluated to calculate three independent simplicial characterizers and maximum dimensionality, that define the structural anatomy and connectivity of the entire network at different topological levels. The maximum values of all the measures acquired are processed to differentiate normal signals against pathological EEG signals, using support vector machine through 10-fold cross validation. The classification analysis is performed on all the possible normal versus epileptic (ictal and inter-ictal) combinations, using two different EEG databases. In the case of University of Bonn database, the results are compared to that of conventional network parameters namely average degree, Global efficiency, Average path length and Assortativity. The classification results of both databases achieved very good accuracies, indicating that the proposed algorithm is an efficient and reliable approach for detecting epileptic EEG signals.

Weak Saturation of Multipartite Hypergraphs

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weaklyH-saturated in F if the edges in E(F)setminus E(G) admit an ordering e_1,ldots , e_k so that for all iin [k] the hypergraph Gcup {e_1,ldots ,e_i} contains an isomorphic copy of H which in turn contains the edge e_i. The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.

Multi-Modal Knowledge Hypergraph for Diverse Image Retrieval

The task of keyword-based diverse image retrieval has received considerable attention due to its wide demand in real-world scenarios. Existing methods either rely on a multi-stage re-ranking strategy based on human design to diversify results, or extend sub-semantics via an implicit generator, which either relies on manual labor or lacks explainability. To learn more diverse and explainable representations, we capture sub-semantics in an explicit manner by leveraging the multi-modal knowledge graph (MMKG) that contains richer entities and relations. However, the huge domain gap between the off-the-shelf MMKG and retrieval datasets, as well as the semantic gap between images and texts, make the fusion of MMKG difficult. In this paper, we pioneer a degree-free hypergraph solution that models many-to-many relations to address the challenge of heterogeneous sources and heterogeneous modalities. Specifically, a hyperlink-based solution, Multi-Modal Knowledge Hyper Graph (MKHG) is proposed, which bridges heterogeneous data via various hyperlinks to diversify sub-semantics. Among them, a hypergraph construction module first customizes various hyperedges to link the heterogeneous MMKG and retrieval databases. A multi-modal instance bagging module then explicitly selects instances to diversify the semantics. Meanwhile, a diverse concept aggregator flexibly adapts key sub-semantics. Finally, several losses are adopted to optimize the semantic space. Extensive experiments on two real-world datasets have well verified the effectiveness and explainability of our proposed method.

Exact Wirelength of Embedding 3-Ary n-Cubes into Certain Cylinders and Trees

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems, the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of networks and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

K-Planar Placement and Packing of Δ-Regular Caterpillars

This paper studies a packing problem in the so-called beyond-planar setting, that is when the host graph is “almost-planar” in some sense. Precisely, we consider the case that the host graph is [Formula: see text]-planar, i.e., it admits an embedding with at most [Formula: see text] crossings per edge, and focus on families of [Formula: see text]-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree [Formula: see text]. We study the dependency of [Formula: see text] from the number [Formula: see text] of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called placement) and when they are not. We give necessary and sufficient conditions for the placement of [Formula: see text] [Formula: see text]-regular caterpillars and sufficient conditions for the packing of a set of [Formula: see text]-, [Formula: see text]-, [Formula: see text], [Formula: see text]-regular caterpillars such that the degree [Formula: see text] and the degree [Formula: see text] of the non-leaf vertices can differ from one caterpillar to another, for [Formula: see text], [Formula: see text].

Human activity recognition from UAV videos using a novel DMLC-CNN model

Human Activity Recognition (HAR) remains a challenging issue that requires to be resolved. Utilizing images, smart phones, or sensors, HAR could be done. Recent studies have explored HAR utilizing Unmanned Aerial Vehicle (UAV) videos. But, owing to numerous restrictions correlated to the platform, HAR from videos captured by drones remains a challenge. To resolve this issue, a novel mechanism for HAR from UAV videos utilizing the Diminutive Multi-Dimensional Locality Coding based Convolutional Neural Network (DMLC-CNN) model is proposed. Primarily, the input video is converted into frames; also, Bounding Boxes (BBs) are produced for humans. For attaining the number of clusters grounded on human presence, the BB is given to the Robust Lucrative Sensitive Amalgam K-Nearest Neighbor (RoLSA-KNN) clustering. After that, utilizing the LR-fit-centric Superior Photogrammetric Silhouette (LR-SPS) segmentation, the humans' silhouettes are segmented. The attained output is divided into patches; then, it is converted into stacked layer frames. Afterward, these frames are converted into a 3D kernel. The Diminutive Multi-Dimensional Hyper Graph (DMHG) extracted the features; then, the Locality-constrained Orthonormal Feature Space Coding (LOFSC) Algorithm encoded the extracted features. Lastly, for recognizing Human Activity (HA), DMLC-CNN is wielded. The proposed framework is contrasted with the conventional techniques. After experimental evaluation, the proposed mechanism was found to be more efficient in HAR.

