Social Network Generation

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Issues on Social Network Generation for Evaluating Visualizations

Watts and Strogatz first described in (Watts, D. J. and S. H. Strogatz (1998). "Collective dynamics of 'small-world' networks." Nature 393: 440 - 442) the concept of small-world networks. They formalized these networks as graphs with three properties: power-law degree distribution, high clustering coefficient and small average shortest path. In the same paper they propose a basic model fitting these properties consisting in a grid (fixed local neighborhood) with additional links simulating some unexpected relations support to the six degrees of separation discovered by Milgram (Milgram, S. (1967). "The small world problem." Psychology Today: 60-67). Barabási and Albert proposed an incremental model to improve it (Barabási, A.-L. and R. Albert (1999). "Emergence of Scaling in Random Networks." Science 286(5439): 509 - 512. ). Since Watts and Strogatz’ model, several have been proposed each generating networks with one or two of the described properties (power-law) but none combine the three of them.

Here are some results of available generators present in the JUNG package. Let's note that for each network generated we only keep the biggest component. Generators present in Pajek[1] and Geomi[2] are incremental scale-free networks generators such as the Barabasi and Albert model.

Small-World Generators

WattsBetaSmallWorldGenerator

Parameters: numVertices (the number of nodes in the ring lattice), beta (the probability of an edge being rewired randomly; the proportion of randomly rewired edges in a graph) and degree( the number of edges connected to each vertex; the local neighborhood size). Degree must be even.

Parameters and Resulting Graph characteristics
graphs W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12
numVertices 47 47 47 47 47 47 47 47 47 47 47 94
beta 0.1 0.3 0.5 0.7 0.9 0.3 0.3 0.3 0.3 0.7 0.1 0.1
degree 6 6 6 6 6 2 4 8 10 4 8 8
numVertices 47 47 47 47 47 47 47 47 47 47 47 94
numEdges 282 282 282 282 282 94 188 376 470 188 376 752
components 1 1 1 1 1 2 1 1 1 1 1 1
density 0.36 0.36 0.36 0.36 0.36 0.21 0.29 0.41 0.46 0.29 0.41 0.29
clusteringCoefficient 0.51 0.25 0.15 0.09 0.12 0.23 0.25 0.32 0.38 0.07 0.53 0.52
diameter 6 4 4 4 4 - 6 4 3 5 5 6
averageShortestDistance 2.97 2.4 2.32 2.3 2.29 - 3.24 2.15 1.98 2.83 2.56 3.15
minDegree 5 4 4 3 4 1 2 5 8 2 7 6
maxDegree 8 9 9 9 9 4 6 10 13 8 10 10

GraphMl files and Pictures:

  • Node-Link diagrams are ordered with the linLog algorithm of Andreas Noack [Graph Drawing 2005] (with edge-repulsion coefficient of 2.5f).
  • Matrices are shown both with the initial order (middle image) and reordered with the TSP-Based algorithm (right image) described by Henry and Fekete [Infovis 2006].

W1 SmallWorld_47_0.1_6


W2 SmallWorld_47_0.3_6


W3 SmallWorld_47_0.5_6

W4 SmallWorld_47_0.7_6

W5 SmallWorld_47_0.9_6

W6 SmallWorld_47_0.3_2

W7 SmallWorld_47_0.3_4

W8 SmallWorld_47_0.3_8


W9 SmallWorld_47_0.3_10


W10 SmallWorld_47_0.7_4

W11 SmallWorld_47_0.1_8

W12 SmallWorld_94_0.1_8


KleinbergSmallWorldGenerator

Parameters:latticeSize (the lattice size (length of row or column dimension)) and clusteringExponent (the clustering exponent parameter).

Parameters and Resulting Graph characteristics
graphs W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11
latticeSize 7 7 7 7 7 7 7 10 10 10 10
clusteringExponent 0.1 0.5 1 2 2.5 4 8 2 4 8 12
numVertices 49 49 49 49 49 49 49 100 100 100 100
numEdges 490 490 490 490 490 490 490 1000 1000 1000 1000
components 1 1 1 1 1 1 1 1 1 1 1
density 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.32 0.32 0.32 0.32
clusteringCoefficient 0.08 0.09 0.14 0.19 0.19 0.26 0.32 0.18 0.23 0.32 0.33
diameter 4 4 4 4 4 5 5 5 6 7 7
averageShortestDistance 2.38 2.36 2.37 2.44 2.48 2.54 2.73 3.1 3.57 3.65 3.68
minDegree 9 9 9 9 9 9 9 9 9 9 9
maxDegree 14 12 13 12 12 13 12 13 13 14 12

