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1
Joint work with
Junteng Jia (Cornell),
Michael T.Schaub (MIT),&
Santiago Segarra (Rice)
Graph-based semi-supervised
learning for edge flows
Austin R. Benson · Cornell University
NetSci HONS
May 28, 2019
Slides. bit.ly/arb-HONS-19
2http://www.vidiani.com/road-map-of-manhattan/
Traffic flow sensors
Two major questions in semi-supervised learning.
3
1. Interpolation. Given the measurements, how do I
interpolate to locations where I don’t know have data.
2. Active learning. Where are the best locations to make
my measurements, knowing step 1?
Background. Classical graph-based semi-supervised
learning interpolates from labels on a few vertices.
4
Key idea.
My label is similar to the
labels of my connections.
minimize
labels x
X
(i,j)2E
(xi xj)2
subject to x matches given labels<latexit 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In the higher-order case of edge flows,we have a
different type of objective.
5
Key idea (“divergence-free”).
Net flow into a node should be
similar to net flow out of a node.
An edge flow represents net flow along an edge.
6
• As an alternating function: F(i, j) = -F(j, i)
• For the linear algebra, first orient each edge i → j if i < j.
Then vector f gives flows on these oriented edges.
• If fi,j > 0, if net flow aligns with orientation
• If fi,j < 0, net flow is opposite of orientation.
1
2
3 4
5
6
7
f1,3 > 0 f5,7 < 0
In the higher-order case of edge flows,we have a
different type of objective.
7
Key idea (“divergence-free”).
Net flow into a node should be
similar to net flow out of a node.
minimize
flows f
X
i
2
4
X
j>i,(i,j)2E
fij
X
k<i,(k,i)2E
fki
3
5
2
subject to f matches labels<latexit 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There is a close relationship between node-based SSL
and edge-based SSLobjective functions.
8
X
(i,j)2E
(xi xj)2
= xT
Lx = xT
BBT
x = kBT
xk2
2
<latexit 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sha1_base64="ySnVyFu/bHlEXTQVOztKcP12UE4=">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</latexit>
X
i
2
4
X
j>i,(i,j)2E
fij
X
k<i,(k,i)2E
fki
3
5
2
= fT
BT
Bf = kBfk2
<latexit sha1_base64="8Wots+j4DG5+yjap6jJuU1BS0Yo=">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</latexit><latexit 