Synchronisation Games on Hypergraphs

We study a strategic game model on hypergraphs where players, modelled by nodes, try to coordinate or anti-coordinate their choices within certain groups of players, modelled by hyperedges. We show this model to be a strict generalisation of symmetric additively separable hedonic games to the hypergraph setting and that such games always have a pure Nash equilibrium, which can be computed in pseudo-polynomial time. Moreover, in the pure coordination setting, we show that a strong equilibrium exists and can be computed in polynomial time when the game possesses a certain acyclic structure.</jats:p

Modelling Players' Interactions in Football: A Multilevel Hypernetworks Approach.

Na presente tese procura-se avançar com fundamentação teórica e prática, assim como com demonstrações empíricas referentes à reconceptualização das equipas de futebol enquanto redes sociais complexas. Estas redes evidenciam comportamentos sinérgicos emergentes e auto-organizados cuja complexidade, enraizada nas redes de interações dos jogadores, pode ser discernida através da análise de redes sociais. Não obstante, as técnicas tradicionais de rede exibem algumas limitações que podem levar a dados imprecisos e falaciosos. Essas limitações estão relacionadas com a exagerada ênfase que é colocada nos comportamentos de ataque das equipas, negligenciando-se as ações defensivas. Tal leva a que: a troca de informações incida maioritariamente nos comportamentos de passe; a variabilidade do comportamento dos jogadores seja, na maioria dos casos, desconsiderada; e a maioria das métricas usadas para modelar as interações dos jogadores se baseiem em distâncias geodésicas. Assim, as hiperredes multiníveis são aqui propostas enquanto nova abordagem metodológica capaz de superar aquelas limitações. Esta abordagem multinível caracteriza-se por um conjunto de conceitos e ferramentas metodológicas coerentes com a análise da dinâmica relacional subjacente aos processos sinergísticos evidenciados durante a competição. Por um lado, estes processos foram capturados na dinâmica de alteração das configurações táticas exibidas pelas equipas durante a competição, pela quantificação do tipo de simplices (interações de grupos de jogadores, e.g., 2vs.1) atendendo à localização da bola, e na dinâmica de interação, transformação dos simplices em determinados eventos do jogo. Por outro lado, a aplicação das hiperredes multiníveis permitiu, de igual modo, capturar as tendências de sincronização local (nível meso) emergentes em contextos de prática. Esta tese destacou o valor da adoção de uma abordagem de hiperredes multiníveis para melhorar a compreensão sobre os processos sinérgicos dos jogadores e equipas de futebol emergentes durante a prática e a competição. Estas poderão vir a revelar-se ferramentas promissoras na análise da performance desportiva, tendo igualmente um papel relevante na monitorização e controlo do treino.PALAVRAS-CHAVE: FUTEBOL, CIÊNCIA DAS REDES, HIPERREDES MULTINÍVEL, DINÂMICA DA EQUIPA, ANÁLISE DA PERFORMANCEThis thesis aims to advance practical and theoretical understanding, as well as empirical evidence regarding the re-conceptualisation of Football teams as complex social networks. These networks display synergetic, emergent and self-organised behaviour and the complexity rooted in the networks of players' interactions can be discerned through analysis of social networks. Notwithstanding, traditional network techniques display some limitations that can lead to inaccurate and misleading data. Such limitations are related with an over-emphasis on network attacking behaviours thus neglecting the defensive actions of the opposing team. This leads to: information exchange mainly analysed through passing behaviours; the variability of players' performance is in most cases disregarded; most metrics used to model players' interactions are based on geodesic distances. Thus, multilevel hypernetworks are proposed as a novel methodological approach capable of overriding such limitations. This multilevel approach is characterised by a set of conceptual and methodological tools consistent with analysis of the relational dynamics underlying the synergistic processes evidenced during competition. On the one hand, these processes were captured in the changing dynamics of tactical configurations of teams during competition, by the quantification of the type of simplices (interactions between sub-groups of players, e.g., 2vs.1) in relation to ball location, and in the dynamics of simplices' interactions and transformations in certain game events. On the other hand, the application of multilevel hypernetworks allowed to capture local (meso level) synchronisation tendencies in practice contexts. This thesis highlighted the value of adopting a multilevel hypernetworks approach for enhancing understanding about the synergistic processes of players and football teams emerging during practice and competition. These tools may prove to be promising in the analysis of sports performance, also having an important role in the monitoring and control of training

Innocent strategies as presheaves and interactive equivalences for CCS

Seeking a general framework for reasoning about and comparing programming languages, we derive a new view of Milner's CCS. We construct a category E of plays, and a subcategory V of views. We argue that presheaves on V adequately represent innocent strategies, in the sense of game semantics. We then equip innocent strategies with a simple notion of interaction. This results in an interpretation of CCS. Based on this, we propose a notion of interactive equivalence for innocent strategies, which is close in spirit to Beffara's interpretation of testing equivalences in concurrency theory. In this framework we prove that the analogues of fair and must testing equivalences coincide, while they differ in the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014

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

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems

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

Multiorder Laplacian for synchronization in higher-order networks

Traditionally, interaction systems have been described as networks, where links encode information on the pairwise influences among the nodes. Yet, in many systems, interactions take place in larger groups. Recent work has shown that higher-order interactions between oscillators can significantly affect synchronization. However, these early studies have mostly considered interactions up to 4 oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this scope, we introduce a multi-order Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multi-order Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multi-order Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way towards a general treatment of dynamical processes beyond pairwise interactions.Comment: Was "A multi-order Laplacian framework for the stability of higher-order synchronization