Spectral clustering with motifs and higher-order structures

Spectral graph clustering
with motifs and
higher-order structures
David F. Gleich
Purdue University
Code & Data github.com/arbenson/higher-order-organization-julia
github.com/dgleich/motif-ssbm
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Austin Benson (Stanford -> Cornell)
Jure Leskovec (Stanford) NAConf'17David Gleich · Purdue
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Graphs and matrices have a long and
intertwined history.
Matrices and graphs represent
relationships among a group of
objects.
To study the relationships
• centrality
• reachability
• clustering
• … and more …
often use matrix computations
• e.g. Estrada & Higham, SIREV
• e.g. Network analysis, Brandes & Erlebach
Helen Bott, Observation of play
activities in a nursery school, 1928
Ax = b Ax = x

… a suggestion based on our work …

given a graph G = (V, E)
and its adjacency matrix A
consider using the weighted matrix W = A2
A
Hadamard /
element-wise

and its non-symmetric
adjacency matrix A
consider using a symmetric
weighted matrix from
given a directed graph G = (V, E)
Motif Matrix computations W =
M1 C = (U · U) UT
C + CT
M2 C = (B · U) UT
+ (U · B) UT
+ (U · U) B C + CT
M3 C = (B · B) U + (B · U) B + (U · B) B C + CT
M4 C = (B · B) B C
M5 C = (U · U) U + (U · UT
) U + (UT
· U) U C + CT
M6 C = (U · B) U + (B · UT
) UT
+ (UT
· U) B C
M7 C = (UT
· B) UT
+ (B · U) U + (U · UT
) B C
M8 C = (U · N) U + (N · UT
) UT
+ (UT
· U) N C

Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
what
why
how
(A*A).*A Matlab
(A*A).*A Julia
np.dot(A,A)*A Python (A is array)
When clustering based on triangles, we often
have better numerical properties (e.g. eigenvalue
gaps) in model partitioning problems (stochastic
block models) and better real-world results.
The matrix W = A2
A arises from our motif and
higher-order clustering framework when using
triangles as the motif.

Networks are sets of nodes and edges (graphs)
that model real world systems
Key insight. [Flake et al., Newman et al., and hundreds more!]
Networks—for real-world systems—have modules, communities, clusters
This structure has traditionally been exposed with node and edge based
clustering metrics. Density, modularity, conductance, cut, ratio cuts, etc.
NAConf'17David Gleich · Purdue
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Co-author network
8
Background network clustering is a fundamental network
analysis for finding coherent groups of nodes based on edges
§ Real-world networks have modular organization [Newman 2004, Newman 2006].
§ We want to automatically find the modules in the system.
Co-author network
§ Old idea Find groups of nodes with high internal edge density and low
external edge density [Newman 2004, Danon 2005, Leskovec+ 2009].
Brain network, de Reus et al., RSTB, 2014.
Brain network, de Reus et al., RSTB, 2014.

Similar tools are used to partition
computations for parallelism
Comanche mesh from Alex Pothen,
from Sparse Matrix Collection NAConf'17David Gleich · Purdue
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There is abundant evidence that higher-order
connectivity patterns drive complex systems.
11
4
es
order
drive
Mangan et al., 2003
Alon, 2007
Signed feed-forward loops
in genetic transcription.A C
B
CC
A C
B
A B
C
ks.
4
Thinking beyond nodes and edges
connectivity patterns, or network motifs, drive
complex systems [Milo+02, Yaveroğlu+14].
Mangan et al., 2003
Alon, 2007
in genetic transcription.A
B
A C
B
A C
B
A
B
A B
C
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Bi-directed length-2
paths in brain networks.
Sporns-Kötter, 2004
Sporns et al., 2007
Honey et al., 2007 4
Mangan et al., 2003
Alon, 2007
B
A C
B
A C
B
A
B
A B
C
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007
4
Mangan et al., 2003
Alon, 2007
B
A C
B
A C
B
A
B
A B
C
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007
Mangan et al., 2003
Alon, 2007
Triangles in
social networks
Bi-directed paths
in brain networks
in genetic transcription
Key Insight. [Milo et al. (Science 2002)]
Certain subgraphs were far more
common than expected.
We call any small subgraph a motif.

