Hypergraph Of Order Research Articles

DeHIN: A Decentralized Framework for Embedding Large-Scale Heterogeneous Information Networks

Modeling heterogeneity by extraction and exploitation of high-order information from heterogeneous information networks (HINs) has been attracting immense research attention in recent times. Such heterogeneous network embedding (HNE) methods effectively harness the heterogeneity of small-scale HINs. However, in the real world, the size of HINs grow exponentially with the continuous introduction of new nodes and different types of links, making it a billion-scale network. Learning node embeddings on such HINs creates a performance bottleneck for existing HNE methods that are commonly centralized, i.e., complete data and the model are both on a single machine. To address large-scale HNE tasks with strong efficiency and effectiveness guarantee, we present <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Decentralized Embedding Framework for Heterogeneous Information Network</i> (DeHIN) in this paper. In DeHIN, we generate a distributed parallel pipeline that utilizes hypergraphs in order to infuse parallelization into the HNE task. DeHIN presents a context preserving partition mechanism that innovatively formulates a large HIN as a hypergraph, whose hyperedges connect semantically similar nodes. Our framework then adopts a decentralized strategy to efficiently partition HINs by adopting a tree-like pipeline. Then, each resulting subnetwork is assigned to a distributed worker, which employs the deep information maximization theorem to locally learn node embeddings from the partition it receives. We further devise a novel embedding alignment scheme to precisely project independently learned node embeddings from all subnetworks onto a common vector space, thus allowing for downstream tasks like link prediction and node classification. As shown from our experimental results, DeHIN significantly improves the efficiency and accuracy of existing HNE models as well as outperforms the large-scale graph embedding frameworks by efficiently scaling up to large-scale HINs.

The α-normal labelling method for computing the p-spectral radii of uniform hypergraphs

ABSTRACT Let G be an r-uniform hypergraph of order n. For each , the p-spectral radius is defined as The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and is called the spectral radius of G. The α-normal labelling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is 1 while the product of all corner labels at any edge is α. Since then, this method has been used by many researchers in studying . In this paper, we extend Lu and Man's α-normal labelling method to the p-spectral radii of uniform hypergraphs for ; and find some applications.

Hypergraph-based connectivity measures for signaling pathway topologies.

Characterizing cellular responses to different extrinsic signals is an active area of research, and curated pathway databases describe these complex signaling reactions. Here, we revisit a fundamental question in signaling pathway analysis: are two molecules "connected" in a network? This question is the first step towards understanding the potential influence of molecules in a pathway, and the answer depends on the choice of modeling framework. We examined the connectivity of Reactome signaling pathways using four different pathway representations. We find that Reactome is very well connected as a graph, moderately well connected as a compound graph or bipartite graph, and poorly connected as a hypergraph (which captures many-to-many relationships in reaction networks). We present a novel relaxation of hypergraph connectivity that iteratively increases connectivity from a node while preserving the hypergraph topology. This measure, B-relaxation distance, provides a parameterized transition between hypergraph connectivity and graph connectivity. B-relaxation distance is sensitive to the presence of small molecules that participate in many functionally unrelated reactions in the network. We also define a score that quantifies one pathway's downstream influence on another, which can be calculated as B-relaxation distance gradually relaxes the connectivity constraint in hypergraphs. Computing this score across all pairs of 34 Reactome pathways reveals pairs of pathways with statistically significant influence. We present two such case studies, and we describe the specific reactions that contribute to the large influence score. Finally, we investigate the ability for connectivity measures to capture functional relationships among proteins, and use the evidence channels in the STRING database as a benchmark dataset. STRING interactions whose proteins are B-connected in Reactome have statistically significantly higher scores than interactions connected in the bipartite graph representation. Our method lays the groundwork for other generalizations of graph-theoretic concepts to hypergraphs in order to facilitate signaling pathway analysis.

Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$ without isolated vertices. If $\deg(u)+\deg(v) &gt; 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.

