Graph Mining in Social Network Analysis

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Graph Mining in Social Network Analysis

• Student Dušan Ristic

Professor Veljko Milutinovic
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Graphs

• A graph G (V,E) is a set of vertices V and a
  set (possibly empty) E of pairs of vertices e1
  (v1, v2), where e1 ? E and v1, v2 ? V.
• Edges may contain weights or labels and have
  direction
• Nodes may contain additional labels

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Motivation

• Many domains produce data that can be
  intuitively represented by graphs.
• We would like to discover interesting facts and
  informationabout these graphs.
• Real world graph datasets are too large for
  humans to make any sense of.
• We would like to discover patterns that have
  structural information present.

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Application Domains

• Web structure graphs
• Social networks
• Protein interaction networks
• Chemical compound
• Program Flow
• Transportation networks

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Social Network Analysis

• Network (Graph)
• Nodes Things (people, places, etc.)
• Edges Relationships (friends, visited, etc.)

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Existing solutions

• Existing graph pattern mining algorithms
• FSG
• gSpan
• Problems
• Fuzzy patterns
• Too many frequent subgraphs generated for large
  graphs

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Proximity Pattern

• Overcomes the above two issues
• Subset of frequently repeating labels in tightly
  connected subgraphs

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Example 1
Proximity pattern a,b,c
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Definitions

• Definition 1 (Support)The support sup(I) of an
  itemset I ? L (label set) is the number of
  transactions in the data set that contain I.
• Definition 2 (Downward Closure)For a frequent
  itemset, all of its subsets are frequent and
  thus for an infrequent itemset, all of its
  supersets must be infrequent.
• Definition 3 (Embedding and Mapping)Given a
  label subset I, subset of vertices p is called an
  embedding of I if I ? L(p). A mapping f between I
  and the vertices in p is a function f I ? p

v1, v2, v3 is an embedding of l1, l2, l5
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Neighbor Association Model

• Find all the embeddings, p1, p2, . . . , pm of an
  itemset I in the graph
• For each embedding p, measure its strength f(p)
• Aggregate the strength of the embeddings, Take
  F(I) as the support of I
• Create an overlapping graph with embeddings as
  nodes

overlapping graph
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Information Propagation Model

• G0 ? G1 ? . . . ? Gn
• G0 starting graph
• Gn stable graph
• G stable graph approximation
• The probability of observing L(u) and l is
• P(L ? l) P(Ll)P(l),
• P(l) is the probability of l in us neighbors
• P(Ll) is the probability that l is successfully
  propagated to u
• For multiple labels, l1, l2, . . . , lm
• P(L?l1, l2, . . . , lm) P(Ll1). . .
  P(Llm)P(l1). . . P(lm).

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Nearest Probabilistic Association

• Definition 4.Au(l) P(L(u)l) e-ad, where
  l? is a label present in a vertex v? closest to
  u?, d? is the distance from v? to u? (1
  for unweighted graph), a? is the decay
  constant (a gt 0)
• Probabilistic Support of I
• If I l1, l2, . . . ,m and J l1, l2, . . .
  , lm, lm1 . . . , n, then since,
• sup(I) sup(J)

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Normalized Probabilistic Association (Improved
NPA)

• Normalized Probabilistic Association of label l?
  at vertex u?
• NMPA can break ties when
• two vertices have the same number of neighbors
• two vertices have different number of neighbors
  but the same number of neighbors having label
  l?.
• The update rule of Algorithm 1 is changed
• The normalizing factor will give more
  association strength for the labels that are
  contained by many neighbors.
• Complexity same as NPA, however propagation
  decays faster due to normalization.

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Conclusion

• With growth in popularity of social
  networksgraph data mining will be more needed
  than ever
• Probabilistic itemset mining discovers new kind
  of patterns

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Literature

• http//cs.ucsb.edu/xyan/papers/sigmod10-associati
  on.pdf
• A. Khan, X. Yan, and K.-L.Wu. Towards proximity
  pattern mining in large graphs. In SIGMOD, 2010.
• C.C. Aggarwal, Y. Li, J. Wang, J. Wang. Frequent
  Pattern Mining with Uncertain Data. KDD 2009.
• Presentation on Frequent Pattern Growth
  (FP-Growth) Algorithm An Introduction by Florian
  Verhein
• J. Han, J. Pei, Y. Yin. Mining frequent patterns
  without candidate generation. SIGMOD, 2000.

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Questions?

• Dušan Ristic 3020/2014
• E-mail rd143020m_at_etf.rs