Nodes and edges may not be
the basis elements of these networks.
Why should we look for module structure
in terms of nodes and edges?
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Idea Find clusters of motifs
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Higher-order organization of
complex networks
We generalize spectral clustering, a classic
technique to find clusters or communities in a
graph, to use motifs to cluster the graph.
• Uses motif conductance instead of node & edge conductance
• We also bound the conductance in terms of the optimal solution
Outline
1. So we’ll briefly review how spectral clustering works
2. Then see how to adapt it to work with network motifs
3. Then see this procedure on real-world & model data

We can do motif-based clustering by
generalizing spectral clustering
Spectral clustering is a classic technique to partition
graphs by looking at eigenvectors.
M. Fiedler, 1973,
Algebraic
connectivity of
graphs
Graph Laplacian Eigenvector
15
Earlier work by Simon, Ando, Courtois
dealt with a related decomposability idea
eigenvalueentry
A L = D 1/2
(D A)D 1/2
(D A)x = Dx

Spectral clustering works based on
conductance with node and edge cuts
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Conductance is one of the most important quality scored used to
identify network modules, clusters or communities [Schaeffer 2007]
used in Markov chain theory, bioinformatics, vision, etc.
(edges leaving the set)
(total edges
in the set)
(S) =
cut(S)
min vol(S), vol( ¯S)
S S
vol(S) =
P
i2S degree of i
(conductance)
cut(S) = # edges between S, ¯S

Spectral clustering works based on
conductance with node and edge cuts
17
Conductance is one of the most important quality scored used to
identify network modules, clusters or communities [Schaeffer 2007]
used in Markov chain theory, bioinformatics, vision, etc.
(edges leaving the set)
(total edges
in the set)
(S) =
cut(S)
S S
(conductance)
cut(S) = 7 cut( ¯S) = 7
|S| = 15 | ¯S| = 20
vol(S) = 85 vol( ¯S) = 151
cut(S) = 7 cut( ¯S) = 7
|S| = 15 | ¯S| = 20
vol(S) = 85 vol( ¯S) = 151
(S) = 7/85
= 0.082
Small conductance ó Good set

Spectral clustering has theoretical
guarantees
Cheeger Inequality
Finding the best conductance set
is NP-hard. L
• Cheeger realized the eigenvalues of the
Laplacian provided a bound in manifolds
• Alon and Milman independently realized
the same thing for a graph!
J. Cheeger, 1970,
A lower bound on
the smallest
eigenvalue of the
Laplacian
N. Alon, V. Milman
1985. λ1 isoperi-
metric inequalities
for graphs and
superconcentrators
Laplacian 2
⇤/2  2  2 ⇤
0 = 1  2  ...  n  2
Eigenvalues of the Laplacian
⇤ = set of smallest conductance
18

The sweep cut algorithm realizes the
guarantee
We can find a set S that achieves
the Cheeger bound.
1. Compute the eigenvector
associated with λ2 (e.g. ARPACK)
2. Sort the vertices by their values
in the eigenvector: σ1, σ2, … σn
3. Let Sk = {σ1, …, σk} and
compute the conductance of
each Sk: φk = φ(Sk)
4. Pick the minimum φm of φk .
M. Mihail, 1989
Conductance and
convergence of
Markov chains
F. C. Graham,
1992, Spectral
Graph Theory.
19
m  2
p
⇤

The sweep cut visualized
0 20 40
0
0.2
0.4
0.6
0.8
1
S
i
φi
(S) =
cut(S)
20

But current problems are much more rich
than where spectral is justified
Spectral clustering is theoretically justified for undirected graphs
• Various extensions to multiple clusters [Dhillon et al.; Gharan et al.; Jordan et al.]
• Weighted graphs are okay
• Approximate eigenvectors are okay [Mihail]
Current network models are more richly annotated
• directed, signed, colored, layered, multiplex, etc.
R. Milo, 2002, Science
X causes Y to be expressed
Z represses Y
X
Z
Y
+
–
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Nice recent work by [Fairbanks
et al. arXiv] on better
numerical stopping criteria!

