Visual Analysis Algebra

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Visual Analysis Algebra

• Anna Shaverdian, Hao Zhou
• H. V. Jagadish, George Michailidis
• University of Michigan

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Find a criminal network within a network? 50
different solutions, 5 minute videos to explain
process, pages of text
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Desired Features in Visual Analysis

• Mix and match ideas from multiple projects
• Compare/Validate tools and techniques
• Document and reproduce results from anothers
  visual analysis
• Not ambiguous
• Not wordy
• Optimize techniques

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Visual Analysis Algebra

• Graph Model
• Predicate/ Witness
• Graph Matching Function
• Operators
• Selection
• Labeling
• Aggregation
• Helper Functions
• Visual Operators

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Graph Model

• Attributed Graph D G,X
• Graph G (V,E)
• Each node assigned unique id through ?(vertex)
  function
• Allows directed, multi-edge graphs
• (Direction captured as an edge attribute)
• Attributes X (XV, XE, XG)
• Each attribute has a name, type, and value
• Attributes can be intrinsic or computed
• Intrinsic independent features which stay
  constant if graph topology changes
• Computed Created through composition functions
• Examples degree, betweeness, centrality

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Example Graph Model

• Cell Phone Network node represents a phone and
  an edge represents a call between two phones
• D G (V,E), X (Xv XphoneID, XE
  Xdate, Xduration, Xtower, XcallerID, XG )
• Initial data set with intrinsic attributes
• Perform operations on sets of attributed graphs
  (closed algebra)
• Dday1, Dday2, , Dday10

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Predicate Definition

• p (V, E, XV, XE, XG, !E)
• V,E describe the graph structure
• XV, XE, XG describe the conditions on the
  attributes in V, E
• Example Xv.weight.node12 lt Xv.weight.node10 in
  XV
• !E describe the excluded edges
• An edge e1 in !E doesnt exist in the graph G and
  given a closed universe U, for all S where G is a
  subgraph of S, then e1 doesnt exist in S either

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Witness

• An attributed graph where there exists
• Bijection mapping between nodes
• The predicates conditions all hold on its node,
  edge, and graph attributes

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Example Excluded Edges Witness

• Predicate Attributed Graph

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Graph Matching Function (? ) Subroutine used by
operators

• Inputs an attributed graph D and predicate p
• Outputs
• A list of witnesses W
• Attributes of the nodes, edges, and graph of
  witness include all attributes of those
  respective elements in D
• If one or more witnesses share ids attributes
  (ex. same but different rotation) combine to
  arbitrary one
• A model witness X
• Set of mapping lists of the witnesses in W to X

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Graph Matching Function (?) Example
Predicate
Attributed Graph
Age 12
Age 22
Age 16
?
Mappings
Age 12
Witness found
Model witness
ID Model ID
1 6
2 7
3 8
4 9
Age 16
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Selection Operator s

• There are two types of selection operators
• Work at the attributed graph level
• Work at the element (nodes edges level)
• Both operate on a set of attributed graphs and
  output a set of graphs

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Set Selection sset

• Given a set of attributed graphs D and a
  predicate p
• Set Selection outputs the set of graphs where
  there exists a witness for the predicate
• Example
• The graphs with an average degree greater than 42
  
• p (V , E , XV , XE ,
  XG Xg.averageDegree
  gt 42,!E )
• sset, p(D1, D2, , D10) D
• where D subset of D, for any Di in D,
  Xg.averageDegree gt 42

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Element Selection selement

• Given a set of attributed graphs and a predicate
  p
• s element,p (D1, D2, , Dn) Ui Di
• where each Di Wi.p.1, , Wi.p.k the k
  witnesses of predicate p found in Di
• An attributed graph for each witness found in the
  set of graphs

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Example Element Selection selement

• Select a subgraph from a set of graphs
• p (V 1, 2, 3, E e12, e23, XV , XE
  , XG ,!E )
• D1 (V(1,2,3,4), E e12, e14 , e23 , e34,
  X (,,)

s element,p (D1)
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Labeling Operator

• During graph analysis, need a way to select
  nodes of interest, mark them somehow, and
  continue analysis, sometimes referring to the
  marked nodes
• We do this by labeling
• Given a set of graphs and a predicate
• We modify each graph to remember its match to the
  predicate

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Labeling Operator

• For each attributed graph Di where there exists a
  witness for the predicate (using ? function)
• Create the model witness structure x within Di
• Label it with a unique group id
• For each witness wj found in Di
• Use the mapping lists to create directed edges
  between the wj and x

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Labeling Example
Labeling
Predicate
Each edge has a group id to say its an edge to a
model witness and a structure id, to say its one
witness found
Attributed Graph
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Example Labeling Visual Analysis

• Given a Social Network
• We have a suspected terrorist subnetwork and some
  features of interest
• Analyze the subgraphs that match the suspected
  subnetwork
• Predicate structure isnt the final structure
  were looking for, its an intermediate step
• VAST 2009 challenge

