8,945 research outputs found
Discovering Functional Communities in Dynamical Networks
Many networks are important because they are substrates for dynamical
systems, and their pattern of functional connectivity can itself be dynamic --
they can functionally reorganize, even if their underlying anatomical structure
remains fixed. However, the recent rapid progress in discovering the community
structure of networks has overwhelmingly focused on that constant anatomical
connectivity. In this paper, we lay out the problem of discovering_functional
communities_, and describe an approach to doing so. This method combines recent
work on measuring information sharing across stochastic networks with an
existing and successful community-discovery algorithm for weighted networks. We
illustrate it with an application to a large biophysical model of the
transition from beta to gamma rhythms in the hippocampus.Comment: 18 pages, 4 figures, Springer "Lecture Notes in Computer Science"
style. Forthcoming in the proceedings of the workshop "Statistical Network
Analysis: Models, Issues and New Directions", at ICML 2006. Version 2: small
clarifications, typo corrections, added referenc
Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions
We consider a family of Cuntz-Pimsner algebras associated to self-similar
group actions, and their Toeplitz analogues. Both families carry natural
dynamics implemented by automorphic actions of the real line, and we
investigate the equilibrium states (the KMS states) for these dynamical
systems.
We find that for all inverse temperatures above a critical value, the KMS
states on the Toeplitz algebra are given, in a very concrete way, by traces on
the full group algebra of the group. At the critical inverse temperature, the
KMS states factor through states of the Cuntz-Pimsner algebra; if the
self-similar group is contracting, then the Cuntz-Pimsner algebra has only one
KMS state. We apply these results to a number of examples, including the
self-similar group actions associated to integer dilation matrices, and the
canonical self-similar actions of the basilica group and the Grigorchuk group.Comment: The paper has been updated to agree with the published versio
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