There is a literature on directed spectral
graph partitioning, but it is hard to interpret
Markov chains
• Stewart (numerical solution to Markov chains)
• Chung (Random walks and cuts in dir graphs )
Nonlinear Laplacian
• Yoshida WSDM2016
Asymmetric Laplacian
• Boley et al. LAA2011 (commute times)
Gleich, Klymko, Kolda ASE BigData 2014
D 1
Ax = x
(D A)x = x
1
2 ⇧(D 1
A) + 1
2 (AT
D 1
)⇧
X
(u,v)2E
(
(xu xv )2
xu xv 0
0 otherwise
22

Our contributions
1. A generalized conductance metric
for motifs
2. A “new” spectral clustering algorithm to
minimize the generalized conductance.
3. AND an associated Cheeger inequality.
(which handles directed graphs)
4. Aquatic layers in food webs
5. Hub structure in transportation
This talk, still preliminary!
23
Some studies in stochastic block modelsNew!

Motif-based conductance generalizes
edge-based conductance
Need notions of cut and volume
S
S
S¯S
¯S
vol(S) = #(edge end points in S)
24
cut(S) = #(edges cut by S) cutM (S) = #(motifs cut by S)
volM (S) = #(motif
end points in S)
M (S) =
cutM (S)
min(volM (S), volM ( ¯S))
(S) =
cut(S)
min(vol(S), vol( ¯S))
vol(S) =
P
i2S degree of i

An example of motif-conductance
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¯S
S
Motif
M (S) =
motifs cut
motif volume
=
1
10
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How can we optimize motif conductance?
We thought that motif conductance would spark new tensor and
hypermatrix methods based on the motif adjacency tensor.
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1
3
2
A
We were wrong!
A(i, j, k) =
(
1 if motif involves nodes i, j, k
0 otherwise
Benson, Gleich, Leskovec, SDM 2016

There is a symmetric matrix that serves as the
appropriate tool to study motif conductance
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A
W(M)
ij = counts co-occurrences of motif pattern between i, j
W(M)
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Going from motifs back to a matrix for
spectral clustering
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W(M)
ij = counts co-occurrences of motif pattern between i, j
W(M)
KEY INSIGHT
Spectral clustering on
W(M) yields results on
the new motif notion
of conductance
M (S) =
motifs cut
motif volume
=
1
10
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Here is a quick illustration of how this works.
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M (S) =
motifs cut
motif volume
=
1
10
cut(S) = 2
vol(S) = 6 + 8 + 2 + 2 + 2
=
1
10

A motif-based clustering algorithm
1. Form weighted graph W(M)
2. Compute the Fiedler vector associated with λ2 of the
motif-normalized Laplacian
3. Run a sweep cut on f
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W(M)
D = diag(W(M)
e)
L(M)
= D 1/2
(D W(M)
)D 1/2
L(M)
z = 2z
f(M)
= D 1/2
z
30

The sweep cut results
2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1
2
0
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1
2
0
4
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Best higher-
order cluster
2nd best higher-
order cluster
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(Order from the Fiedler vector)
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There are nice matrix computations
for three-node motifs
32
W = A2
A
4
Mangan et al., 2003
Alon, 2007
Signed feed-forward loop
B
A
B
A C
B
D
A
B
A B
C
Figure 1: Higher-order network str
framework. A: Higher-order structur
13 connected three-node directed motif
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007

