The model of scientific paradigms spreading throughout the community of agents with memory is analyzed using the master equation. The case of two competing ideas is considered for various networks of interactions, including agents placed at Erdős-Rényi graphs or complete graphs. The pace of adopting a new idea by a community is analyzed, along with the distribution of periods after which a new idea replaces the old one. The approach is extended for the chain topology onto the more general case when more than two ideas compete. Our analytical results are in agreement with numerical simulations.
We analyze composed quantum systems consisting of k subsystems, each described by states in the n-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order n 2 . The global evolution operator is represented by a unitary matrix of size N = n k . We investigate statistical properties of such matrices and show that they display spectral properties characteristic to the Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size n 2 one can construct a fair approximation of large random unitary matrices of size n k . Graph-structured random unitary matrices investigated here allow one to dene the corresponding structured ensembles of random pure states.
The model of community isolation was extended to the case when individuals are randomly placed at nodes of hierarchical modular networks. It was shown that the average number of blocked nodes (individuals) increases in time as a power function, with the exponent depending on network parameters. The distribution of time when the first isolated cluster appears is unimodal, non-gaussian. The developed analytical approach is in a good agreement with the simulation data.PACS numbers: 89.75. Hc, 89.75.Da, 89.75.Fb (1)
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