Web and Social Computing

1. Books on Politics Network-
2004

2. Objective: To analyse a graphical
representation of a network
Coral reef
food web,
Cuba
233 vertices
3,753 edges
Data: http://datadryad.org/resource/doi:10.5061/dryad.c213h

3. Objective: To analyse a graphical
representation of a network
CMT111
Students,
Cardiff
26 vertices
30 edges

4. Objective: To analyse a graphical
representation of a network
Purchase of political books,
USA
105 vertices
441 edges
Data courtesy of Valdis Kreb available at: http://www-personal.umich.edu/~mejn/netdata/

5. Limitations
• Number of purchases missing from database
• No weights : Number of buyers co-purchased the
books

6. Mathematica
• Importing data
• Will not read as wide a variety of CSV formats as Gephi
• Can read .gml, .gv, .dot, .graphml, .gxl, .col, .g6, .s6, .gw, .net, .tgf
• Use Map or create a rule to show links (->) from one column of a CSV onto
another
• Other attributes
• More difficult to show vertex/edge attributes than in Gephi but still ppssiblt to
highlight using HighlightGraph[g, x]
• Built-in functions
• Very intuitive and well documented:
https://reference.wolfram.com/language/guide/GraphsAndNetworks.html

7. Book
customer who
has purchased both
books (endpoints)
NODE
LINK

8. This graph is
unweighted: edges do not have associated weights
undirected: all edges travel in both directions
contains loops: no vertex is linked directly to itself
simple: undirected, unweighted, loop-free and lacks multiple edges
incomplete: each vertex is not connected to every other vertex
cyclic: contains at least one cycle
not bipartite: vertices cannot be divided into two disjoint sets
UndirectedGraphQ[books]
WeightedGraphQ[books]
CompleteGraphQ[books]
SimpleGraphQ[books]
BipartiteGraphQ[books]
LoopFreeGraphQ[books]
AcyclicGraphQ[books]

9. {6, 4, 4, 23, 8, 7, 11, 8, 25, 16, 15, 18, 25, 13, 9, 5, 3, 5, 3, 5, 10, 5, 7, 9, 9,
5, 9, 9, 3, 4, 20, 11, 5, 9, 5, 10, 5, 7, 7, 8, 18, 8, 6, 5, 5, 6, 4, 18, 4, 8, 3, 6,
5, 5, 6, 4, 4, 5, 13, 5, 6, 4, 6, 4, 9, 7, 21, 6, 4, 5, 7, 15, 22, 21, 16, 16, 13, 7,
5, 8, 4, 4, 9, 8, 23, 5, 14, 5, 5, 6, 5, 8, 3, 7, 7, 4, 6, 6, 5, 12, 12, 5, 4, 2, 3}
VertexDegree[books]
Vertex degrees:
Histogram[VertexDegree[books], ChartStyle -> Blue,
AxesLabel -> {HoldForm[Vertex degree], HoldForm[Frequency]},
PlotLabel -> None, LabelStyle -> {14, GrayLevel[0]}]

10. MatrixForm[GraphDistanceMatrix[books]]
Distance matrix

11. Array plot: distance matrix
ArrayPlot[GraphDistanceMatrix[books]]

12. Max[Flatten[GraphDistanceMatrix[books]]]
OR
GraphDiameter[books]
Diameter = 7
Radius = 4
GraphRadius[books]
Maximum and minimum eccentricities of any vertex:

13. Eccentricity Centrality
HighlightGraph[books, VertexList[books],
VertexSize ->
Thread[VertexList[books] ->
Rescale[EccentricityCentrality[books]]]]
Increases as maximum distances to every other reachable node increases

14. Closeness Centrality
HighlightGraph[books, VertexList[books],
VertexSize ->
Thread[VertexList[books] ->
Rescale[ClosenessCentrality[books]]]]
Increases as average distance to other nodes increases

15. Betweenness Centrality
HighlightGraph[books, VertexList[books],
VertexSize ->
Thread[VertexList[books] ->
Rescale[BetweennessCentrality[books]]]]
Increases as node lies on more shortest paths between other node-pairs

16. Degree Centrality
HighlightGraph[books, VertexList[books],
VertexSize ->
Thread[VertexList[books] ->
Rescale[DegreeCentrality[books]]]]
Increases as vertex degree increases

17. a = ListPlot[EccentricityCentrality[books], Filling -> Axis, PlotStyle -> Red]
b = ListPlot[ClosenessCentrality[books], Filling -> Axis, PlotStyle -> Magenta]
c = ListPlot[RadialityCentrality[books], Filling -> Axis, PlotStyle -> Cyan]
d = ListPlot[DegreeCentrality[books], Filling -> Axis]
gg = GraphicsGrid[{{a, b}, {c, d}}]
Export["CentralityGrid.png", gg]
Eccentricity Closeness
Radiality Degree

18. Dual Hub
HighlightGraph[books, GraphHub[books]]
Nodes with the highest vertex degree are returned

19. Hub Neighbours
HighlightGraph[books, NeighborhoodGraph[books, GraphHub[books]]]

20. CommunityGraphPlot[books]
Communities: Small world graph (Modular)

21. Graph partition:
minimises number of endpoints having edges in each part
HighlightGraph[books,FindGraphPartition[books]]

22. Graph communities:
maximises edges joining nodes within communities
with relatively fewer edges joining to nodes in other
communities
HighlightGraph[books, FindGraphCommunities[books]]

25. Cliques
Largest set of connected vertices
HighlightGraph[books, Subgraph[books, FindClique[books]]]

26. Cliques
Largest set of connected vertices within 2 edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 2]]]

27. Cliques
Largest set of connected vertices within 3 edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 3]]]

28. Cliques
Largest set of connected vertices within 4 edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 4]]]

29. Cliques
Largest set of connected vertices within 5 edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 5]]]

30. Cliques
Largest set of connected vertices within 6 edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 6]]]

31. Cliques
Largest set of connected vertices within 7 (=diameter)
edges of each other
HighlightGraph[books, Subgraph[books, FindKClique[books, 7]]]

32. Lessons & Conclusions
• Mathematica Vs Gephi on Data Visualization
• Gephi struggles with larger datasets, crashes on OS X, cannot ‘undo’
• Gephi good a pulling apart larger datasets for easier visualisation, takes a wider range of input
formats, can visualise ‘multiple graphs’ more easily
• All the other functions within Mathematica at your disposal to aid network analysis e.g. Plot
• Source of Data Sets
• Working with a dataset of sufficient size but not so big that it cannot be comprehended.
• Analysis of sub-networks