Graph rewriting and relabeling with PBPO+: A unifying theory for quasitoposes

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO+, allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO+ can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO+ can define a strict superset of the rewrite relations definable by PBPO, AGREE and DPO. Additionally, we show that PBPO+ is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.

Self-supervised Dual Hypergraph learning with Intent Disentanglement for session-based recommendation

Existing works on session-based recommendation have shown the advantage in enhancing the prediction ability of recommendation with various deep learning techniques. However, the following challenges need to be addressed: (1) the hierarchy of item transition patterns is overlooked; (2) existing works fail to distinguish various factors of item transition within a single session for disentangling user intents. To cope with the above challenges, we propose a novel session based recommendation model called Self-supervised Dual Hypergraph learning with Intent Disentanglement model (SDHID). Specifically, we first propose a disentangled capsule hypergraph convolutional channel for ne-grained intent learning to capture the intra-session pattern. Accordingly, we introduce the hypergraph and capsule networks in disentangling to learn the item embedding for different factors, and then the representation of the intra-session pattern is obtained by aggregating item embedding with attention weights. Moreover, we build a novel dual–primal hypergraph convolutional channel by mapping the hypergraph to a dual–primal graph for learning the item transition pattern of inter-session. In addition, the above two channels are combined into a self-supervised contrastive learning framework by maximizing mutual information between the learned session representations. We unify the recommendation and the self-supervised tasks under a primary and auxiliary learning framework. The combined optimization of two tasks leads to a hierarchical joint learning item transition for intra- and inter-session. Extensive experiments on real datasets show that the proposed model outperforms several state-of-the-art models.

A Research on Cancer’s Recognition and Neutrosophic Super Hypergraph by Eulerian Super Hyper Cycles and Hamiltonian Sets as Hyper Covering Versus Super separations

In this research, assume a Super Hyper Graph. Then some notions on Extreme Super Hyper Cycle based on the backgrounds of Eulerian and Hamiltonian styles are proposed. Some results are directed in the way that, the starting definitions make sense on the motivation and the continuous approaches for this research. A basic familiarity with neutrosophic SuperHyperCycle theory, SuperHyperGraphs, and neutrosophic SuperHyperGraphs theory are proposed.

Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs and Super Hyper Graphs Alongside Applications in Cancer’s Treatments

