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

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

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

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

Identify the largest clique /k/ /p/ /t/ /d/ 1 2

Find the underlying BNW /s/ /n/ /k/ /p/ /t/ /d/ 1 2

Find the underlying BNW /s/ /p/ /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2

Is this the only possible way? /k/ /p/ /t/ /d/ 1 2

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2 l3

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2 l3

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2 l3

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 2 l3 l4

Is this the only possible way? /k/ /d/ /t/ /n/ /s/ /n/ /k/ /p/ /t/ /d/ 1 l3 l4 l5

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!

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)

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

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

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) ]