Tootfinder

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Xu He, Junpeng Gong, Zhaohui Li
https://arxiv.org/abs/2506.04582

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Computer simulations serve as powerful tools for scientists and engineers to gain insights into complex systems. Less costly than physical experiments, computer experiments sometimes involve large number of trials. Conventional design optimization and model fitting methods for computer experiments are inefficient for large-scale problems. In this paper, we propose new methods to optimize good lattice point sets, using less computation to construct designs with enhanced space-filling properties …

Nondegenerate hyperplane covers of the hypercube
Lisa Sauermann, Zixuan Xu
https://arxiv.org/abs/2507.00773 https://arxiv.org/pdf/250…

Nondegenerate hyperplane covers of the hypercube
We consider collections of hyperplanes in $\mathbb{R}^n$ covering all vertices of the $n$-dimensional hypercube $\{0,1\}^n$, which satisfy the following nondegeneracy condition: For every $v\in \{0,1\}^n$ and every $i=1,\dots,n$, we demand that there is a hyperplane $H$ in the collection with $v\in H$ such that the variable $x_i$ appears with a non-zero coefficient in the hyperplane equation describing $H$. We prove that every collection $\mathcal{H}$ of hyperplanes in $\mathbb{R}^n$ covering $…

Quantum walk search for exceptional configurations
Pulak Ranjan Giri
https://arxiv.org/abs/2507.02457 https://arxiv.org/pdf/2507.0245…

Quantum walk search for exceptional configurations
There exist two types of configurations of marked vertices on a two-dimensional grid, known as the {\it exceptional configurations}, which are hard to find by the discrete-time quantum walk algorithms. In this article, we provide a comparative study of the quantum walk algorithm with different coins to search these {\it exceptional configurations} on a two-dimensional grid. We further extend the analysis to the hypercube, where only one type of {\it exceptional configurations} are present. Our …

LHS in LHS: A new expansion strategy for Latin hypercube sampling in simulation design
Matteo Boschini, Davide Gerosa, Alessandro Crespi, Matteo Falcone
https://arxiv.org/abs/2509.00159

LHS in LHS: A new expansion strategy for Latin hypercube sampling in simulation design
Latin Hypercube Sampling (LHS) is a prominent tool in simulation design, with a variety of applications in high-dimensional and computationally expensive problems. LHS allows for various optimization strategies, most notably to ensure space-filling properties. However, LHS is a single-stage algorithm that requires a priori knowledge of the targeted sample size. In this work, we present LHS in LHS, a new expansion algorithm for LHS that enables the addition of new samples to an existing LHS-dist…

Crosslisted article(s) found for cs.DS. https://arxiv.org/list/cs.DS/new
[1/1]:
- LHS in LHS: A new expansion strategy for Latin hypercube sampling in simulation design
Matteo Boschini, Davide Gerosa, Alessandro Crespi, Matteo Falcone

Crosslisted article(s) found for astro-ph.HE. https://arxiv.org/list/astro-ph.HE/new
[1/1]:
- LHS in LHS: A new expansion strategy for Latin hypercube sampling in simulation design
Matteo Boschini, Davide Gerosa, Alessandro Crespi, Matteo Falcone

Modeling the Effective Elastic Modulus and Thickness of Corrugated Boards Using Gaussian Process Regression and Expected Hypervolume Improvement
Ricardo Fitas
https://arxiv.org/abs/2507.02208

Modeling the Effective Elastic Modulus and Thickness of Corrugated Boards Using Gaussian Process Regression and Expected Hypervolume Improvement
This work aims to model the hypersurface of the effective elastic modulus, \( E_{z, \text{eff}} \), and thickness, \( th_{\text{eff}} \), in corrugated boards. A Latin Hypercube Sampling (LHS) is followed by Gaussian Process Regression (GP), enhanced by EHVI as a multi-objective acquisition function. Accurate modeling of \( E_{z, \text{eff}} \) and \( th_{\text{eff}} \) is critical for optimizing the mechanical properties of corrugated materials in engineering applications. LHS provides an effi…

Replaced article(s) found for cs.DM. https://arxiv.org/list/cs.DM/new
[1/1]:
- Distribution of codewords on the faces of a hypercube and new combinatorial identities
Jamolidin K. Abdurakhmanov

A note on the strength of a hypercube
Melissa A. Huggan, M. E. Messinger, Dylan Pearson
https://arxiv.org/abs/2507.21908 https://arxiv.org/pdf/2507.21908…

