On the equivalence between nonlinear graph-based dynamics and linear dynamics on higher-order networks
Lacasa, Lucas
Submitted (2026)
Submitted (2026)
In network science, collective dynamics of complex systems are typically modelled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (i.e. hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this work we discuss such structural--dynamical representation duality, and investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. Using Carleman linearisation arguments, we show that finite polynomial dynamics defined in the |V| vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order |V|, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.
Related research projects
CSxAI
Complexity science for understanding AI: from the dynamics of complex networks to collective effects of interacting neural networks
I.P.: Lucas Lacasa, Víctor M. Eguíluz
The CSxAI project engages researchers at IFISC in a (non-oriented research project) on the interdisciplinary challenge of applying tools and concepts of complexity science (dynamical systems, statistical physics, and network science) to …
NEHIL
Neuromorphic-Enhanced Heterogeneously-Integrated FMCW LiDAR
I.P.: Miguel C. Soriano
The NEHIL project, an EU-Korea partnership, is set to transform the landscape of digital technologies through groundbreaking neuromorphic architectures and advanced heterogeneous integration such as LiDAR systems. This collaborative initiative aims to …
MdM-IFISC-2
Maria de Maeztu 2023-2026
I.P.: Ernesto Estrada, Ingo Fischer, Emilio Hernández-García, Rosa Lopez, Claudio Mirasso, Jose Javier Ramasco, Raúl Toral, Roberta Zambrini
After 15 years of its existence, IFISC can point to a proven track record of impactful research. The previous 2018-2022 MdM award has significantly enhanced the institute's capabilities, as demonstrated by an …