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Multiperspective Graph-Theoretic Similarity Measure
ABSTRACT
1 INTRODUCTION
similar
CIKM ’18, October 22–26, 2018, Torino, Italy
Problem. graph-theoretic
similarity
(i,j)similar
k i l j
similar
uniperspective
naive
Proposed Approach. multi-
perspective
inter-object
inter-perspective
Contributions.
First
Second
ird
Fourth
Fih
Finally
2 RELATED WORK
Graph-eoretic Similarity.
G(V,E)
S(a,b) a,b
S(a,b) =
C
|N(a)| |N(b)|
|N(a)|
i=1
|N(b)|
j=1
S(Ni(a),Nj(b)), a,b,
1, a=b
C N(a) N(b)
a b
a,b
a
S(a,b) = 0 b,a
Multiperspective Similarity.
3 OVERVIEW
3.1 Problem Formulation
O={o1,o2, . . . ,on}
m P={p1,p2, . . . ,pm}
O p∈ P
Gp(O,Ep) Ep⊆ O × O
p
G={G1,G2, . . . ,Gm}
Gp
O
m
Figure 1: Illustration of the Hypergraph Representation
perspective
object
G H=
(X,E) X=P ∪ O
E={(pk,oi,oj) : 1 ≤k≤m; 1 ≤i,j≤n}
(pk,oi,oj)∈ E oi oj
pk (oi,oj)∈Epk Gpk
P={p1,p2}
O={o1,o2,o3,o4} (p1,o1,o4)
(p1,o2,o3) p1 o1
o4 o2 o3
p2o1 o2 o3 o4
H(X,E)
oi,oj∈ O p∈ P
Sp(oi,oj) [0,1]
Sp(oi,oj) = 1 i=j
Problem 1 (Multiperspective Similarity). Given a multiper-
spective hypergraph H, determine the similarity score Sp(oi,oj)for
each perspective p∈ P and pair of objects oi,oj∈ O.
3.2 Framework for Multiperspective Solutions
Gp
Disjoint-SimRank
Gp
Table 1: List of Notations
Symbols Description
P {p1,p2, . . . ,pm}
O {o1,o2, . . . , om}
X P ∪ O
E
{(p,oi,oj) : oi oj p}
m
n
Np(oi){oj∈ O|(p,oi,oj)∈ E}
Sp(oi,oj) oi ok
p
Sp[Sp(oi,oj)]n×n
Wp n×n
p∈ P
sim(p,p′) p,p′
C
n n
sim(p,p′)∈[0,1]
p,p′∈ P sim(p,p′)=1
p=p′ p,p′
Sp(oi,oj) oi oj
p
sim(p,p′)
Sp(oi,oj)
p′
Np(oi) oi Gp
Sp(oi,oj) = C
| P |
p′∈P
sim(p,p′)
ok∈Np′(oi)
ol∈Np′(oj)
Sp′(ok,ol)
|Np′(oi)| |Np′(oj)|,
Sp= [Sp(oi,oj)]n×n
Wp
p∈ P
Sp=C
|P |
p′∈P
sim(p,p′).Wp′TSp′Wp′
sim(p,p′)
4 STRAIGHTFORWARD SOLUTION:
PIPELINED-SIMRANK
sim(p,p′) Sp(oi,oj)
sim(p,p′)
H
4.1 Inter-Perspective Similarity
p
Gp
p p′
Gp Gp′ p
p′
H= ({P ,O},E)
B
P O × O
p oij
B (p,oi,oj)∈ E H
Figure 2: : Bipartite graph for computing
similarity between perspective nodes
B
sim(p,p′)
Sp(oi,oj)
4.2 Learning Algorithm
sim(p,p′) p,p′∈ P
sim(p,p′)
S(0)
p(∗,∗)∀p∈ P
S(0)
p(oi,oj) = 0 i,j 1 i=j
Algorithm 1
Require: H
—- create bipartite graph from hypergraph —-
B ← bipartiteTransform(H)
—- compute the similarity between perspectives —-
{sim(∗)(p,p′)}∀p,p′∈P ←bipartiteSimRank(B)
Initialize S(0)
p←n,∀p∈ P
while do
S(t+1)
p(oi,oj) = C
|P |
p′∈P
sim(∗)(p,p′)
×
ok∈Np′(oi)
ol∈Np′(oj)
S(t)
p′(ok,ol)
|Np′(oi)||Np′(oj)|,
( 1≤i,j≤n)
S(t+1)
p(oi,oi) = 1( 1≤i≤n)
Return {Sconverged
p(oi,oj),∀p∈ P,oi,oj∈ O }
{sim(∗)(p,p′),∀p,p′∈ P}.
