Tensor Spectral Clustering

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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.

Data & Analytics

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

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

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

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

Background: sweep cut
5
zTP = λ2zT
2 ϕ({2})
1 ϕ({2,1})
3 ϕ({2,1,3})
4 ϕ({2,1,3,4})
11 ϕ({2,1,3,4,11})
6 ϕ({2,1,3,4,11,6})
8 ϕ({2,1,3,4,11,6, 8})
10 ϕ({2,1,3,4,11,6,8,10})
9 ϕ({2,1,3,4,11,6,8,10,9})
7 ϕ({2,1,3,4,11,6,8,10,9,7})
5 ϕ({2,1,3,4,11,6,8,10,9,7,5})
0 2 4 6
0
0.5
1
size of community
conductance
Cheeger inequality
guarantee on the
conductance.

Problem: clustering methods are based on
edges and do not use higher-order relations
or motifs, which can better model problems.
6
Edges Motifs

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.

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

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

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

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)

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
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
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?

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
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.

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

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.

Tensor Spectral Clustering

1. TENSOR SPECTRAL CLUSTERING FOR PARTITIONING HIGHER-ORDER NETWORK STRUCTURES 1 Austin Benson ICME, Stanford University [email protected] 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

5. Background: sweep cut 5 zTP = λ2zT 2 ϕ({2}) 1 ϕ({2,1}) 3 ϕ({2,1,3}) 4 ϕ({2,1,3,4}) 11 ϕ({2,1,3,4,11}) 6 ϕ({2,1,3,4,11,6}) 8 ϕ({2,1,3,4,11,6, 8}) 10 ϕ({2,1,3,4,11,6,8,10}) 9 ϕ({2,1,3,4,11,6,8,10,9}) 7 ϕ({2,1,3,4,11,6,8,10,9,7}) 5 ϕ({2,1,3,4,11,6,8,10,9,7,5}) 0 2 4 6 0 0.5 1 size of community conductance Cheeger inequality guarantee on the conductance.

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.

19. Some motifs on large networks 19

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.

21. Thanks! 21 [email protected] github.com/arbenson/tensor-sc Tensor Spectral Clustering for partitioning higher-order network structures

