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Model of Communities Isolation at Hierarchical Modular Networks

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Kondratiuk P., Hołyst J.

Acta Physica Polonica A

2012

vol. 121

issue 2B

B-67-B-71

The model of community isolation was extended to the case when individuals are randomly placed at the 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 the network parameters. The distribution of the time when the first isolated cluster appears is unimodal, non-gaussian. The developed analytical approach is in a good agreement with the simulation data.

Random Unitary Matrices Associated to a Graph

100%

Kondratiuk P., Życzkowski K.

Acta Physica Polonica A

2013

vol. 124

issue 6

1098-1105

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 define the corresponding structured ensembles of random pure states.

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2 Acta Physica Polonica A

1 2013

1 2012

2 Kondratiuk P.

1 Hołyst J.

1 Życzkowski K.

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Model of Communities Isolation at Hierarchical Modular Networks

Random Unitary Matrices Associated to a Graph