Multiscale Visualization of Small World Networks

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Multiscale Visualization of Small World Networks

• David Auber LaBRI, Bordeaux, FR
• Yves Chiricota UQAC, Chicoutimi, CA
• Fabien Jourdan
• Guy Melançon

LIRMM, Montpellier, FR
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Motivations

• Small World Networks are common in Information
  Visualization
• Social Networks
• Power Grids, Computer Networks
• Food webs
• Software reverse engineering
• Semantic networks, word association networks

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Motivations

• Common tasks performed on SW networks
• Identify  Social  Groups
• How does the network split ?
• Social subgroups
• Determine Social Positions
• How much control over the flows in the network ?
• Access how easily can other actors be reached ?
• Hence questions such as
• Identify group(s) accessing all others
• Describe organization (network structure) of the
  subgroups
• Address these questions by visual inspection

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Observation

• SW network admit sub-components themselves being
  SW
• Movie actors
• Web sites Adamic
• Software Reverse Engineering (call graphs,
  includes, access graphs)

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Goal

• Compute the network split based on the network
  structure
• Avoid complex heuristics
• Apply recursively to get a hierarchy of subgroups
• Sub-networks
• Meta-networks (subgroups themselves organize into
  a network)
• Offer a tool to navigate the hierarchy

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Just what is a SW network ?

• Requires two structural properties
• Nodes have high cluster coefficients (on average)
• Average path length is low
• When compared to random graphs

Strogatz et al.
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Just what is a SW network ?

• Cluster coefficient of a node

Nv set of neighbours of v
v
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Just what is a SW network ?

• Cluster coefficient of a node

Nv set of neighbours of v
e(Nv) number of edges between vertices in Nv
v
v
c(v) 6/10
Measures the edge density in the subgraph
generated by the set Nv
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Strength metric of an edge

• Extend the cluster coefficient to edges
• How much an edge e is likely to separate highly
  connected subgraphs
• In other words, measure the strenth of edges (in
  relation to cluster cohesion)
• Related to edge density in the neighborhoods of
  end vertices of e
• Should be 0 for an isthmus

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Strength metric of an edge

• Extend the cluster coefficient to edges

Local edge density s(U,V)
Let U,V be subsets of vertices of G
e(U,V) number of edges between vertices of U
and V
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Strength metric of an edge

• Extend the cluster coefficient to edges

Local edge density s(U,V)
Let U,V be subsets of vertices of G
e(U,V) number of edges between vertices of U
and V
e(U,V) 4
(s 4/9 ? 0.44)
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Strength metric of an edge

• The strength metric is based on a partition of
  vertices incident to the endpoints of e

e
u
v
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Strength metric of an edge

• The strength metric is based on a partition of
  vertices incident to the endpoints of e
• This partition help to classify 3-cycles and
  4-cycles through e

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Strength metric of an edge

• Edge density corresponding to 4-cycles

?4(e)
s(Mu, Wuv)
s(Mv, Wuv)
s(Mu, Mv)
s(Wuv, Wuv)
s(Mu, Wuv)
s(Mv, Wuv)
s(Mu, Mv)
s(Wuv, Wuv)
?4(e)
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Strength metric of an edge

• Edge density corresponding to 3-cycles

Wuv
?3(e)
Mu Mu Wuv
Strenght metric
?(e) ?3(e) ?4(e)
s(Mu, Wuv)
s(Mv, Wuv)
s(Mu, Mv)
s(Wuv, Wuv)
?4(e)
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Strength metric of an edge

• Strenght metric
• Assuming constant degree of nodes (on average),
  the strength of all edges can be computed in
  O(E) time

?(e) ?3(e) ?4(e)
s(Mu, Wuv)
s(Mv, Wuv)
s(Mu, Mv)
s(Wuv, Wuv)
?4(e)
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Example application

• Resyn Assistant API access graph

Visualizing the split, the designers were able to
identify design problems
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Repeating the process

• The procedure can be iterated over components
  that are themselves small world

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Repeating the process

• The procedure can be iterated over components
  that are themselves small world

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Example / Video

• SW network extracted from IMDB

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Example / Video

• SW network extracted from IMDB

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Example / Video

• SW network extracted from IMDB

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Automating the process

• Computing the partition boils down to choosing a
  threshold value on edge strength
• Want to chose the best possible partition at each
  stage
• Apply objective quality measure
• Should help find cohesive subsystems that are
  loosely interconnected (whenever possible)

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Automating the process

• For each possible threshold value, evaluate the
  quality

• What we get is a 2D map
• Select optimal threshold

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Example Map

• Resyn Assistant API
• Optimal threshold gives 0.75 quality
• Just how good is that ?

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MQ / Definition

• C (C1, C2, , Cp) is a clustering of a graph G

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MQ / Quality control

• Roughly 10 of all partition have a value greater
  than 0.05
• MQ reaches values 0.75 with probability 10-6
• MQ varies according to a Gaussian distribution

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MQ / Metric performance

• Our metric behaves well with respect to MQ

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MQ / Comparison

• The threshold is selected along a path in the
  whole lattice of partitions

Coarsest partition

• Comparison with a random upward path in the
  lattice

Finest partition
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Conclusion / Future work

• More focused class of SW networks
• Examples we studied have high cluster index 0.9
• Restrict SW properties ?
• Shape / properties of quotient graphs
• Star-like ? Preferential attachment graphs ?
• Refine the automated process
• Fight against undesirable isolated nodes or
  components
• Refine statistical properties of MQ
• Would need to have MQ depend on the size of
  clusters

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Conclusion
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