438 research outputs found
Universality in Complex Networks: Random Matrix Analysis
We apply random matrix theory to complex networks. We show that nearest
neighbor spacing distribution of the eigenvalues of the adjacency matrices of
various model networks, namely scale-free, small-world and random networks
follow universal Gaussian orthogonal ensemble statistics of random matrix
theory. Secondly we show an analogy between the onset of small-world behavior,
quantified by the structural properties of networks, and the transition from
Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody
parameter characterizing a spectral property. We also present our analysis for
a protein-protein interaction network in budding yeast.Comment: 4+ pages, 4 figures, to appear in PRE, major change in the paper
including titl
Note on Branching
It has been demonstrated that the spectrum of the molecular
graph contains information about the extent of branching of the
molecular skeleton. In particular, the largest eigenvalue, xi, in the
spectrum has been shown to be closely related to the total number
of walks in the graph (eqs. (11) and (15)). Thus, a justification of
the recent empirical finding that x1 is a measure of branching9,in
has been obtained
Spectral Analysis of Protein-Protein Interactions in Drosophila melanogaster
Within a case study on the protein-protein interaction network (PIN) of
Drosophila melanogaster we investigate the relation between the network's
spectral properties and its structural features such as the prevalence of
specific subgraphs or duplicate nodes as a result of its evolutionary history.
The discrete part of the spectral density shows fingerprints of the PIN's
topological features including a preference for loop structures. Duplicate
nodes are another prominent feature of PINs and we discuss their representation
in the PIN's spectrum as well as their biological implications.Comment: 9 pages RevTeX including 8 figure
The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs
The number of self-adjoint extensions of a symmetric operator acting on a
complex Hilbert space is characterized by its deficiency indices. Given a
locally finite unoriented simple tree, we prove that the deficiency indices of
any discrete Schr\"odinger operator are either null or infinite. We also prove
that almost surely, there is a tree such that all discrete Schr\"odinger
operators are essentially self-adjoint. Furthermore, we provide several
criteria of essential self-adjointness. We also adress some importance to the
case of the adjacency matrix and conjecture that, given a locally finite
unoriented simple graph, its the deficiency indices are either null or
infinite. Besides that, we consider some generalizations of trees and weighted
graphs.Comment: Typos corrected. References and ToC added. Paper slightly
reorganized. Section 3.2, about the diagonalization has been much improved.
The older section about the stability of the deficiency indices in now in
appendix. To appear in Journal of Mathematical Physic
Quantum transport on small-world networks: A continuous-time quantum walk approach
We consider the quantum mechanical transport of (coherent) excitons on
small-world networks (SWN). The SWN are build from a one-dimensional ring of N
nodes by randomly introducing B additional bonds between them. The exciton
dynamics is modeled by continuous-time quantum walks and we evaluate
numerically the ensemble averaged transition probability to reach any node of
the network from the initially excited one. For sufficiently large B we find
that the quantum mechanical transport through the SWN is, first, very fast,
given that the limiting value of the transition probability is reached very
quickly; second, that the transport does not lead to equipartition, given that
on average the exciton is most likely to be found at the initial node.Comment: 8 pages, 8 figures (high quality figures available upon request
Spectral analysis of deformed random networks
We study spectral behavior of sparsely connected random networks under the
random matrix framework. Sub-networks without any connection among them form a
network having perfect community structure. As connections among the
sub-networks are introduced, the spacing distribution shows a transition from
the Poisson statistics to the Gaussian orthogonal ensemble statistics of random
matrix theory. The eigenvalue density distribution shows a transition to the
Wigner's semicircular behavior for a completely deformed network. The range for
which spectral rigidity, measured by the Dyson-Mehta statistics,
follows the Gaussian orthogonal ensemble statistics depends upon the
deformation of the network from the perfect community structure. The spacing
distribution is particularly useful to track very slight deformations of the
network from a perfect community structure, whereas the density distribution
and the statistics remain identical to the undeformed network. On
the other hand the statistics is useful for the larger deformation
strengths. Finally, we analyze the spectrum of a protein-protein interaction
network for Helicobacter, and compare the spectral behavior with those of the
model networks.Comment: accepted for publication in Phys. Rev. E (replaced with the final
version
Enhancing the spectral gap of networks by node removal
Dynamics on networks are often characterized by the second smallest
eigenvalue of the Laplacian matrix of the network, which is called the spectral
gap. Examples include the threshold coupling strength for synchronization and
the relaxation time of a random walk. A large spectral gap is usually
associated with high network performance, such as facilitated synchronization
and rapid convergence. In this study, we seek to enhance the spectral gap of
undirected and unweighted networks by removing nodes because, practically, the
removal of nodes often costs less than the addition of nodes, addition of
links, and rewiring of links. In particular, we develop a perturbative method
to achieve this goal. The proposed method realizes better performance than
other heuristic methods on various model and real networks. The spectral gap
increases as we remove up to half the nodes in most of these networks.Comment: 5 figure
Random matrix analysis of complex networks
We study complex networks under random matrix theory (RMT) framework. Using
nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the
eigenvalues of adjacency matrix of various model networks, namely, random,
scale-free and small-world networks. These distributions follow Gaussian
orthogonal ensemble statistic of RMT. To probe long-range correlations in the
eigenvalues we study spectral rigidity via statistic of RMT as well.
It follows RMT prediction of linear behavior in semi-logarithmic scale with
slope being . Random and scale-free networks follow RMT
prediction for very large scale. Small-world network follows it for
sufficiently large scale, but much less than the random and scale-free
networks.Comment: accepted in Phys. Rev. E (replaced with the final version
Spectral Measures of Bipartivity in Complex Networks
We introduce a quantitative measure of network bipartivity as a proportion of
even to total number of closed walks in the network. Spectral graph theory is
used to quantify how close to bipartite a network is and the extent to which
individual nodes and edges contribute to the global network bipartivity. It is
shown that the bipartivity characterizes the network structure and can be
related to the efficiency of semantic or communication networks, trophic
interactions in food webs, construction principles in metabolic networks, or
communities in social networks.Comment: 16 pages, 1 figure, 1 tabl
Unsupervised online activity discovery using temporal behaviour assumption
We present a novel unsupervised approach, UnADevs, for discovering activity clusters corresponding to periodic and stationary activities in streaming sensor data. Such activities usually last for some time, which is exploited by our method; it includes mechanisms to regulate sensitivity to brief outliers and can discover multiple clusters overlapping in time to better deal with deviations from nominal behaviour. The method was evaluated on two activity datasets containing large number of activities (14 and 33 respectively) against online agglomerative clustering and DBSCAN. In a multi-criteria evaluation, our approach achieved significantly better performance on majority of the measures, with the advantages that: (i) it does not require to specify the number of clusters beforehand (it is open ended); (ii) it is online and can find clusters in real time; (iii) it has constant time complexity; (iv) and it is memory efficient as it does not keep the data samples in memory. Overall, it has managed to discover 616 of the total 717 activities. Because it discovers clusters of activities in real time, it is ideal to work alongside an active learning system
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