4.3 Convergence Property
e sequence of perspective-specic similarity score
produced by Algorithm 1 is non-decreasing and bounded by [0,1],
i.e., for p∈ P,oi,oj∈ O,t≥0.
1≥S(t+1)
p(oi,oj)≥S(t)
p(oi,oj)≥0,
Proof: (5)
S(1)
p(oi,oj)≥0 = S(0)
p(oi,oj),∀p∈ P,oi,oj∈ O
S(1)
p(oi,oi) = 1 = S(0)
p(oi,oi),∀p∈ P,oi∈ O .
t= 0
∀t≥1
{S(t)
p(oi,oj)}t≥0
{S(t)
p(oi,oj)}t≥0 Sp(oi,oj)∈[0,1]
{Sp(oi,oj)} {sim(∗)(p,p′)}
5 JOINT SOLUTION: MP-SIMRANK
joint
sim(p,p′)
Sp(oi,oj)
5.1 Inter-Perspective Similarity
sim(p,p′)
Sp
sim(p,p′) Sp
sim(p,p′) = 1 −
Sp− S′
p
F
n,
p,p′
Sp Sp′
Sp−S′
p
F
n sim(p,p′)
Sp Sp′
Sp−S′
p
F
n
sim(p,p′)
5.2 Learning Algorithm
S(0)
p(∗,∗)∀p∈
Algorithm 2
Require: H
S(0)
p←In,∀p∈ P
sim(0)(p,p′) = 1 p=p′ 0 p,p′
while do
S(t+1)
p(oi,oj) = C
|P |
p′∈P
sim(t)(p,p′)
×
ok∈Np′(oi)
ol∈Np′(oj)
S(t)
p′(ok,ol)
|Np′(oi)||Np′(oj)|,
( 1≤i,j≤n)
S(t+1)
p(oi,oi) = 1 ( 1≤i≤n)
sim(t+1)(p,p′) = 1 −
S(t+1)
p−S(t+1)
p′
F
n,∀p,p′∈ P
Return {Sconverged
p(oi,oj),∀p∈ P,oi,oj∈ O }
{simconverged(p,p′),∀p,p′∈ P}
P
sim(0)(p,p′)∀p,p′∈ P
sim(0)(p,p′) = 0 p,p′ 1 p=p′
5.3 Convergence Property
e sequence of similarity between perspectives pro-
duced by Algorithm 2 is non-decreasing and bounded by [0,1], i.e.,
for t≥1,
1≥sim(t+1)(p,p′)≥sim(t)(p,p′)≥0,∀p,p′∈ P.
Proof: t≥0
S(t+1)
p− S(t+1)
p′
F≤
S(t)
p− S(t)
p′
F,
∀p,p′∈ P
S(t+1)
p− S(t+1)
p′
F
=
C
|P |
p′′∈ P
sim(t)(p,p′′)−sim(t)(p′,p′′)WT
p′′ · S(t)
p′′ ·Wp′′
F
≤C
|P |
p′′∈ P
sim(t)(p,p′′)−sim(t)(p′,p′′).
WT
p′′ · S(t)
p′′ ·Wp′′
F
=C
|P |
p′′∈ P
S(t)
p− S(t)
p′′
F−
S(t)
p′− S(t)
p′′
F.
WT
p′′ · S(t)
p′′ ·Wp′′
F
n
=C
|P |
p′′∈ P
(S(t)
p− S(t)
p′′ )−(S(t)
p′− S(t)
p′′ )
F.
WT
p′′ · S(t)
p′′ ·Wp′′
F
n
≤C
|P |
p′′∈ P
S(t)
p− S(t)
p′
F<
S(t)
p− S(t)
p′
F
⇒sim(t+1)(p,p′)≥sim(t)(p,p′).
0≤
S(t)
p−S(t)
p′
F
n≤1,∀t≥1 p,p′∈ P
sim(t)(p,p′)∈[0,1],∀t≥0 p,p′∈ P
sim(t)(p,p′) sim(p,p′)
{S(t)
p(oi,oj)}t≥0
Sp(oi,oj)∈[0,1] {sim(t+1)(p,p′)}t≥0
sim(p,p′) Sp(oi,oj) sim(p,p′)
6 EXPERIMENTS ON EFFECTIVENESS
6.1 Experimental Settings
Datasets.
Zoo
legstype
(p,oi,oj)
oi oj p
legselephantgirae
Congressional Voting Records (or HouseVote)
Zoo
Paris Aractions
i j
m∗n2
Task and Metrics.
Recall: p∈ P
E
p E
p Recall
E
p E
p
Recall =1
m
p∈P
|E
p∩ E
p|
|E
p|
PRES:
PRES
n
ri
Nmax
PRES = 1 −ri
n−n+1
2
Nmax
Methods.
