Namur Institute for Complex Systems

Abstract Non-reciprocal interactions play a crucial role in many social and biological complex systems. While directionality has been thoroughly accounted for in networks with pairwise interactions, its effects in systems with higher-order interactions have not yet been explored as deserved. Here, we introduce the concept of M -directed hypergraphs, a general class of directed higher-order structures, which allows to investigate dynamical systems coupled through directed group interactions. As an application we study the synchronization of nonlinear oscillators on 1-directed hypergraphs, finding that directed higher-order interactions can destroy synchronization, but also stabilize otherwise unstable synchronized states.

The evaluation complexity of general nonlinear, possibly nonconvex, constrained optimization is analyzed. It is shown that, under suitable smoothness conditions, an $\epsilon$-approximate first-order critical point of the problem can be computed in order $O(\epsilon^{1-2(p+1)/p})$ evaluations of the problem's functions and their first $p$ derivatives. This is achieved by using a two-phase algorithm inspired by Cartis, Gould, and Toint [SIAM J. Optim., 21 (2011), pp. 1721--1739; SIAM J. Optim., 23 (2013), pp. 1553--1574]. It is also shown that strong guarantees (in terms of handling degeneracies) on the possible limit points of the sequence of iterates generated by this algorithm can be obtained at the cost of increased complexity. At variance with previous results, the $\epsilon$-approximate first-order criticality is defined by satisfying a version of the KKT conditions with an accuracy that does not depend on the size of the Lagrange multipliers.

We consider random walks on dynamical networks where edges appear and disappear during finite time intervals. The process is grounded on three independent stochastic processes determining the walker's waiting time, the up time, and the down time of the edges. We first propose a comprehensive analytical and numerical treatment on directed acyclic graphs. Once cycles are allowed in the network, non-Markovian trajectories may emerge, remarkably even if the walker and the evolution of the network edges are governed by memoryless Poisson processes. We then introduce a general analytical framework to characterize such non-Markovian walks and validate our findings with numerical simulations.

An adaptive regularization algorithm using high-order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian $\ell_q$-norm regularization terms for $q\in (0,1)$. It is shown that the algorithm using a $p$th-order Taylor model for $p$ odd needs in general at most $O(\epsilon^{-(p+1)/p})$ evaluations of the objective function and its derivatives (at points where they are defined) to produce an $\epsilon$-approximate first-order critical point. This result is obtained either with Taylor models, at the price of requiring the feasible set to be “kernel centered” (which includes bound constraints and many other cases of interest), or with non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [C. Cartis, N. I. M. Gould, and Ph. L. Toint, IMA J. Numer. Anal., 32 (2012), pp. 1662--1695], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem's partially separable structure (if present) does not affect the complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.

This paper deals with the analysis of the nonisothermal axial dispersion tubular reactor. The existence of equilibrium profiles is investigated. In particular, for equal Peclet numbers, it is shown that one or three equilibria can be exhibited, depending on the parameters of the system, especially on the diffusion coefficient. In addition, different and close Peclet numbers are also considered. In these cases, it is also shown that the reactor has one or three equilibrium profiles. Some numerical simulations support the theoretical results.

Abstract When the novel coronavirus disease SARS-CoV2 (COVID-19) was officially declared a pandemic by the WHO in March 2020, the scientific community had already braced up in the effort of making sense of the fast-growing wealth of data gathered by national authorities all over the world. However, despite the diversity of novel theoretical approaches and the comprehensiveness of many widely established models, the official figures that recount the course of the outbreak still sketch a largely elusive and intimidating picture. Here we show unambiguously that the dynamics of the COVID-19 outbreak belongs to the simple universality class of the SIR model and extensions thereof. Our analysis naturally leads us to establish that there exists a fundamental limitation to any theoretical approach, namely the unpredictable non-stationarity of the testing frames behind the reported figures. However, we show how such bias can be quantified self-consistently and employed to mine useful and accurate information from the data. In particular, we describe how the time evolution of the reporting rates controls the occurrence of the apparent epidemic peak, which typically follows the true one in countries that were not vigorous enough in their testing at the onset of the outbreak. The importance of testing early and resolutely appears as a natural corollary of our analysis, as countries that tested massively at the start clearly had their true peak earlier and less deaths overall.

