My talk from KDD2012 about vertex neighborhoods and low conductance cuts. See the paper here: http://arxiv.org/abs/1112.0031 and http://dl.acm.org/citation.cfm?id=2339628

1. Vertex Neighborhoods, !
Low Conductance Cuts, !
and Good Seeds for Local
Community Methods
DAVID F. GLEICH
PURDUE
C. SESHADHRI
SANDIA - LIVERMORE
KDD2012
David Gleich · Purdue

2. Neighborhoods are good communities

3. Neighborhoods are good communities
^
conductance
^
Vertex

4. Neighborhoods are good communities
^
conductance
^
A Vertex
(4-4 𝜅)/(3-2 𝜅)

5. Neighborhoods are good communities
^
conductance
^
A Vertex
(4-4 𝜅)/(3-2 𝜅)
where 𝜅 is
the clustering
coefficient
and
the graph has a heavy tailed degree distribution

6. Neighborhoods are good communities
^
conductance
^
A Vertex
(4-4 𝜅)/(3-2 𝜅)
where 𝜅 is
the clustering
coefficient
and
the graph has a heavy tailed degree distribution
is a
y

7. A vertex neighborhood is a
“good” conductance community
in a graph with a heavy-tailed
degree distribution and large
clustering coefficient.

8. Our contributions
1. The previous theorem and its proof. This
shows that good communities are expected
and easy to find in modern networks with
heavy-tailed degrees and large clustering.
2. An empirical evaluation of neighborhood
communities that shows vertex
neighborhoods are the “backbone” of the
network community profile.
KDD2012
David Gleich · Purdue

9. Formal background for the theorem
1. Vertex neighborhoods
2. Low conductance cuts
3. Clustering coefficients
KDD2012
David Gleich · Purdue

10. Vertex neighborhoods
The set of a vertex and"
all its neighborhood
Also called an “egonet”
Prior research on egonets of social networks from
the “structural holes” perspective [Burt95,Kleinberg08].
Used for anomaly detection [Akoglu10], "
community seeds [Huang11,Schaeffer11], "
overlapping communities [Schaeffer07,Rees10].
KDD2012
David Gleich · Purdue

11. Conductance communities
Conductance is one of the most
important community scores [Schaeffer07]
The conductance of a set of vertices is
the ratio of edges leaving to total edges:
Equivalently, it’s the probability that a
random edge leaves the set.
Small conductance ó Good community
(S) =
cut(S)
min vol(S), vol( ¯S)
(edges leaving the set)
(total edges
in the set)
KDD2012
David Gleich · Purdue
cut(S) = 7
vol(S) = 33
vol( ¯S) = 11
(S) = 7/11

12. Clustering coefficients
Wedge
Global clustering coefficient
 =
number of closed wedges
number of wedges
center of wedge
closed wedge
Probability that a
random wedge
is closed
KDD2012
David Gleich · Purdue

13. Simple version of theorem
If global clustering coefficient = 1, then "
the graph is a disjoint union of cliques.
Vertex neighborhoods are optimal communities!
KDD2012
David Gleich · Purdue

14. Theorem
Condition Let graph G have
clustering coefficient 𝜅 and "
have vertex degrees bounded "
by a power-law function with
exponent 𝛾 less than 3.
Theorem Then there exists a vertex
neighborhood with conductance
log degree
logprobability
↵1n/d
↵2n/d
 4(1 )/(3 2)
KDD2012
David Gleich · Purdue

15. Proof Sketch
1) Large clustering coefficient "
⇒ many wedges are closed
2) Heavy tailed degree dist "
⇒ a few vertices have a very large degree
3) Large degree ⇒ O(d 2) wedges ⇒ “most” of wedges
Thus, there must exist a vertex with a high edge density ⇒
“good” conductance
Use the probabilistic method to formalize
10
0
10
1
10
2
10
3
10
4
0
0.2
0.4
0.6
0.8
1
CDFofNumberofWedges
Degree
KDD2012
David Gleich · Purdue

