arXiv subjects

Jeannette Janssen

Publications and source records attributed to Jeannette Janssen.

At least 19 recordsLinked to original sources

arXiv2026

Unified Fourier transform on graphs sampled from stochastic block models

Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we show how a Fourier basis can be easily calculated from the block sizes and the block probability matrix. Using perturbation theory, we derive bounds on the sensitivity of the basis with respect to variations in the block sizes. We then consider SBMs constructed from weighted Cayley graphs. When block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. When block sizes are nearly uniform, we demonstrate that this Fourier basis closely approximates the SBM Fourier basis. For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.

eess.SP↗

arXiv2026

Convergence Rates of Ordering, Testing and Estimation Procedures for Graphons With Fast Boundary Decay Rates

In latent-position random graph models (LPMs), latent vertex positions $U_{1},\ldots,U_{n}$ are sampled from some distribution on a latent space $Ω$, then edges of an observed graph $G = ([n],E)$ are sampled with some probability $\mathbb{P}[(i,j) \in E ]=w(U_i,U_j)$ that depends on the unobserved latent positions. LPMs are ubiquitous in the statistical analysis of networks, offering models that have good empirical performance, strong theoretical guarantees, and tractable algorithms. The special case $Ω= [0,1]$ is important, as it corresponds to graphs with temporal or preference-based structure. In this paper, we study three problems related to LPMs with latent space $[0,1]$: \textit{ordering} the vertices according to the latent positions, \textit{estimating} the generating graphon $w$, and \textit{testing} whether an observed graph $G$ could have come from an LPM with state space $[0,1]$. Our results on the ordering problem greatly generalize two observations of Janssen/Smith (2022): (i) for \textit{some} families of graphons, the best estimate of the ordering converges much faster than the usual statistical rate of $\frac{1}{\sqrt{n}}$, and (ii) this occurs even though, for the same families of graphons, the best estimate of the latent positions still occurs at the usual $\frac{1}{\sqrt{n}}$ rate. As a main consequence, we develop a computationally-efficient graphon-estimation algorithm and show that it has the same convergence rate as the non-explicit optimal algorithm of Gao et al (2015). We also derive and analyze a testing procedure.

math.ST↗

arXiv2025

The Zero Forcing Number of Twisted Hypercubes

Twisted hypercubes are graphs that generalize the structure of the hypercube by relaxing the symmetry constraint while maintaining degree-regularity and connectivity. We study the zero forcing number of twisted hypercubes. Zero forcing is a graph infection process in which a particular colour change rule is iteratively applied to the graph and an initial set of vertices. We use the alternative framing of forcing arc sets to construct a family of twisted hypercubes of dimension k$\geq 3$ with zero forcing sets of size $2^{k-1}-2^{k-3}+1$, which is below the minimum zero forcing number of the hypercube.

math.CO↗

arXiv2024

An approximation algorithm for zero forcing

We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of $\text{pw}(G)+1$, where $\text{pw}(G)$ is the pathwidth of $G$. Starting from a path decomposition, the algorithm runs in $O(nm)$ time, where $n$ and $m$ are the order and size of the graph, respectively. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.

math.CO↗

arXiv2023

Graph sequences sampled from Robinson graphons

The function $Γ$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function $Γ$ is compatible with the cut-norm $\|\cdot \|_\Box$, in the sense that graphons close in cut-norm have similar $Γ$-values. Here we show the converse, by proving that every graphon $w$ can be approximated by a Robinson graphon $R_w$ so that $\|w-R_w\|_\Box$ is bounded in terms of $Γ(w)$. We then use classical techniques from functional analysis to show that a converging graph sequence $\{G_n\}$ converges to a Robinson graphon if and only if $Γ(G_n)\rightarrow 0$. Finally, using probabilistic techniques we show that the rate of convergence of $Γ$ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly $w$ exhibits the Robinson property.

math.CO↗

arXiv2023

Signal processing on large networks with group symmetries

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is subject to error or natural changes over time. This motivated a new perspective on GSP, where the signal processing framework is developed for an entire class of graphs with similar structures. This approach can be formalized via the theory of graph limits, where graphs are considered as random samples from a distribution represented by a graphon. When the network under consideration has underlying symmetries, they may be modeled as samples from Cayley graphons. In Cayley graphons, vertices are sampled from a group, and the link probability between two vertices is determined by a function of the two corresponding group elements. Infinite groups such as the 1-dimensional torus can be used to model networks with an underlying spatial reality. Cayley graphons on finite groups give rise to a Stochastic Block Model, where the link probabilities between blocks form a (edge-weighted) Cayley graph. This manuscript summarizes some work on graph signal processing on large networks, in particular samples of Cayley graphons.

