Social Network Generation

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Research work and images have been realised by Nathalie Henry and Jean-Daniel Fekete, using MatrixExplorer, built with the Infovis Toolkit.

Contents

[edit] Social Network Characterization

Social networks involve persons or groups called actors and relationship between them, with a lot of variety in the kind of actors and relationships. As described in Wasserman and Faust, actors can be people, subgroups, organizations or collectivities; relations may be friendship (relationships), interactions, communications, transactions, movement or kinship. However, the nature of actors and relations does not really matter: we focus on their structure. We can classify the social networks studied in the literature in three categories:

  • Tree-like are trees with additional links forming cycles with a specified probability. This category includes genealogy data and very sparse graphs such as Sexually-Transmitted Disease (STD) transmission patterns. We call them “almost trees” because they have are mostly acyclic and nodes have very few parents.
  • Almost complete graphs are complete graphs with missing relations. For example, data about trade between countries, cities or companies are almost complete graphs. They are interesting to study as valued graphs; since they usually carry values on their edges.
  • Small-world networks (also scale-free or power-law degree-distribution networks) have been studied intensely since they were first described in Watts and Strogatz. They defined them as graphs with three properties: power-law degree distribution, high clustering coefficient and small average shortest path. They are locally dense (sparse with dense sub-graphs).


Three methods exist to select datasets for assessing the quality of analysis systems in the context of social networks: selecting one or two real datasets hoping they are representative, selecting several datasets or generating random datasets with well-known characteristics shared by social networks. With this last method, one should generate datasets with a controlled set of properties and evaluate the systems knowing the properties in advance. It should then eliminate biases linked to a particular dataset and eases the replication of experiments. Unfortunately, while generating tree-like and almost-complete graphs is relatively straightforward, generating graphs with a small-world network structure is still a research topic for computer scientists and physicists. This page shows the results of popular and available network generators. In light of the real social networks we present in the #Real Social Networks, we consider them unsuitable for evaluations since users can easily notice their artifical nature.


[edit] 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.

[edit] About datasets and representations

  • All datasets are downloadable in GraphMl format.
  • 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].

[edit] 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

W1 SmallWorld_47_0.1_6.xml

WSmallWorld 47 0.1 6.JPG WSmallWorldMInit 47 0.1 6.PNG WSmallWorldM 47 0.1 6.JPG


W2 SmallWorld_47_0.3_6.xml

WSmallWorld 47 0.3 6.JPG WSmallWorldMInit 47 0.3 6.PNG WSmallWorldM 47 0.3 6.PNG


W3 SmallWorld_47_0.5_6.xml

WSmallWorld 47 0.5 6.JPG WSmallWorldMInit 47 0.5 6.PNG WSmallWorldM 47 0.5 6.PNG

W4 SmallWorld_47_0.7_6.xml

WSmallWorld 47 0.7 6.JPG WSmallWorldMInit 47 0.7 6.PNG WSmallWorldM 47 0.7 6.PNG

W5 SmallWorld_47_0.9_6.xml


WSmallWorld 47 0.9 6.JPG WSmallWorldMInit 47 0.9 6.PNG WSmallWorldM 47 0.9 6.PNG

W6 SmallWorld_47_0.3_2.xml

WSmallWorld 47 0.3 2.JPG WSmallWorldMInit 47 0.3 2.PNG WSmallWorldM 47 0.3 2.PNG

W7 SmallWorld_47_0.3_4.xml


WSmallWorld 47 0.3 4.JPG WSmallWorldMInit 47 0.3 4.PNG WSmallWorldM 47 0.3 4.PNG

W8 SmallWorld_47_0.3_8.xml


WSmallWorld 47 0.3 8.JPG WSmallWorldMInit 47 0.3 8.PNG WSmallWorldM 47 0.3 8.PNG


W9 SmallWorld_47_0.3_10.xml

WSmallWorld 47 0.3 10.JPG WSmallWorldMInit 47 0.3 10.PNG WSmallWorldM 47 0.3 10.PNG


