1,212 research outputs found
A measure of centrality based on the spectrum of the Laplacian
We introduce a family of new centralities, the k-spectral centralities.
k-Spectral centrality is a measurement of importance with respect to the
deformation of the graph Laplacian associated with the graph. Due to this
connection, k-spectral centralities have various interpretations in terms of
spectrally determined information.
We explore this centrality in the context of several examples. While for
sparse unweighted networks 1-spectral centrality behaves similarly to other
standard centralities, for dense weighted networks they show different
properties. In summary, the k-spectral centralities provide a novel and useful
measurement of relevance (for single network elements as well as whole
subnetworks) distinct from other known measures.Comment: 12 pages, 6 figures, 2 table
Dynamics-based centrality for general directed networks
Determining the relative importance of nodes in directed networks is
important in, for example, ranking websites, publications, and sports teams,
and for understanding signal flows in systems biology. A prevailing centrality
measure in this respect is the PageRank. In this work, we focus on another
class of centrality derived from the Laplacian of the network. We extend the
Laplacian-based centrality, which has mainly been applied to strongly connected
networks, to the case of general directed networks such that we can
quantitatively compare arbitrary nodes. Toward this end, we adopt the idea used
in the PageRank to introduce global connectivity between all the pairs of nodes
with a certain strength. Numerical simulations are carried out on some
networks. We also offer interpretations of the Laplacian-based centrality for
general directed networks in terms of various dynamical and structural
properties of networks. Importantly, the Laplacian-based centrality defined as
the stationary density of the continuous-time random walk with random jumps is
shown to be equivalent to the absorption probability of the random walk with
sinks at each node but without random jumps. Similarly, the proposed centrality
represents the importance of nodes in dynamics on the original network supplied
with sinks but not with random jumps.Comment: 7 figure
Adjustable reach in a network centrality based on current flows
Centrality, which quantifies the "importance" of individual nodes, is among
the most essential concepts in modern network theory. Most prominent centrality
measures can be expressed as an aggregation of influence flows between pairs of
nodes. As there are many ways in which influence can be defined, many different
centrality measures are in use. Parametrized centralities allow further
flexibility and utility by tuning the centrality calculation to the regime most
appropriate for a given network. Here, we identify two categories of centrality
parameters. Reach parameters control the attenuation of influence flows between
distant nodes. Grasp parameters control the centrality's potential to send
influence flows along multiple, often nongeodesic paths. Combining these
categories with Borgatti's centrality types [S. P. Borgatti, Social Networks
27, 55-71 (2005)], we arrive at a novel classification system for parametrized
centralities. Using this classification, we identify the notable absence of any
centrality measures that are radial, reach parametrized, and based on acyclic,
conservative flows of influence. We therefore introduce the ground-current
centrality, which is a measure of precisely this type. Because of its unique
position in the taxonomy, the ground-current centrality has significant
advantages over similar centralities. We demonstrate that, compared to other
conserved-flow centralities, it has a simpler mathematical description.
Compared to other reach centralities, it robustly preserves an intuitive rank
ordering across a wide range of network architectures. We also show that it
produces a consistent distribution of centrality values among the nodes,
neither trivially equally spread (delocalization), nor overly focused on a few
nodes (localization). Other reach centralities exhibit both of these behaviors
on regular networks and hub networks, respectively
Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths
We introduce several novel and computationally efficient methods for
detecting "core--periphery structure" in networks. Core--periphery structure is
a type of mesoscale structure that includes densely-connected core vertices and
sparsely-connected peripheral vertices. Core vertices tend to be well-connected
both among themselves and to peripheral vertices, which tend not to be
well-connected to other vertices. Our first method, which is based on
transportation in networks, aggregates information from many geodesic paths in
a network and yields a score for each vertex that reflects the likelihood that
a vertex is a core vertex. Our second method is based on a low-rank
approximation of a network's adjacency matrix, which can often be expressed as
a tensor-product matrix. Our third approach uses the bottom eigenvector of the
random-walk Laplacian to infer a coreness score and a classification into core
and peripheral vertices. We also design an objective function to (1) help
classify vertices into core or peripheral vertices and (2) provide a
goodness-of-fit criterion for classifications into core versus peripheral
vertices. To examine the performance of our methods, we apply our algorithms to
both synthetically-generated networks and a variety of networks constructed
from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on
"Network Analysis and Modelling" (available at
https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB
Graphs and networks theory
This chapter discusses graphs and networks theory
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