The role of modularity in self-organisation dynamics in biological networks 
Siebert et al., 2020

Interconnected ensembles of biological entities are perhaps some of the most complex systems that modern science has encountered so far. In particular, scientists have concentrated on understanding how the complexity of the interacting structure between different neurons, proteins or species influences the functioning of their respective systems. It is well-established that many biological networks are constructed in a highly hierarchical way with two main properties: short average paths that join two apparently distant nodes (neuronal, species, or protein patches) and a high proportion of nodes in modular aggregations. Although several hypotheses have been proposed so far, still little is known about the relation of the modules with the dynamical activity in such biological systems. Here we show that network modularity is a key ingredient for the formation of self-organising patterns of functional activity, independently of the topological peculiarities of the structure of the modules. In particular, we show that macroscopic spatial patterns at the modular scale can develop in this case, which may explain how spontaneous order in biological networks follows their modular structural organisation. Our results also show that Turing patterns on biological complex networks can be a signature of the presence of modular structure and consequently a possible protocol for community detection. We test our results on real-world networks to confirm the important role of modularity in creating macro-scale patterns. 

Modular vs. Small-world topology in Turing pattern formation. a) A Newman{Watts (NW) network with N = 125 nodes, and 660 edges, where patterns are absent. The colour of the nodes represents the concentration of the activator, ui(t), at long time. b) The dispersion relation of the NW network (red stars) overlain on the dispersion relation of the continuous case (blue curve), i.e. if the system was on a continuous domain and not on a network. Notice the absence of the unstable eigenvalues (inset) and the gap between the zero eigenvalue and the second smallest 2, known as the spectral gap. c) A modular network of the same size (same number of nodes and edges) as in a) where indeed Turing patterns are present. The ve modules are of the Erdos-R enyi (ER) family. The colour of the nodes again represents the concentration of the activator, ui(t), at long time. d) The dispersion relation of the modular network (red stars) overlain on the dispersion relation of the continuous domain (blue curve). Notice here the presence of unstable eigenvalues (inset) and that the eigenvalues are separated in two sets by an important gap, between the rst and second set of eigenvalues. The rst four non-zero one are denoted as the modular eigenvalues and the rest non-negative ones as the non-modular eigenvalues. The parameters of the Fitzhugh-Nagumo model are in both cases Du = 1, = 5:5, a = 0:7, b = 0:05; c = 1:7. Finally, note the di erent colormaps used between panels a) and c) to highlight the lack of patterns in the former.

https://arxiv.org/abs/2003.12311
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