16. Confession!
The theory is weak
(S)  4(1 )/(3 2)
Collaboration
networks "
𝜅 ~ [0.1 – 0.5]
Social networks "
𝜅 ~ [0.05 – 0.1]
Graph Verts Edges Avg.
Deg.
Max
Deg.
 ¯C
ca-AstroPh 17903 196972 22.0 504 0.318 0.633
email-Enron 33696 180811 10.7 1383 0.085 0.509
cond-mat-2005 36458 171735 9.4 278 0.243 0.657
arxiv 86376 517563 12.0 1253 0.560 0.678
dblp 226413 716460 6.3 238 0.383 0.635
hollywood-2009 1069126 56306653 105.3 11467 0.310 0.766
fb-Penn94 41536 1362220 65.6 4410 0.098 0.212
fb-A-oneyear 1138557 4404989 7.7 695 0.038 0.060
fb-A 3097165 23667394 15.3 4915 0.048 0.097
soc-LiveJournal1 4843953 42845684 17.7 20333 0.118 0.274
oregon2-010526 11461 32730 5.7 2432 0.037 0.352
p2p-Gnutella25 22663 54693 4.8 66 0.005 0.005
as-22july06 22963 48436 4.2 2390 0.011 0.230
itdk0304 190914 607610 6.4 1071 0.061 0.158
Verts Edges Avg.
Deg.
Max
Deg.
 ¯C
17903 196972 22.0 504 0.318 0.633
n 33696 180811 10.7 1383 0.085 0.509
2005 36458 171735 9.4 278 0.243 0.657
86376 517563 12.0 1253 0.560 0.678
226413 716460 6.3 238 0.383 0.635
2009 1069126 56306653 105.3 11467 0.310 0.766
41536 1362220 65.6 4410 0.098 0.212
ar 1138557 4404989 7.7 695 0.038 0.060
3097165 23667394 15.3 4915 0.048 0.097
urnal1 4843953 42845684 17.7 20333 0.118 0.274
10526 11461 32730 5.7 2432 0.037 0.352
la25 22663 54693 4.8 66 0.005 0.005
6 22963 48436 4.2 2390 0.011 0.230
190914 607610 6.4 1071 0.061 0.158
Tech. networks "
𝜅 ~ [0.005 – 0.05]
This bound is useless
unless 𝜅 ≥ 1/2
KDD2012
David Gleich · Purdue

17. We view this theory as "
“intuition for the truth”
KDD2012
David Gleich · Purdue

18. Empirical Evaluation using
Network Community Profiles
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Community Size
Minimum
conductance for
any community of
the given size
Canonical shape
found by
Leskovec, Lang,
Dasgupta, and
Mahoney
Holds for a variety
of approximations
to conductance.
KDD2012
David Gleich · Purdue

19. Empirical Evaluation using
Network Community Profiles
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Community Size"
(Degree + 1)
Minimum
conductance for
any community
neighborhood of
the given size
“Egonet
community
profile” shows
the same
shape, 3 secs
to compute.
1.1M verts, 4M edges
The Fiedler
community
computed from
the normalized
Laplacian is a
neighborhood!
KDD2012
David Gleich · Purdue
Facebook data
from Wilson et
al. 2009

20. Not just one graph
10
5
10
5
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
max
deg
ver t s
2
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
max
deg
ver t s
2
arxiv
10
5
10
5
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
ca-AstroPh
10
0
10
0
t any procedure to compute
o compute the conductance
he graph. Most of the work
e cient is performed when
the vertex. We can express
s:
{v})/2
ighbors produces a triangle
double-counts). Note also
dges(N1(v))/2 dv. Then
ges(N1(v)). And so, given
compute the cut given the
well. This is easy to do with
tly stores the degrees.
hood communities
network community plot to
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
max
deg
1
10
−4
10
−3
1
10
−4
10
−3
soc-LiveJournal1
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
1
10
−2
10
−1
10
0
1
10
−2
10
−1
10
0
Number of vertices in cluster N
Figure 2: The best neighbo
ductance at each size (black
arXiv – 86k verts, 500k edges
soc-LiveJournal – 5M verts, 42M edges
15 more graphs available
www.cs.purdue.edu/~dgleich/codes/neighborhoods
KDD2012
David Gleich · Purdue