A note on the strength of a hypercube
As a generalization of super magic strength, the strength of a graph was introduced in [R. Ichishima, F.A. Muntaner-Batle, A. Oshima, Bounds for the strength of graphs, Austral. J. of Combin. 72(3) (2018) 492-508]. For a vertex ordering $f$ of graph $G$, the strength of $f$ is the maximum sum of the labels on any pair of adjacent vertices. The strength of $G$ is defined as the minimum strength of $f$, taken over all vertex orderings of $G$. The strength of the hypercube is unknown, but bounded.…

On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences
Hoang Anh Tran, Toh Kim-Chuan
https://arxiv.org/abs/2507.00572

On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences
The moment-SOS hierarchy is a widely applicable framework to address polynomial optimization problems over basic semi-algebraic sets based on positivity certificates of polynomial. Recent works show that the convergence rate of this hierarchy over certain simple sets, namely, the unit ball, hypercube, and standard simplex, is of the order $O(1/r^2)$, where r denotes the level of the moment-SOS hierarchy. This paper aims to provide a comprehensive understanding of the convergence rate of the mom…

Quantitative Obata's theorem in discrete setting
Shiping Liu, Chiyu Zhou
https://arxiv.org/abs/2508.20815 https://arxiv.org/pdf/2508.20815

Quantitative Obata's theorem in discrete setting
Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-Émery and the $d$-th non-zero Laplacian eigenvalue $λ_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in …

EHCube4P: Learning Epistatic Patterns Through Hypercube Graph Convolution Neural Network for Protein Fitness Function Estimation
Muhammad Daud, Philippe Charton, Cedric Damour, Jingbo Wang, Frederic Cadet
https://arxiv.org/abs/2506.16921

EHCube4P: Learning Epistatic Patterns Through Hypercube Graph Convolution Neural Network for Protein Fitness Function Estimation
Understanding the relationship between protein sequences and their functions is fundamental to protein engineering, but this task is hindered by the combinatorially vast sequence space and the experimental noise inherent in fitness measurements. In this study, we present a novel framework that models the sequence landscape as a hypercube $H(k,2)$ and integrates wavelet-based signal denoising with a graph convolutional neural network (GCN) to predict protein fitness across rugged fitness landsca…

Replaced article(s) found for cond-mat.dis-nn. https://arxiv.org/list/cond-mat.dis-nn/new
[1/1]:
- How to realize compact and non-compact localized states in disorder-free hypercube networks
Ievgen I. Arkhipov, Fabrizio Minganti, Franco Nori

Random cubic graph embedded in a hypercube: Entanglement spectrum and many-body localization
Frantisek Slanina
https://arxiv.org/abs/2508.13789 https://arx…

Random cubic graph embedded in a hypercube: Entanglement spectrum and many-body localization
The schematic model of interacting spins is introduced, which combines the symmetry of hypercube with the simplicity of random regular graph with degree three, i.e. the random cubic graph. We study the localization transition in this model, which shares essential characteristics with the systems exhibiting many-body localization. Namely, we investigate the transition in terms of the entanglement entropy and entanglement spectrum. We also show that the most significant indicator of the localizat…

Triangle-free subsets of the Hypercube
Padmini Mukkamala
https://arxiv.org/abs/2506.18782 https://arxiv.org/pdf/2506.18782

Triangle-free subsets of the Hypercube
We study the problem of determining the largest subset of vertices of the $n$ dimensional hypercube without three vertices at a Hamming distance of exactly $r$ from each other. In particular, we provide a lower bound of $\frac{c2^n}{e^r2^{\frac r2} (\frac nr)^{\frac{3r}{4}}}$ and an upper bound of $O(\frac{r2^n}{n+1})$ when $2r \le n$. In particular, when $r$ is a constant, the lower bound is $\frac{c2^n}{n^{3r/4}}$, while the upper bound is $\frac{c'2^n}{n}$.

Optimal Young's convolutions inequality and its reverse form on the hypercube
David Beltran, Paata Ivanisvili, Jos\'e Madrid, Lekha Patil
https://arxiv.org/abs/2507.06115 …

Optimal Young's convolutions inequality and its reverse form on the hypercube
We establish sharp forms of Young's convolution inequality and its reverse on the discrete hypercube $\{0,1\}^d$ in the diagonal case $p=q$. As applications, we derive bounds for additive energies and sumsets. We also investigate the non-diagonal regime $p\neq q$, providing necessary conditions for the inequality to hold, along with partial results in the case $r = 2$.

