Tensor Spectral Clustering is an algorithm that generalizes graph partitioning and spectral clustering methods to account for higher-order network structures. It defines a new objective function called motif conductance that measures how partitions cut motifs like triangles in addition to edges. The algorithm represents a tensor of higher-order random walk transitions as a matrix and computes eigenvectors to find a partition that minimizes the number of motifs cut, allowing networks to be clustered based on higher-order connectivity patterns. Experiments on synthetic and real networks show it can discover meaningful partitions by accounting for motifs that capture important structural relationships.
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Tensor Spectral Clustering
1. TENSOR SPECTRAL CLUSTERING
FOR PARTITIONING HIGHER-ORDER NETWORK
STRUCTURES
1
Austin Benson
ICME, Stanford University
arbenson@stanford.edu
Joint work with
David Gleich, Purdue
Jure Leskovec, Stanford
SIAM Data Mining 2015
Vancouver, BC
2. Background: graph partitioning and applications
2
Goal: find a ``balanced” partition of a graph that does
not cut many edges.
Applications: community structure in social networks,
decompose networks into functional modules
3. Background: graph partitioning and clustering
3
A popular measure of the quality of a cut is conductance:
vol(S) is the number of edge end points in the set S
NP-hard in general, but there are approximation algorithms
4. Background: spectral clustering and random
walks
4
P is a transition matrix representing
the random walk Markov chain.
Entries of z used to partition graph.
Central computation:
P43 = Pr(3 4) = 1/3
zTP = λ2zT
P = ATD-1
6. Problem: clustering methods are based on
edges and do not use higher-order relations
or motifs, which can better model problems.
6
Edges Motifs
7. Problem: current methods only consider edges
… and that is not enough to model many problems
7
In social networks, we want to penalize cutting
triangles more than cutting edges. The triangle motif
represents stronger social ties.
8. Problem: current methods only consider edges
… and that is not enough to model many problems
8
SPT16
HO
CLN1
CLN2
In transcription networks, the ``feedforward loop” motif
represents biological function. Thus, we want to look for
clusters of this structure.
SWI4_SWI6
9. Our contributions
9
1. We generalize the definition of conductance for motifs.
2. We provide an algorithm for optimizing this objective:
Tensor Spectral Clustering (TSC) Algorithm:
Input: set of motifs and weights
Output: Partition of graph that does not cut the motifs
corresponding to the weights (and some normalization).
1 1 1 2
10. Roadmap of Tensor Spectral Clustering
10
G
Random walk
transition matrix
P
Eigenvector
zTP = λ2zT
z S
Objective Φ
Motifs and weights
that model problem
Random walk
transition tensor
P
Represent tensor
by a random walk matrix
PG++
New objective Φ’
Sweep cut
11. Motif-based conductance
11
Our algorithm is a heuristic for minimizing this objective
based on the random walk interpretation of spectral
Edges cut Triangles cut
vol(S) =
#(edge end
points in S)
vol3(S) =
#(triangle end
points in S)
12. First-order second-order Markov chain
12
k
ji
r
1/3
1/3
1/3
k
ji
r
1/2
1/2
Prob(i j) = 1/3 Prob((i, j) (j, k)) = 1/2
P P
13. 13
P
k
ji
r
1/2
1/2
Representing the transition tensor
Idea: Represent the tensor as a matrix, respecting the
motif transitions of the data. Then we can compute
eigenvectors.
P
Problem 1: Even stationary distribution of second-
order Markov chain is O(n2) storage.
Problem 2: Tensor eigenvectors are hard to compute.
14. 14
Representing the transition tensor
P(:, :,
1)
P
Each slice of transition tensor is a transition matrix.
Convex combinations of these slices is a transition matrix.
Which combination should we use?
15. Transition tensor transition matrix
15
k
ji
r
1/2
1/2
1. Compute tensor PageRank vector [Gleich+14]
2. Collapse back to probability matrix
Convex combination
of slices P(:, :, k)
16. 16
Theorem
Suppose there is a partition of the graph that
does not cut any of the motifs of interest. Then
the second left eigenvector of the matrix P[x]
properly partitions the graph.
17. Layered flow network
17
The network “flows” downward
Use directed 3-cycles to model flow:
Tensor spectral clustering: {0,1,2,3}, {4,5,6,7}, {8,9,10,11}
Standard spectral: {0,1,2,3,4,5,6,7}, {8,10,11}, {9}
1 1 1 2
18. Planted motif communities
18
Tensor spectral clustering: {0,1,2,3,4,5,12,13,16}
Standard spectral: {0,1,4,5,9,11,16,17,19,20}
0
2
4
Plant a group of 6 nodes
with high motif frequency
into a random graph.
20. Summary of results
20
1. New objective function: motif conductance
2. Tensor Spectral Clustering algorithm that is a
heuristic for minimizing motif conductance.
Input: different motifs and weights
Output: partition minimizing the number of motifs cut
corresponding to the weights
More recent work: algorithm with Cheeger-like
inequality for motif conductance.