1. Monojit Choudhury Microsoft Research India
Bipartite Networks - 2
Monojit Choudhury
Microsoft Research India

2. BNWs: What’s so Special?
BNW  2-colorability  Triangle freeness
Generalization: k-partite graphs
k = 1: unipartite (ambiguous)
k = 2: BNW
k > 2: not very interesting
The chromatic number of a k-partite graph is less than or equal to k

3. One-mode Projection /s/ /n/ /k/ /p/ /t/ /d/ 1 2 l1 l2 l3 l4 /s/ /p/A
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/n/ 1 2 B
/s/ /p/ /k/ /t/ /d/ /n/
l l l l4 l1 l2 l3 l4 1 3 2 B′
ATA – D′
AAT – D

4. Bipartite Structure of all Complex Networks
Jean-Loup Guillaume, Matthieu Latapy (2004) Bipartite structure of all complex networks.
Information Processing Letters 90

5. Identify the largest clique
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6. Find the underlying BNW
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7. Find the underlying BNW
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8. Is this the only possible way?
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9. Is this the only possible way?
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10. Is this the only possible way?
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11. Is this the only possible way?
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12. Is this the only possible way?
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13. Is this the only possible way?
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14. Is this the only possible way?
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15. Is this the only possible way?
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16. Is this the only possible way?
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17. Conclusion
Although we can conceive of an underlying bipartite network for every general graph, the underlying BNW need not be unique.
Open Problem:
Under what constraints it might be possible to define the underlying BNW uniquely?
Known Result: Degree distributions doesn’t help!

18. Degree distribution of the one-mode projection
Given the degree distributions of the actor – Q(q) and tie – S(n) nodes, find the degree distribution P(k) of the actor nodes.
Suppose S(n) is impulse distribution at n = , what is the degree, k, of a node v in the one-mode, whose degree is q in the bipartite network?
k = q(-1)

19. DD of one-mode
k = q(-1)
P(k) ~ Q(k/(-1))
P(k)  (-1)-1 Q(k/(-1))

20. CC of one-mode
Assumption: If we consider a particular actor, v, who has played in q movies, in the thermo-dynamic limit none of his co-actors repeats twice in different films
Each movie gives rise to (-1)(-2)/2 triangles
Total observed triangles: q(-1)(-2)/2
Total possible triangles: k(k-1)/2
= q(-1)[q(-1)-1]/2

21. CC of one-mode Therefore, local CC c(q) = (-2)/[q(-1)-1]
c(k) = (-2)/(k-1)
c = P(k)c(k) = (-2)  P(k)/(k-1)
= Q(q)c(q) = (-2)/(-1)  Q(q)/[q-1/(-1) ]