Light extraction from light-emitting materials is fundamentally limited by internal reflections due to the high dielectric-constant contrast between the material that produces the light and the emergent medium. These internal reflections can however be reduced significantly by a well-designed texturation of the surface of the emitting material. We used a genetic algorithm to determine optimal geometrical and material parameters for this texturation, the objective being to maximize the extraction of light of a GaN light-emitting diode. This study, which was restricted to two-dimensional texturations, shows that symmetric triangles actually correspond to the optimal shape. The dielectric constant of the material used for this texturation should ideally have the same dielectric constant as the GaN. The optimal texturation determined in this work leads to a light-extraction efficiency of 11.1%, which improves significantly the value of 3.7% obtained with a flat surface and the value of 5.7% achieved in previous work.

Abstract Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the co-existence of different timescales in the system. Here, we introduce random walks on general stochastic temporal networks allowing for lasting interactions, with up to three competing timescales. We then compare the mean resting time and stationary state of different models. We also discuss the accuracy of the mathematical analysis depending on the random walk model and the structure of the underlying network, and pay particular attention to the emergence of non-Markovian behaviour, even when all dynamical entities are governed by memoryless distributions.

Individual‐level trait diversity has been identified as an essential component of trait diversity (TD), influencing community assembly and structure. Traditionally, one employs trait diversity indices to measure facets of individual‐level trait diversity (divergence, richness and evenness). However, the application of species‐level trait diversity indices to individual‐level traits data and their implications have not been adequately studied. Thus, we examined the possible challenges of using four commonly used multi‐trait TD indices: Rao's quadratic entropy (Rao), functional dispersion (FDis), functional evenness (FEve) and functional richness (FRic); two indices primarily developed to measure individual‐level trait diversity: trait evenness distribution (TED‐for evenness) and trait onion peeling (TOP‐for richnness); and a modified version of TED (TEDM‐for evenness). Additionally, we considered an index that integrates both evenness and richness by generalizing ordinary Hill indices for traits (coined HIT). We measured individual‐level trait diversity with these indices using simulated traits data and experimental data from a growth experiment with cyanobacteria. Comparing the observed trends from the indices with the expected trends, we observed that only the trait divergence indices (FDis and Rao) produced the expected trends in the simulation scenarios and experimental data. TED and TEDM are not robust against the number of individuals used, and FEve is not sensitive to some changes in the location of individuals in the trait space. Also, TOP proved to be a discontinuous function dependent on the number of individuals, and FRic did not produce the anticipated trend when changes in the trait space did not affect the edges of the trait space. HIT did produce the anticipated changes, but it was only reliable when many individuals were sampled. In summary, applying these individual‐level trait diversity indices to quantify anything except trait divergence may lead to misinterpretation of the original situation of trait distribution in the trait space if their specific properties are not adequately considered.

FUNDP, GRT)Par consquent, le besoin s'est fait sentir de raliser au niveau fdral une nouvelle enqute du mme type que MOBEL.Celle-ci constituerait un outil permettant aux dcideurs et, plus gnralement, tous les acteurs de la mobilit en Belgique, de disposer d'un portrait rcent des comportements de mobilit de la population belge.Elle devrait aussi permettre de percevoir, par comparaison avec MOBEL, quelles sont les volutions qui font jour dans la mobilit en Belgique.C'est dans cette perspective qu'a t mis sur pied ds 2008 le projet BELDAM (BELgian Daily Mobility).Le SPF Mobilit & Transports, conscient du besoin d'une telle enqute a sollicit la Politique scientifique fdrale (BELSPO) pour, dans le cadre du programme AGORA destin soutenir le dveloppement des donnes publiques fdrales, monter ensemble un financement afin de mener bien cette entreprise.Cela a t accept par le Comit interdpartemental de Coordination AGORA et BELSPO s'est engag financer la partie scientifique du projet, savoir la prparation de l'enqute et l'exploitation des rsultats tandis que le SPF dgageait le budget pour l'excution de l'enqute proprement dite.Un appel d'offres a donc t lanc en mars 2008 pour dsigner l'quipe scientifique en charge du projet.C'est un consortium coordonn par le GRT 2 (FUNDP -Universit de Namur) qui a t retenu.Outre le GRT, qui avait dj coordonn l'enqute MOBEL, ce consortium comprendrait l'IMOB 3 (Universit de Hasselt) et le CES 4 (Facults Universitaires Saint Louis, Bruxelles), deux centres de recherches prsentant galement une grande exprience dans les questions de mobilit (par exemple, l'IMOB coordonne les enqutes rgionales flamandes (OVG) et le CES est partie prenante dans l'Observatoire de la Mobilit en Rgion bruxelloise).

