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 Generator

WattsBetaSmallWorldGenerator

Parameters: numNodes (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
numNodes	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
numNodes	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
numNodes (sqrt)	7	7	7	7	7	7	7	10	10	10	10
clustering exponent	0.1	0.5	1	2	2.5	4	8	2	4	8	12
numNodes	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
numStartingNodes	4	4	4	4	2	2	2	4
numAdditionnalEdges	2	2	2	1	1	1	2	4
numSteps	10	50	100	100	100	50	50	50
numNodes	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

Name	Infovis Comp4
Type	Co-authoring network
File
numNodes	32
numEdges	109
components	1
density	0.33
clusteringCoefficient	0.81
diameter	6
averageShortestDistance	2.60
minDegree
maxDegree