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Discrete Calculus pp 267–289Cite as

Measuring Networks

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Abstract

We have adopted the view of graphs and, more generally, cell complexes as a domain upon which we may apply the tools of calculus to formulate differential equations and to analyze data. An important aspect of the discrete differential operators is that the operators are defined by the topology of the domain itself. Therefore, in an effort to provide a complete treatment of these differential operators, we examine in this chapter the properties of the network which may be extracted from the structure of these operators. In addition to the network properties extracted directly from the differential operators, we also review other methods for measuring the structural properties of a network. Specifically, the properties of the network that we consider are based on distances, partitioning, geometry, and topology. Our particular focus will be on the measurement of these properties from the graph structure. Applications will illustrate the use of these measures to predict the importance of nodes and to relate these measures to other properties of the subject being modeled by the network.

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Notes

  1. 1.

    In the circuit theory analogy, for the two indices to be compatible the prescribed edge weights of a weighted graph are interpreted as resistances when measuring the Wiener index, and the same prescribed edge weights on the weighted graph are interpreted as conductances when measuring the Kirchhoff index.

  2. 2.

    The exterior face is a device to enable a finite graph to be defined such that it has no boundary and is therefore closed. This imparts a global topology on the graph—that of a sphere or a sphere with handles—which may be interpreted as a finite graph including a face “at infinity”, in analogy to projections of the sphere into the plane (e.g., by stereographic projection) in which the coordinate at the pole of the sphere opposite the origin is mapped to the point at infinity on the flat plane.

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Authors and Affiliations

  1. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540-6632, USA

    Dr. Leo J. Grady

  2. Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA

    Dr. Jonathan R. Polimeni

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  1. Dr. Leo J. Grady
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  2. Dr. Jonathan R. Polimeni
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Correspondence to Leo J. Grady .

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Grady, L.J., Polimeni, J.R. (2010). Measuring Networks. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_8

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  • DOI: https://doi.org/10.1007/978-1-84996-290-2_8

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