586 research outputs found
Secluded Connectivity Problems
Consider a setting where possibly sensitive information sent over a path in a
network is visible to every {neighbor} of the path, i.e., every neighbor of
some node on the path, thus including the nodes on the path itself. The
exposure of a path can be measured as the number of nodes adjacent to it,
denoted by . A path is said to be secluded if its exposure is small. A
similar measure can be applied to other connected subgraphs, such as Steiner
trees connecting a given set of terminals. Such subgraphs may be relevant due
to considerations of privacy, security or revenue maximization. This paper
considers problems related to minimum exposure connectivity structures such as
paths and Steiner trees. It is shown that on unweighted undirected -node
graphs, the problem of finding the minimum exposure path connecting a given
pair of vertices is strongly inapproximable, i.e., hard to approximate within a
factor of for any (under an
appropriate complexity assumption), but is approximable with ratio
, where is the maximum degree in the graph. One of
our main results concerns the class of bounded-degree graphs, which is shown to
exhibit the following interesting dichotomy. On the one hand, the minimum
exposure path problem is NP-hard on node-weighted or directed bounded-degree
graphs (even when the maximum degree is 4). On the other hand, we present a
polynomial algorithm (based on a nontrivial dynamic program) for the problem on
unweighted undirected bounded-degree graphs. Likewise, the problem is shown to
be polynomial also for the class of (weighted or unweighted) bounded-treewidth
graphs
Recurrence-based time series analysis by means of complex network methods
Complex networks are an important paradigm of modern complex systems sciences
which allows quantitatively assessing the structural properties of systems
composed of different interacting entities. During the last years, intensive
efforts have been spent on applying network-based concepts also for the
analysis of dynamically relevant higher-order statistical properties of time
series. Notably, many corresponding approaches are closely related with the
concept of recurrence in phase space. In this paper, we review recent
methodological advances in time series analysis based on complex networks, with
a special emphasis on methods founded on recurrence plots. The potentials and
limitations of the individual methods are discussed and illustrated for
paradigmatic examples of dynamical systems as well as for real-world time
series. Complex network measures are shown to provide information about
structural features of dynamical systems that are complementary to those
characterized by other methods of time series analysis and, hence,
substantially enrich the knowledge gathered from other existing (linear as well
as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos
(2011
Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus Hurst index
Nonlinear time series analysis aims at understanding the dynamics of
stochastic or chaotic processes. In recent years, quite a few methods have been
proposed to transform a single time series to a complex network so that the
dynamics of the process can be understood by investigating the topological
properties of the network. We study the topological properties of horizontal
visibility graphs constructed from fractional Brownian motions with different
Hurst index . Special attention has been paid to the impact of Hurst
index on the topological properties. It is found that the clustering
coefficient decreases when increases. We also found that the mean
length of the shortest paths increases exponentially with for fixed
length of the original time series. In addition, increases linearly
with respect to when is close to 1 and in a logarithmic form when
is close to 0. Although the occurrence of different motifs changes with ,
the motif rank pattern remains unchanged for different . Adopting the
node-covering box-counting method, the horizontal visibility graphs are found
to be fractals and the fractal dimension decreases with . Furthermore,
the Pearson coefficients of the networks are positive and the degree-degree
correlations increase with the degree, which indicate that the horizontal
visibility graphs are assortative. With the increase of , the Pearson
coefficient decreases first and then increases, in which the turning point is
around . The presence of both fractality and assortativity in the
horizontal visibility graphs converted from fractional Brownian motions is
different from many cases where fractal networks are usually disassortative.Comment: 12 pages, 8 figure
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