Sets In Hypergraphs Research Articles

Filtration evolution of hypergraphs: A novel approach to studying multidimensional datasets.

The rapid growth of large datasets has led to a demand for novel approaches to extract valuable insights from intricate information. Graph theory provides a natural framework to model these relationships, but standard graphs may not capture the complex interdependence between components. Hypergraphs are a powerful extension of graphs that can represent higher-order relationships in the data. In this paper, we propose a novel approach to studying the structure of a dataset using hypergraph theory and a filtration method. Our method involves building a set of hypergraphs based on a variable distance parameter, enabling us to infer qualitative and quantitative information about the data structure. We apply our method to various sets of points, dynamical systems, signal models, and real electrophysiological data. Our results show that the proposed method can effectively differentiate between varying datasets, demonstrating its potential utility in a range of scientific applications.

An application of node and edge nonlinear hypergraph centrality to a protein complex hypernetwork.

The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins. One of the key advantages of applying this model to a protein complex hypernetwork is that it also provides a classification method for protein complexes, unlike previous approaches which are only concerned with classifying proteins.

Inapproximability of counting independent sets in linear hypergraphs

It is shown in this note that approximating the number of independent sets in a k-uniform linear hypergraph with maximum degree at most Δ is NP-hard if Δ≥5⋅2k−1+1. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon et al. (2019) [5], the perfect sampler by Qiu et al. (2022) [6], and the deterministic approximation scheme by Feng et al. (2023) [7].

Compression with wildcards: All exact or all minimal hitting sets

Abstract Our objective is the compressed enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author’s best knowledge, the only previous attempt towards compression, due to Toda, is based on binary decision diagrams and much different from our techniques. Traditional one-by-one enumeration schemes cannot compete when the number of minimal hitting sets is large and the degree of compression is high. Our method works particularly well in these two cases: either compressing all minimum cardinality hitting sets, or compressing all exact hitting sets.

Hypergraphs with edge-dependent vertex weights: p-Laplacians and spectral clustering.

We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW.

Coloring Bipartite Graphs with Semi-small List Size

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex’s list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most of the parameter space. By connecting this quantity to a simple question on independent sets of hypergraphs, we strengthen bounds when a part has list size 2. Finally, we state via our framework a conjecture on general bipartite graphs, unifying three conjectures of Alon–Cambie–Kang.

Core-Periphery Detection in Hypergraphs

Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks modeled as hypergraphs, and we propose a method for computing a core-score vector that quantifies how close each node is to the core. In particular, we show that this method solves the corresponding nonconvex core-periphery optimization problem globally to an arbitrary precision. This method turns out to coincide with the computation of the Perron eigenvector of a nonlinear hypergraph operator, suitably defined in terms of the incidence matrix of the hypergraph, generalizing recently proposed centrality models for hypergraphs. We perform several experiments on synthetic and real-world hypergraphs showing that the proposed method outperforms alternative core-periphery detection algorithms, in particular those obtained by transferring established graph methods to the hypergraph setting via clique expansion.

Disjoint maximal independent sets in graphs and hypergraphs

Disjoint maximal independent sets in graphs and hypergraphs

Tropicalization of graph profiles

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its “combinatorial shadow”. We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities.

Uniformity norms, their weaker versions, and applications

We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence:

Independent sets in hypergraphs omitting an intersection

AbstractA k‐uniform hypergraph with n vertices is an ‐omitting system if it has no two edges with intersection size . If in addition it has no two edges with intersection size greater than , then it is an ‐system. Rödl and Šiňajová proved a sharp lower bound for the independence number of ‐systems. We consider the same question for ‐omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number , where F is the k‐uniform Fan. The behavior is quite different than the case which is the classical Ramsey number .

Efficiently enumerating hitting sets of hypergraphs arising in data profiling

The transversal hypergraph problem asks to enumerate the minimal hitting sets of a hypergraph. If the solutions have bounded size, Eiter and Gottlob [SICOMP'95] gave an algorithm running in output-polynomial time, but whose space requirement also scales with the output. We improve this to polynomial delay and space. Central to our approach is the extension problem, deciding for a set X of vertices whether it is contained in any minimal hitting set. We show that this is one of the first natural problems to be W[3]-complete. We give an algorithm for the extension problem running in time O(m|X|+1n) and prove a SETH-lower bound showing that this is close to optimal. We apply our enumeration method to the discovery problem of minimal unique column combinations from data profiling. Our empirical evaluation suggests that the algorithm outperforms its worst-case guarantees on hypergraphs stemming from real-world databases.

Number of [formula omitted] solutions in abelian groups and application to counting independent sets in hypergraphs

The paper deals with a problem of Additive Combinatorics. Let G be a finite abelian group of order N. We prove that the number of subset triples A,B,C⊂G such that for any x∈A, y∈B and z∈C one has x+y≠z equals 3⋅4N+N3N+1+O((3−c∗)N) for some absolute constant c∗>0. This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and disproves the conjecture of Cohen et al. (0000). concerning the maximal possible number of independent sets in such hypergraphs on n vertices.