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sha1_base64="DzXqcCT2r5jJ8wnUxXYU8xa/QFc=">AAAHrnicfVVtb9s2EFa6re60l6brlw37wi4w0BWyY6fIkgzLYGxdsQItls1OWsB0XUo6WYxJSiOpxK6mT/uV+wn7FztazmK72QhIoo73PA95dyTDXHBjO52/tm699/4Htxt3PvQ/+viTT+9u3/vszGSFjuA0ykSmX4XMgOAKTi23Al7lGpgMBbwMpz+68ZcXoA3P1MDOcxhJNlE84RGzaBpv/0lNIcclrwgVkNhh/XtOvic8IA95cP41oVyRnyqSoNd5RVqkdpmS7xYu04DwVZ8pr6jmk9SOXu8R/5jQi+T1gMofFi/8IWj6g2DfjWB3vOePt3c67c6ikXc73WVnx1u2k/G92/dpnEWFBGUjwYwZdju5HZVMWx4JqHxaGMhZNGUTGGJXMQlmVC6iVZEmWmKSZBofZcnC6q9CkEez+RpLaVlYCKZn69Ywy6Y4YirfX9e0yeGo5CovLKiolkwKQWxGXBJIzDVEVszJuq7l07eB4hEkmkUBk0YymwY5d/MM7PRta6JZngaSTSECIa5N9awcXPBQMz13S8guTRAi80RnhYpNkDNrQSuDeKv5LDApy8EECbdBxETk/mOHyUVmJdNT81+sbQmW4eAicgJsOSgSC79BXJUa4geHnQehQN1VD5vCRAOoqlx8nM9lyi1s+ISigKp07xUPv0lSa3Pz7e6uhVnbWOSGWZQyNYF2lMnd3wswrprNbveb/aO9o10DkmPRh1jjsnXJbdpyi2hx1Qpxa4Be+D0+2Kk/PnUBZbh1XHx8OhFZyARuB0sdrAfKFBp6cSawAHq4caIshmOqQbDZFTbDya8X0XDQHZUuca4A1rJ8Mugz5YKrQcElLkAyFZc0YZKLeQwJK4StSmqSq/56kZjEVUXlN1fFDGYQ4uNO+yiIJEdRLAuBJY8CdmYSR7G+SOSmys4cVa8Gl+bREPfa/qjaXNQTwE2moT+XYSae4pLKmsVU5S8vnlelchKSV6WsSo7TpX2wNzmjId6EhEvIUsMB+kWI6bSFS+nNApsK/acvXEiuBAbdtfCV4awqjbgWcc41unyGni4GTOQpq66n+ubZRtTjiQAepa069jeNYKINHi/r54N0NKtZln0+kahE66pydCUNZUlre/VOWcjneJjHNyGWA9W6xCM6C5keYvHRNMxmJb1w76ZPU10IICm4sxlP14P93JImGaRAWGQLJgjCfDrFE6LT3tuHWZNctSZ5ghcRUxGQEOwl7l/nS1CMmEUY/Vqq6ROyIGh12l2QzSt0P800RoerCckUwaIi7qohhsfgECvr2ulW/5LgBfD4f0kWt0zNUrko4DXS3bw03u2c7bW7OL1fOzu9w+WFcsf70vvKe+h1vQOv5/3snXinXuT9vXV36/OtLxqdxllj1BjXrre2lpj73lprpP8AUtKs/Q==</latexit>
Bk,(i,j) =
8
><
>:
1 k = i, i < j
1 k = j, i < j
0 otherwise<latexit 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One catch.
• Having labels in node case gives unique answer.
• Having labels in edge case is under-constrained.
We add regularization to get a nice sparse linear least
squares problem.
9
minimize
flows f
kBfk2
2 + kfk2
2
subject to f matches labels<latexit 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sha1_base64="FX1ilveIMUE873uv0A2QVMsGL2A=">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</latexit>
• We use iterative solvers
LSQR or LSMR to compute
the solution efficiently.
Key Idea
My label is similar
to the labels of my
connections.
Objective
Net in flow =
net out flow
at all nodes.
minimize
vertex values x
kBT
xk2
2
subject to x matches labels<latexit 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sha1_base64="loGbjI0bxqRY8lMfhZLYiEbFdFU=">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</latexit><latexit sha1_base64="loGbjI0bxqRY8lMfhZLYiEbFdFU=">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</latexit>
minimize
edge flows f
kBfk2
2 + kfk2
2
subject to f matches labels<latexit 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11
0.0
0.2
0.4
0.6
0.8
1.0
correlation
Anaheim Barcelona
0.0 0.2 0.4 0.6 0.8 1.0
ratio labeled (| L
|/| |)
0.0
0.2
0.4
0.6
0.8
1.0
correlation
0.0 0.2 0.4 0.6 0.8 1.0
ratio labeled (| L
|/| |)
Winnipeg Chicago
ZeroFill
LineGraph
FlowSSL
Why do we do so poorly on Chicago?
