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Discrete Calculus pp 125–154Cite as

Building a Weighted Complex from Data

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Abstract

In some applications, both the neighborhood structure of the data and the weights of the cells are naturally and directly defined by the problem at hand (e.g., road networks, social networks, communication networks, chemical graph theory or surface simplification). However, in many other applications the appropriate representation of the data to be analyzed is not provided (e.g., machine learning). Therefore, to use the tools of discrete calculus, a practitioner must determine the topology and weights of the graph or complex from the data that is most appropriate for solving the problem. In this chapter, we will discuss different techniques for generating a meaningful weighted complex from an embedding or from the data itself. Our focus will be primarily on generating weighted edges and faces from node and/or edge data, but we additionally demonstrate how these techniques may be applied to weighting higher-order structures.

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Notes

  1. 1.

    A parallel machine, with one processor devoted to one node or to a small number of nodes, can potentially be used to address fully connected graphs, since each processor must do work only on the order of \( |\mathcal{V}| \). Although such machines are becoming increasingly common, the computers of today are still predominantly unable to process fully connected graphs of any size.

  2. 2.

    In graph theory, a bridge is an edge whose deletion would result in increasing the number of connected components in the graph. Thus an edge is a bridge if (and only if) it is not contained in a cycle.

  3. 3.

    Note that when the graph genus equals zero then, by the relationship between the Euler Characteristic and genus, \( |\mathcal{F}| = |\mathcal{E}| - |\mathcal{V}| + 2 \). In contrast, the number of linearly independent cycles (spanning the nullspace of \(\boldsymbol{\mathsf {A}}\)) is given by \( |\mathcal{F}| = |\mathcal{E}| - |\mathcal{V}| + 1 \). Therefore, a planar graph implicitly has an extra face that is not linearly independent. This is the “outside face” described in Chap. 2.

  4. 4.

    Recall that, for a connected graph with \( \mathcal{V} \) and \( \mathcal{E} \), a tree subgraph is identified by the connected subgraph comprised of \( \mathcal{V} \) and \( \mathcal{E}_{{\rm tree}} \subseteq \mathcal{E} \), where \( |\mathcal{E}_{{\rm tree}} | = |\mathcal{V}| - 1 \). The cotree subgraph corresponding to a tree subgraph is the (possibly disconnected) subgraph consisting of \( \mathcal{V} \) and the set of the remaining edges \( \mathcal{E}_{{\rm cotree}} = \mathcal{E} - \mathcal{E}_{{\rm tree}} \). Thus the union of the edge set from any given tree with the edge set from its cotree is equivalent to the original graph.

  5. 5.

    This concept appears again in Chap. 5 and Appendix B.

  6. 6.

    Based on the available evidence at the time of writing, the authors would suggest using a Cauchy function for most applications, although the Welsch function and Tukey functions also seem to perform well. The remaining robust functions have yet to prove as generically useful.

  7. 7.

    When \(\boldsymbol{\mathsf {G}}^{-1}_{2}\) and \(\boldsymbol {\mathsf{G}}_{0}\) are matrices, these viscosities are linear and therefore the flow field may be viewed as a Newtonian fluid. The more complicated analysis of non-Newtonian fluids would describe the behavior of the flows when \(\boldsymbol{\mathsf {G}}^{-1}_{2}\) and \(\boldsymbol{\mathsf{G}}_{0}\) are replaced by nonlinear functions.

  8. 8.

    Desbrun et al. [102] distinguish between a local metric which is defined only for pairs of nodes which share an edge and a global metric which is defined for any pair of nodes in a connected graph. In this terminology, our treatment will focus exclusively on global metrics.

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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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Grady, L.J., Polimeni, J.R. (2010). Building a Weighted Complex from Data. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_4

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