There are nice matrix computations
for three-node directed motifs
Given a (directed) adjacency matrix A, let B = A AT
and U = A B
bidirectional unidirectional
Motif Matrix computations W =
M1 C = (U · U) UT
C + CT
M2 C = (B · U) UT
+ (U · B) UT
+ (U · U) B C + CT
M3 C = (B · B) U + (B · U) B + (U · B) B C + CT
M4 C = (B · B) B C
M5 C = (U · U) U + (U · UT
) U + (UT
· U) U C + CT
M6 C = (U · B) U + (B · UT
) UT
+ (UT
· U) B C
M7 C = (UT
· B) UT
+ (B · U) U + (U · UT
) B C
M8 C = (U · N) U + (N · UT
) UT
+ (UT
· U) N C
N
= ee
T
B
U
U
T
33

The three-node
motif-based Cheeger inequality
THEOREM
If the motif has three nodes, then the
sweep procedure on the weighted graph
finds a set S of nodes for which
M(G) = {instances of M in G}
Key Proof Step
34
cutM (S, G) =
X
{i,j,k}2M(G)
Indicator[xi , xj , xk not the same]
= 1
4 (x2
i + x2
j + x2
k xi xj xj xk xi xk )
= quadratic in x
M (S)  2
q
⇤
M
IMPLICATION
Just run spectral clustering
on those weighted matrices.

Awesome advantages
Works for arbitrary non-neg. combos of motifs too
We inherit 40+ years of research!
• Fast algorithms (ARPACK, etc.)!
• Local methods!
Yin, Benson, Leskovec, Gleich,
KDD2017
• Overlapping!
• Easy to implement
(20 lines of Matlab/Julia)
• Scalable (1.4B edges graphs
are not a prob.)
35
17 elseif motif == "M5"
18 C = (U * U) .* U + (U * U’) .* U + (U’ * U) .* U
19 W = C + C’
21 W = (U * B) .* U + (B * U’) .* U’ + (U’ * U) .* B
23 W = (U’ * B) .* U’ + (B * U) .* U + (U * U’) .* B
24 else
25 error("Motif must be one of M1, M2, M3, M4, M5, M6, or M7.")
26 end
27
28 # Get Fiedler eigenvector
29 dinvsqrt = spdiagm(1.0 ./ sqrt.(vec(sum(W, 1))))
30 LM = I - dinvsqrt * W * dinvsqrt
31 lambdas, evecs = eigs(LM, nev=2, which=:SM)
32 z = dinvsqrt * real(evecs[:, 2])
33
34 # Sweep cut
35 sigma = sortperm(z)
36 C = W[sigma, sigma]
37 Csums = sum(C, 1)’
38 motifvolS = cumsum(Csums)
39 motifvolSbar = sum(W) * ones(length(sigma)) - motifvolS
40 conductances = cumsum(Csums - 2 * sum(triu(C), 1)’) ./ min.(motif
41 split = indmin(conductances)
42 if split <= length(size(A, 1) / 2)
43 return sigma[1:split]
44 else
45 return sigma[(split + 1):end]
46 end
47 end
Figure 2.3 – Julia implementation of the motif-based spectral clusteri

Case study 1
Motifs partition the food webs
Food webs model
energy exchange
in species of an
ecosystem.
means i’s energy
goes to j
(or j eats i)
36
i j

Case study 1
Food webs model
energy exchange
in species of an
ecosystem.
means i’s energy
goes to j
(or j eats i)
Via Cheeger, motif
conductance is
better than edge
conductance.
37
i j

Demo and reproducibility
https://github.com/arbenson/higher-order-organization-julia
38
# form W0 … W4
sc0 = spectral_cut(W0)
plt = x ->
semilogx(x.sweepcut_profile
.conductance)
plt(sc0)
plt(sc1)
plt(sc2)
plt(sc3)
plt(sc4)