12
ftruth = y z, y 2 R, z 2 C<latexit 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Our divergence-free assumption says that ftruth ⇡ z.<latexit 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sha1_base64="xvKzv/3dvMkp8PffS78cmxHikj4=">AAAHdnicfVVdb9s2FFW7Lem0r3R9HDCwS40Nhe3YKbIkAwoYWFGsWItmc9IWiIyMkq4kwqSkkVeJHEI/ar9m2Nv2L/a4S9tZLCebAFsUec89vIeHZFhKYXAw+OPO3ffe/2Bj896H/kcff/LpZ1v3P39jikpHcBIVstDvQm5AihxOUKCEd6UGrkIJb8Pp92787TloI4r8GGclTBRPc5GIiCN1nW39+LrSLBYUkkIeQS/RAIwbU6nSBTDDZ4ZhxpE9Cs6TMxsg1GhRV5g1DQt4WeqiZsH55aM+O9vaHvQH84fdbAyXjW1v+Ryd3d94EMRFVCnIMZLEejoclDixXKOIJDR+UBkoeTTlKZxSM+cKzMTOq25Yh3pilhSafjmyea+/CqE8ms9aWSzysJJc1+3esCimNGIa329zYnIwsSIvKyR1FpRJJRkWzIlJwmmIUM5YmxfF9LKbiwgSzaMuV0ZxzLqlcPPs4vSyl2peZl3FpxCBlNddi1k5uBSh5nrmSiguTDekzKkuqjw23ZIjgs4N4VGLumsyXoLpJgK7EZeR+44dppQFKq6n5r+y9hUgp8G5chLQHlcJws8QN1ZD/PBg8DCUxLsagRmk5JC8sfOXi7nIBMJaTCgraKz7X4nwOyxDLM13Oztkor5Byg11lPE8hX5UqJ1fKzDOdGZn+O3e4e7hjgHlnBmSV1XvQmDWc0X0RN4LyeKg53FP9rcXLz9wgnLaAk4fP0hlEXIZ0GfgYCPITaVhFBeSDDCiDRAVMTwNNEheX2ELmnzbRKfHw4l1C+cM0Frlo+Mxz524GnK4oAIUz2MbJFwJOYsh4ZXExgYmuWq3TWIS54rG76ySGVpBiJ8O+ofdSAkiJVtIsjwRYG0Sl6JdJOUOcqxdqtECbM3jU9pre5NmvahnQJtMw3imwkI+p5LsIotp7OtXLxubOwolGqsaK2i6wRjwtmDqiNch4RKy5HCAcRXScmLllvR2gnWG8fNXTpIrguNhSz4b1o018prEBS/Q9gVFOg24LDPeXE/1lxdrqsepBBFlvYX2t43QQhs6Xtrng3JpVldZjUWqiClYuMqls0GobLDob27YQr2kQzm+DbEcaNoUj4M65PqUzBdkYVHb4Nz9d/wg05UEloFIM6TTdX+vRNZhxxkd2xFWXDKC+cGUTohBf3cP6g67ejrsGV0onI55FgJe0P51sYzImJnL6C+oOj5j8wS9QX8IqnOFHmeFJnVEnjK6GchUTEKCzIgYHGKlru1h828SugCe/G8SPa9knqVxKtA1Mly/NG423uz2hzS9n3a3RwfLC+We94X3lfeNN/T2vZH3g3fknXiR95v3u/en99fG35tfbnY2v16E3r2zxDzwWs/m4B+9paDJ</latexit><latexit sha1_base64="xvKzv/3dvMkp8PffS78cmxHikj4=">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</latexit>
Does this actually hold in our data?
13
0.0
0.2
0.4
0.6
0.8
1.0
correlation
Anaheim Barcelona
0.0 0.2 0.4 0.6 0.8 1.0
ratio labeled (| L
|/| |)
0.0
0.2
0.4
0.6
0.8
1.0
correlation
0.0 0.2 0.4 0.6 0.8 1.0
ratio labeled (| L
|/| |)
Winnipeg Chicago
ZeroFill
LineGraph
FlowSSL
kyk2
kzk2
⇡ 0.2.
<latexit 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kyk2
kzk2
⇡ 0.5.
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kyk2
kzk2
⇡ 0.8.
<latexit 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sha1_base64="w4FIuq2wqROL7e9rX/Ax8PmPD9c=">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</latexit><latexit 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kyk2
kzk2
⇡ 1.7.
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We have some theoretical guarantees if the true flow is
indeed nearly divergence-free.
14
Theorem.
• Let VC be a basis for the divergence-free space ker(B).
• Suppose the true flow is a divergence-free flow perturbed by d.
• For any m – n + 1 labels corresponding to linearly independent rows
of VC, denoted , the relative reconstruction error is bounded by
minimize
edge flows f
kBfk2
2 + kfk2
2
subject to f matches labels<latexit 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h
1
min(VL
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Two major questions in semi-supervised learning.
15
1. Interpolation. Given the measurements, how do I
interpolate to locations where I don’t know have data.
