378 research outputs found
Quantifying loopy network architectures
Biology presents many examples of planar distribution and structural networks
having dense sets of closed loops. An archetype of this form of network
organization is the vasculature of dicotyledonous leaves, which showcases a
hierarchically-nested architecture containing closed loops at many different
levels. Although a number of methods have been proposed to measure aspects of
the structure of such networks, a robust metric to quantify their hierarchical
organization is still lacking. We present an algorithmic framework, the
hierarchical loop decomposition, that allows mapping loopy networks to binary
trees, preserving in the connectivity of the trees the architecture of the
original graph. We apply this framework to investigate computer generated
graphs, such as artificial models and optimal distribution networks, as well as
natural graphs extracted from digitized images of dicotyledonous leaves and
vasculature of rat cerebral neocortex. We calculate various metrics based on
the Asymmetry, the cumulative size distribution and the Strahler bifurcation
ratios of the corresponding trees and discuss the relationship of these
quantities to the architectural organization of the original graphs. This
algorithmic framework decouples the geometric information (exact location of
edges and nodes) from the metric topology (connectivity and edge weight) and it
ultimately allows us to perform a quantitative statistical comparison between
predictions of theoretical models and naturally occurring loopy graphs.Comment: 17 pages, 8 figures. During preparation of this manuscript the
authors became aware of the work of Mileyko at al., concurrently submitted
for publicatio
Topological self-similarity on the random binary-tree model
Asymptotic analysis on some statistical properties of the random binary-tree
model is developed. We quantify a hierarchical structure of branching patterns
based on the Horton-Strahler analysis. We introduce a transformation of a
binary tree, and derive a recursive equation about branch orders. As an
application of the analysis, topological self-similarity and its generalization
is proved in an asymptotic sense. Also, some important examples are presented
Hierarchical ordering of reticular networks
The structure of hierarchical networks in biological and physical systems has
long been characterized using the Horton-Strahler ordering scheme. The scheme
assigns an integer order to each edge in the network based on the topology of
branching such that the order increases from distal parts of the network (e.g.,
mountain streams or capillaries) to the "root" of the network (e.g., the river
outlet or the aorta). However, Horton-Strahler ordering cannot be applied to
networks with loops because they they create a contradiction in the edge
ordering in terms of which edge precedes another in the hierarchy. Here, we
present a generalization of the Horton-Strahler order to weighted planar
reticular networks, where weights are assumed to correlate with the importance
of network edges, e.g., weights estimated from edge widths may correlate to
flow capacity. Our method assigns hierarchical levels not only to edges of the
network, but also to its loops, and classifies the edges into reticular edges,
which are responsible for loop formation, and tree edges. In addition, we
perform a detailed and rigorous theoretical analysis of the sensitivity of the
hierarchical levels to weight perturbations. We discuss applications of this
generalized Horton-Strahler ordering to the study of leaf venation and other
biological networks.Comment: 9 pages, 5 figures, During preparation of this manuscript the authors
became aware of a related work by Katifori and Magnasco, concurrently
submitted for publicatio
Community analysis in social networks
We present an empirical study of different social networks obtained from
digital repositories. Our analysis reveals the community structure and provides
a useful visualising technique. We investigate the scaling properties of the
community size distribution, and that find all the networks exhibit power law
scaling in the community size distributions with exponent either -0.5 or -1.
Finally we find that the networks' community structure is topologically
self-similar using the Horton-Strahler index.Comment: Submitted to European Physics Journal
- …