The emergence of Large Language Models (LLMs) with increasingly sophisticated natural language understanding and generative capabilities has sparked interest in the Agent-based Modelling (ABM) community. With their ability to summarize, generate, analyze, categorize, transcribe and translate text, answer questions, propose explanations, sustain dialogue, extract information from unstructured text, and perform logical reasoning and problem-solving tasks, LLMs have a good potential to contribute to the modelling process. After reviewing the current use of LLMs in ABM, this study reflects on the opportunities and challenges of the potential use of LLMs in ABM. It does so by following the modelling cycle, from problem formulation to documentation and communication of model results, and holding a critical stance.

The Canonical Polyadic (CP) tensor decomposition is frequently used as a model in applications in a variety of different fields. Using jackknife resampling to estimate parameter uncertainties is often desirable but results in an increase of the already high computational cost. Upon observation that the resampled tensors, though different, are nearly identical, we show that it is possible to extend the recently proposed Concurrent ALS (CALS) technique to a jackknife resampling scenario. This extension gives access to the computational efficiency advantage of CALS for the price of a modest increase (typically a few percent) in the number of floating point operations. Numerical experiments on both synthetic and real-world datasets demonstrate that the new workflow based on a CALS extension can be several times faster than a straightforward workflow where the jackknife submodels are processed individually.

Hyperedge replacement (HR) grammars can generate NP-complete graph languages, which makes parsing hard even for fixed HR languages. Therefore, we study predictive shift-reduce (PSR) parsing that yields efficient parsers for a subclass of HR grammars, by generalizing the concepts of SLR(1) string parsing to graphs. We formalize the construction of PSR parsers and show that it is correct. PSR parsers run in linear space and time, and are more efficient than the predictive top-down (PTD) parsers recently developed by the authors.

In recent breakthrough results, Saxton--Thomason and Balogh--Morris--Samotij have developed powerful theories of hypergraph containers. These theories have led to a large number of new results on transference, and on counting and characterising typical graphs in hereditary properties. In a different direction, Hatami--Janson--Szegedy proved results on the entropy of graph limits which count and characterise graphs in dense hereditary properties. In this paper, we make a threefold contribution to this area of research: 1) We generalise results of Saxton--Thomason to obtain container theorems for general, dense hereditary properties of multicoloured graphs. Our main tool is the adoption of an entropy-based framework. As corollaries, we obtain general counting, characterization and transference results. We further extend our results to cover a variety of combinatorial structures: directed graphs, oriented graphs, tournaments, multipartite graphs, multi-graphs, hypercubes and hypergraphs. 2) We generalise the results of Hatami--Janson--Szegedy on the entropy of graph limits to the setting of decorated graph limits. In particular we define a cut norm for decorated graph limits and prove compactness of the space of decorated graph limits under that norm. 3) We explore a weak equivalence between the container and graph limit approaches to counting and characterising graphs in hereditary properties. In one direction, we show how our multicolour container results may be used to recover decorated versions of the results of Hatami--Janson--Szegedy. In the other direction, we show that our decorated extensions of Hatami--Janson--Szegedy's results on graph limits imply counting and characterization applications. Similar container results were recently obtained independently by Terry.

Deep graph learning and network science both analyze graphs but approach similar problems from different perspectives. Whereas network science focuses on models and measures that reveal the organizational principles of complex systems with explicit assumptions, deep graph learning focuses on flexible and generalizable models that learn patterns in graph data in an automated fashion. Despite these differences, both fields share the same goal: to better model and understand patterns in graph-structured data. Early efforts to integrate methods, models, and measures from network science and deep graph learning indicate significant untapped potential. In this position, we explore opportunities at their intersection. We discuss open challenges in deep graph learning, including data augmentation, improved evaluation practices, higher-order models, and pooling methods. Likewise, we highlight challenges in network science, including scaling to massive graphs, integrating continuous gradient-based optimization, and developing standardized benchmarks.

We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified Hunter--Saxton equation, and the two-component Hunter--Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.

In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colourings of very general objects, which include both vertex- and edge-colourings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterisation and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity and multicoloured graphs amongst others. Similar results were recently and independently obtained by Terry.

The level of abstraction at which application experts reason about linear algebra computations and the level of abstraction used by developers of high-performance numerical linear algebra libraries do not match. The former is conveniently captured by high-level languages and libraries such as Matlab and Eigen, while the latter expresses the kernels included in the BLAS and LAPACK libraries. Unfortunately, the translation from a high-level computation to an efficient sequence of kernels is a task, far from trivial, that requires extensive knowledge of both linear algebra and high-performance computing. Internally, almost all high-level languages and libraries use efficient kernels; however, the translation algorithms are too simplistic and thus lead to a suboptimal use of said kernels, with significant performance losses. In order to both achieve the productivity that comes with high-level languages, and make use of the efficiency of low level kernels, we are developing Linnea, a code generator for linear algebra problems. As input, Linnea takes a high-level description of a linear algebra problem and produces as output an efficient sequence of calls to high-performance kernels. In 25 application problems, the code generated by Linnea always outperforms Matlab, Julia, Eigen and Armadillo, with speedups up to and exceeding 10x.

Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on a common vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a triangle in $V$. In this paper we consider the following question: what triples of minimum degree conditions $(\delta(G_1), \delta(G_2), \delta(G_3))$ guarantee the existence of a rainbow triangle? This may be seen as a minimum degree version of a problem of Aharoni, DeVos, de la Maza, Montejanos and Šámal on density conditions for rainbow triangles, which was recently resolved by the authors. We establish that the extremal behaviour in the minimum degree setting differs strikingly from that seen in the density setting, with discrete jumps as opposed to continuous transitions. Our work leaves a number of natural questions open, which we discuss.

We consider the problem of ranking a set of objects based on their performance when the measurement of said performance is subject to noise. In this scenario, the performance is measured repeatedly, resulting in a range of measurements for each object. If the ranges of two objects do not overlap, then we consider one object as 'better' than the other, and we expect it to receive a higher rank; if, however, the ranges overlap, then the objects are incomparable, and we wish them to be assigned the same rank. Unfortunately, the incomparability relation of ranges is in general not transitive; as a consequence, in general the two requirements cannot be satisfied simultaneously, i.e., it is not possible to guarantee both distinct ranks for objects with separated ranges, and same rank for objects with overlapping ranges. This conflict leads to more than one reasonable way to rank a set of objects. In this paper, we explore the ambiguities that arise when ranking with ties, and define a set of reasonable rankings, which we call partial rankings. We develop and analyse three different methodologies to compute a partial ranking. Finally, we show how performance differences among objects can be investigated with the help of partial ranking.

Fuzzing is a well-established technique for detecting bugs and vulnerabilities. With the surge of fuzzers and fuzzer platforms being developed such as AFL and OSSFuzz rises the necessity to benchmark these tools' performance. A common problem is that vulnerability benchmarks are based on bugs in old software releases. For this very reason, Magma introduced the notion of forward-porting to reintroduce vulnerable code in current software releases. While their results are promising, the state-of-the-art lacks an update on the maintainability of this approach over time. Indeed, adding the vulnerable code to a recent software version might either break its functionality or make the vulnerable code no longer reachable. We characterise the challenges with forward-porting by reassessing the portability of Magma's CVEs four years after its release and manually reintroducing the vulnerabilities in the current software versions. We find the straightforward process efficient for 17 of the 32 CVEs in our study. We further investigate why a trivial forward-porting process fails in the 15 other CVEs. This involves identifying the commits breaking the forward-porting process and reverting them in addition to the bug fix. While we manage to complete the process for nine of these CVEs, we provide an update on all 15 and explain the challenges we have been confronted with in this process. Thereby, we give the basis for future work towards a sustainable forward-ported fuzzing benchmark.

We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is $(a:b)$-colorable if every edge is properly colored. We prove that for any $2\leq b\leq a-2\leq n/\ln n$, every $n$-uniform hypergraph $H$ with $ |E(H)| \leq (ab^3)^{-1/2}\left(\frac{n}{\log n}\right)^{1/2} \left(\frac{a}{b}\right)^{n-1} $is proper$(a:b)$-colorable. We also address specific cases, including $(a:a-1)$-colorability.

An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ is called the weak saturation number $\mathrm{wsat}(n, F)$. We obtain exact values and new bounds for $\mathrm{wsat}(n, K_{s,t})$ in the previously unaddressed range s+t < n < 3t-3, where $3\leq s\leq t$. To prove lower bounds, we introduce a new method that takes into account connectivity properties of subgraphs of a complement $G'$ to a weakly saturated graph $G$. We construct an auxiliary hypergraph and show that a linear combination of its parameters always increases in the process of the deletion of edges of $G'$. This gives a lower bound which is tight, up to an additive constant.

We consider the random right-angled Coxeter group $W_{\Gamma}$ whose presentation graph $\Gamma\sim \mathcal{G}_{n,p}$ is an Erd{\H o}s--Rényi random graph on $n$ vertices with edge probability $p=p(n)$. We establish that $p=1/\sqrt{n}$ is a threshold for relative hyperbolicity of the random group $W_{\Gamma}$. As a key step in the proof, we determine the minimal number of pairs of generators that must commute in a right-angled Coxeter group which is not relatively hyperbolic, a result which is of independent interest. We also show that there is an interval of edge probabilities of width $\Omega(1/\sqrt{n})$ in which the random right-angled Coxeter group has precisely cubic divergence. This interval is between the thresholds for relative hyperbolicity (whence exponential divergence) and quadratic divergence. Moreover, a simple random walk on any Cayley graph of the random right-angled Coxeter group for $p$ in this interval satisfies a central limit theorem.