2. Active learning. Where are the best locations to make
my measurements, knowing step 1?
Active learning strategies in vertex-based and edge-
based SSLare similar.
16
Active vertex-based SSL [Guillory-Bilmes 17]
1. Cluster the graph (e.g., using spectral clustering).
2. Pick points from each cluster.
Active edge-based SSL
1. Cluster the graph (e.g., using spectral clustering).
2. Pick edges that cross cluster boundaries.
17
Winnipeg Chicago
18
We can extend our framework beyond looking for
divergence-free flows.
19
f|{z}
edge flow
=
gradient flow
z}|{
B|
y
divergence-free flow
z }| {
Cw|{z}
curl flow
h|{z}
harmonic flow<latexit 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Hodge decomposition [Lim 15, others]
• So far, we have penalized gradient flow via .
• Could penalize other types of flows.
kBfk2
2<latexit 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j k
i
fij + fjk + fki<latexit 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sha1_base64="FM/CzNQyWw1iZJChcVVaAdty+9o=">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</latexit>
We can extend our framework beyond looking for
divergence-free flows.
20
argmin
flows f
kC|
fk2
+ 2
· kf fmid
k2
subject to fbid
 f  fask
<latexit 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• Data is currency exchange rates (fully connected graphs).
• Buyers willing to buy at “bid” price.
• Sellers willing to sell at “ask” price.
• Settle on some price in the middle (usually mid point).
• Want prices that have no cyclic flow (arbitrage).
Optimal flows eliminate arbitrage opportunities.
21
Graph-based semi-supervised learning for edge flows.
22
• We have a framework for semi-supervised learning in the edge space
with a natural connection to classical vertex-based SSL.
• We also have a practical and efficient active learning method.
• Can extend the SSL framework to other types of edge flows through
results in combinatorial Hodge theory.
23
THANKS! Austin R. Benson
Slides. bit.ly/arb-HONS-19
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
Graph-based semi-supervised
learning for edge flows
bit.ly/ssl-flow-code
(code, reproducibility, and data)
To appear in KDD 2019.

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Semi-supervised learning of edge flows

  • 1. 1 Joint work with Junteng Jia (Cornell), Michael T.Schaub (MIT),& Santiago Segarra (Rice) Graph-based semi-supervised learning for edge flows Austin R. Benson · Cornell University NetSci HONS May 28, 2019 Slides. bit.ly/arb-HONS-19
  • 3. Two major questions in semi-supervised learning. 3 1. Interpolation. Given the measurements, how do I interpolate to locations where I don’t know have data. 2. Active learning. Where are the best locations to make my measurements, knowing step 1?
  • 4. Background. Classical graph-based semi-supervised learning interpolates from labels on a few vertices. 4 Key idea. My label is similar to the labels of my connections. minimize labels x X (i,j)2E (xi xj)2 subject to x matches given labels<latexit 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  • 5. In the higher-order case of edge flows,we have a different type of objective. 5 Key idea (“divergence-free”). Net flow into a node should be similar to net flow out of a node.
  • 6. An edge flow represents net flow along an edge. 6 • As an alternating function: F(i, j) = -F(j, i) • For the linear algebra, first orient each edge i → j if i < j. Then vector f gives flows on these oriented edges. • If fi,j > 0, if net flow aligns with orientation • If fi,j < 0, net flow is opposite of orientation. 1 2 3 4 5 6 7 f1,3 > 0 f5,7 < 0
  • 7. In the higher-order case of edge flows,we have a different type of objective. 7 Key idea (“divergence-free”). Net flow into a node should be similar to net flow out of a node. minimize flows f X i 2 4 X j>i,(i,j)2E fij X k<i,(k,i)2E fki 3 5 2 subject to f matches labels<latexit 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  • 8. There is a close relationship between node-based SSL and edge-based SSLobjective functions. 8 X (i,j)2E (xi xj)2 = xT Lx = xT BBT x = kBT xk2 2 <latexit 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X i 2 4 X j>i,(i,j)2E fij X k<i,(k,i)2E fki 3 5 2 = fT BT Bf = kBfk2 <latexit 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Bk,(i,j) = 8 >< >: 1 k = i, i < j 1 k = j, i < j 0 otherwise<latexit 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One catch. • Having labels in node case gives unique answer. • Having labels in edge case is under-constrained.