W10 SmallWorld_47_0.7_4.xml

WSmallWorld 47 0.7 4.JPG WSmallWorldMInit 47 0.7 4.PNG WSmallWorldM 47 0.7 4.PNG

W11 SmallWorld_47_0.1_8.xml

WSmallWorld 47 0.1 8.JPG WSmallWorldMInit 47 0.1 8.PNG WSmallWorldM 47 0.1 8.PNG

W12 SmallWorld_94_0.1_8.xml

WSmallWorld 94 0.1 8.JPG WSmallWorldMInit 94 0.1 8.PNG WSmallWorldM 94 0.1 8.PNG


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.xml

KSmallWorld 49 0.1.PNG KSmallWorldMInit 49 0.1.PNG KSmallWorldM 49 0.1.PNG

W2 SmallWorld_49_0.5.xml


KSmallWorld 49 0.5.PNG KSmallWorldMInit 49 0.5.PNG KSmallWorldM 49 0.5.PNG

W3 SmallWorld_49_1.0.xml


KSmallWorld 49 1.0.PNG KSmallWorldMInit 49 1.0.PNG KSmallWorldM 49 1.0.PNG

W4 SmallWorld_49_2.0.xml


KSmallWorld 49 2.0.PNG KSmallWorldMInit 49 2.0.PNG KSmallWorldM 49 2.0.PNG

W5 SmallWorld_49_2.5.xml


KSmallWorld 49 2.5.PNG KSmallWorldMInit 49 2.5.PNG KSmallWorldM 49 2.5.PNG

W6 SmallWorld_49_4.0.xml


KSmallWorld 49 4.0.PNG KSmallWorldMInit 49 4.0.PNG KSmallWorldM 49 4.0.PNG

W7 SmallWorld_49_8.0.xml


KSmallWorld 49 8.0.PNG KSmallWorldMInit 49 8.0.PNG KSmallWorldM 49 8.0.PNG

W8 SmallWorld_100_2.0.xml


KSmallWorld 100 2.0.PNG KSmallWorldMInit 100 2.0.PNG KSmallWorldM 100 2.0.PNG

W9 SmallWorld_100_4.0.xml

KSmallWorld 100 4.0.PNG KSmallWorldMInit 100 4.0.PNG KSmallWorldM 100 4.0.PNG

W10 SmallWorld_100_8.0.xml

KSmallWorld 100 8.0.PNG KSmallWorldMInit 100 8.0.PNG KSmallWorldM 100 8.0.PNG

W11 SmallWorld_100_12.0.xml

KSmallWorld 100 12.0.PNG KSmallWorldMInit 100 12.0.PNG KSmallWorldM 100 12.0.PNG

[edit] 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.xml

BaScaleFree 4 2 10.PNG BaScaleFreeMInit 4 2 10.PNG BaScaleFreeM 4 2 10.PNG

W2 ScaleFree_4_2_50.xml


BaScaleFree 4 2 50.PNG BaScaleFreeMInit 4 2 50.PNG BaScaleFreeM 4 2 50.PNG

W3 ScaleFree_4_2_100.xml


BaScaleFree 4 2 100.PNG BaScaleFreeMInit 4 2 100.PNG BaScaleFreeM 4 2 100.PNG

W4 ScaleFree_4_1_100.xml

BaScaleFree 4 1 100.PNG BaScaleFreeMInit 4 1 100.PNG BaScaleFreeM 4 1 100.PNG

W5 ScaleFree_2_1_100.xml

BaScaleFree 2 1 100.PNG BaScaleFreeMInit 2 1 100.PNG BaScaleFreeM 2 1 100.PNG

W6 ScaleFree_2_1_50.xml

BaScaleFree 2 1 50.PNG BaScaleFreeMInit 2 1 50.PNG BaScaleFreeM 2 1 50.PNG

W7 ScaleFree_2_2_50.xml

BaScaleFree 2 2 50.PNG BaScaleFreeMInit 2 2 50.PNG BaScaleFreeM 2 2 50.PNG

W8 ScaleFree_4_4_50.xml


BaScaleFree 4 4 50.PNG BaScaleFreeMInit 4 4 50.PNG BaScaleFreeM 4 4 50.PNG


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).

