72 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
MODELLING AND PROCESSING THE DYNAMICS OF TOPOLOGICAL SIGNALS WITH THE DIRAC OPERATOR
Networks provide a powerful description of the complex systems that surround us. These structures support dynamical processes such as diffusion, synchronisation or percolation. The study of networks has remained limited to the pairwise formalism, where interactions take place between two participants. Simplicial complexes capture interactions between any number of elements. They are naturally equipped with a topological description, and support generalised dynamical variables called topological signals. Such signals can be found in many systems, yet have received little interest so far. Where studied, signals supported by structures of different dimension have been typically treated independently. This thesis explores the use of the Dirac operator as a tool rooted in topology to model the coupled dynamics of topological signals, and capture the interplay between signals of different orders. We apply this framework to describe the synchronisation dynamics of coupled phase os- cillators supported by nodes and edges in networks. We show through analytical and numerical investigations that the system undergoes explosive transitions between syn- chronised and incoherent states, while the synchronised state is characterised by coherent emergent oscillations. In the context of reaction-diffusion systems, we demonstrate how the Dirac operator is key to the formation of spatial heterogeneous patterns in topological signals supported by nodes and edges of a network. As a final illustration, we investi- gate two methods to filter and process real topological signals supported by nodes, edges and triangles in a simplicial complex. With the Dirac operator, we capture the interplay between signals of different orders and successfully jointly filter topological signals from artificial noise. This thesis illustrates how the Dirac operator provides a convenient formalism rooted in topology to consider the coupled dynamics of topological signals. This opens many further research avenues at the intersections of applied topology, dynamical systems, simplicial signal processing and higher-order network science
Learning the effective order of a hypergraph dynamical system
Dynamical systems on hypergraphs can display a rich set of behaviours not
observable for systems with pairwise interactions. Given a distributed
dynamical system with a putative hypergraph structure, an interesting question
is thus how much of this hypergraph structure is actually necessary to
faithfully replicate the observed dynamical behaviour. To answer this question,
we propose a method to determine the minimum order of a hypergraph necessary to
approximate the corresponding dynamics accurately. Specifically, we develop an
analytical framework that allows us to determine this order when the type of
dynamics is known. We utilize these ideas in conjunction with a hypergraph
neural network to directly learn the dynamics itself and the resulting order of
the hypergraph from both synthetic and real data sets consisting of observed
system trajectories
The complementary contribution of each order topology into the synchronization of multi-order networks
ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 12072262).Peer reviewedPostprin
Communities in temporal networks: from theoretical underpinnings to real-life applications
Static aggregations of network activity can unravel attributes of the complex systems they represent. However, they fall short when the structure of the systems changes over time. In some cases, changes are sluggish, such as in power grids, where lines enjoy a lengthy temporal permanence. In others, a high frequency of change is observed, such as on a network of online messages, social contacts, pathogen transmission or ball passing in a soccer game. In these cases, reducing what is inherently a temporal network to a static one, leads necessarily to a loss of information, such as causal relationships, precedence or reachability rules. Temporal networks are thus the main subject of this thesis, centered on the study of network evolution from the point of view of its clusters as significant meso-structures. The thesis has two interrelated parts. In the first, theoretical challenges are addressed and original algorithms, methods and tools are developed that can further the study of network theory. In the second, these developments are applied to the analysis of team invasion sports. A measurement of game dynamics was created based on a temporal network representation of a match, with nodes clustered by spatial proximity. These measurements were found to correlate with match events of known dynamics. Moreover, they reveal unique, multi-level, aspects of the game, from the individual players contributions, to the clusters of interacting players, to their teams and their matches, which is useful for game analysis, training and strategy development.As agregações estáticas das ligações de uma rede podem revelar atributos dos sistemas complexos que representam. Todavia, são insuficientes quando a estrutura dos sistemas se altera com o tempo. Em alguns casos, as transformações são lentas, tais como em redes de transmissão de eletricidade em que as linhas se mantêm inalteráveis por largos perÃodos de tempo. Noutras, regista-se uma taxa elevada de mudança, como por exemplo numa rede de mensagens em linha, contatos sociais, transmissão de patógenos ou passes num jogo de futebol. Nestes casos, reduzir o que é inerentemente uma rede temporal a uma rede estática, leva a uma perda de informação, tais como relações causais, regras de precedência ou de acessibilidade. Redes temporais são assim o tema desta tese, centrada nos seus agrupamentos, como meso-estruturas significantes. A tese está dividida em duas partes. Na primeira, são considerados problemas teóricos, e são desenvolvidos algoritmos, métodos e ferramentas que avançam o estudo da teoria de redes. Na segunda, estes desenvolvimentos são aplicados à análise de jogos desportivos coletivos de invasão. Foi criada uma medida de dinâmica do jogo, baseada na representação da partida através de uma rede temporal de nós agrupados por proximidade espacial. Os resultados obtidos correlacionam-se com eventos do jogo de dinâmica conhecida. Adicionalmente, esta medida revela aspetos únicos e multi-nÃvel da dinâmica do jogo, desde a contribuição individual do jogador, até aos agrupamentos de jogadores, da equipa e das partidas, útil para a análise do jogo, de treino e de desenvolvimento estratégico
Monoidal Width
We introduce monoidal width as a measure of complexity for morphisms in
monoidal categories. Inspired by well-known structural width measures for
graphs, like tree width and rank width, monoidal width is based on a notion of
syntactic decomposition: a monoidal decomposition of a morphism is an
expression in the language of monoidal categories, where operations are
monoidal products and compositions, that specifies this morphism. Monoidal
width penalises the composition operation along ``big'' objects, while it
encourages the use of monoidal products. We show that, by choosing the correct
categorical algebra for decomposing graphs, we can capture tree width and rank
width. For matrices, monoidal width is related to the rank. These examples
suggest monoidal width as a good measure for structural complexity of processes
modelled as morphisms in monoidal categories.Comment: Extended version of arXiv:2202.07582 and arXiv:2205.0891
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
- …