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Synchronization is a widespread emergent feature of complex systems. Here, the authors investigate the optimization of synchronization in phase oscillators with higher-order interactions, and find that optimized networks are more homogeneous in the nodes’ degree for undirected interactions, and for directed interactions they are generally structurally asymmetric, but can be symmetric, which differs from the pairwise case.

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset, that is, establishing the integrated taxonomic relation among points, lines, and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.

Characterising the structure of real-world complex networks is of crucial importance to understand the emerging dynamics taking place on top of them. In this work the authors investigate the cycle organization of synthetic and real systems, and use such information to define a centrality measure that is more informative than traditional indexes to the end of understanding network dismantling, synchronization, and spreading processes

The study of high-order networks as attracted significant attention recently. The authors introduce the concept of computability for searching maximum clique in large networks and an optimized algorithm for finding cavities with different orders.

In this paper, we give a new proof of the transition frequency formula for the state transition counting process in a vector Markov process (VMP). The formula was first presented by the author in the early eighties. We also derive the absorbing distribution, the renewal distribution and the entering probability formulas from the formula. As an application, we use derived results to reduce the analysis of a fluid model with random disruptions. [ABSTRACT FROM AUTHOR]