Triadic interactions in the brain are general mechanisms by which a node, e.g. a neuron or a glia cell such as the astrocyte, can regulate directly the link, e.g. synapse between other two nodes. The regulation takes place in a familiar way by either depressing or facilitating synaptic transmission. Such interactions are ubiquitous in neural systems, accounting both for axo-axonic and tripartite synapses mediated by astrocytes, for instance, and have been related to neuronal and synaptic processes at different time-scales, including short- and long-term synaptic plasticity. In the field of network science, triadic interactions have been shown to produce complex spatio-temporal patterns of connectivity. Here, we investigate the emergent behavior of an in silico neural medium constituted by a population of leaky integrate-and-fire neurons with triadic interactions. We observe that, depending on relevant parameters defining triadic interactions, different activity patterns emerge. These include (i) a silent phase, (ii) a low-activity phase in which complex spatio-temporal patterns of low neuronal firing rate emerge that propagate through the medium, (iii) a high-activity phase characterized by complex spatio-temporal patterns of high neuronal firing rate that propagate through the neural medium as waves of high firing activity over a bulk of low activity neurons, and (iv) a pseudo-blinking phase in which the neural medium switches between high and low activity states. Here we analyze in depth the features of such patterns and relate our findings to the recently proposed model of triadic percolation.

Higher-order networks capture the many-body interactions present in complex systems, shedding light on the interplay between topology and dynamics. The theory of higher-order topological dynamics, which combines higher-order interactions with discrete topology and nonlinear dynamics, has the potential to enhance our understanding of complex systems, such as the brain and the climate, and to advance the development of next-generation AI algorithms. This theoretical framework, which goes beyond traditional node-centric descriptions, encodes the dynamics of a network through topological signals—variables assigned not only to nodes but also to edges, triangles and other higher-order cells. Recent findings show that topological signals lead to the emergence of distinct types of dynamical state and collective phenomena, including topological and Dirac synchronization, pattern formation and triadic percolation. These results offer insights into how topology shapes dynamics, how dynamics learns topology and how topology evolves dynamically. This Perspective primarily aims to guide physicists, mathematicians, computer scientists and network scientists through the emerging field of higher-order topological dynamics, while also outlining future research challenges.

Quantum networks (QNs) exhibit stronger connectivity than predicted by classical percolation, yet the origin of this phenomenon remains unexplored. We apply a statistical physics model—concurrence percolation—to uncover the origin of stronger connectivity on hierarchical scale-free networks, the (U, V) flowers. These networks allow full analytical control over path connectivity through two adjustable path-length parameters, ≤V. This precise control enables us to determine critical exponents well beyond current simulation limits, revealing that classical and concurrence percolations, while both satisfying the hyperscaling relation, fall into distinct universality classes. This distinction arises from how they “superpose” parallel, nonshortest path contributions into overall connectivity. Concurrence percolation, unlike its classical counterpart, is sensitive to nonshortest paths and shows higher resilience to detours as these paths lengthen. This enhanced resilience is also observed in real-world hierarchical, scale-free internet networks. Our findings highlight a crucial principle for QN design: When nonshortest paths are abundant, they notably enhance QN connectivity beyond what is achievable with classical percolation.

In this work, we propose a comprehensive theoretical framework combining percolation theory with nonlinear dynamics to study hypergraphs with a time-varying giant component. We consider in particular hypergraphs with higher-order triadic interactions. Higher-order triadic interactions occur when one or more nodes up-regulate or down-regulate a hyperedge. For instance, enzymes regulate chemical reactions involving multiple reactants. Here we propose and investigate higher-order triadic percolation on hypergraphs showing that the giant component can have a nontrivial dynamics. Specifically, we show that the fraction of nodes in the giant component undergoes a route to chaos in the universality class of the logistic map. In hierarchical higher-order triadic percolation, we extend this paradigm in order to treat hierarchically nested higher-order triadic interactions. We demonstrate the nontrivial effects of their increased combinatorial complexity on the critical phenomena and the dynamical properties of the process. Finally, we consider other generalizations of the model studying the effect of adopting interdependencies and node regulation instead of hyperedge regulation. The comprehensive theoretical framework presented here sheds light on possible scenarios for climate networks, biological networks, and brain networks, where the hypergraph connectivity changes over time.

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical, and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2⁢D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2⁢D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.

Triadic interactions are higher-order interactions which 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 axo-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.

We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract-through a conventional finite-size scaling analysis with modest lattice sizes-the critical temperature with less than 1% error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.