  Triadic interactions are higher-order interactions that occur when a set of
nodes affects the interaction between two other nodes. Examples of triadic
interactions are present in the brain when glia modulate the synaptic signals
among neuron pairs or when interneuron axon-axonic synapses enable presynaptic
inhibition and facilitation, and in ecosystems when one or more species can
affect the interaction among two other species. On random graphs, triadic
percolation has been recently shown to turn percolation into a fully-fledged
dynamical process in which the size of the giant component undergoes a route to
chaos. However, in many real cases, triadic interactions are local and occur on
spatially embedded networks. Here we show that triadic interactions in spatial
networks induce a very complex spatio-temporal modulation of the giant
component which gives rise to triadic percolation patterns with significantly
different topology. We classify the observed patterns (stripes, octopus, and
small clusters) with topological data analysis and we assess their information
content (entropy and complexity). Moreover, we illustrate the multistability of
the dynamics of the triadic percolation patterns and we provide a comprehensive
phase diagram of the model. These results open new perspectives in percolation
as they demonstrate that in presence of spatial triadic interactions, the giant
component can acquire a time-varying topology. Hence, this work provides a
theoretical framework that can be applied to model realistic scenarios in which
the giant component is time-dependent as in neuroscience.

Adaptation and Self-Organizing Systems;Disordered Systems and Neural Networks;Statistical Mechanics;Chaotic Dynamics;Physics and Society