VLDB 2015 Tutorial: On Uncertain Graph Modeling and Queries

Arijit Khan
Systems Group
ETH Zurich
Lei Chen
Hong Kong University of
Science and Technology

Social Network Transportation Network
Chemical Compound Biological Network
Graphs are Everywhere
Graphs in Machine Learning
Program Flow Images
1/ 160

3
Big-Data as Big-Graph
2/ 160
Bill Gates
Sergey Brin
Maryland
Harvard
Microsoft
Stanford
Jane Stanford
Seattle
Steve
Woznaik
Jerry Yang
Apple
NeXT
Ajim Premji
Wipro
Yahoo!
Silicon Valley
Google
Founded in
founded
founded
nationality
Knowledge Graph

“… the real world is always certain; it is our knowledge
of it that is sometimes uncertain. ”
Uncertainty
Amihai Motro [Management of Uncertainty in Database Systems]
3/ 160

Uncertainty in
Graph Data
Uncertain Graph
(Edge Uncertainty)
T
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
Social Networks
Traffic Networks
Ad-hoc Mobile Networks
Protein-interaction Networks
Knowledge Bases Constructed
from Diverse Sources
4/ 160

Sources of Uncertain
Graphs
5/ 160
Biological Networks
Interaction network of Mic17
obtained from the STRING database.
All interactions are derived from
experimental evidence
Gabriele Cavallaro [Genome-wide analysis of eukaryotic twin CX9C proteins]
http://string-db.org/
BIOMINE
https://www.cs.helsinki.fi/group/biomine/
http://www.ncbi.nlm.nih.gov//

Sources of Uncertain
Graphs
6/ 160
Social Networks
Probability of an edge (u, v) represents
the likelihood that some action of u will
be adopted by v
David Clarke [http://mashable.com/2012/04/03/twitter-changes-for-brands/]
0.2
0.3
0.7

Other Sources of
Uncertain Graphs
Sensor Networks
Traffic Networks
Knowledge Bases
Entity Resolution via
Crowd-Sourcing
Uncertain Query
Explicit Manipulation
due to privacy
purposes
Link Prediction
Jiawei
Han
Wei
Wang
Wei
Wang
0.3
Identity Uncertainty
[ICDE 2014]
Packet Delivery Probability in
Sensor Network
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6
Crowd-Sourced Entity Resolution
[VLDB 2012]
7/ 160

Why Consider Uncertainty
8/ 160
Considering the edge probabilities as weights
- no meaningful way to perform such a casting
- no easy way to additionally encode normal weights on the edges
Setting a threshold value to the edge probabilities and ignore any edge
below that value
- deciding what the right value of the threshold
Often we are interested in the probability that a certain property holds,
rather than a binary Yes/No answer

Challenges with
Uncertain Graphs
9/ 160
Uncertainty Semantics
Computational Complexity

Semantics: Shortest Path
in Uncertain Graphs
10/ 160
Social Networks
M. Potamias et. al. [VLDB 2010]
T
S
A
B1
B2
Bn
1.0
1.0
1.0
What is the shortest path from S to T?
[Assume independent edge probabilities]

in Uncertain Graphs
11/ 160
T
S
A
B1
B2
Bn
1.0
1.0
1.0
The probability of the
shortest path (S-T) might
be arbitrarily small

in Uncertain Graphs
12/ 160
T
S
A
B1
B2
Bn
1.0
1.0
1.0
The probability that the most
probable path (S-B1-B2 … Bn-T) is
indeed the shortest path might
be arbitrarily small
The most probable path (S-B1-
B2 … Bn-T) might still have an
arbitrarily small probability

in Uncertain Graphs
Social Networks
T
S
A
B1
B2
Bn
1.0
1.0
1.0
13/ 160
 

dd ts
ts
E
p
dp
dtsd
| ,
,
)(1
)(
),(
Expected Shortest-Path Distance:

Semantics: Frequent Subgraphs
in Uncertain Graphs
14/ 160
A
B C DF
A
B C
E
D
A
B C
E
DF
A
B C
E
D
A
B C
E
DF
A
B C
0.1 0.2
0.3 0.51.0
0.2 0.3
0.2 1.0
0.1 0.1
0.2 0.20.5
0.3 0.1
0.1 0.8
0.2 0.2
0.3 0.50.8
0.1 0.1
0.3
0.1
1.0
1.0
0.9
0.2
1.0
G1
G2
G3
G4
G5
G6
Is sub-graph (ABC) frequent?
Support = 6
Expected Support = 0.038
[Zou et. al., CIKM 2009; Papapetrou et. al., EDBT 2011]