Connectivity measures for signaling pathway topologies

Abstract Characterizing cellular responses to different extrinsic signals is an active area of research, and curated pathway databases describe these complex signaling reactions. Here, we revisit a fundamental question in signaling pathway analysis: are two molecules “connected” in a network? This question is the first step towards understanding the potential influence of molecules in a pathway, and the answer depends on the choice of modeling framework. We examined the connectivity of Reactome signaling pathways using four different pathway representations. We find that Reactome is very well connected as a graph, moderately well connected as a compound graph or bipartite graph, and poorly connected as a hypergraph (which captures many-to-many relationships in reaction networks). We present a novel relaxation of hypergraph connectivity that iteratively increases connectivity from a node while preserving the hypergraph topology. This measure, B-relaxation distance, provides a parameterized transition between hypergraph connectivity and graph connectivity. B-relaxation distance is sensitive to the presence of small molecules that participate in many functionally unrelated reactions in the network. We also define a score that quantifies one pathway’s downstream influence on another, which can be calculated as B-relaxation distance gradually relaxes the connectivity constraint in hypergraphs. Computing this score across all pairs of 34 Reactome pathways reveals pairs of pathways statistically significant influence. We present two such case studies, and we describe the specific reactions that contribute to the large influence score. Finally, we investigate the ability for connectivity measures to capture functional relationships among proteins, and use the evidence channels in the STRING database as a benchmark dataset. STRING interactions whose proteins are B-connected in Reactome have statistically significantly higher scores than interactions connected in the bipartite graph representation. Our method lays the groundwork for other generalizations of graph-theoretic concepts to hypergraphs in order to facilitate signaling pathway analysis. Author summary Signaling pathways describe how cells respond to external signals through molecular interactions. As we gain a deeper understanding of these signaling reactions, it is important to understand how molecules may influence downstream responses and how pathways may affect each other. As the amount of information in signaling pathway databases continues to grow, we have the opportunity to analyze properties about pathway structure. We pose an intuitive question about signaling pathways: when are two molecules “connected” in a pathway? This answer varies dramatically based on the assumptions we make about how reactions link molecules. Here, examine four approaches for modeling the structural topology of signaling pathways, and present methods to quantify whether two molecules are “connected” in a pathway database. We find that existing approaches are either too permissive (molecules are connected to many others) or restrictive (molecules are connected to a handful of others), and we present a new measure that offers a continuum between these two extremes. We then expand our question to ask when an entire signaling pathway is “downstream” of another pathway, and show two case studies from the Reactome pathway database that uncovers pathway influence. Finally, we show that the strict notion of connectivity can capture functional relationships among proteins using an independent benchmark dataset. Our approach to quantify connectivity in pathways considers a biologically-motivated definition of connectivity, laying the foundation for more sophisticated analyses that leverage the detailed information in pathway databases.

Some Ore-type Results for Matching and Perfect Matching in k-uniform Hypergraphs

Let S1 and S2 be two (k − 1)-subsets in a k-uniform hypergraph H. We call S1 and S2 strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S1 ∩ e ≠ ∅ and S2 ∩ e ≠ ∅ or e ⊆ S1 ∪ S2 or e ⊇ S1 ∪ S2, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k2 − k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k − 1)-sets equal to 2n−4(k−1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k−2)k. If the degree sum of any two middle independent (k−1)-subsets is larger than 2(d−1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k − 1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.

D-matching in k-uniform hypergraphs

A matching of a [Formula: see text]-uniform hypergraph is a set of pairwise disjoint edges. A [Formula: see text]-matching in a [Formula: see text]-uniform hypergraph [Formula: see text] is a matching of size [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-uniform hypergraph of order [Formula: see text] and [Formula: see text]. If [Formula: see text] for any two adjacent vertices [Formula: see text], and [Formula: see text], then [Formula: see text] contains a [Formula: see text]-matching. This result is an extension of a work of Bollobás, Daykin and Erdős [Sets of independent edges of a hypergraph, Quart. J. Math. Oxford 21 (1976) 25–32].

SIS Epidemic Propagation on Hypergraphs.

