Abstract: We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes.

Abstract: In this paper we study functions on the Boolean hypercube that have the property that after applying certain random restrictions, the restricted function is correlated to a linear function with non-negligible probability. If the given function is correlated with a linear function then this property clearly holds. Furthermore, the property also holds for low-degree functions as low-degree functions become a constant function under a random restriction with a non-negligible probability. We show that this essentially is the only possible reason. More specifically, we show that the function must be correlated to a product of a linear function and a low-degree function. One of the main motivations of studying this question comes from the recent work of the authors towards understanding approximability of satisfiable Constraint Satisfaction Problems. Towards proving our structural theorem, we analyze a 2-query direct product test for the table F: [n] qn → {0,1}qn where q∈ (0,1). We show that, for every constant ε>0, if the test passes with probability ε>0, then there is a global function g: [n]→ {0,1} such that for at least δ(ε) fraction of sets, the global function g agrees with the given table on all except α(ε) many locations. The novelty of this result lies in the fact that α(ε) is independent of the set sizes. Prior to our work, such a conclusion (in fact, a stronger conclusion with α = 0) was shown by Dinur, Filmus, and Harsha albeit when the test accepts with probability 1−ε for a small constant ε>0. The setting of parameters in our direct product tests is fundamentally different compared to the previous results and hence our analysis involves new techniques, including the use of the small-set expansion property of graphs defined on multi-slices. As one application of our structural result, we give a 4-query linearity test under the p-biased distribution. More specifically, for any p∈ (1/3,2/3), we give a test that queries a given function f: {0,1}n → {0,1} at 4 locations, where the marginal distribution of each query is µp⊗ n. The test has perfect completeness and soundness 1/2+ε – in other words, for every constant ε>0, if the test passes with probability at least 1/2+ε, then the function f is correlated to a linear function under the µp⊗ n measure. This qualitatively improves the results on the linearity testing under the p-biased distribution from the previous work where the authors studied the test with soundness 1−ε, for ε close to 0.

Abstract: Abstract For a given graph G , the general position problem asks for the largest set of vertices $$S \subseteq V(G)$$ S ⊆ V ( G ) , such that no three distinct vertices of S belong to a common shortest path of G . The general position problem for Cartesian products of two cycles as well as for hypercubes is considered. The problem is completely solved for the first family of graphs, while for the hypercubes, some partial results based on reduction to SAT are given.

Abstract: The problem of testing monotonicity for Boolean functions on the hypergrid, f:[n]d → {0,1} is a classic topic in property testing. When n=2, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making O(ε−2√d) queries. Up to polylog d and ε factors, this bound matches the Ω(√d)-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any n > 2, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a O(d5/6)-query upper bound (SODA 2020), quite far from the √d bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant n, up to poly(ε−1logd) factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making O(ε−2n√d) queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid [n]d. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.

Abstract: The generalized connectivity of a graph G, denoted as κk(G), is a generalization of the traditional connectivity and is a parameter to measure the capability of connecting multiple vertices in G. The divide-and-swap cube DSCn is a variant of the hypercube with a nice hierarchical structure and plentiful properties. This paper investigates the generalized connectivity on DSCn. We obtain the result κ4(DSCn)=d, where d=log2⁡n≥1, by the construction showing that there are d internally disjoint trees connecting any four arbitrary vertices of DSCn. As a corollary, one can directly obtain κ3(DSCn)=d.

Abstract: We developed a physics-informed neural network based on a mixture of Cartesian grid sampling and Latin hypercube sampling to solve forward and backward modified diffusion equations. We optimized the parameters in the neural networks and the mixed data sampling by considering the squeeze boundary condition and the mixture coefficient, respectively. Then, we used a given modified diffusion equation as an example to demonstrate the efficiency of the neural network solver for forward and backward problems. The neural network results were compared with the numerical solutions, and good agreement with high accuracy was observed. This neural network solver can be generalized to other partial differential equations.