In this research, new setting is introduced for new notions, namely, Super Hyper Degree and Co-Super Hyper Degree. Two different types of definitions are debut for them but the research goes further and the Super Hyper Notion, Super Hyper Uniform, and Super Hyper Class based on that are well-defined and well-reviewed. The literature review is implemented in the whole of this research. For shining the elegancy and the significancy of this research, the comparison between this Super Hyper Notion with other Super Hyper Notions and fundamental Super Hyper Numbers are featured. The definitions are followed by the examples and the instances thus the clarifications are driven with different tools. The applications are figured out to make sense about the theoretical aspect of this ongoing research. The cancer’s treatments are the under research to figure out the challenges make sense about ongoing and upcoming research. The special case is up. The cells are viewed in the deemed ways. There are different types of them. Some of them are individuals and some of them are well-modeled by the group of cells. These types are all officially called “Super Hyper Vertex” but the relations amid them all officially called “Super Hyper Edge”. The frameworks “Super Hyper Graph” and “neutrosophic Super Hyper Graph” are chosen and elected to research about “cancer’s treatments”. Thus, these complex and dense Super Hyper Models open up some avenues to research on theoretical segments and “cancer’s treatments”. Some avenues are posed to pursue this research. It’s also officially collected in the form of some questions and some problems. If there’s a Super Hyper Edge between two Super Hyper Vertices, then these two Super Hyper Vertices are called Super Hyper Neighbors. The number of Super Hyper Neighbors for a given Super Hyper Vertex is called Super Hyper Degree. The number of common Super Hyper Neighbors for some Super Hyper Vertices is called Co-Super Hyper Degree for them and used Super Hyper Vertices are called Co-Super Hyper Neighbors. A graph is Super Hyper Uniform if it’s Super Hyper Graph and the number of elements of Super Hyper Edges are the same. Assume a neutrosophic Super Hyper Graph. There are some Super Hyper Classes as follows. It’s SuperHyperPath if it’s only one SuperVertex as intersection amid two given Super Hyper Edges with two exceptions; it’s SuperHyperCycle if it’s only one SuperVertex as intersection amid two given Super Hyper Edges; it’s Super Hyper Star it’s only one SuperVertex as intersection amid all Super Hyper Edges; it’s SuperHyperBipartite it’s only one SuperVertex as intersection amid two given Super Hyper Edges and these SuperVertices, forming two separate sets, has no Super Hyper Edge in common; it’s SuperHyperMultiPartite it’s only one SuperVertex as intersection amid two given Super Hyper Edges and these SuperVertices, forming multi separate sets, has no Super Hyper Edge in common; it’s SuperHyperWheel if it’s only one SuperVertex as intersection amid two given Super Hyper Edges and one SuperVertex has one Super Hyper Edge with any common SuperVertex. The number of Super Hyper Edges for a given Super Hyper Vertex is called Super Hyper Degree.

Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer's Treatments

Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer's Treatments

Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes

Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes

Robust capped norm dual hyper-graph regularized non-negative matrix tri-factorization.

Non-negative matrix factorization (NMF) has been widely used in machine learning and data mining fields. As an extension of NMF, non-negative matrix tri-factorization (NMTF) provides more degrees of freedom than NMF. However, standard NMTF algorithm utilizes Frobenius norm to calculate residual error, which can be dramatically affected by noise and outliers. Moreover, the hidden geometric information in feature manifold and sample manifold is rarely learned. Hence, a novel robust capped norm dual hyper-graph regularized non-negative matrix tri-factorization (RCHNMTF) is proposed. First, a robust capped norm is adopted to handle extreme outliers. Second, dual hyper-graph regularization is considered to exploit intrinsic geometric information in feature manifold and sample manifold. Third, orthogonality constraints are added to learn unique data presentation and improve clustering performance. The experiments on seven datasets testify the robustness and superiority of RCHNMTF.

Rainbow spanning structures in graph and hypergraph systems

Abstract We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$ of not necessarily distinct k-graphs on the same vertex set $[n]$ , a (sub)graph H on $[n]$ is rainbow if there exists an injection $\varphi : E(H)\rightarrow [m]$ , such that $e\in E(G_{\varphi (e)})$ for each $e\in E(H)$ . Note that if $|E(H)|=m$ , then $\varphi $ is a bijection, and thus H contains exactly one edge from each $G_i$ . Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following: (1) A rainbow analogue of an asymptotical version of the Hajnal–Szemerédi theorem, namely, if $t\mid n$ and $\delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$ for each $i\in [\frac {n}{t}\binom {t}{2}]$ , then $\mathit {\mathbf {G}}$ contains a rainbow $K_t$ -factor; (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for $d\in [k-1]$ . The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.

1‐independent percolation on ℤ2×Kn

AbstractA random graph model on a host graph is said to be 1‐independent if for every pair of vertex‐disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . For an infinite connected graph , the 1‐independent critical percolation probability is the infimum of the such that every 1‐independent random graph model on in which each edge is present with probability at least almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that tends to a limit in as , and they asked for the value of this limit. We make progress on a related problem by showing that In fact, we show that the equality above remains true if the sequence of complete graphs is replaced by a sequence of weakly pseudorandom graphs on vertices with average degree . We conjecture the answer to Balister and Bollobás's question is also .

Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

Dual Dynamic Spatial-Temporal Graph Convolution Network for Traffic Prediction

Recently, Graph Convolution Network (GCN) and Temporal Convolution Network (TCN) are introduced into traffic prediction and achieve state-of-the-art performance due to their good ability for modeling the spatial and temporal property of traffic data. In spite of having good performance, the current methods generally focus on the traffic measurement of road segments, i.e. the nodes of traffic flow graph, while the edges of the graph, which represent the correlation of traffic data of different road segments and form the affinity matrix for GCN, are usually constructed according to the structure of road network, but the spatial and temporal properties are not well exploited in their theories. In this paper, we propose a Dual Dynamic Spatial-Temporal Graph Convolution Network (DDSTGCN), which not only models the dynamic property of the nodes of the traffic flow graph but also captures the dynamic spatial-temporal feature of the edges by transforming the traffic flow graph into its dual hypergraph. The traffic prediction is enhanced by the collaborative convolutions on the traffic flow graph and its dual hypergraph. The proposed method is evaluated by extensive traffic prediction experiments on six real road datasets and the results show that it outperforms state-of-the-art related methods. Source codes are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/j1o2h3n/DDSTGCN</uri> .

Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes

In this research article, the notions of Super Hyper Dominating and Super Hyper Resolving are defined in the setting of neutrosophic Super Hyper Graphs. Some ideas are introduced on both notions of Super Hyper Dominating and Super Hyper Resolving, simultaneously and as the same with each other. Some neutrosophic Super Hyper Classes are defined based on the notion, Super Hyper Resolving. The terms of duality, totality, perfectness, connectedness, and stable, are added to basic framework and initial notions, Super Hyper Dominating and Super Hyper Resolving but the concentration is on the “perfectness” to figure out what’s going on when for all targeted Super Hyper Vertices, there’s only one Super Hyper Vertex in the intended set. There are some instances and some clarifications to make sense about what’s happened and what’s done in the starting definitions. The key point is about the minimum sets. There are some questions and some problems to be taken as some avenues to pursue this study and this research. A basic familiarity with Super Hyper Graph theory and neutrosophic Super Hyper Graph theory is proposed.

A Multi-Dimensional Matrix Product—A Natural Tool for Parameterized Graph Algorithms

We introduce the concept of a k-dimensional matrix product D of k matrices A1,…,Ak of sizes n1×n,…,nk×n, respectively, where D[i1,…,ik] is equal to ∑ℓ=1nA1[i1,ℓ]×…×Ak[ik,ℓ]. We provide upper bounds on the time complexity of computing the product and solving related problems of computing witnesses and maximum witnesses of the Boolean version of the product in terms of the time complexity of rectangular matrix multiplication. The multi-dimensional matrix product framework is useful in the design of parameterized graph algorithms. First, we apply our results on the multi-dimensional matrix product to the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a reduction to triangle detection/counting. We provide an alternative simple method of detection/counting copies of fixed size cliques based on the multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of K4 and K5. Next, we show an immediate reduction of the k-dominating set problem to the multi-dimensional matrix product. It implies the W[2] hardness of the problem of computing the k-dimensional Boolean matrix product. Finally, we provide an efficient reduction of the problem of finding the lowest common ancestors for all k-tuples of vertices in a directed acyclic graph to the problem of finding witnesses of the Boolean variant of the multi-dimensional matrix product. Although the time complexities of the algorithms resulting from the aforementioned reductions solely match those of the known algorithms, the advantage of our algorithms is simplicity. Our algorithms also demonstrate the versatility of the multi-dimensional matrix product framework.

Host-graph-sensitive RETE nets for incremental graph pattern matching with nested graph conditions

BackgroundEfficient querying of large graph structures is a problem at the heart of several application domains such as social networks and model driven engineering. In particular in the context of model driven engineering, where the same query is executed frequently over an evolving graph structure, incremental techniques based on RETE nets are a popular solution. However, the construction of adequate RETE nets for a specific problem instance is a challenge in and of itself. MethodsIn this article, we propose an approach to RETE net construction for queries in the form of graph patterns equipped with nested graph conditions that considers not only the structure of the query, but also the structure of the host graph to improve the net's performance regarding execution time and memory consumption. Furthermore, we suggest a technique for adapting the net structure to changing host graph characteristics. We evaluate the presented concepts empirically based on queries and data from two independent benchmarks, using a mature tool for incremental graph pattern matching as a reference. ResultsOur evaluation results indicate that considering host graph structure during RETE net construction can improve performance of the resulting net. The experiments also demonstrate that adapting the net structure to changing host graph characteristics can be performed with an acceptable execution time overhead for our examples and may yield more adequate RETE nets. Overall, our approach never performed worse than the reference by more than factor 1.2 regarding execution time and achieved an improvement by up to factor 20.