247 research outputs found
Network synchronization: Optimal and Pessimal Scale-Free Topologies
By employing a recently introduced optimization algorithm we explicitely
design optimally synchronizable (unweighted) networks for any given scale-free
degree distribution. We explore how the optimization process affects
degree-degree correlations and observe a generic tendency towards
disassortativity. Still, we show that there is not a one-to-one correspondence
between synchronizability and disassortativity. On the other hand, we study the
nature of optimally un-synchronizable networks, that is, networks whose
topology minimizes the range of stability of the synchronous state. The
resulting ``pessimal networks'' turn out to have a highly assortative
string-like structure. We also derive a rigorous lower bound for the Laplacian
eigenvalue ratio controlling synchronizability, which helps understanding the
impact of degree correlations on network synchronizability.Comment: 11 pages, 4 figs, submitted to J. Phys. A (proceedings of Complex
Networks 2007
The spectral dimension of random trees
We present a simple yet rigorous approach to the determination of the
spectral dimension of random trees, based on the study of the massless limit of
the Gaussian model on such trees. As a byproduct, we obtain evidence in favor
of a new scaling hypothesis for the Gaussian model on generic bounded graphs
and in favor of a previously conjectured exact relation between spectral and
connectivity dimensions on more general tree-like structures.Comment: 14 pages, 2 eps figures, revtex4. Revised version: changes in section
I
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
Preferential attachment of communities: the same principle, but a higher level
The graph of communities is a network emerging above the level of individual
nodes in the hierarchical organisation of a complex system. In this graph the
nodes correspond to communities (highly interconnected subgraphs, also called
modules or clusters), and the links refer to members shared by two communities.
Our analysis indicates that the development of this modular structure is driven
by preferential attachment, in complete analogy with the growth of the
underlying network of nodes. We study how the links between communities are
born in a growing co-authorship network, and introduce a simple model for the
dynamics of overlapping communities.Comment: 7 pages, 3 figure
A New Comparative Definition of Community and Corresponding Identifying Algorithm
In this paper, a new comparative definition for community in networks is
proposed and the corresponding detecting algorithm is given. A community is
defined as a set of nodes, which satisfy that each node's degree inside the
community should not be smaller than the node's degree toward any other
community. In the algorithm, the attractive force of a community to a node is
defined as the connections between them. Then employing attractive force based
self-organizing process, without any extra parameter, the best communities can
be detected. Several artificial and real-world networks, including Zachary
Karate club network and College football network are analyzed. The algorithm
works well in detecting communities and it also gives a nice description for
network division and group formation.Comment: 11 pages, 4 fihure
Network theory approach for data evaluation in the dynamic force spectroscopy of biomolecular interactions
Investigations of molecular bonds between single molecules and molecular
complexes by the dynamic force spectroscopy are subject to large fluctuations
at nanoscale and possible other aspecific binding, which mask the experimental
output. Big efforts are devoted to develop methods for effective selection of
the relevant experimental data, before taking the quantitative analysis of bond
parameters. Here we present a methodology which is based on the application of
graph theory. The force-distance curves corresponding to repeated pulling
events are mapped onto their correlation network (mathematical graph). On these
graphs the groups of similar curves appear as topological modules, which are
identified using the spectral analysis of graphs. We demonstrate the approach
by analyzing a large ensemble of the force-distance curves measured on:
ssDNA-ssDNA, peptide-RNA (system from HIV1), and peptide-Au surface. Within our
data sets the methodology systematically separates subgroups of curves which
are related to different intermolecular interactions and to spatial
arrangements in which the molecules are brought together and/or pulling speeds.
This demonstrates the sensitivity of the method to the spatial degrees of
freedom, suggesting potential applications in the case of large molecular
complexes and situations with multiple binding sites
Comparing the reliability of networks by spectral analysis
We provide a method for the ranking of the reliability of two networks with
the same connectance. Our method is based on the Cheeger constant linking the
topological property of a network with its spectrum. We first analyze a set of
twisted rings with the same connectance and degree distribution, and obtain the
ranking of their reliability using their eigenvalue gaps. The results are
generalized to general networks using the method of rewiring. The success of
our ranking method is verified numerically for the IEEE57, the
Erd\H{o}s-R\'enyi, and the Small-World networks.Comment: 7 pages, 3 figure
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