Operators on random hypergraphs and random simplicial complexes

Random hypergraphs and random simplicial complexes have potential applications in computer science and engineering. Various models of random hypergraphs and random simplicial complexes on n-points have been studied. Let L be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of L. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some operators on the space of probability functions on sub-hypergraphs of L. We study the compositions of these operators as well as their actions on the space of probability functions. As applications in computer science, we give algorithms generating large sparse random hypergraphs and large sparse random simplicial complexes.Comment: 22 page

Weighted (Co)homology and Weighted Laplacian

In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $\phi$-weighted coboundary operator induced by a weight function $\phi$. Our weight function $\phi$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.Comment: 22 page

Comparative genomics: multiple genome rearrangement and efficient algorithm development

Multiple genome rearrangement by signed reversal is discussed: For a collection of genomes represented by signed permutations, reconstruct their evolutionary history by using signed reversals, i.e. find a bifurcating tree where sampled genomes are assigned to leaf nodes and ancestral genomes (i.e. signed permutations) are hypothesized at internal nodes such that the total reversal distance summed over all edges of the tree is minimized. It is equivalent to finding an optimal Steiner tree that connects the given genomes by signed reversal paths. The key for the problem is to reconstruct all optimal Steiner nodes/ancestral genomes.;The problem is NP-hard and can only be solved by efficient approximation algorithms. Various algorithms/programs have been designed to solve the problem, such as BPAnalysis, GRAPPA, grid search algorithm, MGR greedy split algorithm (Chapter 1). However, they may have expensive computational costs or low inference accuracy. In this thesis, several new algorithms are developed, including nearest path search algorithm (Chapter 2), neighbor-perturbing algorithm (Chapter 3), branch-and-bound algorithm (Chapter 3), perturbing-improving algorithm (Chapter 4), partitioning algorithm (Chapter 5), etc. With theoretical proofs, computer simulations, and biological applications, these algorithms are shown to be 2-approximation algorithms and more efficient than the existing algorithms

Stability of persistent homology for hypergraphs

In topological data analysis, the stability of persistent diagrams gives the foundation for the persistent homology method. In this paper, we use the embedded homology and the homology of associated simplicial complexes to define the persistent diagram for a hypergraph. Then we prove the stability of this persistent diagram. We generalize the persistent diagram method and define persistent diagrams for a homomorphism between two modules. Then we prove the stability of the persistent diagrams of the pull-back filtration and the push-forward filtration on hypergraphs, induced by a morphism between two hypergraphs.Comment: 22 page