ConiQ: Enabling Concatenated Quantum Error Correction on Neutral Atom Arrays
Pengyu Liu, Mingkuan Xu, Hengyun Zhou, Hanrui Wang, Umut A. Acar, Yunong Shi
https://arxiv.org/abs/2508.05779

ConiQ: Enabling Concatenated Quantum Error Correction on Neutral Atom Arrays
Recent progress on concatenated codes, especially many-hypercube codes, achieves unprecedented space efficiency. Yet two critical challenges persist in practice. First, these codes lack efficient implementations of addressable logical gates. Second, the required high degree of parallelism and long-range interactions pose significant challenges for current hardware platforms. In this paper, we propose an efficient compilation approach for concatenated codes, specifically many-hypercube codes, ta…

Cycle lengths in the percolated hypercube
Michael Anastos, Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich, Lyuben Lichev
https://arxiv.org/abs/2506.16858

Cycle lengths in the percolated hypercube
Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2…

Delta-matroids and toric degenerations in OG(n,2n 1)
Grace Chen, Carl Lian
https://arxiv.org/abs/2507.16133 https://arxiv.org/pdf/250…

Delta-matroids and toric degenerations in OG(n,2n+1)
We construct an explicit, embedded degeneration of the general torus orbit closure in the maximal orthogonal Grassmannian OG(n,2n+1) into a union of Richardson varieties. In particular, we deduce a formula for the cohomology class of the torus orbit closure, as a sum of products of Schubert classes. The moment map images of the degenerate pieces are the base polytopes of their underlying delta-matroids, and give a polyhedral decomposition of the unit hypercube.

Accelerating RF Power Amplifier Design via Intelligent Sampling and ML-Based Parameter Tuning
Abhishek Sriram, Neal Tuffy
https://arxiv.org/abs/2507.11928 …

Accelerating RF Power Amplifier Design via Intelligent Sampling and ML-Based Parameter Tuning
This paper presents a machine learning-accelerated optimization framework for RF power amplifier design that reduces simulation requirements by 65% while maintaining $\pm0.3$ to $\pm0.4$ dBm accuracy. The proposed method combines MaxMin Latin Hypercube Sampling with CatBoost gradient boosting to intelligently explore multidimensional parameter spaces. Instead of exhaustively simulating all parameter combinations to achieve target P2dB compression specifications, our approach strategically selec…

Dynamical Phase Diagram of the REM under independent spin-flips
Chokri Manai, Simone Warzel
https://arxiv.org/abs/2506.15353 https://…

Dynamical Phase Diagram of the REM under independent spin-flips
We study the energy landscape of the Random Energy model (REM) integrated along trajectories of the simple random walk on the hypercube. We show that the quenched cumulant generating function of the time integral of the REM energy undergoes phase transitions in the large $N$ limit for trajectories of any time extent, and identify phases distinguished by the activity and value of the time integral. This is achieved by relating the dynamical behavior to the spectral properties of Hamiltonians ass…

An Intrinsic Barrier for Resolving P = NP (2-SAT as Flat, 3-SAT as High-Dimensional Void-Rich)
M. Alasli
https://arxiv.org/abs/2508.13200 https://arxiv.org…

An Intrinsic Barrier for Resolving P = NP (2-SAT as Flat, 3-SAT as High-Dimensional Void-Rich)
We present a topological barrier to efficient computation, revealed by comparing the geometry of 2 SAT and 3 SAT solution spaces. Viewing the set of satisfying assignments as a cubical complex within the Boolean hypercube, we prove that every 2 SAT instance has a contractible solution space, topologically flat, with all higher Betti numbers bk equals 0 for k greater than or equal 1, while both random and explicit 3 SAT families can exhibit exponential second Betti numbers, corresponding to expo…

A second-order cone representable class of nonconvex quadratic programs
Santanu S. Dey, Aida Khajavirad
https://arxiv.org/abs/2508.18435 https://arxiv.org/…

A second-order cone representable class of nonconvex quadratic programs
We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the linearized problem is SOC-representable. While the propo…

On the fault diameter and wide diameter of the exchanged 3-ary $n$-cube
Rongshuan Geng, Wantao Ning
https://arxiv.org/abs/2508.07174 https://arxiv.org/pdf/…

On the fault diameter and wide diameter of the exchanged 3-ary $n$-cube
Fault diameter and wide diameter are two critical parameters for evaluating communication performance in interconnection networks. They measure the fault tolerance and transmission efficiency of networks. The exchanged 3-ary $n$-cube is a recently proposed variant of the hypercube, denoted by $E3C(r, s, t)$. In this work, we obtain that the $(2r + 1)$-fault diameter and $(2r + 2)$-wide diameter of $E3C(r, s, t)$ are bounded between $n + 3$ and $n + 5$ for $1 \leq r \leq s \leq t$.