C
uniperspective Merged-
SimRank
Average-
SimRank
Disjoint-SimRank
Personalized Collaborative Clustering PCC
6.2 Comparison to Baselines
Recall
Disjoint-SimRank Recall Zoo
HouseVote Paris Aractions
Merged-SimRank Recall Disjoint-
SimRank Zoo HouseVote
Paris Aractions
Average-SimRank
Recall Zoo
HouseVote Paris Aractions
00.150.30.450.60.75
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
Recall
Zoo
00.150.30.450.60.75
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
Recall
HouseVote
00.010.020.030.040.0
5
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
Recall
Paris Attractions
Figure 3: Recall values of all models
00.20.40.60.81
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
PRES
Zoo
00.20.40.60.81
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
PRES
Paris Attractions
00.20.40.60.81
Merged-S imRank
Average-SimRank
Disjoint-Si mRank
PCC
Pipelined-SimRank
MP-SimRan k
PRES
HouseVote
Figure 4: PRES values of all models
Recall Zoo HouseVote
Paris Aractions Recall
PCC
Zoo HouseVote Paris Aractions
Disjoint-SimRank
Merged-
SimRank Average-SimRank
PRES
Recall
Recall PRES
PRES
Paris Aractions
Recall PRES
6.3 Inter-Perspective Similarities
sim(p,p′)∀p,p′∈ P
p,p′
sim(p,p′)
sim(p,p′)
Zoo HouseVote
p∈ P
P Paris Aractions
Table 2: Correlation between NMI scores
and inter-perspective similarities for Zoo (17 perspectives)
p1
p2
p3
p3
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
p16
p17
6.4 Illustrative Case Study
Paris Aractions
| |
Paris Aractions
7 DISCUSSION ON EFFICIENCY
Table 3: Correlation between NMI scores and
inter-perspective similarities for HouseVote (16
perspectives)
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
p16
Table 4: Cluster data of four users from Paris Aractions
30 50 62 76 88
30 50 62 88
62 88 50
88 76
30
Table 5: Complexity analysis (per iteration) of all
SimRank-based methods
Methods Storage Time
On2On2dmax
On2Omn2dmax
Omn2Omn2dmax
Om2+n4+mn2O(m2+n4+mn2)dbi +m2n2dmax
Om2+mn2Om2n2dmax
7.1 Complexity Analysis
Merged-
SimRank
n2
dp
|Np(oi)|.|Np(oj)| oi,oj∈ O dmax
∀p∈ P
Average-SimRank Disjoint-SimRank
m
50
62
30
88
50
62
30
88
50
62
88
76
76
88
30
id:
id:
id:
id:
Figure 5: Illustrative example of multiperspective similarity from Paris Aractions dataset.
mn2
m2 n4
dbi B
m2n2dmax
n4
convergence rate
Dt=1
m
p∈P
S(t+1)
p− S(t)
p
F
n,
Dt
t
Zoo HouseVote
Paris Aractions
7.2 Heuristic for More Ecient MP-SimRank
Disjoint-SimRank
Sdisjoint
p,∀p∈ P
O(mn2dmax)
k-medoids
k
O(m2+km)
Hc cluster-specic inter-object Sc
O(k2n2dmax)
O(mn2dmax+m2+km+k2n2dmax)
Om2n2dmax
k≤m
k
Recall PRES
k
k=m k
8 CONCLUSION
Running Time (second)
HouseVote
PRES Recall
k = 1
910 11 12 13 14 15 16 k=17
k=1
910 11
12 13
14 15 16
k=17
Running Time (second)
Zoo
PRES Recall
Running Time (second)
P
ar
i
s
A
ttract
i
ons
PRES Recall
Figure 6: PRES, Recall, and running time of with dierent number of clusters k.
Algorithm 3
Require: H k
– Step 1: run disjoint-simrank on each perspective graph –
Sdistjoint
p←Disjoint −SimRank(Gp),∀p∈ P.
- Step 2: compute Frobenius distances between perspectives -
F=[F(p,p′)]p,p′∈P
F(p,p′) =
Sdistjoint
p−Sdistjoint
p′
F
– Step 3: cluster perspectives and merge graphs –
C ← K−Medoids(F,k); Hc←merge −graph(H,C)
– Step 4: run on the new hypergraph Hc–
{Sc}c∈C ←(Hc)
– Step 5: assign each perspective the inter-object similarity –
15: – of the cluster it belongs to–
Sp←Sc,∀p∈ P,c∈ C, p∈c
{Sp,∀p∈ P}
ACKNOWLEDGMENTS
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