21 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
Spectral Representations of Graphons in Very Large Network Systems Control
Graphon-based control has recently been proposed and developed to solve
control problems for dynamical systems on networks which are very large or
growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper,
spectral representations, eigenfunctions and approximations of graphons, and
their applications to graphon-based control are studied. First, spectral
properties of graphons are presented and then approximations based on Fourier
approximated eigenfunctions are analyzed. Within this framework, two classes of
graphons with simple spectral representations are given. Applications to
graphon-based control analysis are next presented; in particular, the
controllability of systems distributed over very large networks is expressed in
terms of the properties of the corresponding graphon dynamical systems.
Moreover, spectral analysis based upon real-world network data is presented,
which demonstrates that low-dimensional spectral approximations of networks are
possible. Finally, an initial, exploratory investigation of the utility of the
spectral analysis methodology in graphon systems control to study the control
of epidemic spread is presented.Comment: 8 pages, 58th IEEE Conference on Decision and Control (CDC 2019
Learning Graphons via Structured Gromov-Wasserstein Barycenters
We propose a novel and principled method to learn a nonparametric graph model
called graphon, which is defined in an infinite-dimensional space and
represents arbitrary-size graphs. Based on the weak regularity lemma from the
theory of graphons, we leverage a step function to approximate a graphon. We
show that the cut distance of graphons can be relaxed to the Gromov-Wasserstein
distance of their step functions. Accordingly, given a set of graphs generated
by an underlying graphon, we learn the corresponding step function as the
Gromov-Wasserstein barycenter of the given graphs. Furthermore, we develop
several enhancements and extensions of the basic algorithm, , the
smoothed Gromov-Wasserstein barycenter for guaranteeing the continuity of the
learned graphons and the mixed Gromov-Wasserstein barycenters for learning
multiple structured graphons. The proposed approach overcomes drawbacks of
prior state-of-the-art methods, and outperforms them on both synthetic and
real-world data. The code is available at
https://github.com/HongtengXu/SGWB-Graphon