Three easy pieces for Hodge Laplacian and higher order interactions

Abstract: Firstly, we present a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. The renormalization group is a fundamental concept in the physics theory of scaling, scale-invariance, and universality. An RG scheme was recently introduced for complex networks with dyadic interactions based on diffusion dynamics. However, we still lack a general RG scheme for higher-order networks despite the mounting evidence of the importance of polyadic interactions. Our approach uses a diffusion process to group nodes or simplices, where information can flow between nodes and between simplices (higher-order interactions).

Secondly, we discuss simplicial Kuramoto models, which have emerged as a diverse and intriguing model that describes oscillators on simplices rather than nodes. We present a unified framework to describe different variants of these models, which are categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. We explore a potential application in reconstructing brain functional connectivity from structural connectomes. We find that simple edge-based Kuramoto models perform competitively or outperform complex extensions of node-based models.

Lastly, we consider associated games in cooperative game theory, which allows for the meaningful characterization of solution concepts. Moreover, generalized values allow computing each coalition's influence or power index in a game. We view associated games through the lens of game maps and graph Laplacian, thus defining the novel Hodge Generalized Value (HGV). We characterize HGV via an axiomatic approach as a generalized value. Finally, we show how HGV is linked to the solution of the Poisson equation derived from the Hodge decomposition of the direct graph associated with the poset of coalitions in the game.

References and coauthor list:

Nurisso, M., Morandini, M., Lucas, M., Vaccarino, F., Gili, T., & Petri, G. (2024). Higher-order Laplacian Renormalization. arXiv preprint arXiv:2401.11298.

Nurisso, M., Arnaudon, A., Lucas, M., Peach, R. L., Expert, P., Vaccarino, F., & Petri, G. (2023). A unified framework for Simplicial Kuramoto models. arXiv e-prints, arXiv-2305.

Mastropietro, Antonio, and Francesco Vaccarino. "The Shapley-Hodge Associated Game." arXiv preprint arXiv:2303.17151(2023).

geometric topology

Audience: researchers in the topic

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GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology

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