175,143 research outputs found
Centrality measures for graphons: Accounting for uncertainty in networks
As relational datasets modeled as graphs keep increasing in size and their
data-acquisition is permeated by uncertainty, graph-based analysis techniques
can become computationally and conceptually challenging. In particular, node
centrality measures rely on the assumption that the graph is perfectly known --
a premise not necessarily fulfilled for large, uncertain networks. Accordingly,
centrality measures may fail to faithfully extract the importance of nodes in
the presence of uncertainty. To mitigate these problems, we suggest a
statistical approach based on graphon theory: we introduce formal definitions
of centrality measures for graphons and establish their connections to
classical graph centrality measures. A key advantage of this approach is that
centrality measures defined at the modeling level of graphons are inherently
robust to stochastic variations of specific graph realizations. Using the
theory of linear integral operators, we define degree, eigenvector, Katz and
PageRank centrality functions for graphons and establish concentration
inequalities demonstrating that graphon centrality functions arise naturally as
limits of their counterparts defined on sequences of graphs of increasing size.
The same concentration inequalities also provide high-probability bounds
between the graphon centrality functions and the centrality measures on any
sampled graph, thereby establishing a measure of uncertainty of the measured
centrality score. The same concentration inequalities also provide
high-probability bounds between the graphon centrality functions and the
centrality measures on any sampled graph, thereby establishing a measure of
uncertainty of the measured centrality score.Comment: Authors ordered alphabetically, all authors contributed equally. 21
pages, 7 figure
Correlation Between Student Collaboration Network Centrality and Academic Performance
We compute nodal centrality measures on the collaboration networks of
students enrolled in three upper-division physics courses, usually taken
sequentially, at the Colorado School of Mines. These are complex networks in
which links between students indicate assistance with homework. The courses
included in the study are intermediate Classical Mechanics, introductory
Quantum Mechanics, and intermediate Electromagnetism. By correlating these
nodal centrality measures with students' scores on homework and exams, we find
four centrality measures that correlate significantly with students' homework
scores in all three courses: in-strength, out-strength, closeness centrality,
and harmonic centrality. These correlations suggest that students who not only
collaborate often, but also collaborate significantly with many different
people tend to achieve higher grades. Centrality measures between simultaneous
collaboration networks (analytical vs. numerical homework collaboration)
composed of the same students also correlate with each other, suggesting that
students' collaboration strategies remain relatively stable when presented with
homework assignments targeting different skills. Additionally, we correlate
centrality measures between collaboration networks from different courses and
find that the four centrality measures with the strongest relationship to
students' homework scores are also the most stable measures across networks
involving different courses. Correlations of centrality measures with exam
scores were generally smaller than the correlations with homework scores,
though this finding varied across courses.Comment: 10 pages, 4 figures, submitted to Phys. Rev. PE
The impact of partially missing communities~on the reliability of centrality measures
Network data is usually not error-free, and the absence of some nodes is a
very common type of measurement error. Studies have shown that the reliability
of centrality measures is severely affected by missing nodes. This paper
investigates the reliability of centrality measures when missing nodes are
likely to belong to the same community. We study the behavior of five commonly
used centrality measures in uniform and scale-free networks in various error
scenarios. We find that centrality measures are generally more reliable when
missing nodes are likely to belong to the same community than in cases in which
nodes are missing uniformly at random. In scale-free networks, the betweenness
centrality becomes, however, less reliable when missing nodes are more likely
to belong to the same community. Moreover, centrality measures in scale-free
networks are more reliable in networks with stronger community structure. In
contrast, we do not observe this effect for uniform networks. Our observations
suggest that the impact of missing nodes on the reliability of centrality
measures might not be as severe as the literature suggests
Spectral centrality measures in complex networks
Complex networks are characterized by heterogeneous distributions of the
degree of nodes, which produce a large diversification of the roles of the
nodes within the network. Several centrality measures have been introduced to
rank nodes based on their topological importance within a graph. Here we review
and compare centrality measures based on spectral properties of graph matrices.
We shall focus on PageRank, eigenvector centrality and the hub/authority scores
of HITS. We derive simple relations between the measures and the (in)degree of
the nodes, in some limits. We also compare the rankings obtained with different
centrality measures.Comment: 11 pages, 10 figures, 5 tables. Final version published in Physical
Review
Robustness of Network Measures to Link Errors
In various applications involving complex networks, network measures are
employed to assess the relative importance of network nodes. However, the
robustness of such measures in the presence of link inaccuracies has not been
well characterized. Here we present two simple stochastic models of false and
missing links and study the effect of link errors on three commonly used node
centrality measures: degree centrality, betweenness centrality, and dynamical
importance. We perform numerical simulations to assess robustness of these
three centrality measures. We also develop an analytical theory, which we
compare with our simulations, obtaining very good agreement.Comment: 9 pages, 9 figure
Network centrality: an introduction
Centrality is a key property of complex networks that influences the behavior
of dynamical processes, like synchronization and epidemic spreading, and can
bring important information about the organization of complex systems, like our
brain and society. There are many metrics to quantify the node centrality in
networks. Here, we review the main centrality measures and discuss their main
features and limitations. The influence of network centrality on epidemic
spreading and synchronization is also pointed out in this chapter. Moreover, we
present the application of centrality measures to understand the function of
complex systems, including biological and cortical networks. Finally, we
discuss some perspectives and challenges to generalize centrality measures for
multilayer and temporal networks.Comment: Book Chapter in "From nonlinear dynamics to complex systems: A
Mathematical modeling approach" by Springe
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