Abstract: We develop a framework for sampling from discrete distributions µ on the hypercube {± 1}n by sampling from continuous distributions supported on ℝn obtained by convolution with spherical Gaussians. We show that for well-studied families of discrete distributions µ, the result of the convolution is well-conditioned log-concave, whenever the Gaussian’s variance is above an O(1) threshold. We plug off-the-shelf continuous sampling methods into our framework to obtain novel discrete sampling algorithms. Additionally, we introduce and study a crucial notion of smoothness for discrete distributions that we call transport stability, which we use to control the propagation of error in our framework. We expect transport stability to be of independent interest, as we connect it to constructions of optimally mixing local random walks and concentration inequalities. As our main application, we resolve open questions raised by Anari, Hu, Saberi, and Schild on the parallel sampling of distributions which admit parallel counting. We show that determinantal point processes can be sampled via RNC algorithms, that is in time log(n)O(1) using nO(1) processors. For a wider class of distributions, we show our framework yields Quasi-RNC sampling, i.e., log(n)O(1) time using nO(logn) processors. This wider class includes non-symmetric determinantal point processes and random Eulerian tours in digraphs, the latter nearly resolving another open question raised by prior work.

Abstract: We demonstrate a new geometric method for fast template placement for searches for gravitational waves from the inspiral, merger and ringdown of compact binaries. The method is based on a binary tree decomposition of the template bank parameter space into nonoverlapping hypercubes. We use a numerical approximation of the signal overlap metric at the center of each hypercube to estimate the number of templates required to cover the hypercube and determine whether to further split the hypercube. As long as the expected number of templates in a given cube is greater than a given threshold, we split the cube along its longest edge according to the metric. When the expected number of templates in a given hypercube drops below this threshold, the splitting stops and a template is placed at the center of the hypercube. Using this method, we generate aligned-spin template banks covering the mass range suitable for a search of Advanced LIGO, Advanced Virgo and KAGRA data. The aligned-spin bank was generated in $\ensuremath{\sim}24\text{ }\text{ }\mathrm{hours}$ using a single CPU core and produced 2 million templates. Our primary motivation for developing this algorithm is to produce a bank with useful geometric properties in the physical parameter space coordinates. Such properties are useful for population modeling and parameter estimation.

Abstract: Classical connectivity is a vital metric to explore fault tolerance and reliability of network-based multiprocessor systems. The component connectivity is a more advanced metric to assess the fault tolerance of network structures beyond connectivity and has gained great progress. For a non-complete graph G=(V(G),E(G)), a subset T⊆V(G) is called an r-component cut of G, if G−T is disconnected and has at least r components (r≥2). The r-component connectivity of G, denoted by cκr(G), is the cardinality of the minimum r-component cut. The component connectivities of some networks for small r have been determined, while some progresses for large r only focus on the networks which take hypercube as their modules. In this paper, we determine the (r+1)-component connectivity of augmented cubes cκr+1(AQn)=2nr−4r−(r2)+3, for n≥13, 6≤r≤⌊n−12⌋, and particularly cκr+1(AQn)=2nr−4r−(r2)+2 for n≥5, r∈{4,5}.

Abstract: With the increasing demand of data center interconnection in the network cloud era, data center networks (DCNs) play key roles in computing and communication, becoming the center of a great deal of resources and business. Therefore, how to design a DCN with good performance has been a significant issue in networks. The hypercube is a popular interconnection network topology with low diameter, symmetry and recursive structure, and scalability. As a variant of the hypercube, the crossed cube not only retains the excellent properties of the hypercube, but also performs better in terms of Hamiltonian-connectivity, embeddability, diameter, etc. In this paper, we first propose a novel and server-centric DCN, called CSDC, which is based on the crossed cube network. The performance of CSDC can be characterized by the properties of its logical structure, so we then give its logical structure $C_{n}$ , and determine the connectivity and edge-connectivity of $C_{n}$ . Furthermore, we explore the fault-tolerant paths and disjoint paths between any two distinct nodes in $C_{n}$ , and obtain the diameter of $C_{n}$ . Moreover, we propose a method to construct two completely independent spanning trees (CISTs) in $C_{n}$ . Extensive evaluations demonstrate that CSDC is a magnetic DCN for building large-scale data centers.

Abstract: It is well known that the behaviour of a random subgraph of a d-dimensional hypercube, where we include each edge independently with probability p, undergoes a phase transition when p is around 1d. More precisely, standard arguments show that just below this value of p all components of this graph have order O(d) with probability tending to one as d→∞ (whp for short), whereas Ajtai, Komlós and Szemerédi (Combinatorica 2 (1982) 1–7) showed that just above this value, in the supercritical regime, whp there is a unique “giant” component of order Θ(2d). We show that whp the vertex expansion of the giant component is inverse polynomial in d. As a consequence, we obtain polynomial in d bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollobás, Kohayakawa and Łuczak (Random Structures and Algorithms 5 (1994) 627–648) and of Pete (Electron. Commun. Probab. 13 (2008) 377–392). Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of p, which are tight up to polynomial factors in d.