Mathematical modelling of epidemic propagation on networks is extended to hypergraphs in order to account for both the community structure and the nonlinear dependence of the infection pressure on the number of infected neighbours. The exact master equations of the propagation process are derived for an arbitrary hypergraph given by its incidence matrix. Based on these, moment closure approximation and mean-field models are introduced and compared to individual-based stochastic simulations. The simulation algorithm, developed for networks, is extended to hypergraphs. The effects of hypergraph structure and the model parameters are investigated via individual-based simulation results.

A characterization of hypergraphs with large domination number

Let H = (V,E) be a hypergraph with vertex set V and edge set E. A dominating set in H is a subset of vertices D ⊆ V such that for every vertex v ∈ V \D there exists an edge e ∈ E for which v ∈ e and e ∩D 6= ∅. The domination number γ(H) is the minimum cardinality of a dominating set in H. It is known ([1]) that for k ≥ 5, if H is a hypergraph of order n and size m with all edges of size at least k and with no isolated vertex, then γ(H) ≤ (n + b(k − 3)/2cm)/(b3(k − 1)/2c). In this talk, we characterize the hypergraphs achieving equality in this bound.

Hamiltonian decompositions of prisms over complete 3-uniform hypergraphs

We consider prisms over complete 3-uniform hypergraphs and decompose them into Hamiltonian cycles of Katona–Kierstead type. We show that prisms over complete 3-uniform hypergraphs of order 4, 5 and 8 can be decomposed into Hamiltonian cycles and we suggest that prisms over complete 3-uniform hypergraphs of order in the form 6t + 2 may have Hamiltonian decompositions.

A New Lower Bound on the Independence Number of a Graph and Applications

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

Lower Bounds on the Size of Maximum Independent Sets and Matchings in Hypergraphs of Rank Three

AbstractIn this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let and be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3‐uniform hypergraph of order at least 8 and with maximum degree at most three, then and there is a connected 3‐uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on vertices, where denotes the maximum degree in H, then and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory

Hypergraphs with large domination number and with edge sizes at least three

Let H=(V,E) be a hypergraph with vertex set V and edge set E. A dominating set in H is a subset of vertices D⊆V such that for every vertex v∈V∖D there exists an edge e∈E for which v∈e and e∩D≠0̸. The domination number γ(H) is the minimum cardinality of a dominating set in H. It is known that if H is a hypergraph of order n with edge sizes at least three and with no isolated vertex, then γ(H)≤n/3. In this paper, we characterize the hypergraphs achieving equality in this bound.

On packing of two copies of a hypergraph

Graphs and Algorithms A 2-packing of a hypergraph H is a permutation sigma on V (H) such that if an edge e belongs to epsilon(H), then sigma(e) does not belong to epsilon(H). Let H be a hypergraph of order n which contains edges of cardinality at least 2 and at most n - 2. We prove that if H has at most n - 2 edges then it is 2-packable.

Vertex-Transitive $q$-Complementary Uniform Hypergraphs

For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b &lt; k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t &lt; k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

Linear time analysis of properties of conflict-free and general Petri nets

We introduce the notion of a T - path within Petri nets, and propose to adopt the model of directed hypergraphs in order to determine properties of nets; in particular, we study the relationships between T -paths and firable sequences of transitions. Let us consider a Petri net P = 〈 P , T , A , M 0 〉 and the set of places with a positive marking in M 0 , i.e., P 0 = { p ∣ M 0 ( p ) > 0 } . If we regard the net as a directed graph, the existence of a simple path from any place in P 0 to a transition t is, of course, a necessary condition for the potential firability of t . This is sufficient only if the net is a state machine, where | • t | = | t • | = 1 for all t ∈ T . In this paper we show that the existence of a T -path from any subset of P 0 to a transition t is a more restrictive condition and is, again, a necessary condition for the potential firability of t . But, in this case: (a) if P is a conflict-free Petri net, this is also a sufficient condition, (b) if P is a general Petri net, t is potentially firable by increasing the number of tokens in P 0 . For conflict-free nets ( CFPN) we consider the following problems: (a) determining the set of firable transitions, (b) determining the set of coverable places, (c) determining the set of live transitions, (d) deciding the boundedness of the net. For all these problems we provide algorithms requiring linear space and time, i.e., O ( | P | + | T | + | A | ) , for a net P = 〈 P , T , A , M 0 〉 . Previous results for this class of networks are given by Howell et al. (1987) [20], providing algorithms for solving problems in conflict-free nets in O ( | P | × | T | ) time and space. Given a Petri net and a marking M , the well-known coverability problem consists in finding a reachable marking M ′ such that M ′ ≥ M ; this problem is known to be EXPSPACE hard (Rackoff (1978) [33]). For general Petri nets we provide a partial answer to this problem. M is coverable by augmentation if it is coverable from an augmented marking M 0 ′ of the initial marking M 0 : M 0 ′ ≥ M 0 and, for all p ∈ P , M 0 ′ ( p ) = 0 if M 0 ( p ) = 0 . We solve this problem in linear time. The algorithms for computing T -paths are incremental: it is possible to modify the network (adding new places, transitions, arcs, tokens), and update the set of potentially firable transitions and coverable places without recomputing them from scratch. This feature is meaningful when used during the interactive design of a system.