Case study 1
39
B D
Micronutrient
sources
Pelagic fishes
and benthic
prey
Benthic macro-
invertebrates
Benthic Fishes
Motif M6 reveals
aquatic layers
A
61% accuracy vs.
48% with edge-
based methods
24
Application 1 Food webs

Case study 2
Hub structure in the air transportation network
North American air
transport network
Nodes are airports
Edges reflect
reachability, and
are unweighted.
(Based on Frey
et al.’s 2007)
40

The weighed adjacency matrix already
reveals hub-like structure
41
Accepted pending

B
A
Counts length-two walks

The motif embedding shows this structure
and splits into east-west
Top 10
U.S. hubs
East coast non-hubs
West coast non-hubs
Primary spectral coordinate
Atlanta, the top hub, is
next to Salina, a non-hub.
MOTIF SPECTRAL
EMBEDDING
EDGE SPECTRAL
EMBEDDING
42

Case study 3: the stochastic block model
shows numerical advantages to motif-matrices
Model problems are useful in because they are simple and we often
“know” everything about them. They may not reflect real-world issues.
Mouse picture from Wikipedia Мышь_2.jpg,
Fly from oregonstateuniversity/11179958483
Biology Matrix Computations Clustering
r2
u = f
43

The stochastic block model is extremely well
understood in theory
The symmetric stochastic block model (SSBM)
• k blocks, each of size m-by-m
• within-block edges exist with prob p
• between-block edges with prob q
symmetric stochastic block model
m = 200, k = 5,
p = 0.3, q = 0.13
m m m
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
m p q · · · q
m q p
...
...
...
m q q p
.
Reminescnt of Simon & Ando.
44

The stochastic block model is extremely well
understood in theory
The symmetric stochastic block model (SSBM)
• k blocks, each of size m-by-m
• within-block edges exist with prob p
• between-block edges with prob q
m m m
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
m p q · · · q
m q p
...
...
...
m q q p
.
The task
Given a graph that is an SSBM and
given m, k, p, q
Find the k blocks.
Theory
E. Abbe, community
detection and the
stochastic block model
(In prep, on webpage)
• Necessary p > q
• Exact recovery (get all correct)
• Detectability (find a non-trivial portion)
• Uses non-backtracking random walk.
m = 200, k = 5,
p = 0.3, q = 0.13
45

4
Mangan et al., 2003
Alon, 2007
B
D
Figure S8: Higher-order organization of the S. cerevisiae transcriptional regulation network.
A: The four higher-order structures used by our higher-order clustering method, which can
model signed motifs. These are coherent feedfoward loop motifs, which act as sign-sensitive
delay elements in transcriptional regulation networks (46). The edge signs refer to activation
(positive) or repression (negative). B: Six higher-order clusters revealed by the motifs in (A).
Clusters show functional modules consisting of several motifs (coherent feedforward loops),
which were previously studied individually (45). The higher-order clustering framework identi-
fies the functional modules with higher accuracy (97%) than existing methods (68–82%). C–D:
Two higher-order clusters from (B). In these clusters, all edges have positive sign. The func-
tionality of the motifs in the modules correspond to drug resistance (C) or cell cycle and mating
type match (D). The clustering suggests that coherent feedforward loops function together as a
single processing unit rather than as independent elements.
S48
A C
B
D
S48
A C
B
D
S48
A C
B
D
S48
A B
C
Figure 1: Higher-order network structures and the higher-order network clustering
framework. A: Higher-order structures are captured by network motifs. For example, all
13 connected three-node directed motifs are shown here. B: Clustering of a network based on
motif M7. For a given motif M, our framework aims to find a set of nodes S that minimizes
motif conductance, M (S), which we define as the ratio of the number of motifs cut (filled
triangles cut) to the minimum number of nodes in instances of the motif in either S or ¯S (13).
In this case, there is one motif cut. C: The higher-order network clustering framework. Given a
graph and a motif of interest (in this case, M7), the framework forms a motif adjacency matrix
(WM ) by counting the number of times two nodes co-occur in an instance of the motif. An
eigenvector of a Laplacian transformation of the motif adjacency matrix is then computed. The
ordering of the nodes provided by the components of the eigenvector (15) produces nested sets
Sr = { 1, . . . , r} of increasing size r. We prove that the set Sr with the smallest motif-based
conductance, M (Sr), is a near-optimal higher-order cluster (13).
7
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007
Our study is to look at SSBM model using
our motif-weighting based on triangles.
4
Mangan et al., 2003
Alon, 2007
B
D
A C
B
D
A C
B
D
A C
B
D
A B
C
Figure 1: Higher-order network structures and the
Triangles in social
relationships.
Simmel, 1908
Rapoport, 1953
Granovetter, 1973
Sporns et al., 2007
Honey et al., 2007
W = A2
A
46
Based also on Tsourakakis, Pachocki, & Mitzenmacher. WWW, 2017
who showed -conductance < edge-conductance in a model.