W1 SmallWorld_49_0.1

W2 SmallWorld_49_0.5

W3 SmallWorld_49_1.0

W4 SmallWorld_49_2.0

W5 SmallWorld_49_2.5

W6 SmallWorld_49_4.0

W7 SmallWorld_49_8.0

W8 SmallWorld_100_2.0

W9 SmallWorld_100_4.0

W10 SmallWorld_100_8.0

W11 SmallWorld_100_12.0

Scale-Free Networks Generator

BarabasiAlbertGenerator

Parameters: init_vertices (number of vertices that the graph should start with), numEdgesToAttach (the number of edges that should be attached from the new vertex to pre-existing vertices at each time step) and numSteps (number of time steps). init_vertices must be superior or equal to numEdgesToAttach.

Parameters and Resulting Graph characteristics
graphs W1 W2 W3 W4 W5 W6 W7 W8
init_vertices 4 4 4 4 2 2 2 4
numEdgesToAttach 2 2 2 1 1 1 2 4
numSteps 10 50 100 100 100 50 50 50
numVertices 14 53 104 80 76 51 52 54
numEdges 40 200 400 158 150 100 200 400
components 1 1 1 1 1 1 1 1
density 0.45 0.27 0.19 0.16 0.16 0.2 0.27 0.37
clusteringCoefficient 0.15 0.2 0.07 0.51 0.51 0.66 0.16 0.23
diameter 4 6 6 11 14 8 5 4
averageShortestDistance 2.24 2.81 3.18 5.26 5.7 3.74 2.8 2.15
minDegree 2 1 2 1 1 1 2 4
maxDegree 5 16 19 8 12 16 17 26

W1 ScaleFree_4_2_10

W2 ScaleFree_4_2_50

W3 ScaleFree_4_2_100

W4 ScaleFree_4_1_100

W5 ScaleFree_2_1_100

W6 ScaleFree_2_1_50

W7 ScaleFree_2_2_50

W8 ScaleFree_4_4_50


EppsteinPowerLawGenerator

Parameters: numVertices (the number of vertices for the generated graph), numEdges (the number of edges the generated graph will have, should be Theta(numVertices)) and r (the model parameter).

Issues on using real Social Networks for Evaluating Visualizations

Here is a panel of undirected networks issued from scientific articles, benchmarks or contests. Social network visualization or analysis tools provide also some real datasets: Pajek [3] and UCINet [4].

Parameters and Resulting Graph characteristics for Email Communication within our research group (per person and per team)
Name emailDay per person emailWeek per person emailMonth per person emailYear per person emailDay per team emailWeek per team emailMonth per team emailYear per team
Source Collected Collected Collected Collected Collected Collected Collected Collected
numVertices 134 200 242 447 30 33 35 42
numEdges 442 1676 3514 11462 183 410 564 980
components 1 1 1 1 1 1 1 1
density 0.16 0.2 0.24 0.24 0.45 0.61 0.68 0.75
clusteringCoefficient 0.52 0.55 0.62 0.71 0.62 0.78 0.83 0.84
diameter 9 7 6 6 5 3 3 3
averageShortestDistance 4.29 2.92 2.52 2.42 2.17 1.71 1.57 1.45
minDegree 1 1 1 1 1 1 1 3
maxDegree 15 51 86 195 16 26 34 40


Parameters and Resulting Graph characteristics for Co-Authoring Networks
Name Team Collaboration Team Collaboration (within) Infovis component 1 Infovis component 2 Infovis component 3 Infovis component 4
Source Collected Collected Contest Contest Contest Contest
numNodes 146 96 135 48 47 32
numEdges 540 316 321 91 114 109
components 1 1 1 1 1 1
density 0.16 0.19 0.13 0.2 0.23 0.33
clusteringCoefficient 0.91 0.91 0.82 0.79 0.83 0.81
diameter 4 5 11 7 10 6
averageShortestDistance 2.65 2.81 4.4 3.71 3.84 2.6
minDegree 1 1 1 1 1 1
maxDegree 57 29 22 11 15 15