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Example Labeling Visual Analysis
Degree 40
Geographic size small island
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Helper Functions

• Visual Operators
• Ex. Feed values into a histogram, layouts,
  presentation
• Creating/Deleting
• Create/Delete a set of nodes/edges/attributes
• Copy a graph

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Phone Record Case Study

• In an attempt to characterize the entire network,
  we loaded the entire data set into MobiVis, which
  links people (blue nodes) if they had a phone
  conversation. Unfortunately, the tight
  connectivity of the resulting network made it
  impossible to find interesting patterns.
  Following the lead that person 200 is likely to
  be FerdinandoCatalano, we filtered the data to
  visualize only its closest nodes. Figure 1 shows
  the social network of person 200. Figure 1.
  Overview of the social network of
  FerdinandoCatalano (id 200). This reflects the
  general social structure over, at least, the
  first seven days. We can further characterize
  this network by looking at the links between the
  immediate neighbors of person 200. Persons 5,
  200, 97 and 137 seem to form a clique, whereas
  persons 1,2 and 3 form another. Looking at the
  amount of communication between those, which is
  depicted as the thickness of the edges, we
  discovered that 200 and 5 talk a lot among
  themselves. The color coding of the edges helps
  visualize the symmetry of the calls. For example,
  a warm color (orange) in the middle indicates a
  symmetric connection (both parties call each
  other frequently), whereas a biased orange color
  indicates more calls in the direction of the
  bias. We then characterized the network as being
  the connection of the two families the
  Catalanos, represented in persons 200
  (FerdinandoCatalano), 5 (which we believe is
  EstabanCatalano, since its tight connection to
  200), 97 and 137. And the Vidros, represented in
  persons 1,2 and 3. We can further characterize
  the substructure of the Vidrosas hierarchical.
  Although it was not evident at first, person 1
  always calls persons 2 and 3, which led us to
  believe that he has a role of coordinator. We
  validated this with another capability of
  MobiVis, which allows us to display people in the
  social network according to some semantic
  filtering criteria. In Figure 2(a), we display
  the people called by 1 and people who called
  person 1 . Those people who called person 1 are
  connected to an orange node, while people who
  where called by person 1 are connected to a red
  node. We can see that person 1 had a
  bi-directional communication with
  FerdinandoCatalano, but only in one direction
  with 2,3 and 5. Figure 2(b) shows the same
  analysis for person 5. We noticed an inverse
  behavior 1, 2 and 3 always call 5, but not vice
  versa. Furthermore, it helped us characterized
  the social structure better. The high symmetry of
  communication between 200 and 5 validates our
  claim about their identities being of
  Ferdinandoand EstabanCatalano, respectively.
  Person 1, however, seems to coordinate the
  efforts of 2,3 and 5, which suggests that he can
  be associated to David

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Phone Record Case Study

• Original data set (10 days)
• D G (V,E), X (Xv XphoneID, XE Xdate,
  Xduration, Xtower, XcallerID, XG )
• View Entire Graph
• Create 10 graphs (per day)
• Predicate for day i calls
• pday_i(V v1, v2, E e12 , XV , XE
  Xe.12.day i, XG ,!E )
• Labeling by day
• µday_iD
• Element Selection on day_igroup
• ?element,day_iD D1, D2, D3, D4, D5, D6,
  D7,, D8,, D9,, D10
• View Each Graph

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Phone Record Case Study

• Look at pattern change in node 200s neighborhood
• Predicate for node 200 neighbor
• p200Neighbor(V v1, v2, E e12 , XV
  Xv.1.callerID 200, XE , XG ,!E )
• Labeling by day
• µ200NeighborD1, D2, , D10
• Selection on 200 neighbor group
• ?element,200NeighborD1, D2, , D10 D1,
  D2, D3, D4, D5, D6, D7, D8, D9, D10
• Aggregate days 1-7 and days 8-10 graphs
• Set Aggregation
• ?set, pdays1-7. pdays8-10(D1, D2, , D10)
  Dday1-7, Dday8-10
• Element Aggregation on CallerID
• ?element, pdays1-7. pdays8-10(Dday1-7,
  Dday8-10 )

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Algebraic Visual Analysis The Catalano Phone
Call Data Set Case Study

• Anna Shaverdian, Hao Zhou, George Michailidis,
  and H.V. Jagadish, VAKD 09
• Simulate many existing analytical workflows with
  operators from visual analytic algebra
• Ability to do analysis beyond existing workflows

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Multiple Step Social Structure Analysis with
Cytoscape

• Hao Zhou, Anna Shaverdian, H.V. Jagadish, George
  Michailidis, VAST 09
• VAST 09 Flitter Mini Challenge Award Good Tool
  Adaption
• Demonstrates Cytoscapes utility in identifying
  the structure in a social network