1,307 research outputs found
Community detection in networks via nonlinear modularity eigenvectors
Revealing a community structure in a network or dataset is a central problem
arising in many scientific areas. The modularity function is an established
measure quantifying the quality of a community, being identified as a set of
nodes having high modularity. In our terminology, a set of nodes with positive
modularity is called a \textit{module} and a set that maximizes is thus
called \textit{leading module}. Finding a leading module in a network is an
important task, however the dimension of real-world problems makes the
maximization of unfeasible. This poses the need of approximation techniques
which are typically based on a linear relaxation of , induced by the
spectrum of the modularity matrix . In this work we propose a nonlinear
relaxation which is instead based on the spectrum of a nonlinear modularity
operator . We show that extremal eigenvalues of
provide an exact relaxation of the modularity measure , however at the price
of being more challenging to be computed than those of . Thus we extend the
work made on nonlinear Laplacians, by proposing a computational scheme, named
\textit{generalized RatioDCA}, to address such extremal eigenvalues. We show
monotonic ascent and convergence of the method. We finally apply the new method
to several synthetic and real-world data sets, showing both effectiveness of
the model and performance of the method
Super-resolution community detection for layer-aggregated multilayer networks
Applied network science often involves preprocessing network data before
applying a network-analysis method, and there is typically a theoretical
disconnect between these steps. For example, it is common to aggregate
time-varying network data into windows prior to analysis, and the tradeoffs of
this preprocessing are not well understood. Focusing on the problem of
detecting small communities in multilayer networks, we study the effects of
layer aggregation by developing random-matrix theory for modularity matrices
associated with layer-aggregated networks with nodes and layers, which
are drawn from an ensemble of Erd\H{o}s-R\'enyi networks. We study phase
transitions in which eigenvectors localize onto communities (allowing their
detection) and which occur for a given community provided its size surpasses a
detectability limit . When layers are aggregated via a summation, we
obtain , where is the number of
layers across which the community persists. Interestingly, if is allowed to
vary with then summation-based layer aggregation enhances small-community
detection even if the community persists across a vanishing fraction of layers,
provided that decays more slowly than . Moreover,
we find that thresholding the summation can in some cases cause to decay
exponentially, decreasing by orders of magnitude in a phenomenon we call
super-resolution community detection. That is, layer aggregation with
thresholding is a nonlinear data filter enabling detection of communities that
are otherwise too small to detect. Importantly, different thresholds generally
enhance the detectability of communities having different properties,
illustrating that community detection can be obscured if one analyzes network
data using a single threshold.Comment: 11 pages, 8 figure
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
On Spectral Graph Embedding: A Non-Backtracking Perspective and Graph Approximation
Graph embedding has been proven to be efficient and effective in facilitating
graph analysis. In this paper, we present a novel spectral framework called
NOn-Backtracking Embedding (NOBE), which offers a new perspective that
organizes graph data at a deep level by tracking the flow traversing on the
edges with backtracking prohibited. Further, by analyzing the non-backtracking
process, a technique called graph approximation is devised, which provides a
channel to transform the spectral decomposition on an edge-to-edge matrix to
that on a node-to-node matrix. Theoretical guarantees are provided by bounding
the difference between the corresponding eigenvalues of the original graph and
its graph approximation. Extensive experiments conducted on various real-world
networks demonstrate the efficacy of our methods on both macroscopic and
microscopic levels, including clustering and structural hole spanner detection.Comment: SDM 2018 (Full version including all proofs
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