1,949 research outputs found
Approximating Spectral Impact of Structural Perturbations in Large Networks
Determining the effect of structural perturbations on the eigenvalue spectra
of networks is an important problem because the spectra characterize not only
their topological structures, but also their dynamical behavior, such as
synchronization and cascading processes on networks. Here we develop a theory
for estimating the change of the largest eigenvalue of the adjacency matrix or
the extreme eigenvalues of the graph Laplacian when small but arbitrary set of
links are added or removed from the network. We demonstrate the effectiveness
of our approximation schemes using both real and artificial networks, showing
in particular that we can accurately obtain the spectral ranking of small
subgraphs. We also propose a local iterative scheme which computes the relative
ranking of a subgraph using only the connectivity information of its neighbors
within a few links. Our results may not only contribute to our theoretical
understanding of dynamical processes on networks, but also lead to practical
applications in ranking subgraphs of real complex networks.Comment: 9 pages, 3 figures, 2 table
Harmonic analysis on Cayley Trees II: the Bose Einstein condensation
We investigate the Bose-Einstein Condensation on non homogeneous non amenable
networks for the model describing arrays of Josephson junctions on perturbed
Cayley Trees. The resulting topological model has also a mathematical interest
in itself. The present paper is then the application to the Bose-Einstein
Condensation phenomena, of the harmonic analysis aspects arising from additive
and density zero perturbations, previously investigated by the author in a
separate work. Concerning the appearance of the Bose-Einstein Condensation, the
results are surprisingly in accordance with the previous ones, despite the lack
of amenability. We indeed first show the following fact. Even when the critical
density is finite (which is implied in all the models under consideration,
thanks to the appearance of the hidden spectrum), if the adjacency operator of
the graph is recurrent, it is impossible to exhibit temperature locally normal
states (i.e. states for which the local particle density is finite) describing
the condensation at all. The same occurs in the transient cases for which it is
impossible to exhibit locally normal states describing the Bose--Einstein
Condensation at mean particle density strictly greater than the critical
density . In addition, for the transient cases, in order to construct locally
normal temperature states through infinite volume limits of finite volume Gibbs
states, a careful choice of the the sequence of the finite volume chemical
potential should be done. For all such states, the condensate is essentially
allocated on the base--point supporting the perturbation. This leads that the
particle density always coincide with the critical one. It is shown that all
such temperature states are Kubo-Martin-Schwinger states for a natural
dynamics. The construction of such a dynamics, which is a very delicate issue,
is also done.Comment: 28 pages, 6 figures, 1 tabl
Spectral dimension reduction of complex dynamical networks
Dynamical networks are powerful tools for modeling a broad range of complex
systems, including financial markets, brains, and ecosystems. They encode how
the basic elements (nodes) of these systems interact altogether (via links) and
evolve (nodes' dynamics). Despite substantial progress, little is known about
why some subtle changes in the network structure, at the so-called critical
points, can provoke drastic shifts in its dynamics. We tackle this challenging
problem by introducing a method that reduces any network to a simplified
low-dimensional version. It can then be used to describe the collective
dynamics of the original system. This dimension reduction method relies on
spectral graph theory and, more specifically, on the dominant eigenvalues and
eigenvectors of the network adjacency matrix. Contrary to previous approaches,
our method is able to predict the multiple activation of modular networks as
well as the critical points of random networks with arbitrary degree
distributions. Our results are of both fundamental and practical interest, as
they offer a novel framework to relate the structure of networks to their
dynamics and to study the resilience of complex systems.Comment: 16 pages, 8 figure
Social structure contains epidemics and regulates individual roles in disease transmission in a group-living mammal
This is the final version. Available from Wiley via the DOI in this record. Data accessibility: The original weighted adjacency matrix for the high‐density population of European badgers, as well as code used for simulating networks and disease simulations can be found online https://doi.org/10.5061/dryad.49n3878.Population structure is critical to infectious disease transmission. As a result, theoretical and empirical contact network models of infectious disease spread are increasingly providing valuable insights into wildlife epidemiology. Analyzing an exceptionally detailed dataset on contact structure within a high-density population of European badgers Meles meles, we show that a modular contact network produced by spatially structured stable social groups, lead to smaller epidemics, particularly for infections with intermediate transmissibility. The key advance is that we identify considerable variation among individuals in their role in disease spread, with these new insights made possible by the detail in the badger dataset. Furthermore, the important impacts on epidemiology are found even though the modularity of the Badger network is much lower than the threshold that previous work suggested was necessary. These findings reveal the importance of stable social group structure for disease dynamics with important management implications for socially structured populations.Natural Environment Research Council (NERC
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