Editor's Notes

#2: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\\ \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \path[draw, ultra thick] (0) edge [->] (2); % i -> j \end{tikzpicture}
#3: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [-] (1); % i -> k \path[draw, ultra thick] (1) edge [-] (2); % k -> j \path[draw, ultra thick] (0) edge [-] (2); % i -> j \end{tikzpicture} \tikzset{first node/.style={circle,fill=yellow!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (0) edge [->] (2); % k -> j \path[draw, ultra thick] (1) edge [->] (2); % i -> j \path[draw, ultra thick] (2) edge [->] (1); % i -> j \end{tikzpicture}
#4: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [-] (1); % i -> k \path[draw, ultra thick] (1) edge [-] (2); % k -> j \path[draw, ultra thick] (0) edge [-] (2); % i -> j \end{tikzpicture} \min_{S} \phi(S) = \min_{S} \frac{\#(\text{edges cut})}{\min(\text{vol}(S), \text{vol}(\bar{S})) } \tikzset{first node/.style={circle,fill=yellow!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (0) edge [->] (2); % k -> j \path[draw, ultra thick] (1) edge [->] (2); % i -> j \path[draw, ultra thick] (2) edge [->] (1); % i -> j \end{tikzpicture}
#5: \begin{aligned} & \underset{\vect{z} \in \mathbb{R}^n}{\text{minimize}} & & \vect{z}^{\Tra\right(}\mat{D} - \mat{A}\left)\vect{z} / \vect{z}^{\Tra}\mat{D}\vect{z} \\ & \text{subject to} & & \allones^{\Tra}\mat{D}\vect{z} = 0, \quad \| \vect{z} \| = 1 \end{aligned} \vect{z}^T\mat{P} = \lambda_2\vect{z}^T
#6: \begin{aligned} & \underset{\vect{z} \in \mathbb{R}^n}{\text{minimize}} & & \vect{z}^{\Tra\right(}\mat{D} - \mat{A}\left)\vect{z} / \vect{z}^{\Tra}\mat{D}\vect{z} \\ & \text{subject to} & & \allones^{\Tra}\mat{D}\vect{z} = 0, \quad \| \vect{z} \| = 1 \end{aligned} \vect{z}^T\mat{P} = \lambda_2\vect{z}^T
#7: \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [right = \sqdist of 0] {$\knode$}; \path[draw, ultra thick] (0) edge [-] (1); \end{tikzpicture} \\ \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (0) edge [->] (2); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [<->] (1); % i -> k \path[draw, ultra thick] (1) edge [<->] (2); % k -> j \path[draw, ultra thick] (2) edge [<->] (0); % j -> k \end{tikzpicture}\\\vspace{0.3cm} \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); \path[draw, ultra thick] (0) edge [->] (2); \end{tikzpicture}\quad \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} % open 0 recip out \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [<->] (1); \path[draw, ultra thick] (0) edge [<->] (2); \path[draw, ultra thick] (1) edge [<->] (0); \path[draw, ultra thick] (2) edge [<->] (0); \end{tikzpicture}\quad \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} % open 0 recip out \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (1) edge [-] (0); \path[draw, ultra thick] (0) edge [-] (2); \path[draw, ultra thick] (1) edge [-] (2); \end{tikzpicture}\quad \tikzset{first node/.style={circle,fill=blue!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [right = \sqdist of 0] {$\knode$}; \node[first node] (2) [below = \sqdist of 0] {$\jnode$}; \node[first node] (3) [below = \sqdist of 1] {$\lnode$}; \path[draw, ultra thick] (0) edge [-] (1); \path[draw, ultra thick] (0) edge [-] (2); \path[draw, ultra thick] (0) edge [-] (3); \path[draw, ultra thick] (1) edge [-] (2); \path[draw, ultra thick] (1) edge [-] (3); \end{tikzpicture}
#8: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [-] (1); % i -> k \path[draw, ultra thick] (1) edge [-] (2); % k -> j \path[draw, ultra thick] (0) edge [-] (2); % i -> j \end{tikzpicture} \tikzset{first node/.style={circle,fill=yellow!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (0) edge [->] (2); % k -> j \path[draw, ultra thick] (1) edge [->] (2); % i -> j \path[draw, ultra thick] (2) edge [->] (1); % i -> j \end{tikzpicture}
#9: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [-] (1); % i -> k \path[draw, ultra thick] (1) edge [-] (2); % k -> j \path[draw, ultra thick] (0) edge [-] (2); % i -> j \end{tikzpicture} \tikzset{first node/.style={circle,fill=yellow!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (0) edge [->] (2); % k -> j \path[draw, ultra thick] (1) edge [->] (2); % i -> j \path[draw, ultra thick] (2) edge [->] (1); % i -> j \end{tikzpicture}
#12: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$\knode$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$\jnode$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture} \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$\inode$}; \node[first node] (1) [below right = \tridist of 0] {$\knode$}; \path[draw, ultra thick] (0) edge [-] (1); % i -> k \end{tikzpicture} \phi(S) = \frac{\#(\text{edges cut})}{\min(\text{vol}(S), \text{vol}(\bar{S})) }, \quad \text{vol}(S) = \#(\text{edge points in $S$}) \phi_3(S)&=& \frac{\#(\text{triangles cut})}{\min(\text{vol}_3(S), \text{vol}_3(\bar{S})) }, \\ \text{vol}_3(S) &=& \#(\text{triangle end points in $S$}) \phi(S) = \frac{\#(\text{edges cut})}{\min(\text{vol}(S), \text{vol}(\bar{S})) } vol(S) =
#13: \begin{aligned} &\mat{P}_{ij} &= 1 \\ &\mat{A}_{ik}&= 1 \\ &\mat{A}_{il} &= 1 \\ &\mat{A}_{jk} &= 1 \\ \end{aligned} \begin{aligned} \tens{A}(i, j, k) &=& 1 \\ \tens{A}(i, j, l) &=& 1 \\ \end{aligned} P_{ji} \\ = A_{ij} / \sum_{k} A_{ik} \tens{P}_{kji} \\ = \tens{A}_{ijk} / \sum_{l}A_{ijl} \\ \tens{P}(i, j, k) = \\ \text{Pr}(S_{t+1} = k \;\mid\; S_t = j, \; S_{t-1} = i) \mat{P}(i, j) =
