**Authors:** Jonathan Mei,JosÃ© M. F. Moura

**ArXiv:** 1806.01455
**Document:**
PDF
DOI
**Abstract URL:** http://arxiv.org/abs/1806.01455v2
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Many applications donot have the benefit of the laws of physics to derive
succinct descriptive models for observed data. In alternative,
interdependencies among $N$ time series $\{ x_{nk}, k>0 \}_{n=1}^{N}$ are
nowadays often captured by a graph or network $G$ that in practice may be very
large. The network itself may change over time as well (i.e., as $G_k$).
Tracking brute force the changes of time varying networks presents major
challenges, including the associated computational problems. Further, a large
set of networks may not lend itself to useful analysis. This paper approximates
the time varying networks $\left\{G_k\right\}$ as weighted linear combinations
of eigennetworks. The eigennetworks are fixed building blocks that are
estimated by first learning the time series of graphs $G_k$ from the data $\{
x_{nk}, k>0 \}_{n=1}^{N}$, followed by a Principal Network Analysis procedure.
The weights of the eigennetwork representation are eigenfeatures and the time
varying networks $\left\{G_k\right\}$ describe a trajectory in eigennetwork
space. These eigentrajectories should be smooth since the networks $G_k$ vary
at a much slower rate than the data $x_{nk}$, except when structural network
shifts occur reflecting potentially an abrupt change in the underlying
application and sources of the data. Algorithms for learning the time series of
graphs $\left\{G_k\right\}$, deriving the eigennetworks, eigenfeatures and
eigentrajectories, and detecting changepoints are presented. Experiments on
simulated data and with two real time series data (a voting record of the US
senate and genetic expression data for the \textit{Drosophila Melanogaster} as
it goes through its life cycle) demonstrate the performance of the learning and
provide interesting interpretations of the eigennetworks.
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