6 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
Network evolution towards optimal dynamical performance
Understanding the mutual interdependence between the behavior of dynamical
processes on networks and the underlying topologies promises new insight for a
large class of empirical networks. We present a generic approach to investigate
this relationship which is applicable to a wide class of dynamics, namely to
evolve networks using a performance measure based on the whole spectrum of the
dynamics' time evolution operator. As an example, we consider the graph
Laplacian describing diffusion processes, and we evolve the network structure
such that a given sub-diffusive behavior emerges.Comment: 5 pages, 4 figure
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
Comparing community structure identification
We compare recent approaches to community structure identification in terms
of sensitivity and computational cost. The recently proposed modularity measure
is revisited and the performance of the methods as applied to ad hoc networks
with known community structure, is compared. We find that the most accurate
methods tend to be more computationally expensive, and that both aspects need
to be considered when choosing a method for practical purposes. The work is
intended as an introduction as well as a proposal for a standard benchmark test
of community detection methods.Comment: 10 pages, 3 figures, 1 table. v2: condensed, updated version as
appears in JSTA
Improved community structure detection using a modified fine tuning strategy
The community structure of a complex network can be determined by finding the
partitioning of its nodes that maximizes modularity. Many of the proposed
algorithms for doing this work by recursively bisecting the network. We show
that this unduely constrains their results, leading to a bias in the size of
the communities they find and limiting their effectivness. To solve this
problem, we propose adding a step to the existing algorithms that does not
increase the order of their computational complexity. We show that, if this
step is combined with a commonly used method, the identified constraint and
resulting bias are removed, and its ability to find the optimal partitioning is
improved. The effectiveness of this combined algorithm is also demonstrated by
using it on real-world example networks. For a number of these examples, it
achieves the best results of any known algorithm.Comment: 6 pages, 3 figures, 1 tabl