Uniform forcing and immune sets in graphs and hypergraphs

Zero forcing is an iterative coloring process on a graph that has been widely used in such different areas as the modeling of propagation phenomena in networks and the study of minimum rank problems in matrices and graphs. This paper deals with zero forcing on hypergraphs. (Representing a network by a hypergraph allows us to account for its community structure and for more general interactions between different subsets of nodes.)We consider two natural generalizations to hypergraphs of zero forcing on graphs (one of them already known) and, for each one of these generalizations we look into two clutters that play a significant role in the forcing process: the clutter of minimal forcing sets and the one of minimal immune sets. A formulation of immune sets in terms of neighborhoods (hence without making reference to the iterative zero forcing process) is presented, highlighting the different behavior of the distinct forcing rules. Moreover, we obtain the families of minimal forcing and minimal immune sets in the case of complete hypergraphs and we provide a full characterization of forcing and immune uniform clutters, both in the graph and in the hypergraph case.

Epidemics on hypergraphs: spectral thresholds for extinction.

Epidemic spreading is well understood when a disease propagates around a contact graph. In a stochastic susceptible–infected–susceptible setting, spectral conditions characterize whether the disease vanishes. However, modelling human interactions using a graph is a simplification which only considers pairwise relationships. This does not fully represent the more realistic case where people meet in groups. Hyperedges can be used to record higher order interactions, yielding more faithful and flexible models and allowing for the rate of infection of a node to depend on group size and also to vary as a nonlinear function of the number of infectious neighbours. We discuss different types of contagion models in this hypergraph setting and derive spectral conditions that characterize whether the disease vanishes. We study both the exact individual-level stochastic model and a deterministic mean field ODE approximation. Numerical simulations are provided to illustrate the analysis. We also interpret our results and show how the hypergraph model allows us to distinguish between contributions to infectiousness that (i) are inherent in the nature of the pathogen and (ii) arise from behavioural choices (such as social distancing, increased hygiene and use of masks). This raises the possibility of more accurately quantifying the effect of interventions that are designed to contain the spread of a virus.

Independent sets in algebraic hypergraphs

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given algebraic hypergraph is “dense” in a certain sense, then a generic low-dimensional subset of its vertices induces a subhypergraph that is also “dense.” (For technical reasons, we only consider low-dimensional subsets that are parameterized by rational functions.) Our proof approach is inspired by the hypergraph containers method, developed by Balogh, Morris, and Samotij and independently by Saxton and Thomason (although adapting this method to the algebraic setting presents some unique challenges that do not occur when working with finite hypergraphs). Along the way, we establish a natural generalization of the classical dimension of fibers theorem in algebraic geometry, which is interesting in its own right.

Independent sets in hypergraphs with a forbidden link

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erd\H{o}s and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.

Counting independent sets in regular hypergraphs

Amongst d-regular r-uniform hypergraphs on n vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is to propose such a generalisation. Lending credence to our conjecture, we verify it within the class of ‘quasi-bipartite’ hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropic approach of Kahn.

MEMPHA: Model of Exascale Message-Passing Programs on Heterogeneous Architectures

Delivering optimum performance on a parallel computer is highly dependant on the efficiency of the scheduling and mapping procedure. If the composition of the parallel application is known a prior, the mapping can be accomplished statically on the compilation time. The mapping algorithm uses the model of the parallel application and maps its tasks to processors in a way to minimize the total execution time. In this article, current modeling approaches have discussed. Later, a new modeling schema named Model of Exascale Message-Passing Programs on Heterogeneous Architectures (MEMPHA) has proposed. A comparative study has been performed between MEMPHA and existing models. To exhibit the efficiency of the MEMPHA, experiments have performed on a set of data-set hypergraphs. The results obtained from the experiments show that deploying the MEMPHA helps to optimize metrics, including the congestion, total communication volume and maximum volume of data being sent or received. These improvements vary from 76 to 1 percent, depending on the metric and benchmark model. Moreover, MEMPHA supports the modeling of applications with multiple producers for a single data transmission, where the rest of the approaches fail.

Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric Neutrosophic Hypergraphic Approach

Neutrosophic sets are comprehensively used in decision-making environment. The manifestation of neutrosophic sets in concentric hypergraphs is proposed in this research work. The intention of developing a decision-making model using the combination of Fuzzy Cognitive Maps and concentric neutrosophic hypergraph is to rank the core factors of decision-making problem and find the inter-relational impacts. This proposed model is validated with the exploration of the causative factors of autoimmune diseases. The proposed model is highly compatible as it assists in determining the core factors and their inter-association. This model will certainly benefit the decision-maker at all managerial levels to design optimal decisions.