Abstract: We consider the averaging process on a graph, that is the evolution of a mass distribution undergoing repeated averages along the edges of the graph at the arrival times of independent Poisson processes. We establish cutoff phenomena for both the L1 and L2 distance from stationarity when the graph is a discrete hypercube and when the graph is complete bipartite. Some general facts about the averaging process on arbitrary graphs are also discussed.

Abstract: Abstract The undirected graph, exchanged hypercube $EH(s,t)$, is a variant of hypercube proposed by Loh et al. It is obtained by removing some links from $(s+t+1)$-dimensional hypercube. It retains many excellent properties, so many people have studied its reliability and fault tolerance. In this paper, combining the structure connectivity and substructure connectivity of graphs proposed not long ago, we obtain its $P_k$-path, $C_{2l}$-cycle and $K_{1,r}$-star structure connectivity and substructure connectivity where $2\le k,r\le s-1\le t-1$ and $6\le 2l\le s-1\le t-1$; we also establish $\kappa ^s(EH(s,t);C_4)$ for $5\le s\le t$ and the upper bound of $\kappa (EH(s,t);C_4)$ for $4\le s\le t$.

Abstract: Motivated by a multitude of practical applications, many distinct vulnerability parameters of multiprocessor systems have been explored. Traditional connectivity and diagnosability are undoubtedly the most well investigated of these metrics, but often fail to capture the most subtle differences of a multiprocessor system. Subsequently, it is necessary to take into account the minimum degree of components, the size of components or the number of components. However, the structure of the components is ignored in these circumstances. In this work, we propose a novel diagnostic strategy based on cyclic connectivity, namely the cyclic diagnosability. The cyclic diagnosability, denoted by $ct(G)$, is the maximum size of the faulty vertex set $F$ of $G$ such that the self-diagnosable system $G$ can identify all the vertices in $F$ under the condition that at least two connected components of $G-F$ contain a cycle. Furthermore, we investigate the cyclic diagnosability of hypercube $Q_{n}$ under the PMC model and the MM* model, and show that $ct(Q_{n})=5n-10$ for $n\geq 7$.

Abstract: Multi-dimensional classification (MDC) task can be considered the most inclusive description of all classifications tasks, as it joins multiple class spaces and their multiple class members into a single compound classification problem. The challenges in MDC arise from the possible class dependencies across different class spaces, as well as the imbalance of labels in training datasets due to lack of all possible combinations. In this paper, we propose a straightforward, yet efficient, MDC deep learning classifier, named Deep Self-Organizing Cube (DSOC) that can model dependencies among classes in multiple class spaces, while consolidating its ability to classify rare combinations of labels. DSOC is formed of two n-dimensional components, namely the Hypercube Classifier and the multiple DSOC Neural Networks connected to the hypercube. The multiple neural networks component is responsible for feature selection and segregation of classes, while the Hypercube classifier is responsible for creating the semantics among multiple class spaces and accommodate the model for rare sample classification. DSOC is a multiple-output learning algorithm that successfully classify samples across all class spaces simultaneously. To challenge the proposed DSOC model, we conducted an assessment on seventeen benchmark datasets in the four types of classification tasks, binary, multi-class, multi-label and multi-dimensional. The obtained results were compared to four standard classifiers and eight competitive state-of-the-art approaches reported in literature. The DSOC has achieved superior performance over standard classifiers as well as the state-of-the-art approaches in all the four classification tasks. Moreover, in terms of Exact Match accuracy metrics, DSOC has outperformed all state-of-the-art approaches in 77.8% of the cases, which reflects the superior ability of DSOC to model dependencies and successfully classify rare samples across all dimensions simultaneously.