Connecting hypercube 1-factors
Lawrence Hollom, Benedict Randall Shaw
https://arxiv.org/abs/2508.15698 https://arxiv.org/pdf/2508.15698

Connecting hypercube 1-factors
A 1-factorisation of a regular graph $G$ is a partition of its edge set $E(G)$ into perfect matchings of $G$. Behague asked for the minimal $r=r(d)$ such that some $1$-factorisation of the $d$-dimensional hypercube $Q_d$ has the property that the union of any $r$ of its 1-factors is connected. Previous work by Laufer on perfect $1$-factorisations implied that $r$ is at least three, and Behague gave a construction with $r=\big\lceil\frac{d}{2}\big\rceil+1$. We improve this upper bound, giving a …

Predictive Hydrodynamic Simulations for Laser Direct-drive Implosion Experiments via Artificial Intelligence
Zixu Wang, Yuhan Wang, Junfei Ma, Fuyuan Wu, Junchi Yan, Xiaohui Yuan, Zhe Zhang, Jie Zhang
https://arxiv.org/abs/2507.16227

Predictive Hydrodynamic Simulations for Laser Direct-drive Implosion Experiments via Artificial Intelligence
This work presents predictive hydrodynamic simulations empowered by artificial intelligence (AI) for laser driven implosion experiments, taking the double-cone ignition (DCI) scheme as an example. A Transformer-based deep learning model MULTI-Net is established to predict implosion features according to laser waveforms and target radius. A Physics-Informed Decoder (PID) is proposed for high-dimensional sampling, significantly reducing the prediction errors compared to Latin hypercube sampling. …

Khovanov-Rozansky cycle calculus for bipartite links
A. Anokhina, E. Lanina, A. Morozov
https://arxiv.org/abs/2506.08721 https://arxi…

Khovanov-Rozansky cycle calculus for bipartite links
Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a straightforward generalization of the Khovanov calculus without a need of the technically complicated matrix factorization used for arbitrary $N$ in the Khovanov-Rozansky (KR) approach. The main object here is the $3^{n}$-dimensional hypercube with $n$ being the n…

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions
Ziang Chen, Liqiang Huang
https://arxiv.org/abs/2508.07559 https://arxi…

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions
We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE…

On the forts and related parameters of the hypercube graph
Boris Brimkov, Thomas R. Cameron, Owen Grubbs
https://arxiv.org/abs/2507.10826 https://

On the forts and related parameters of the hypercube graph
In 2018, forts were defined as non-empty subsets of vertices in a graph where no vertex outside the set has exactly one neighbor in the set. Forts have since been used to characterize zero forcing sets, model zero forcing as an integer program, and provide lower bounds on the zero forcing number. In this article, we give a complete characterization of minimum forts in the hypercube graph, showing that they are automorphic to one of two sets. In contrast, non-automorphic minimum zero forcing set…

Replaced article(s) found for cs.CC. https://arxiv.org/list/cs.CC/new
[1/1]:
- The Randomized Query Complexity of Finding a Tarski Fixed Point on the Boolean Hypercube
Simina Br\^anzei, Reed Phillips, Nicholas Recker

Hamiltonian paths extending a set of matchings in hypercubes
Abid Ali, Lina Ba, Weihua Yang
https://arxiv.org/abs/2506.21432 https://…

Hamiltonian paths extending a set of matchings in hypercubes
The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these paths have \( x \) or \( y \) as internal vertices nor both as endpoints. Dvořák and Gregor answered a problem posed by Caha and Koubek and proved that for every \( n \geq 5 \), there exist vertices \( x \) and \( y \) with a set of \( 2n - 4 \) edges in \( Q_n…

Hamiltonian laceability with a set of faulty edges in hypercubes
Abid Ali, Weihua Yang
https://arxiv.org/abs/2506.21391 https://arxiv…

Hamiltonian laceability with a set of faulty edges in hypercubes
Faulty networks are useful because link or node faults can occur in a network. This paper examines the Hamiltonian properties of hypercubes under certain conditional faulty edges. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and set of faulty edges \( F \) such that \( |F| \leq 4n - 17 \). We prove that a Hamiltonian path exists connecting any two vertices in \( Q_n - F \) from distinct partite sets if they verify the next two conditions: (i) in $Q_n - F$ any vertex has a …

