[…] In Section 3 we consider an application of these concepts in a practical setting, which serves as an illustration of the definitions above. We will consider a graph made up of nodes that represent mathematical concepts with links that reflect conceptual relations between pairs of concepts (inferred from Wikipedia hyperlinks) and study a collection of scientific papers that use these concepts. The occurrence of a subset of concepts in a given paper is translated to a distribution on the relevant nodes (concepts) in the network, and the frequency of pairs of concepts appearing together in a paper is represented as a joint distribution on the network.
## 2.2. (co)variance with respect to node distances.
As introduced, the interpretation of a graph distribution is twofold: we can consider it as a signal (function) on the nodes of a graph (see also [44]), or as representing a random node.
[44] ⇒ SHUMAN, David I, NARANG, Sunil K., FROSSARD, Pascal, ORTEGA, Antonio and VANDERGHEYNST, Pierre, 2013. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine. May 2013. Vol. 30, no. 3, p. 83–98. DOI 10.1109/MSP.2012.2235192.
> In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.