On the Metastability of Quadratic Majority Dynamics on Clustered Graphs and its Biological Implications
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We investigate the behavior of a simple majority dynamics on network topologies that exhibit a clustered structure. By leveraging on recent advancements in the analysis of dynamics, we prove that, when the initial states of the nodes are randomly initialized or when they satisfy a slight bias condition, the network rapidly and stably converges to a configuration in which the clusters maintain internal consensus on two different states. This is the first analytical result on the behavior of dynamics for non-consensus problems on non-complete topologies, based on the first symmetry-breaking analysis in such setting. In the context of the study of Label Propagation Algorithms, a class of widely used heuristics for clustering in data mining, it represents the first theoretical result on the behavior of a label propagation algorithm with quasilinear message complexity. In the context of evolutionary biology, dynamics such as the Moran process have been used to model the spread of mutations in genetic populations. Our results shows that, when the probability of adoption of a given mutation by a node of the evolutionary graph depends super-linearly on the frequency of the mutation in the neighborhood of the node, and the underlying evolutionary graph has a clustered structure, there is a non-negligible probability for species differentiation to occur. Finally, in the context of neuroscience, evolutionary graph dynamics have been used to model the innervation of synapses on muscular junctions during development. Our result, corroborated by numerical simulations on a wider class of dynamics, provide evidence that, in order for the model to comply with experimental evidence on the outcome of the innervation process, either the innervation sites do not exhibit spatial bottlenecks or the dynamics cannot be based on majority-like mechanism.
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