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Statistical Analysis of Network Data with R pp 161–178Cite as

Analysis of Network Flow Data

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Part of the book series: Use R! ((USE R,volume 65))

Abstract

Many networks serve as conduits—either literally or figuratively—for flows , in the sense that they facilitate the movement of something, such as materials, people, or information. For example, transportation networks (e.g., of highways, railways, and airlines) support flows of commodities and people, communication networks allow for the flow of data, and networks of trade relations among nations reflect the flow of capital.

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Notes

  1. 1.

    Data were originally collected by Manfred Fischer and Petra Staufer.

  2. 2.

    More precisely, we can write (9.7) in the form \( \log (\mu ) = \mathbf{M}\gamma \), where M is an (IJ) × (I + J + K) matrix, and \( \gamma = {(\alpha _{1},\ldots,\alpha _{I},\beta _{1},\ldots,\beta _{J},\theta _{1},\ldots,\theta _{K})}^{T} \) is an (I + J + K) × 1 vector. The first I + J columns of M are binary vectors, indicating the appropriate origin and destination for each entry of μ, and are redundant in that both the first I and the next J sum to the unit vector. The last K columns correspond to the K variables defining the c ij . Assuming that the latter are linearly independent of themselves and of the former, the rank of M will be (I + J − 1) + K. See Sen and Smith [130, Chap. 5.2].

  3. 3.

    The AIC statistic for a likelihood-based model, with k-dimensional parameter η, is defined as \( AIC = -2\ell(\hat{\eta }) + 2k \), where \( \ell(\eta ) \) is the log-likelihood evaluated at η, and \( \hat{\eta } \) is the maximum likelihood estimate of η. This statistic, as with others of its type, provides an estimate of the generalization error associated with the fitted model, in this case effectively by off-setting the assessment of how well the model fits the data by a measure of its complexity. See, for example, Hastie, Tibshirani, and Friedman [71, Chap. 7.5] for additional details.

  4. 4.

    If multiple routes are possible, the entries of B are instead fractions representing, for example, the proportion of traffic from i to j that is expected to use the link e.

References

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University Professor, Boston, MA, USA

    Eric D. Kolaczyk

  2. Department of Statistics, Harvard University Research Associate, Cambridge, MA, USA

    Gábor Csárdi

Authors
  1. Eric D. Kolaczyk
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  2. Gábor Csárdi
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Kolaczyk, E.D., Csárdi, G. (2014). Analysis of Network Flow Data. In: Statistical Analysis of Network Data with R. Use R!, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0983-4_9

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