Semantics: Frequent Subgraphs
in Uncertain Graphs
15/ 160
Social Networks
A
B C DF
A
B C
E
D
A
B C
E
DF
A
B C
E
D
A
B C
E
DF
A
B C
0.1 0.2
0.3 0.51.0
0.2 0.3
0.2 1.0
0.1 0.1
0.2 0.20.5
0.3 0.1
0.1 0.8
0.2 0.2
0.3 0.50.8
0.1 0.1
0.3
0.1
1.0
1.0
0.9
0.2
1.0
G1
G2
G3
G4
G5
G6
Expected support of edge (AE)
= Expected support of edge (CD)
= 3
How certain can we be that
those edges are frequent?
Frequentness Probability
[Bernecker et. al., KDD 2009]

18
Tutorial Outline
Data as Uncertain Graphs
 Sources of Uncertain Graphs
 Application and Challenges of Uncertain Graphs
 What is Uncertain
 Modeling of Uncertain Graphs
Open Problems
16/ 160
Queries over Uncertain Graphs
 Reliability Queries: Reachability, Shortest Path,
Nearest Neighbor
 Pattern Matching Queries
 Similarity-based Search
 Influence Maximization

19
Tutorial Outline
Open Problems
16/ 160
Nearest Neighbor

20
This tutorial is not about …
Device Network Reliability: Two-terminal reliability, All-terminal reliability, k-terminal
reliability (Reliability Evaluation: A Comparative Study of Different Techniques. Micro.
Rel., 1975)
Generative Models for Graphs: Preferential attachment, Forest fire, Erdős–Rényi
(Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations.
KDD 2005)
Uncertain Graphs Mining: Frequent pattern mining (CIKM 2009, EDBT 2011),
Clustering/ Community detection (TKDE 2011, ICDM 2012), Classification (SDM 2013), Core
decomposition (KDD 2014)
Uncertain Databases: Incomplete uncertain databases (MUD 2010), MayBMS (ICDE
2008), Probabilistic Queries (SIGMOD 2003), Possibilistic databases (IEEE T. Fuzzy Sys. 2005)
17/ 160
Probabilistic Graphical Models: Bayesian network, Markov random field, Belief
propagation
Uncertainty Theory: Dempster–Shafer theory, Aleatory vs. Epistemic uncertainty,
Possibilistic graphs

21
Tutorial Outline
Open Problems
18/ 160
Nearest Neighbor, Centrality

What is Uncertain?
Edge Uncertainty
 Edge existence probability
 Edge strength based on
edge-attributes
Node Uncertainty
 Node existence probability
 Identity uncertainty
Attribute Uncertainty
 Uncertainty about attribute values
 Unknown attribute values
0.8
0.9
0.7
0.2
Music
Fashion
Politics
Lady Gaga
Edge Existence
Edge Strength based on Attributes
Jiawei
Han
Wei
Wang
Wei
Wang
0.3
Identity Uncertainty

Modeling of Uncertain Graphs
Independent Probability
 Independent probability of existence on graph components
 A graph with m uncertain components generates 2m possible worlds
Conditional Probability
 Probability conditioned on existence of other graph components
 E.g., congestion probabilities on roads in an intersection
20/ 160
0.3 0.8
0.14 0.06 0.56 0.24
Uncertain Graph 22 = 4 Possible Worlds/ Certain Graphs
Uncertain Graph is a generative model for deterministic graphs

Independent Probability Model
21/ 160
0.3 0.8
0.14 0.06 0.56 0.24
Uncertain Graph
(Edge Uncertainty)
22 = 4 Possible Worlds/ Certain Graphs
A graph with m uncertain components generates 2m possible worlds
Probability of observing any possible world G = (V, EG) sampled from
uncertain graph G = (V, E, p) is:
 

GG EEeEe
epepG

))(1()()Pr(

25
Tutorial Outline
Open Problems
22/ 160
Nearest Neighbor

Reliability Query over
Uncertain Graphs
23/ 160
Applications:
 Mobile Ad-hoc Networks: find the
probability of delivering a packet
from a source node to a sink node
 Biological Networks: predicting co-
complex memberships and new
interactions requires to compute all
proteins that are reachable from a
source protein with higher
probability
 Social Networks: find the probability
that a tweet by some user will be
reached to another user
Packet Delivery Probability in
Mobile Ad-hoc Networks
T0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
Two-Terminal Reliability: Find the
probability of reaching a destination
node T from a source node S

Formal Definition of Reliability
24/ 160
Uncertain Graph (G)
T
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
A Certain Graph/ Possible World (G)
T
S
W
U
VSample Edges
 

GG EEeEe
epepG

))(1()()Pr(

GG
G GTSITSR )Pr(),(),(

Complexity of Reliability Computation
25/ 160
Two-terminal reliability computation is a #P-complete problem
Counting Problem: Given a graph G = (V,E) together with node and/or
edge weights, find the number of sub-graphs that satisfy property X.

