7,380 research outputs found
Using Spectral Radius Ratio for Node Degree to Analyze the Evolution of Scale Free Networks and Small World Networks
In this paper, we show the evaluation of the spectral radius for node degree
as the basis to analyze the variation in the node degrees during the evolution
of scale-free networks and small-world networks. Spectral radius is the
principal eigenvalue of the adjacency matrix of a network graph and spectral
radius ratio for node degree is the ratio of the spectral radius and the
average node degree. We observe a very high positive correlation between the
spectral radius ratio for node degree and the coefficient of variation of node
degree (ratio of the standard deviation of node degree and average node
degree). We show how the spectral radius ratio for node degree can be used as
the basis to tune the operating parameters of the evolution models for
scale-free networks and small-world networks as well as evaluate the impact of
the number of links added per node introduced during the evolution of a
scale-free network and evaluate the impact of the probability of rewiring
during the evolution of a small-world network from a regular network.Comment: 8 pages, 8 figures, Second International Conference on Computer
Science and Information Technology, (COSIT-2015), Geneva, Switzerland, March
21-22, 201
Effect of Coupling on the Epidemic Threshold in Interconnected Complex Networks: A Spectral Analysis
In epidemic modeling, the term infection strength indicates the ratio of
infection rate and cure rate. If the infection strength is higher than a
certain threshold -- which we define as the epidemic threshold - then the
epidemic spreads through the population and persists in the long run. For a
single generic graph representing the contact network of the population under
consideration, the epidemic threshold turns out to be equal to the inverse of
the spectral radius of the contact graph. However, in a real world scenario it
is not possible to isolate a population completely: there is always some
interconnection with another network, which partially overlaps with the contact
network. Results for epidemic threshold in interconnected networks are limited
to homogeneous mixing populations and degree distribution arguments. In this
paper, we adopt a spectral approach. We show how the epidemic threshold in a
given network changes as a result of being coupled with another network with
fixed infection strength. In our model, the contact network and the
interconnections are generic. Using bifurcation theory and algebraic graph
theory, we rigorously derive the epidemic threshold in interconnected networks.
These results have implications for the broad field of epidemic modeling and
control. Our analytical results are supported by numerical simulations.Comment: 7 page
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
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