The document proposes a Partially Labeled Stochastic Block Model (PLSBM) as a probabilistic generative model for networks with some labeled nodes. It proves the relationship between label propagation (LP) and the Stochastic Block Model (SBM) through PLSBM. Specifically, it shows that the solution to LP is identical to maximum likelihood estimation under PLSBM when the label ratio is uniform, edge probabilities are uniform within and between labels, and the network is assortative. When these conditions do not hold, LP is shown to fail both theoretically and experimentally.

1. When
Does
Label
Propaga1on
Fail?
A
View
from
a
Network
Genera1ve
Model
Yuto
Yamaguchi
and
Kohei
Hayashi
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IJCAI@Melbourne
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2. Node Classification
Given
Find
Partially labeled
undirected graph
Labels of all nodes
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3. Example:
User profile inference
Friends
Soccer
Soccer
Soccer
Tennis
Baseball
？？？
What’s his hobby?
Node Classification
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4. Label Propagation (1/2)
Propagate neighbors’ labels
Friends
Soccer
Soccer
Soccer
Tennis
Baseline
？？？
Soccer
Soccer
Soccer
Tennis
Baseline
Soccer
[Zhu+, 03], [Zhou+, 03], etc.
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5. Label Propagation (2/2)
Q F;X,Y,λ( )=
1
2
fi − yi 2
2
i=1
N
∑ +
λ
2
xij fi − fj 2
2
j=1
N
∑
i=1
N
∑
Given: adjacency matrix X and labels Y
Find: F = { fi } that minimizes Q
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F ∈ RN x K
Y ∈ {0, 1}N x K
X ∈ {0, 1}N x N
N: # of nodes
K: # of labels
λ ∈ R+ : user parameter
[Zhu+, 03], [Zhou+, 03], etc.

6. Cases
when
LP
fails
(prac1cally
known)
Different labels
are connected
Label ratio is not uniform
Q. So, do we know why LP fails in these cases?
A. No. Since it’s not a probabilistic model, we
don’t know the assumptions behind the model.
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Edge probability is not uniform

7. What
we
do
in
this
work
1. Prove
a
theore1cal
rela1onship
between
LP
and
Stochas(c
Block
Model,
which
is
a
well-­‐
studied
probabilis1c
genera1ve
model
2. Find
the
assump(ons
behind
LP
through
the
assump1ons
behind
SBM
3. Show
when
and
why
LP
fails
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8. NETWORK
GENERATIVE
MODELS
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9. Stochastic Block Model
Generative process
Multinomial
Bernoulli
①
②
①： Generate cluster assignment for each node
(which can be thought of labels)
②： Generate adjacency matrix
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γ ∈ RK
Π ∈ RKxK
Parameters:

10. Proposed:
Partially Labeled SBM (PLSBM)
Generative process
①
②
③
②：Generate labels for “labeled nodes”
(α large à yi is more likely to be the same as zi)
Depends on
parameter α
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γ ∈ RK
Π ∈ RKxK
α ∈ 0,1[ ]
Parameters:

11. Rela1onships
between
models
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SBM
PLSBM
LP
Discre1zed
LP
Main
result
(next
slide)
No
labels
Con1nuous
relaxa1on

12. Main Result
Map estimator Z of PLSBM is identical to the solution of
(discretized) LP when the following conditions hold
Condition 1:
Condition 2:
Condition 3:
Condition 4: （omitted）
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13. Condition 1
Implication （implicit assumption of LP）
• Label ratio is uniform
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Violates this assumption L

14. Condition 2
Implication （Implicit assumptions of LP）
• Edge probs between the same labels are all the same (μ)
• Edge probs between different labels are all the same (ν)
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Violates this assumption L

15. Condition 3
Implication （Implicit assumption of LP）
• Assortative (same labels tend to be connected)
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Violates this assumption L

16. Experimental results
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… Come see full results at the poster session J
Better
Setups:
1. Generate datasets by PLSBM
2. infer labels (Z) by PLSBM, SBM, and LP
3. Report mean accuracy of 20 trials
Assortative
Disassortative
Agree with
theoretical results

17. Summary
• Proposed
Par1ally-­‐Labeled
SBM
(PLSBM)
• Proved
the
rela1onship
between
LP
and
SBM
via
PLSBM
• Showed
cases
when
LP
fails
• Experimental
and
Theore1cal
results
agree
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Github: yamaguchiyuto/plsbm