In this article, Stochastic port-Hamiltonian systems (SPHS) on infinite-dimensional spaces governed by Itô stochastic differential equations (SDEs) are introduced, and some properties of this new class of systems are studied. They are an extension of SPHSs defined on a finite-dimensional state space. The concept of well-posedness in the sense of Weiss and Salamon is generalized to the stochastic context. Under this extended definition, SPHSs are shown to be well posed. The theory is illustrated on an example of a vibrating string subject to a Hilbert space-valued Gaussian white noise process.

Abstract We propose a one-parameter family of random walk processes on hypergraphs, where a parameter biases the dynamics of the walker towards hyperedges of low or high cardinality. We show that for each value of the parameter, the resulting process defines its own hypergraph projection on a weighted network. We then explore the differences between them by considering the community structure associated to each random walk process. To do so, we adapt the Markov stability framework to hypergraphs and test it on artificial and real-world hypergraphs.

Systems of oscillators whose internal phases and spatial dynamics are coupled, swarmalators, present diverse collective behaviors which in some cases lead to explosive synchronization in a finite population as a function of the coupling parameter between internal phases. Near the synchronization transition, the phase energy of the particles is represented by the XY model, and they undergo a transition which can be of the first order or the second depending on the distribution of natural frequencies of their internal dynamics. The first-order transition is obtained after an intermediate state (static wings phase wave state) from which the nodes, in cascade over time, achieve complete phase synchronization at a precise value of the coupling constant. For a particular case of natural frequencies distribution, a new phenomenon, the rotational splintered phase wave state, is observed and leads progressively to synchronization through clusters switching alternatively from one to two and for which the frequency decreases as the phase coupling increases.

The worst-case complexity of the steepest-descent algorithm with exact linesearches for unconstrained smooth optimization is analyzed, and it is shown that the number of iterations of this algorithm which may be necessary to find an iterate at which the norm of the objective function's gradient is less that a prescribed epsilon is, essentially, a multiple of 1/epsilon(superscript)2, as is the case for variants of the same algorithms using inexact linesearches.

We present a genetic algorithm that we developed in order to address computationally expensive optimization problems in optical engineering. The idea consists of working with a population of individuals representing possible solutions to the problem. The best individuals are selected. They generate new individuals for the next generation. Random mutations in the coding of parameters are introduced. This strategy is repeated from generation to generation until the algorithm converges to the global optimum of the problem considered. For computationally expensive problems, one can analyze the data collected by the algorithm in order to infer more rapidly the final solution. The use of a mutation operator that acts on randomly-shifted Gray codes helps the genetic algorithm escape local optima and enables a wider diversity of displacements. These techniques reduce the computational cost of optical engineering problems, where the design parameters have a finite resolution and are limited to a realistic range. We demonstrate the performance of this algorithm by considering a set of 22 benchmark problems in 5, 10 and 20 dimensions that reflect the conditions of these engineering problems. We finally show how these techniques accelerate the determination of optimal structures for the broadband absorption of electromagnetic radiations.

