A coherence theorem for pseudonatural transformations

We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing rewriting techniques based on Squier's Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part of the structure, and we use a new rewriting result to conclude. To this end, we introduce the notions of white-categories and partial coherence

Verifying the Steane code with Quantomatic

In this paper we give a partially mechanized proof of the correctness of Steane's 7-qubit error correcting code, using the tool Quantomatic. To the best of our knowledge, this represents the largest and most complicated verification task yet carried out using Quantomatic.Comment: In Proceedings QPL 2013, arXiv:1412.791

The Diamond Lemma for non-terminating rewriting systems using deterministic reduction strategies

International audienc

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

Do higher-order interactions promote synchronization?

Understanding how nonpairwise interactions alter dynamical processes in networks is of fundamental importance to the characterization and control of many coupled systems. Recent discoveries of hyperedge-enhanced synchronization under various settings raised speculations that such enhancements might be a general phenomenon. Here, we demonstrate that even for simple systems such as Kuramoto oscillators, the effects of higher-order interactions are highly representation-dependent. Specifically, we show numerically and analytically that hyperedges typically enhance synchronization in random hypergraphs, but have the opposite effect in simplicial complexes. As an explanation, we identify higher-order degree heterogeneity as the key structural determinant of synchronization stability in systems with a fixed coupling budget. Our findings highlight the importance of appropriate representations in describing higher-order interactions. In particular, the choice of simplicial complexes or hypergraphs has significant ramifications and should not be purely motivated by technical conveniences.Comment: Comments welcome! Y.Z. and M.L. contributed equally to this work. Code available at https://github.com/maximelucas/higherorder_sync_promote