  • 9. We add regularization to get a nice sparse linear least squares problem. 9 minimize flows f kBfk2 2 + kfk2 2 subject to f matches labels<latexit 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• We use iterative solvers LSQR or LSMR to compute the solution efficiently.
  • 10. Key Idea My label is similar to the labels of my connections. Objective Net in flow = net out flow at all nodes. minimize vertex values x kBT xk2 2 subject to x matches labels<latexit 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minimize edge flows f kBfk2 2 + kfk2 2 subject to f matches labels<latexit 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  • 11. 11 0.0 0.2 0.4 0.6 0.8 1.0 correlation Anaheim Barcelona 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) 0.0 0.2 0.4 0.6 0.8 1.0 correlation 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) Winnipeg Chicago ZeroFill LineGraph FlowSSL
  • 12. Why do we do so poorly on Chicago? 12 ftruth = y z, y 2 R, z 2 C<latexit 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Our divergence-free assumption says that ftruth ⇡ z.<latexit 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Does this actually hold in our data?
  • 13. 13 0.0 0.2 0.4 0.6 0.8 1.0 correlation Anaheim Barcelona 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) 0.0 0.2 0.4 0.6 0.8 1.0 correlation 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) Winnipeg Chicago ZeroFill LineGraph FlowSSL kyk2 kzk2 ⇡ 0.2. <latexit 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kyk2 kzk2 ⇡ 0.5. <latexit 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  • 14. We have some theoretical guarantees if the true flow is indeed nearly divergence-free. 14 Theorem. • Let VC be a basis for the divergence-free space ker(B). • Suppose the true flow is a divergence-free flow perturbed by d. • For any m – n + 1 labels corresponding to linearly independent rows of VC, denoted , the relative reconstruction error is bounded by minimize edge flows f kBfk2 2 + kfk2 2 subject to f matches labels<latexit 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VL C<latexit 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h 1 min(VL C) + 1 i · kdk2 <latexit 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  • 15. Two major questions in semi-supervised learning. 15 1. Interpolation. Given the measurements, how do I interpolate to locations where I don’t know have data. 2. Active learning. Where are the best locations to make my measurements, knowing step 1?
  • 16. Active learning strategies in vertex-based and edge- based SSLare similar. 16 Active vertex-based SSL [Guillory-Bilmes 17] 1. Cluster the graph (e.g., using spectral clustering). 2. Pick points from each cluster. Active edge-based SSL 1. Cluster the graph (e.g., using spectral clustering). 2. Pick edges that cross cluster boundaries.
  • 18. 18
  • 19. We can extend our framework beyond looking for divergence-free flows. 19 f|{z} edge flow = gradient flow z}|{ B| y divergence-free flow z }| { Cw|{z} curl flow h|{z} harmonic flow<latexit 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Hodge decomposition [Lim 15, others] • So far, we have penalized gradient flow via . • Could penalize other types of flows. kBfk2 2<latexit 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j k i fij + fjk + fki<latexit 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  • 20. We can extend our framework beyond looking for divergence-free flows. 20 argmin flows f kC| fk2 + 2 · kf fmid k2 subject to fbid  f  fask <latexit 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• Data is currency exchange rates (fully connected graphs). • Buyers willing to buy at “bid” price. • Sellers willing to sell at “ask” price. • Settle on some price in the middle (usually mid point). • Want prices that have no cyclic flow (arbitrage).
  • 21. Optimal flows eliminate arbitrage opportunities. 21
  • 22. Graph-based semi-supervised learning for edge flows. 22 • We have a framework for semi-supervised learning in the edge space with a natural connection to classical vertex-based SSL. • We also have a practical and efficient active learning method. • Can extend the SSL framework to other types of edge flows through results in combinatorial Hodge theory.
  • 23. 23 THANKS! Austin R. Benson Slides. bit.ly/arb-HONS-19 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Graph-based semi-supervised learning for edge flows bit.ly/ssl-flow-code (code, reproducibility, and data) To appear in KDD 2019.