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An Introduction to Proximity Graphs

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Business and Consumer Analytics: New Ideas

Abstract

Proximity graphs are one of the combinatorial data-miner’s frontline tools. They allow expression of complex proximity relationships and are the basis of many other algorithms. Here we introduce the concept of proximity graphs, present basic definitions and discuss some of the most common types of proximity graphs.

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Notes

  1. 1.

    Reinvented three further times under different names [9, 17], most notably as Sollin’s algorithm [39].

  2. 2.

    Also invented repeatedly [14, 24, 35], first by Vojtĕch Jarník [24]. The name Prim’s algorithm is however almost universally used.

  3. 3.

    Note also that the Voronoi Diagram of a set of points in general position is the dual graph of the Delaunay Triangulation.

  4. 4.

    It is interesting to note that not only was it discovered concurrently, but both articles were published in the same issue of the Computer Journal.

  5. 5.

    In a case of a set of points, of the implicit complete graph.

  6. 6.

    Some general algorithms are Dijkstra’s Algorithm [14], the Bellman–Ford–Moore Algorithm [3, 19, 33], the Floyd–Roy–Warshall Algorithm [18, 37, 46] and Johnson’s Algorithm [26].

  7. 7.

    Given a set of points P, a point q is a nearest point of point p if d(p, q) =minr ∈ P{d(p, r)}.

  8. 8.

    That is, no hyperplane contains more than d points where d is the dimension of the space. In particular, in two dimensions this means that no three points are colinear.

  9. 9.

    c being the length of the “unit” in the graph’s title.

  10. 10.

    Not to be confused with the amenity based star rating system!

  11. 11.

    Note that the figures are 2-dimensional projections of the 5-dimensional proximity graphs.

  12. 12.

    Note that the choice of projective plane here is largely for convenience. The choice of two of the ratings categories to form the x and y dimensions in the image has no additional bearing on the graph itself, only its layout. Certainly other axes could be used; other selections of the ratings categories are of course possible, but we note that given the nature of ratings, they are all essentially pairwise correlated and have sharp distributions, leading to roughly the same point layout. We could also choose a projection unrelated to the underlying data, for example, we could generate the graph, then apply any of a suite of layout algorithms, divorcing the position of the vertices from any intrinsic meaning.

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Acknowledgements

Pablo Moscato acknowledges previous support from the Australian Research Council Future Fellowship FT120100060 and Australian Research Council Discovery Projects DP120102576 and DP140104183.

Author information

Authors and Affiliations

  1. School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW, Australia

    Luke Mathieson & Pablo Moscato

  2. School of Software, University of Technology Sydney, Ultimo, NSW, Australia

    Luke Mathieson

Authors
  1. Luke Mathieson
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  2. Pablo Moscato
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Corresponding author

Correspondence to Luke Mathieson .

Editor information

Editors and Affiliations

  1. School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW, Australia

    Pablo Moscato

  2. School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW, Australia

    Natalie Jane de Vries

Appendix: Example Point Set

Appendix: Example Point Set

Point

x-position

y-position

Point

x-position

y-position

1

0.723212968

0.178106849

26

0.907661273

0.729217676

2

0.920432236

0.485452615

27

0.545987988

0.340721248

3

0.551488565

0.873938297

28

0.181010885

0.597560442

4

0.306689395

0.414440461

29

0.254672002

0.906480972

5

0.252440631

0.386800466

30

0.105705

0.170610555

6

0.326211397

0.990150763

31

0.667526022

0.476145574

7

0.669619173

0.276130067

32

0.784634882

0.565077639

8

0.385344312

0.279983967

33

0.287186647

0.402820584

9

0.695385871

0.066082659

34

0.182059349

0.688221614

10

0.475169562

0.266587733

35

0.637400634

0.602165836

11

0.545891144

0.912037976

36

0.322844825

0.588968292

12

0.017635186

0.333899253

37

0.760253692

0.383415267

13

0.621307759

0.610259379

38

0.25928387

0.955812554

14

0.716927846

0.528621053

39

0.791903241

0.644445644

15

0.207482761

0.976842469

40

0.005358534

0.280421027

16

0.018216165

0.891276263

41

0.419955306

0.763752059

17

0.320578694

0.264128312

42

0.2918794

0.122258146

18

0.420885588

0.702595526

43

0.010059218

0.809975938

19

0.237547534

0.454149704

44

0.008974315

0.354484587

20

0.608278025

0.297481298

45

0.450824119

0.497067255

21

0.830447143

0.512086882

46

0.01188008

0.444212976

22

0.105544439

0.033963298

47

0.892440161

0.2077337

23

0.741506729

0.279505876

48

0.245942889

0.718327626

24

0.99144205

0.866106852

49

0.636074704

0.058158036

25

0.021779104

0.004131444

50

0.133528981

0.009221669

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Mathieson, L., Moscato, P. (2019). An Introduction to Proximity Graphs. In: Moscato, P., de Vries, N. (eds) Business and Consumer Analytics: New Ideas. Springer, Cham. https://doi.org/10.1007/978-3-030-06222-4_4

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  • DOI: https://doi.org/10.1007/978-3-030-06222-4_4

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