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Pattern Formation on Hypergraphs

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Higher-Order Systems

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

We present a general framework that enables one to model high-order interactions among entangled dynamical systems, via hypergraphs. Several relevant processes can be ideally traced back to the proposed scheme. We shall here solely elaborate on the conditions that seed the spontaneous emergence of patterns, spatially heterogeneous solutions resulting from the many-body interaction between fundamental units. In particular we will focus, on two relevant settings. First, we will assume long-ranged mean field interactions between populations, and then turn to considering diffusive-like couplings. Two applications are presented, respectively to a generalised Volterra system and the Brusselator model.

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Notes

  1. 1.

    We will adopt the convention of using roman indexes for nodes and greek ones for edges.

  2. 2.

    Let us introduce \(\mathcal {L}^{ sym }=\mathbf {d}^{-1/2}\mathbf {L}^H\mathbf {d}^{-1/2}\), where \(\mathbf {d}\) is the diagonal matrix containing the \(d_i\)’s on the diagonal and \(\mathbf {L}^H\) is the high-order (combinatorial) Laplace matrix defined in[21]. Then \(\mathcal {L}^{ sym }=D_{ij}/\sqrt{d_i d_j} -\delta _{ij}\) from which it immediately follows that \(\mathcal {L}^{ sym }\) is symmetric and nonpositive definite; indeed take any \(\mathbf {x}\in \mathbb {R}^N\setminus \{0\}\), N standing for the dimension of the matrices, then \((\mathbf {x},\mathcal {L}^{ sym } \mathbf {x})=(\mathbf {d}^{-1/2} \mathbf {x},\mathbf {L}^H\mathbf {d}^{-1/2} \mathbf {x})\le 0\) where the last inequality follows from the fact that \(\mathbf {L}^H\) is nonpositive definite. Finally let us observe that \(\mathcal {L}=\mathbf {d}^{-1}\mathbf {L}^H =\mathbf {d}^{-1/2}\mathcal {L}^{ sym }\mathbf {d}^{1/2}\), hence, \(\mathcal {L}\) is similar to \(\mathcal {L}^{ sym }\) and, thus they display the same non-positive spectrum. Moreover this implies also that \(-2\le \Lambda ^{(\alpha )}\le 0\).

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Author information

Authors and Affiliations

  1. naXys, Namur Institute for Complex Systems, University of Namur, rue de Bruxelles 61, B5000, Namur, Belgium

    Timoteo Carletti

  2. Dipartimento di Fisica e Astronomia, CSDC and INFN, Università degli Studi di Firenze, via G. Sansone 1, 50019, Sesto Fiorentino, Italy

    Duccio Fanelli

Authors
  1. Timoteo Carletti
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  2. Duccio Fanelli
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Correspondence to Timoteo Carletti .

Editor information

Editors and Affiliations

  1. Department of Network and Data Science, Central European University, Vienna, Austria

    Federico Battiston

  2. ISI Foundation, Turin, Italy

    Giovanni Petri

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Carletti, T., Fanelli, D. (2022). Pattern Formation on Hypergraphs. In: Battiston, F., Petri, G. (eds) Higher-Order Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-91374-8_5

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