eess.SP↗

arXiv2023

Orthogonal Colourings of Random Geometric Graphs

In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal colouring using $n^{1-2α}(1+o(1))$ colours with high probability. Then, we show for an infinite number of values of $n$, random geometric graphs with threshold $r<cn^{-\frac{1}{4}}$, $c<1$, have an optimal orthogonal colouring with high probability. We obtain both of these results by constructing orthogonal colourings of the clique grid graph.

math.CO↗

arXiv2022

A noncommutative approach to the graphon Fourier transform

Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon. Theoretically, graphons are limit objects of converging sequences of graphs. Our work extends previous research aiming to provide a common scheme for signal analysis of graphs that are similar in structure to a graphon. We extend a previous definition of graphon Fourier transform, and show that the graph Fourier transforms of graphs in a converging graph sequence converge to the graphon Fourier transform of the limiting graphon. We then apply this convergence result to signal processing on Cayley graphons. We show that Fourier analysis of the underlying group enables the construction of a suitable eigen-decomposition for the graphon, which can be used as a common framework for signal processing on graphs converging to the graphon.

math.CO↗

arXiv2022

Reconstruction of Line-Embeddings of Graphons

Consider a random graph process with $n$ vertices corresponding to points $v_{i} \sim {Unif}[0,1]$ embedded randomly in the interval, and where edges are inserted between $v_{i}, v_{j}$ independently with probability given by the graphon $w(v_{i},v_{j}) \in [0,1]$. Following Chuangpishit et al. (2015), we call a graphon $w$ diagonally increasing if, for each $x$, $w(x,y)$ decreases as $y$ moves away from $x$. We call a permutation $σ\in S_{n}$ an ordering of these vertices if $v_{σ(i)} < v_{σ(j)}$ for all $i < j$, and ask: how can we accurately estimate $σ$ from an observed graph? We present a randomized algorithm with output $\hatσ$ that, for a large class of graphons, achieves error $\max_{1 \leq i \leq n} | σ(i) - \hatσ(i)| = O^{*}(\sqrt{n})$ with high probability; we also show that this is the best-possible convergence rate for a large class of algorithms and proof strategies. Under an additional assumption that is satisfied by some popular graphon models, we break this "barrier" at $\sqrt{n}$ and obtain the vastly better rate $O^{*}(n^ε)$ for any $ε> 0$. These improved seriation bounds can be combined with previous work to give more efficient and accurate algorithms for related tasks, including: estimating diagonally increasing graphons, and testing whether a graphon is diagonally increasing.

math.PR↗

arXiv2021

Uniform Embeddings for Robinson Similarity Matrices

A Robinson similarity matrix is a symmetric matrix where the entry values on all rows and columns increase toward the diagonal. Decompose the Robinson matrix into the sum of k {0, 1}-matrices, then these k {0, 1}-matrices are the adjacency matrices of a set of nested unit interval graphs. Previous studies show that unit interval graphs coincide with indifference graphs. An indifference graph has an embedding that maps each vertex to a real number, where two vertices are adjacent if their embedding is within a fixed threshold distance. In this thesis, consider k different threshold distances, we study the problem of finding an embedding that, simultaneously and with respect to each threshold distance, embeds the k indifference graphs corresponding to the k adjacency matrices. This is called a uniform embedding of a Robinson matrix with respect to the k threshold distances. We give a sufficient and necessary condition on Robinson matrices that have a uniform embedding, which is derived from paths in an associated graph. We also give an efficient combinatorial algorithm to find a uniform embedding or give proof that it does not exist, for the case where k = 2.

math.CO↗

arXiv2021

Geometric random graphs and Rado sets of continuous functions

We prove the existence of Rado sets in the Banach space of continuous functions on [0,1]. A countable dense set S is Rado if with probability 1, the infinite geometric random graph on S, formed by probabilistically making adjacent elements of S that are within unit distance of each other, is unique up to isomorphism. We show that for a suitable measure which we construct, almost all countable dense sets in the subspaces of piecewise linear functions and of polynomials are Rado. Moreover, all graphs arising from such sets are of a unique isomorphism type. For the subspace of Brownian motion paths, almost all countable subsets are Rado (for a suitable measure) and the resulting graphs are of a unique isomorphism type. We show that the graph arising from piecewise linear functions and polynomials is not isomorphic to the graph arising from Brownian motion paths. Moreover, these graphs are non-isomorphic to graphs arising from Rado sets in $\mathbb{R}^n$, or the sequence spaces $c$ and $c_0$.