An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of vertices spans at most $q$ edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry [An extremal problem with a transcendental solution, Combinatorics Probability and Computing 2019] posed the problem of determining the maximum of the product of the edge multiplicities in $(n,s,q)$-graphs. We give a general lower bound construction for this problem for many pairs $(s,q)$, which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the $(s,q)=(4,6a+3)$ case of the problem (for $a\geq2$); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for `sparse' multigraphs (i.e. when $q\leq 2\binom{s}{2}$). Finally we introduce some tools that are likely to be useful for attacking the problem in general.

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erdős, Łuczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a largest full subgraph of $G$. If $p\binom{n}{2}$ is a non-negative integer, define $$f(n,p) = \min\{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} \}.$$ Erdős, Łuczak and Spencer proved that for $n \geq 2$, $$(2n)^{\frac{1}{2}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{\frac{2}{3}}(\log n)^{\frac{1}{3}}.$$ In this paper, we prove the following lower bound: for $n^{-\frac{2}{3}}

The product of a matrix chain consisting of $n$ matrices can be computed in $C_{n-1}$ (Catalan's number) different ways, each identified by a distinct parenthesisation of the chain. The best algorithm to select a parenthesisation that minimises the cost runs in $O(n \log n)$ time. Approximate algorithms run in $O(n)$ time and find solutions that are guaranteed to be within a certain factor from optimal; the best factor is currently $1.155$. In this article, we first prove two results that characterise different parenthesisations, and then use those results to improve on the best known approximation algorithms. Specifically, we show that (a) each parenthesisation is optimal somewhere in the problem domain, and (b) exactly $n + 1$ parenthesisations are essential in the sense that the removal of any one of them causes an unbounded penalty for an infinite number of problem instances. By focusing on essential parenthesisations, we improve on the best known approximation algorithm and show that the approximation factor is at most $1.143$.

We observe a disconnect between the developers and the end users of linear algebra libraries. On the one hand, the numerical linear algebra and the high-performance communities invest significant effort in the development and optimization of highly sophisticated numerical kernels and libraries, aiming at the maximum exploitation of both the properties of the input matrices, and the architectural features of the target computing platform. On the other hand, end users are progressively less likely to go through the error-prone and time consuming process of directly using said libraries by writing their code in C or Fortran; instead, languages and libraries such as Matlab, Julia, Eigen and Armadillo, which offer a higher level of abstraction, are becoming more and more popular. Users are given the opportunity to code matrix computations with a syntax that closely resembles the mathematical description; it is then a compiler or an interpreter that internally maps the input program to lower level kernels, as provided by libraries such as BLAS and LAPACK. Unfortunately, our experience suggests that in terms of performance, this translation is typically vastly suboptimal. In this paper, we first introduce the Linear Algebra Mapping Problem, and then investigate how effectively a benchmark of test problems is solved by popular high-level programming languages. Specifically, we consider Matlab, Octave, Julia, R, Armadillo (C++), Eigen (C++), and NumPy (Python); the benchmark is meant to test both standard compiler optimizations such as common subexpression elimination and loop-invariant code motion, as well as linear algebra specific optimizations such as optimal parenthesization of a matrix product and kernel selection for matrices with properties. The aim of this study is to give concrete guidelines for the development of languages and libraries that support linear algebra computations.

The state-of-the-art methods for drum transcription in the presence of melodic instruments (DTM) are machine learning models trained in a supervised manner, which means that they rely on labeled datasets. The problem is that the available public datasets are limited either in size or in realism, and are thus suboptimal for training purposes. Indeed, the best results are currently obtained via a rather convoluted multi-step training process that involves both real and synthetic datasets. To address this issue, starting from the observation that the communities of rhythm games players provide a large amount of annotated data, we curated a new dataset of crowdsourced drum transcriptions. This dataset contains real-world music, is manually annotated, and is about two orders of magnitude larger than any other non-synthetic dataset, making it a prime candidate for training purposes. However, due to crowdsourcing, the initial annotations contain mistakes. We discuss how the quality of the dataset can be improved by automatically correcting different types of mistakes. When used to train a popular DTM model, the dataset yields a performance that matches that of the state-of-the-art for DTM, thus demonstrating the quality of the annotations.