ACE:A Fast Multiscale Eigenvectors Computation for Drawing Huge Graphs

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ACEA Fast Multiscale Eigenvectors Computation
for Drawing Huge Graphs

• Yehunda Koren
• Liran Carmel
• David Harel

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Fast Graph Drawing Algorithm

• Very large graphs
• Millions of nodes in less than a minute
• Finds optimal drawing by minimizing a quadratic
  energy function
• Minimization generalized eigenvalue problem

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Need for Speed

• Drawing of graph -gt capture its nature
• Most graph drawing methods- lengthy computation
  times
• Fastest alg- require 10 min for 105-node graph
• ACE- 10-20 sec for 105-node graph

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Minimization

• For commonly encountered graphs, in which all
  weights are non-negative, the Laplacian is
  positive semi-definite. ? Key observation that
  this provides for the feasibility of nice drawings

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Minimization

• Drawing expect nodes connected by large positive
  weights to be close together, and those connected
  by large negative weights to be far away from
  each other

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ACE algorithm

• Technique common to algebraic multigrid (AMG)
  algorithm
• Express as originally high-dimensional problem in
  lower and lower dimensions using coarsening
• Problem solved exactly then starts a refinement
  process- the solution is progressively projected
  back into higher and higher dimensions until
  original problem is reproduced

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ACE algorithm
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Coarsening

• Let G be a PSD graph with n nodes
• Replaces G with another PSD graph Gc containing m
  lt n nodes approximating the same entity on
  different scales
• Interpolation matrix- tool that linkes G
  and Gc

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Coarsening

• Given a fine n-node PSD graph, and an nxm
  interpolation matrix, take the coarse m-node PSD
  graph to be Gc (Lc,Mc)

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Coarsening

• Interpolation Matrix
• Retain structure
• Easy to compute in running time
• The sparser the better
• Types
• Edge Contraction Interpolation
• Weighted Interpolation

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Interpolation Types

• Edge Contracting Interpolation
• Interpolate each node of the finer graph from
  exactly one coarse node
• 1. Divide the nodes of fine graph into small
  disjoint connected subsets
• 2. Associate the members of each subset with
  single coarse node
• Sparse and easy to compute
• Only takes into consideration the strongest
  connection of nodes

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Interpolation Types

• Weighted Interpolation
• Interpolate each node of the fine graph from
  possibly several coarse nodes
• 1. Choose a subset of m nodes out of n nodes of G
  representatives
• 2. Fix the coordinates of the representatives by
  their values
• 3. Determine the coordinates of the
  non-representatives as a weighted sum of
  neighboring representatives

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Running Time

• The Sparser the interpolation matrix, the faster
  a single iteration is
• The more accurate the interpolation matrix, the
  less iterations are required
• Edge Contracting fastest structure of
  homogeneous graphs
• Weighted fastest structure of non-homogeneous
  graphs

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Coarsening cont.
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Coarsening cont.
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Coarsening cont.

• Given Gc, the generalized eigenprojection problem
  and the original problem in the same dimension
  is (substituted x Ay)

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Refinement

• Keep coarsening until couse PSD graph is
  sufficiently small lt 100 nodes
• The drawing of G(L,M) is obtained from lowest
  positive generalized eigenvectors of (L,M)
• Since M is diagonal, the problem can be an
  eigenvalue problem Bv µv and since B is small,
  this takes fraction of total running time
• Use power iteration algorithm to find the largest
  eigenvalues of symmetric matrices but reverse
  since we need to find the lowest not the largest

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Examples

• Graphs 105 nodes - few sec
• Graphs 106 nodes 10-20 sec
• Graphs 7.5106 nodes 2 min

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Examples

• Electronic Noses chemical devices to quantify
  odors
• Array of sensors that gives an odor-unique
  response pattern
• N300 measurements
• 30 types of odors
• Odors in up-rt outliers
• Diff clusteres of odors
• separated

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Multiscale Vizualization of Small World Networks

• David Auber
• Yves Chiricota
• Fabien Jourdan
• Guy Melancon

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Problem

• small world networks
• Shown to be relevant models in study social
  networks, software reverse engineering, biology,
  etc.
• Have a multiscale nature
• viewed as a network of groups also small world
  networks

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Small World Networks

• First identified by Milgram studying the
  structure of social networks
• Six degree of separation

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Small World Networks

• Defining Properties
• Average path length
• Same edges as random graph
• Clustering index of nodes
• Orders of magnitude larger than ave

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Internet Movie Database
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Interactive Visualization of Small World Networks

• Common tasks Identify patterns in set of
  connections
• Social groups collections of actors who are
  closely linked to one another
• Social Positions sets of actors who are linked
  into the total social system in similar ways

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Might not need

• Use of small world properties of networks to
  support the visualization
• Compute decomposition of small world network into
  highly connected components also small world

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Technique

• Compute clustering index for nodes and graph
• Where c(v) is the clustering index of node v,
• v is a node in the network,
• r(N(v)) is the number of edges connecting
  neighbors of v ,
• k is the size of vs neighborhood N(v)

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Technique

• Clustering index of graph G
• Take the average
• Where N is the number of nodes in G

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Technique

• Small world networks
• High clustering index
• Small average path length between nodes

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Technique

• Finding a clustering index of an edge
• Denote
• M(u) set of neighbors of u that are not
  neighbors of v
• W(u,v) set of common neighbors to u and v
• R(A,B) for the number of edges linking nodes in a
  set A to nodes in set B
• S(A,B) r(A,B) / AB computes the proportion
  of existing edges among the set of all possible
  edges

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Technique

• Finding a clustering index of an edge
• Given edge(u,v), strength is computed by dividing
  neighborhood of u or v into distinct subsets
• Compute

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Discard weak edges

• Strength of edge measure of its contribution to
  the cohesion of the neighborhood
• Filter out the weak
• If strength of edge lt threshold
• Maximal disconnected subgraphs corresponding to
  clustering of initial graph
• Quotient graph subgraphs as vertices

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Visual Coherence

• Force-directed algorithm
• naturally embeds neighbor nodes close to one
  another
• Not distinguish between edges
• Induce same attractive force for all edges of the
  graph avoid be assigning weights
• Quotient graph laid out using Force-directed
  algorithm
• Weaker edges replaces by single edge

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Navigational Coherence

• Selected component showed in same manner as
  overview panel
• Left click display of overview
• Middle click show same component unfolded as a
  flat graph
• Wheel zoom in/out

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Examples

• Resyn Assistan API
• Software designed for the study of chemical
  components developed in Montpellier (LIRMM)
• IMDB subset
• Central group renowned movie actors
• Periphery movies

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Threshold and Quality Measurements

• Threshold choice determined by good partitioning
  from filtering
• Quality of a partition comput how close a
  partition is to a partion with optimal cost

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Threshold and Quality Measurements

• Cost function
• Where C clustering of graph G
• First term mean value of edge density inside
  each cluster
• Second term mean value of edge density between
  clusters

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Quality

• Optimal threshold approaches 1.6 for MQ close to
  0.80

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MQ scores (really high)
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Partitioning

• Interactive approach to choosing threshold might
  be an alternative
• Clustering techniques aim at finding partitions
  that have a minimal number of outer edges
• Immediate visual feedback provided
• Computing time still low
• Method able to find partitions with a high MQ
  value

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