Abstract
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit.
Highlights
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks
We show that by adopting a global adaptive coupling of dynamics[47,48,49] the coupled synchronization dynamics of topological signals defined on nodes and links is explosive[50], i.e., it occurs at a discontinuous phase transition in which the two topological signals of different dimension synchronize at the same time
In order to define the higher-order synchronization of topological signals we will make use of algebraic topology and we indicate with B[n] the nth incidence matrix representing the nth boundary operator
Summary
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Using simplicial complexes allows the network scientist to formulate new mathematical frameworks for mining data[5,6,7,8,9,10] and for understanding these generalized network structures revealing the underlying deep physical mechanisms for emergent geometry[11,12,13,14,15] and for higher-order dynamics[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] This very vibrant research activity is relevant in neuroscience to analyze real brain data and its profound relation to dynamics[1,6,15,34,35,36,37] and in the study of biological transport networks[10,38]. The analytical results reveal that the investigated transition is discontinuous
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