Burning numbers via eigenpolytopes -- Hamming graphs, Johnson graphs, and halved cubes
Hajime Tanaka, Norihide Tokushige
https://arxiv.org/abs/2508.17559 https://

Burning numbers via eigenpolytopes -- Hamming graphs, Johnson graphs, and halved cubes
We give lower and upper bounds on the burning number of Hamming graphs, Johnson graphs, and halved cube graphs. For the lower bounds, we use the fact that $1$-skeletons of the eigenpolytopes of these graphs are isomorphic to the original graphs. Then, we present a dynamic search algorithm performed on the eigenpolytope to find an unburned vertex. This idea was originally used by Alon (Discrete Appl.\ Math.,\ 1992), who determined the burning number of the hypercube graphs.

$p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes
Michel Mollard (IF)
https://arxiv.org/abs/2507.16387 https://

$p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes
The Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $Γ^{(p)}_n$, which are subgraphs of $Q_n$ induced by strings without p consecutive 1s. We show the link between vertices of $Γ^{(p)}_n$ and compositions of integers with parts in $\{1, 2, \ldots , p\}$. Among other eumerative properties, we study the order, size and cube polynomial of $Γ^{(p)}_n$ as well as their gene…

Rainbow Trees in Hypercubes
Nicholas Crawford, Maya Sankar, Carl Schildkraut, Sam Spiro
https://arxiv.org/abs/2508.14186 https://arxiv.org/pdf/2508.14186…

Rainbow Trees in Hypercubes
We prove that every proper edge-coloring of the $n$-dimensional hypercube $Q_n$ contains a rainbow copy of every tree $T$ on at most $n$ edges. This result is best possible, as $Q_n$ can be properly edge-colored using only $n$ colors while avoiding rainbow cycles.

Hamiltonian connectivity of some base-cobase graphs
Leonardo Mart\'inez-Sandoval, Kolja Knauer
https://arxiv.org/abs/2506.15049 https://

Hamiltonian connectivity of some base-cobase graphs
There has been wide interest in understanding which properties of base graphs of matroids extend to base-cobase graphs of matroids. A significant result of Naddef and Pulleyblank (1984) shows that the $1$-skeleton of any $(0,1)$-polytope is either a hypercube, or Hamiltonian-connected, i.e. there is a Hamiltonian path connecting any two vertices. In particular, this is true for base graphs of matroids. A natural question raised by Farber, Richter, and Shank (1985) is whether this extends to bas…

Graphical Designs find Combinatorial Structures
Zawad Chowdhury, Stefan Steinerberger, Rekha R. Thomas
https://arxiv.org/abs/2507.13327 https://

Graphical Designs find Combinatorial Structures
Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and well-known combinatorial objects: orthogonal arrays in hypercube graphs, combinatorial block designs and extremizers of the Erdos-Ko-Rado theorem in Johnson graphs, and t-wise uniform sets of permutations and symmetric subgroups in normal Cayley graphs on the symmetri…

A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
Da-Wei Yang, Hongyang Zhang, Rong-Xia Hao, Sun-Yuan Hsieh
https://arxiv.org/abs/2507.12834

A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
This paper focuses on the embeddability of hypercubes in an important class of Cayley graphs, known as augmented cubes. An $n$-dimensional augmented cube $AQ_n$ is constructed by augmenting the $n$-dimensional hypercube $Q_n$ with additional edges, thus making $Q_n$ a spanning subgraph of $AQ_n$. Dong and Wang (2019) first posed the problem of determining the number of $Q_n$-isomorphic subgraphs in $AQ_n$, which still remains open. By exploiting the Cayley properties of $AQ_n$, we establish a l…

Majority bootstrap percolation on the permutahedron and other high-dimensional graphs
Maur\'icio Collares, Joshua Erde, Anna Geisler, Mihyun Kang
https://arxiv.org/abs/2507.06597

Majority bootstrap percolation on the permutahedron and other high-dimensional graphs
Majority bootstrap percolation is a model of infection spreading in networks. Starting with a set of initially infected vertices, new vertices become infected once half of their neighbours are infected. Balogh, Bollobás and Morris studied this process on the hypercube and showed that there is a phase transition as the density of the initially infected set increases. Generalising their results to a broad class of high-dimensional graphs, the authors of this work established similar bounds on th…