math.CO↗

arXiv2020

Orthogonal Colourings of Cayley Graphs

Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number of certain circulant graphs is explored. Lastly, orthogonal colourings of product graphs and Hamming graphs are studied.

math.CO↗

arXiv2019

Total Colourings of Direct Product Graphs

A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product graphs are determined. In particular, a sufficient condition is given for direct products of bipartite graphs to have total chromatic number equal to its maximum degree plus one. Partial results towards the total chromatic number of the direct product of complete graphs are also established.

math.CO↗

arXiv2019

High Degree Vertices and Spread of Infections in Spatially Modelled Social Networks

We examine how the behaviour of high degree vertices in a network affects whether an infection spreads through communities or jumps between them. We study two stochastic susceptible-infected-recovered (SIR) processes and represent our network with a spatial preferential attachment (SPA) network. In one of the two epidemic scenarios we adjust the contagiousness of high degree vertices so that they are less contagious. We show that, for this scenario, the infection travels through communities rather than jumps between them. We conjecture that this is not the case in the other scenario, when contagion is independent of the degree of the originating vertex. Our theoretical results and conjecture are supported by simulations.

cs.SI↗

arXiv2019

An optimization parameter for seriation of noisy data

A square symmetric matrix is a Robinson similarity matrix if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinson similarity matrix can be viewed as the affinity matrix between objects arranged in linear order, where objects closer together have higher affinity. We define a new parameter, $Γ_\max$, which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its $Γ_\max$ attains zero, and a matrix with small $Γ_\max$ is close (in the normalized $\ell^1$-norm) to a Robinson similarity matrix. Moreover, both $Γ_\max$ and the Robinson similarity approximation can be computed in polynomial time. Thus, our parameter recognizes Robinson similarity matrices which are perturbed by noise, and can therefore be a useful tool in the problem of seriation of noisy data.

math.CO↗

arXiv2018

Geometric random graphs and Rado sets in sequence spaces

We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability $p$ between pairs within threshold distance $δ$. A countable dense set in a metric space is {\sl Rado} if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that in finite dimensional spaces $\mathbb{R}^n$ equipped with the $\ell_{\infty}$ norm, all countable dense set satisfying a mild non-integrality condition are Rado. In this paper, we extend this result to infinite-dimensional spaces. If the underlying metric space is a separable Banach space, then we show in some cases that we can almost surely recover the Banach space from such a geometric random graph. More precisely, we show that in the sequence spaces $c$ and $c_0$, for measures $μ$ satisfying certain conditions, $μ^\N$-almost all countable sets are Rado. Moreover, with probability 1, in $c$ as in $c_0$, all graphs obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other. Finally, we show that representatives of the isomorphism classes obtained in this way from $c$ and $c_0$ are non-isomorphic to each other, and also non-isomorphic to their counterparts obtained from finite dimensional spaces.

math.CO↗

arXiv2017

The Spread of Cooperative Strategies on Grids with Random Asynchronous Updating

The Prisoner's Dilemma Process on a graph $G$ is an iterative process where each vertex, with a fixed strategy (cooperate or defect), plays the game with each of its neighbours. At the end of a round each vertex may change its strategy to that of its neighbour with the highest pay-off. Here we study the spread of cooperative and selfish behaviours on a toroidal grid, where each vertex is initially a cooperator with probability $p$. When vertices are permitted to change their strategies via a randomized asynchronous update scheme, we find that for some values of $p$ the limiting density of cooperators may be modelled as a polynomial in $p$. Theoretical bounds for this density are confirmed via simulation.

cs.DM↗

arXiv2017

Recovering the Structure of Random Linear Graphs

In a random linear graph, vertices are points on a line, and pairs of vertices are connected, independently, with a link probability that decreases with distance. We study the problem of reconstructing the linear embedding from the graph, by recovering the natural order in which the vertices are placed. We propose an approach based on the spectrum of the graph, using recent results on random matrices. We demonstrate our method on a particular type of random linear graph. We recover the order and give tight bounds on the number of misplaced vertices, and on the amount of drift from their natural positions.

math.CO↗