3,086 research outputs found
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 -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
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