Cyclically [formula omitted]-complementary uniform hypergraphs

A cyclically t - complementary k - hypergraph is a k -uniform hypergraph with vertex set V and edge set E for which there exists a permutation θ ∈ S y m ( V ) such that the sets E , E θ , E θ 2 , … , E θ t − 1 partition the set of all k -subsets of V . Such a permutation θ is called a ( t , k ) - complementing permutation. The cyclically t -complementary k -hypergraphs are a natural and useful generalization of the self-complementary graphs, which have been studied extensively in the past due to their important connection to the graph isomorphism problem. For a prime p , we characterize the cycle type of the ( p r , k ) -complementing permutations θ ∈ S y m ( V ) which have order a power of p . This yields a test for determining whether a permutation in S y m ( V ) is a ( p r , k ) -complementing permutation, and an algorithm for generating all of the cyclically p r -complementing k -hypergraphs of order n , for feasible n , up to isomorphism. We also obtain some necessary and sufficient conditions on the order of these structures. This generalizes previous results due to Ringel, Sachs, Adamus, Orchel, Szymański, Wojda, Zwonek, and Bernaldez.

Hypergraphs with large transversal number and with edge sizes at least 3

AbstractIt is shown in several manuscripts [Discrete Math 86 (1990), 117–126; Combinatorica 12 (1992), 19–26] that every hypergraph of order n and size m with edge sizes at least 3 has a transversal of size at most (n + m)/4. In this article, we characterize the connected such hypergraphs that achieve equality in this bound. As a direct consequence of this bound, the total domination of a graph with minimum degree at least 3 is at most one‐half its order. An elegant graph theoretic proof of this result is presented in Archdeacon et al. [J Graph Theory 46 (2004), 207–210]. Using our hypergraph result, we characterize the connected graphs with minimum degree at least 3 and with total domination number exactly one‐half their order. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 326–348, 2008

Hypernetworks in a directed hypergraph

Recently novel distributed approaches to the formation of collaborative networked organizations have been proposed. Some of them make use of logical–formal structures (i.e., hypernetworks and hyperpaths) based on concepts related to directed hypergraphs in order to formally represent competitive inter-cluster and collaborative intra-cluster relationships. In this paper, we introduce the concepts of the s-hypernetwork, the structure consisting of all hyperpaths from a node s to any node in a directed hypergraph; the (s, d)-hypernetwork, the structure consisting of all hyperpaths from a node s to a node d in a directed hypergraph. Moreover, we present characterization theorems for both structures, a polynomial algorithm to determine the s-hypernetwork and some conditions under which the problem of finding the (s, d)-hypernetwork is solvable in polynomial time.

A note on packing of two copies of a hypergraph

A 2-packing of a hypergraph H is a permutation ae on V (H) such that if an edge e belongs to E(H), then ae(e) does not belong to E(H). We prove that a hypergraph which does not contain neither empty edge ; nor complete edge V (H) and has at most 1 n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1 n edges shows that this result cannot be improved by increasing the size of H.