[edit] Real Social Networks

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].


[edit] Small-World

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

TeamCollaborationExternal TeamCollaborationExternal.xml


TeamCollaborationExternal.PNG TeamCollaborationExternalMInit.PNG TeamCollaborationExternalM.PNG

Infovis Component 1 ivComp1.xml

IvComp1.PNG IvComp1MInit.PNG IvComp1M.PNG

Infovis Component 2 ivComp2.xml


IvComp2.PNG IvComp2MInit.PNG IvComp2M.PNG

Infovis Component 3 ivComp3.xml


IvComp3.PNG IvComp3MInit.PNG IvComp3M.PNG

Infovis Component 4 ivComp4.xml


IvComp4.PNG IvComp4MInit.PNG IvComp4M.PNG

[edit] Tree-like

Parameters and Resulting Graph characteristics for Genealogy and Virus Transmission Name genealogy MSTTransmission1 MSTTransmission2 HIVTransmission
Source Pajek Article [5] Article[6] Article [7]
numVertices 242 38 84 243
numEdges 510 78 182 514
components 1 1 1 1
density 0.09 0.23 0.16 0.09
clusteringCoefficient 0.66 0.53 0.52 0.65
diameter 11 10 9 23
averageShortestDistance 5.78 4.42 4.31 8.27
minDegree 1 1 1 1
maxDegree 14 7 17 20

Gondola Genealogy GondolaGen.xml


GondolaGen.PNG GondolaGenMInit.PNG GondolaGenM.PNG

MSTTransmission 1 Mst1.xml

Mst1.PNG Mst1MInit.PNG Mst1M.PNG

MSTTransmission 2 Mst2.xml


Mst2.PNG Mst2MInit.PNG Mst2M.PNG

HIV Transmission Hiv.xml

Hiv.PNG HivMInit.PNG HivM.PNG

[edit] Almost Complete Graphs

Parameters and Resulting Graph characteristics for Email Communication within a research lab.
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

Email exchange per person during a day emailDay.xml

EmailDay.PNG EmailDayMInit.PNG EmailDayM.PNG

Email exchange per person during a week emailWeek.xml

EmailWeek.PNG EmailWeekMInit.PNG EmailWeekM.PNG

Email exchange per person during a month emailMonth.xml

EmailMonth.PNG EmailMonthMInit.PNG EmailMonthM.PNG

Email exchange per person during a year emailYear.xml


EmailYear.PNG EmailYearMInit.PNG EmailYearM.PNG

Email exchange per research group during a day emailGDay.xml

EmailGDay.PNG EmailGDayMInit.PNG EmailGDayM.PNG

Number of email coded with link width in the nodelink, edge color in the matrix
EmailGDayS.PNG EmailGDayMC.PNG

Email exchange per research group during a week emailGWeek.xml

EmailGWeek.PNG EmailGWeekMInit.PNG EmailGWeekM.PNG

Number of email coded with link width in the nodelink, edge color in the matrix
EmailGWeekS.PNG EmailGWeekMC.PNG

Email exchange per research group during a month emailGMonth.xml

EmailGMonth.PNG EmailGMonthMInit.PNG EmailGMonthM.PNG

Number of email coded with link width in the nodelink, edge color in the matrix
EmailGMonthS.PNG EmailGMonthMC.PNG

Email exchange per research group during a year emailGYear.xml

EmailGYear.PNG EmailGYearMInit.PNG EmailGYearM.PNG

Number of email coded with link width in the nodelink, edge color in the matrix
EmailGYearS.PNG EmailGYearMC.PNG