Just using the motif-weighting highlights the
blocks for a range of parameters.
We introduce a mixing parameter μ to scale q.
µ = 0 $ q = 0, µ = k 1
k $ q = p
W = A2
AA
47

The power method identifies a cluster using
the motif weighting better than the adjacency
Detectability
Exact recovery
Exp. details.
We take the
normalized
Lap and shift
to reverse the
spectrum.
Then we
deflate given
knowledge of
the leading
eigenvector.
Accuracy is
the most
accurate block
in the extremal
m entries
AccuracyW = A2
AA

The power method identifies a cluster using
the motif weighting better than the adjacency
W=A2
A
A
49

We don’t converge faster for the usual
reasons that the power method converges
50

There is a bigger gap deeper in the spectrum,
that could explain what is going on
51

The motif weighting shifts all the eigenvalues
down, but lowest drop the most.
Semi-circle law
Not a Marchenko-
Pastur law!
W = A2
AA

We’d like a numerical understanding of why
we get better results faster with motifs.
Eigenvalues show that we’ll converge to the “cluster subspace” faster.
Conjecture. Higher accuracy for motifs because the eigenvectors are
more localized—or sharper—around the clusters.
• What remains is to understand why they are sharper!
W = A2
AA

Related work.
• Laplacian we propose was originally proposed by Rodríguez [2004]
and again by Zhou et al. [2006]
Our new theory (motif Cheeger inequality) explains why these were good ideas.
• Falls under general strategy of encoding hypergraph partitioning
problem as graph clustering problem [Agarwal+ 06]
• Serrour, Arenas, & Gómez, Detecting communities of triangles in
complex networks using spectral optimization, 2011.
• Arenas et al., Motif-based communities in complex networks, 2008.
• Rohe & Qin, Blessing of transitivity …, arXiv, 2013.
• Klymko, Gleich, Kolda (Using triangles & cycles …, ASE BigData 2014)
• Benson, Gleich, Leskovec (Motifs & Tensors, SIAM Data Mining 2015)
54

Paper
Benson, Gleich, Leskovec
Science, 2016
1. A generalized conductance metric for motifs
2. A new spectral clustering algorithm to
minimize the generalized conductance.
3. AND an associated Cheeger inequality.
4. Aquatic layers in food webs
5. Hub structure in transportation networks
6. Eigenvalues & vectors of motifs in SSBMs.
7. Lots of cool stuff on signed networks.
Joint work with
Austin Benson and Jure
Leskovec, Stanford
Supported by NSF CAREER
CCF-1149756, IIS-1422918
IIS- DARPA SIMPLEX
9 10
2
0
4
3
6
5
1
55
Code & Data
snap.stanford.edu/higher-order
github.com/arbenson/higher-order-organization-julia
github.com/dgleich/motif-ssbm
Open questions
• What is the distribution law
for the Laplacian of A2 ⊙ A
• How to work with element-
wise prods like matvecs for
N = eeT
B U UT
Thank you!

Spectral clustering with motifs and higher-order structures

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