#14: \begin{aligned} &\mat{P}_{ij} &= 1 \\ &\mat{A}_{ik}&= 1 \\ &\mat{A}_{il} &= 1 \\ &\mat{A}_{jk} &= 1 \\ \end{aligned} \begin{aligned} \tens{A}(i, j, k) &=& 1 \\ \tens{A}(i, j, l) &=& 1 \\ \end{aligned} P_{ji} \\ = A_{ij} / \sum_{k} A_{ik} \tens{P}_{kji} \\ = \tens{A}_{ijk} / \sum_{l}A_{ijl}
#15: \begin{aligned} &\mat{P}_{ij} &= 1 \\ &\mat{A}_{ik}&= 1 \\ &\mat{A}_{il} &= 1 \\ &\mat{A}_{jk} &= 1 \\ \end{aligned} \begin{aligned} \tens{A}(i, j, k) &=& 1 \\ \tens{A}(i, j, l) &=& 1 \\ \end{aligned} P_{ji} \\ = A_{ij} / \sum_{k} A_{ik} \tens{P}_{kji} = \tens{A}_{ijk} / \sum_{l}A_{ijl}
#16: \alpha\mat{R}\left( \vect{x} \otimes \vect{x} \right) + (1 - \alpha)\vect{v} = \vect{x}, \; x_k \ge 0, \; \allones^{\Tra}\vect{x} = 1 \mat{P}[\vect{x}] := \sum_{k=1}^{n}x_k\tens{P}(:, :, k) \vect{z}^{\Tra} \mat{P}[\vect{x}] = \lambda_2\vect{z}^{\Tra}
#17: \begin{aligned} &\mat{P}_{ij} &= 1 \\ &\mat{A}_{ik}&= 1 \\ &\mat{A}_{il} &= 1 \\ &\mat{A}_{jk} &= 1 \\ \end{aligned} \begin{aligned} \tens{A}(i, j, k) &=& 1 \\ \tens{A}(i, j, l) &=& 1 \\ \end{aligned} P_{ji} \\ = A_{ij} / \sum_{k} A_{ik} \tens{P}_{kji} = \tens{A}_{ijk} / \sum_{l}A_{ijl}
#18: \tikzset{first node/.style={circle,fill=red!20,draw,minimum size=0.6cm,inner sep=0pt},} \tikzset{second node/.style={circle,fill=blue!20,draw,minimum size=0.6cm,inner sep=0pt},} \tikzset{third node/.style={circle,fill=green!20,draw,minimum size=0.6cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$0$}; \node[first node] (1) [below right = 0.4cm and 2cm of 0] {$1$}; \node[first node] (2) [below right = 0.4cm and 0.4cm of 0] {$2$}; \node[first node] (3) [below right = 0.1cm and 3cm of 0] {$3$}; \node[second node] (4) [below = 0.5cm of 2] {$4$}; \node[second node] (5) [right = 1cm of 4] {$5$}; \node[second node] (6) [below right = 0.3cm and 0.5cm of 4] {$6$}; \node[second node] (7) [below right = 0.3cm and 0.5cm of 5] {$7$}; \node[third node] (8) [below= 0.5cm of 6] {$8$}; \node[third node] (9) [right = 1.7cm of 8] {$9$}; \node[third node] (10) [below left = 0.4cm and 0.5cm of 8] {$10$}; \node[third node] (11) [right = 1.5cm of 10] {$11$}; \path[draw, ultra thick] (0) edge [->] (1); \path[draw, ultra thick] (1) edge [->] (2); \path[draw, ultra thick] (2) edge [->] (0); \path[draw, ultra thick] (1) edge [->] (0); \path[draw, ultra thick] (2) edge [->] (1); \path[draw, ultra thick] (0) edge [->] (2); \path[draw, ultra thick] (0) edge [->] (3); \path[draw, ultra thick] (3) edge [->] (1); \path[draw, ultra thick] (4) edge [->] (5); \path[draw, ultra thick] (5) edge [->] (6); \path[draw, ultra thick] (6) edge [->] (4); \path[draw, ultra thick] (6) edge [->] (7); \path[draw, ultra thick] (7) edge [->] (5); \path[draw, ultra thick] (5) edge [->] (7); \path[draw, ultra thick] (8) edge [->] (9); \path[draw, ultra thick] (9) edge [->] (10); \path[draw, ultra thick] (10) edge [->] (8); \path[draw, ultra thick] (10) edge [->] (11); \path[draw, ultra thick] (11) edge [->] (10); \path[draw, ultra thick] (8) edge [->] (11); % Level 1 -> 2 \path[draw, ultra thick] (1) edge [->] (4); \path[draw, ultra thick] (1) edge [->] (5); \path[draw, ultra thick] (1) edge [->] (7); \path[draw, ultra thick] (2) edge [->] (4); \path[draw, ultra thick] (4) edge [->] (5); \path[draw, ultra thick] (3) edge [->] (5); \path[draw, ultra thick] (3) edge [->] (7); % Level 2 -> 3 \path[draw, ultra thick] (6) edge [->] (8); \path[draw, ultra thick] (6) edge [->] (9); \path[draw, ultra thick] (6) edge [->] (11); \path[draw, ultra thick] (7) edge [->] (8); \path[draw, ultra thick] (7) edge [->] (9); % Level 1 -> 3 \path[draw, ultra thick] (3) edge [->] (9); % Add connectedness to graph \path[draw, ultra thick] (10) edge [->] (0); \path[draw, ultra thick] (8) edge [->] (7); \path[draw, ultra thick] (7) edge [->] (3); \end{tikzpicture} \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$k$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$j$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$k$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$j$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$k$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$j$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture}\quad \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$k$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$j$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (1) edge [->] (0); % k -> i \path[draw, ultra thick] (1) edge [->] (2); % k -> j \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \path[draw, ultra thick] (0) edge [->] (2); % i -> j \end{tikzpicture}
#20: \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$j$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$k$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (2) edge [->] (1); % j -> k \path[draw, ultra thick] (2) edge [->] (0); % j -> i \end{tikzpicture} \tikzset{first node/.style={circle,fill=black!20,draw,minimum size=0.35cm,inner sep=0pt},} \begin{tikzpicture} \node[first node] (0) {$i$}; \node[first node] (1) [below right = \tridist and \tridist of 0] {$j$}; \node[first node] (2) [below left = \tridist and \tridist of 0] {$k$}; \path[draw, ultra thick] (0) edge [->] (1); % i -> k \path[draw, ultra thick] (2) edge [<->] (1); % j -> k \path[draw, ultra thick] (0) edge [->] (2); % j -> i \end{tikzpicture}

Tensor Spectral Clustering

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Tensor Spectral Clustering

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