21. Filling in the !
Network Community Profile
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Minimum
conductance for
any community
neighborhood of
the given size
We are missing
a region of the
NCP when we
just look at
neighborhoods
KDD2012
David Gleich · Purdue
Community Size"
(Degree + 1)

22. Personalized PageRank
Communities [Andersen06]
To find the canonical NCP structure, Leskovec et
al. used a personalized PageRank based
community finder.
These start with a single vertex seed, and then
expand the community based on the solution of a
personalized PageRank problem.
The resulting community satisfies a local Cheeger
inequality.
This needs to run thousands of times for an NCP
KDD2012
David Gleich · Purdue

23. 10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
Filling in the !
Network Community Profile
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Minimum
conductance for
any community of
the given size
7807 seconds
This region
fills when
using the
PPR method
(like now!)
KDD2012
David Gleich · Purdue
Community Size"

24. Vertex Neighborhoods, !
Low Conductance Cuts, !
and Good Seeds for Local
Community Methods
KDD2012
David Gleich · Purdue

25. Am I a good seed?!
Locally Minimal Communities
“My conductance is the best locally.”
(N(v))  (N(w))
for all w adjacent to v
In Zachary’s Karate Club
network, there are four locally
minimal communities, the two
leaders and two peripheral
nodes.
KDD2012
David Gleich · Purdue

26. 10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
Locally minimal communities
capture extremal neighborhoods
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Red dots are
conductance "
and size of a "
locally minimal
community
Usually about 1%
of # of vertices.
The red
circles – the
best local
mins – find
the extremes
in the egonet
profile.
KDD2012
David Gleich · Purdue
Community Size"

27. 10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
Filling in the NCP!
Growing locally minimal comm.
la25 [27]) or as
304 [10]). The
the nodes.
and the edges
].
OD
ure to compute
he conductance
ost of the work
erformed when
We can express
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
max
deg
10
0
10
1
1
10
−4
10
−3
10
0
10
1
1
10
−4
10
−3
fb-A-oneyear
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
10
0
10
1
10
−4
10
−3
10
−2
10
−1
10
0
soc-LiveJournal1 ca
10
0
10
0
10
0
10
0
Growing only
locally minimal
communities
283 seconds
vs.
7807 seconds
Full NCP
Locally min
NCP
Original
Egonet
KDD2012
David Gleich · Purdue
Community Size"

28. 10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
Filling in the NCP!
Growing locally minimal comm.
Growing only
locally minimal
communities
143 seconds
vs.
2211 seconds
Full NCP
Locally min
NCP
Original
Egonets
arXiv – 86k verts, 500k edges
KDD2012
David Gleich · Purdue
Community Size"

29. Recap
A theorem relating clustering,"
heavy-tailed degrees, and"
low-conductance cuts of "
vertex neighborhoods.
Empirical evaluation of "
vertex neighborhoods.
More on k-cores in the paper.
⇒ Many communities are easy to find!
⇒ Explains success of community detection?
Acknowledgements!
David supported by NSF CAREER
award 1149756-CCF.
Sesh supported by the Sandia
LDRD program (project 158477) and
the applied mathematics program at
the Dept. of Energy.
KDD2012
David Gleich · Purdue
Code and results available online
www.cs.purdue.edu/~dgleich/
codes/neighborhoods

30. Two words on computing
Can be done by just
counting the triangles at
each node. Linear
complexity in |E| in a power-
law graph.
It’s possible to do this in
MapReduce too.
KDD2012
David Gleich · Purdue

31. 10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
10
0
10
1
10
2
10
3
10
4
10
5
10
−4
10
−3
10
−2
10
−1
10
0
max
deg
ver t s
2
Filling in the NCP!
Growing k cores
Growing only
locally minimal
communities
and k-cores
143 seconds
vs.
2211 seconds
Full NCP
Locally min
NCP
Original
Egonets
arXiv – 86k verts, 500k edges
KDD2012
David Gleich · Purdue
Community Size"
PPR grown
k-cores
k-cores

32. Clustering coefficients
Wedge
Global clustering coefficient
Local clustering coefficient
 =
number of closed wedges
number of wedges
Cv =
number of closed wedges centered at v
number of wedges centered at v
center of wedge
closed wedge
Probability that a
random wedge
is closed
KDD2012
David Gleich · Purdue