Abstract: We present an overview of combinatorial techniques for studying large data sets with hypercubes and halocarbons as the foci. We outline a variety of combinatorial techniques such as the Möbius inversion and generalization of Sheehan’s enumerative combinatorics for all characters for the equivalence classes for coloring hyperplanes of hypercubes with a given set of color partitions. The techniques are also applicable to the enumeration of stereo and position isomers of polysubstituted halocarbons. Hypercubes are of considerable interest in large data sets such as genetic regulatory networks, potential energy surfaces of molecules, and visualizations. Halocarbons are of interest because they are environmental pollutants and due to their potential toxicity including carcinogenicity. The combinatorial techniques have the capability not only to generate a large data set to provide a platform for further analysis but also to obtain structure–property relations. Furthermore, quantum chemical techniques provide electronic parameters that are of potential use in the toxicity predictions of a large dataset of halocarbons. In particular, the quantum parameters for the Crebellei data set of 55 halocarbons are reviewed to enhance our understanding of the mode toxicity action of these compounds.

Abstract: Abstract Edge connectivity is an important parameter for the reliability of the inter-connection network. A graph $G$ is strong Menger edge-connected ($SM$-$\lambda $ for short) if there exist min$\{\deg _{G}(u),\deg _{G}(v)\}$ edge-disjoint paths between any pair of vertices $u$ and $v$ of $G$. The conditional edge-fault-tolerance strong Menger edge connectivity of $G$, denoted by $sm_{\lambda }^{r}(G)$, is the maximum integer $m$ such that $G-F$ remains $SM$-$\lambda $ for any edge set $F$ with $|F|\leq m$ and $\delta (G-F)\geq r$, where $\delta (G-F)\geq r$ is the minimum degree of $G-F$. Most of the previous papers discussed $sm_{\lambda }^{r}(G)$ in the case of $r\leq 2$. In this paper, we show that $sm_{\lambda }^{r}(FQ_{n})=2^{r}(n-r+1)-(n+1)$ for $1\leq r\leq n-2$, where $n\geq 4$.

Abstract: The XAI community is currently studying and developing symbolic knowledge-extraction (SKE) algorithms as a means to produce human-intelligible explanations for black-box machine learning predictors, so as to achieve believability in human-machine interaction. However, many extraction procedures exist in the literature, and choosing the most adequate one is increasingly cumbersome, as novel methods keep on emerging. Challenges arise from the fact that SKE algorithms are commonly defined based on theoretical assumptions that typically hinder practical applicability. This paper focuses on hypercube-based SKE methods, a quite general class of extraction techniques mostly devoted to regression-specific tasks. We first show that hypercube-based methods are flexible enough to support classification problems as well, then we propose a general model for them, and discuss how they support SKE on datasets, predictors, or learning tasks of any sort. Empirical examples are reported as well –based upon the PSyKE framework –, showing the applicability of hypercube-based methods to actual classification tasks.

Abstract: One of the most central issues in interconnection networks of parallel and distributed systems is finding edge-disjoint paths that transmit information. Finding as many as possible many-to-many edge-disjoint paths is conducive to improving the fault-tolerance of such networks. As an interconnection network topology, (n,k)-enhanced hypercube Qn,k (1≤k≤n−1), is a momentous variant of well-known hypercube. For integers 1≤l≤n−1 and n≥2, let δ=0 if 1≤l≤n−k, and δ=1 if n−k+1≤l≤n−1. This paper offers a unified method to determine the minimum cardinalities of faulty links in Qn,k, whose malfunction divides this network into several connected components such that each processor has at least l+δ neighbors, each component contains no less than 2l processors and the number of average neighbors for all processors is at least l+δ, respectively. Under these three different link-faulty assumptions, but for the condition of k=2 and l=n−2, the minimum cardinalities of such faulty links share the same value (n−l−δ+1)2l. And the value in the exceptional case is (n−l−δ)2l+1=2n−1. In other words, we find the maximum numbers of many-to-many edge-disjoint paths of Qn,k under the above three hypotheses, which offers refined measurements for the reliability and fault-tolerance of interconnection networks.

Abstract: We consider the problem of computing with many coins of unknown bias. We are given access to samples of n coins with unknown biases p1,…,pn and are asked to sample from a coin with bias f(p1,…,pn) for a given function f:[0,1]n→[0,1]. We give a complete characterization of the functions f for which this is possible. As a consequence, we show how to extend various combinatorial sampling procedures (most notably, the classic Sampford sampling for k-subsets) to the boundary of the hypercube.