A natural way of studying connected complex systems is by viewing them through the lens of network science. A network is a set of items, called nodes, connected by edges [1], and network science is the study of complex networks found in biological, physical, social, or communication systems.

The building blocks of many complex systems are known to be higher-order topological structures, called simplices [2, 3]. From a network theoretic point of view, a simplex is a maximal complete subgraph. A simplicial structure of a network is made up of simplices and how they are connected to each other [4]. With the information of the simplicial structure of networks, we can inspect how higher-order structures are created, and how they affect the networks and dynamics occurring on them [5].

We have studied three very different complex systems: the first is time series-networks, which are network representations of the time series of dynamical systems; the second is collective behaviour of interacting phase oscillators; and the third is the social network of scientific collaborations. These systems have been studied in the past using networks and network metrics (that employ dyadic links and interactions) [6-10]. We analyzed these systems by taking into consideration their simplicial structures and metrics. In the case of time series-networks, we studied their complexity by relating correlations in the time series with the simplicial structure of the corresponding networks. Next, for the case of collective behaviour of phase oscillators, we studied the effects of higher-order interactions (based on simplices) on the transition to synchronization, occurring on a special simplicial structure. Finally, in the study of scientific collaboration networks we considered simplices to be basic building blocks, and we analyzed the simplicial structure of these networks and the nature of their connectedness.

References
1. Newman, M. E. J. (2003). The Structure and Function of Complex Networks. SIAM Review, Vol. 45, 45(2), 167–256.
2. Tadic, B., Andjelkovic, M., and Suvakov, M. (2016). The influence of architecture of nanoparticle networks on collective charge transport revealed by the fractal time series and topology of phase space manifolds. J. Coupl. Syst. Multisc. Dynam. 4, 30.
3. Tadic, B. and Gupte, N. (2020). Hidden geometry and dynamics of complex networks: Spin reversal in nanoassemblies with pairwise and triangle-based interactions. EPL, 132, 60008.
4. Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G., and Petri, G. (2020). Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874(June), 1–92.
5. Battiston, F., Amico, E., Barrat, A., Bianconi, G., Arruda, G. F. De, Franceschiello, B., Iacopini, I., & Kefi, S. (2021). The physics of higher-order interactions in complex complex systems. Nature Physics, 17.
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