The process of diffusion is the most elementary stochastic transport process. Brownian motion, the representative model of diffusion, played a important role in the advancement of scientific fields such as physics, chemistry, biology and finance. However, in recent decades, non-diffusive transport processes with non-Brownian statistics were observed experimentally in a multitude of scientific fields. Examples include human travel, in-cell dynamics, the motion of bright points on the solar surface, the transport of charge carriers in amorphous semiconductors, the propagation of contaminants in groundwater, the search patterns of foraging animals and the transport of energetic particles in turbulent plasmas. These examples showed that the assumptions of the classical diffusion paradigm, assuming an underlying uncorrelated (Markovian), Gaussian stochastic process, need to be relaxed to describe transport processes exhibiting a non-local character and exhibiting long-range correlations. This article does not aim at presenting a complete review of non-diffusive transport, but rather an introduction for readers not familiar with the topic. For more in depth reviews, we recommend some references in the following. First, we recall the basics of the classical diffusion model and then we present two approaches of possible generalizations of this model: the Continuous-Time-Random-Walk (CTRW) and the fractional Lévy motion (fLm).

A generalization of the Bernstein matrix concentration inequality to random tensors of general order is proposed. This generalization is based on the use of Einstein products between tensors, from which a strong link can be established between matrices and tensors, in turn allowing exploitation of existing results for the former.

The Koopman operator is a linear, infinite-dimensional operator defined for a nonlinear dynamical system. Through this linear operator, we can exploit established linear techniques (e.g. linear algebra, function analysis, operator theory) to tackle a wide variety of nonlinear problems in systems and control. In this paper, we present some research challenges in Koopman operator approach to nonlinear systems theory and control. Our discussion begins with a review of the current status of this approach—definitions of the Koopman operators for systems without/with inputs. We then pose distinct problems on identification, structural analysis, controller design, and computation related to Koopman operator theory in nonlinear control systems.

Quels sont les déterminants de la mobilité résidentielle en Belgique ? Quand on déménage, quels sont les facteurs de choix d'une nouvelle commune de résidence ? Telles étaient les questions au cœur du projet MOBLOC. Dans ce cadre, un modèle a été développé en deux parties: un modèle de propension à migrer et un modèle de choix de nouvelle localisation. L'objectif étant de travailler avec un niveau de désagrégation spatiale fin (les 589 communes belges), des outils de micro-simulation étaient également nécessaires. La méthode originale développée par le GRT pour créer des populations synthétiques est donc utile de même que des techniques pour faire évoluer ces populations dans le temps. Ainsi, ces outils et les modèles développés permettent de réaliser des études prospectives et d'estimer les évolutions futures des localisations résidentielles en Belgique sur un maillage géographique fin. Dans cet article, nous présentons principalement les méthodologies développés et les modèles mis au point.

The Gillespie algorithm provides statistically exact methods to simulate stochastic dynamics modelled as interacting sequences of discrete events including systems of biochemical reactions or earthquakes, networks of queuing processes or spiking neurons, and epidemic and opinion formation processes on social networks. Empirically, inter-event times of various human activities, in particular human communication, and some natural phenomena are often distributed according to long-tailed distributions. The Gillespie algorithm and its extant variants either assume the Poisson process, which produces exponentially distributed inter-event times, not long-tailed distributions, assume particular functional forms for time courses of the event rate, or works for non-Poissonian renewal processes including the case of long-tailed distributions of inter-event times but at a high computational cost. In the present study, we propose an innovative Gillespie algorithm for renewal processes on the basis of the Laplace transform. It uses the fact that a class of point processes is represented as a mixture of Poisson processes with different event rates. The method allows renewal processes whose survival function of inter-event times is completely monotone functions and works faster than a recently proposed Gillespie algorithm for general renewal processes. We also propose a method to generate sequences of event times with a given distribution of inter-event times and a tunable amount of positive correlation between inter-event times. We demonstrate our algorithm with exact simulations of epidemic processes on networks. We find that positive correlation in inter-event times modulates dynamics but in a quantitatively minor way with the amount of positive correlation comparable with empirical data.