Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer $h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set $V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever ${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}. Let $s(h)$ be the least integer $k$ such that the conclusion above holds and let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)= h+2^{\lfloor \log_2 h\rfloor}$. In the special case $h=2^{\ell}$ or $h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and $v(2)=6$ were obtained in a previous work.Comment: 15 page

Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics

Hypergraphs and graph products extend traditional graph theory by incorporating multi-way and coupled relationships, which are ubiquitous in real-world systems. While the Kronecker product, rooted in matrix analysis, has become a powerful tool in network science, its application has been limited to pairwise networks. In this paper, we extend the coupling of graph products to hypergraphs, enabling a system-theoretic analysis of network compositions formed via the Kronecker product of hypergraphs. We first extend the notion of the matrix Kronecker product to the tensor Kronecker product from the perspective of tensor blocks. We present various algebraic and spectral properties and express different tensor decompositions with the tensor Kronecker product. Furthermore, we study the structure and dynamics of Kronecker hypergraphs based on the tensor Kronecker product. We establish conditions that enable the analysis of the trajectory and stability of a hypergraph dynamical system by examining the dynamics of its factor hypergraphs. Finally, we demonstrate the numerical advantage of this framework for computing various tensor decompositions and spectral properties.Comment: 29 pages, 4 figures, 2 table

network: A Package for Managing Relational Data in R

Effective memory structures for relational data within R must be capable of representing a wide range of data while keeping overhead to a minimum. The network package provides an class which may be used for encoding complex relational structures composed a vertex set together with any combination of undirected/directed, valued/unvalued, dyadic/hyper, and single/multiple edges; storage requirements are on the order of the number of edges involved. Some simple constructor, interface, and visualization functions are provided, as well as a set of operators to facilitate employment by end users. The package also supports a C-language API, which allows developers to work directly with network objects within backend code.

Multilinear Control Systems Theory and its Applications

In biological and engineering systems, structure, function, and dynamics are highly coupled. Such multiway interactions can be naturally and compactly captured via tensor-based representations. Exploiting recent advances in tensor algebraic methods, we develop novel theoretical and computational approaches for data-driven model learning, analysis, and control of such tensor-based representations. In one line of work, we extend classical linear time-invariant (LTI) system notions including stability, reachability, and observability to multilinear time-invariant (MLTI) systems, in which the state, inputs, and outputs are represented as tensors, and express these notions in terms of more standard concepts of tensor ranks/decompositions. We also introduce a tensor decomposition-based model reduction framework which can significantly reduce the number of MLTI system parameters. In another line of work, we develop the notion of tensor entropy for uniform hypergraphs, which can capture higher order interactions between entities than classical graphs. We show that this tensor entropy is an extension of von Neumann entropy for graphs and can be used as a measure of regularity for uniform hypergraphs. Moreover, we employ uniform hypergraphs for studying controllability of high-dimensional networked systems. We propose another tensor-based multilinear system representation for characterizing the multidimensional state dynamics of uniform hypergraphs, and derive a Kalman-rank-like condition to identify the minimum number of control nodes (MCN) needed to achieve full control of the whole hypergraph. We demonstrate these new tensor-based theoretical and computational developments in a variety of biological and engineering examples.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169968/1/canc_1.pd

network: A Package for Managing Relational Data in R

Effective memory structures for relational data within R must be capable of representing a wide range of data while keeping overhead to a minimum. The network package provides an class which may be used for encoding complex relational structures composed a vertex set together with any combination of undirected/directed, valued/unvalued, dyadic/hyper, and single/multiple edges; storage requirements are on the order of the number of edges involved. Some simple constructor, interface, and visualization functions are provided, as well as a set of operators to facilitate employment by end users. The package also supports a C-language API, which allows developers to work directly with network objects within backend code

Relational learning for set value functions

Relational learning is learning in a context where we have a set of items with relationships. For example, in a recommender system or advertising platform, items are grouped into lists to attract user attention, and some items may be more popular than others. We are often interested in learning individual abilities, approximating group performances and making best set selection. However, it could be challenging as we have limited feedback and various uncertainties. We might only observe noisy aggregate feedback at the set level (set level randomness), and each item could be a random variable following some distributions (item level randomness). To tackle the problem, we model the group performance using a set value function, defined as a function of item values within the group of interest. We first study the beta model for hypergraphs. The model treats relational data as hypergraphs where nodes represent items and hyper-edges group items into sets. The goal is to estimate individual beta values from the group outcomes. We study the inference problem under different settings using maximum likelihood estimation (MLE). We move on to consider more general set value functions and the second source of randomness at the item level. The goal is to find good item representations (sketches) for approximation of stochastic valuation functions, defined as the expectation of set value functions of independent random variables. We present an approximation everywhere guarantee for a wide range of stochastic valuation functions. Finally, we study an online variant where an agent can draw samples sequentially. At each time step, the agent chooses a group of items subject to constraints and receives some form of feedbacks. The goal is to select a set of items with maximum performances according to some stochastic valuation functions. We consider the regret minimization setting and address the problem under value-index feedback

Sparse random hypergraphs: Non-backtracking spectra and community detection

We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. (2015). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.Comment: 61 pages, 8figures. To appear in Information and Inferenc

Combinatorial Methods in Coding Theory

This thesis is devoted to a range of questions in applied mathematics and signal processing motivated by applications in error correction, compressed sensing, and writing on non-volatile memories. The underlying thread of our results is the use of diverse combinatorial methods originating in coding theory and computer science. The thesis addresses three groups of problems. The first of them is aimed at the construction and analysis of codes for error correction. Here we examine properties of codes that are constructed using random and structured graphs and hypergraphs, with the main purpose of devising new decoding algorithms as well as estimating the distribution of Hamming weights in the resulting codes. Some of the results obtained give the best known estimates of the number of correctable errors for codes whose decoding relies on local operations on the graph. In the second part we address the question of constructing sampling operators for the compressed sensing problem. This topic has been the subject of a large body of works in the literature. We propose general constructions of sampling matrices based on ideas from coding theory that act as near-isometric maps on almost all sparse signal. This matrices can be used for dimensionality reduction and compressed sensing. In the third part we study the problem of reliable storage of information in non-volatile memories such as flash drives. This problem gives rise to a writing scheme that relies on relative magnitudes of neighboring cells, known as rank modulation. We establish the exact asymptotic behavior of the size of codes for rank modulation and suggest a number of new general constructions of such codes based on properties of finite fields as well as combinatorial considerations