25/ 160
#P: Those counting problems with the property that, given a
candidate sub-graph, testing whether or not it satisfies property X
can be accomplished in polynomial time
The counting version of any problem in NP is in #P

25/ 160
#P: Those counting problems with the property that, given a
candidate sub-graph, testing whether or not it satisfies property X
can be accomplished in polynomial time
#P-Complete: Those problems in #P with the property that if a
polynomial algorithm exists for one of them, then a polynomial
algorithm exists for all members of #P
The counting version of any problem in NP is in #P
#P-Complete problems are at least as hard as NP-Complete problems

26/ 160
Reliability Polynomial:
Proof Sketch
Uncertain Graph (G)
T
p
p
p
p
p
p
p
pS
W
U
V



m
i
iim
i ppfTSR
0
)1(),(
Coefficient fi is the number of subsets of edges of cardinality i, such
that when a subset is deleted, there still remains a path from S to T
By determining fi , we immediately know the number of minimum
cardinality (S, T)-cuts
Counting minimum cardinality (S,T)-cuts is #P-complete
L. G. Valiant [SIAM J. Comp 1979]; M. O. Ball [IEE Tran. Rel. 1986]

27/ 160
Two-terminal reliability on special graph structures
Linear time over tree networks
Linear time over series/ parallel networks
S
U
V
T
G is not series/parallel w.r.t. S and T, but
is series/parallel w.r.t. U and V
#P-complete over planar graphs
#P-complete over directed acyclic
graphs
J. S. Provan et. al. [SIAM J. Comp 1983]

Exact Reliability Computation
State Enumeration
Pathset Enumeration
Cutset Enumeration
A graph with m uncertain edges generates 2m possible worlds  Exponential!
An (S,T)-cutset is a minimal set of edges whose deletion leaves no path
from S to T
C1, C2, …, Ck are cut sets









k
i
iCTSR
1
Pr1),(
An (S,T)-pathset is a minimal set of edges whose existence ensures a
path from S to T
P1, P2, …, Pr are cut sets









r
i
iPTSR
1
Pr),(
28/ 160

Exact Reliability Computation
Inclusion-Exclusion
Principle
 
   r
r
ji
ji
i
i
r
i
i
PPPPP
PPTSR












...Pr)1(...Pr
PrPr),(
21
1

Right-hand-side contains 2r terms
Number of pathsets and cutsets can be exponential in the number of
nodes and edges
Polynomial-time algorithm exists to compute R(S,T) in the number of
(S,T)-cutsets [Provan et. al., Operations Research 1984]
Exploiting special structures [Agrawal et. al., Operations Research , 1984], upper and
lower bounds [Esary et. al., Technometrics , 1966], efficient Monte Carlo methods
[Karp et. al., UC Berkeley Tech. Report , 1983]
29/ 160

Monte Carlo Sampling to
Estimate Reliability
Basic Monte-Carlo/ Hit-and-Miss
Monte-Carlo
Sample K possible graphs, G1, G2, …, GK of uncertain graph G according to
edge probabilities
Compute IS,T(Gi) = 1 if T is reachable from S in Gi, and IS,T(Gi) = 0 otherwise


K
i
iTS GI
K
TSR
1
, )(
1
),(ˆ
Time Complexity
))(( mnK Ο n = # nodes, m = # edges
30/ 160

36
Basic Monte Carlo with
Breadth-First-Search
Only sample the outgoing edges from the currently visited vertex
Do not sample all edges in the beginning
Stop when T is reached, or no new vertex can be reached with the sampled
edges
Uncertain Graph (G)
T
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
31/ 160
Sample + BFS
S
W
U
Start BFS from S

37
Basic Monte Carlo with
Breadth-First-Search
Only sample the outgoing edges from the currently visited vertex
Do not sample all edges in the beginning
Stop when T is reached, or no new vertex can be reached with the sampled
edges
Uncertain Graph (G)
T
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
32/ 160
Sample + BFS
T
S
W
U
V
- Continue BFS from U and W
- Terminate

38
Accuracy Guarantees for
Basic Monte Carlo
33/ 160
Unbiased estimator
Variance due to binomial distribution ~ B(K, R(S,T))
   ),(1),(
1
),(ˆ TSRTSR
K
TSRVar 
G. S. Fishman [IEEE Tran. Rel. 1986]

39
Accuracy Guarantees for
Basic Monte Carlo
34/ 160
Number of trials necessary to achieve an (ɛ, δ) algorithm
Having No of samples ≥ , we ensure
         TSRTSRTSR ,,,ˆPr
 







2
ln
,
3
2
TSR
Follows from Chernoff bound [M. Potamias et. al. VLDB 2010]
One can also apply Chebychev’s inequality [Karp et. al., UC Berkeley Tech. Report ,
1983] or Central Limit Theorem [M. Y. ATA., Applied Math. , 2006] to derive similar
bounds

40
Asking Reliability Query
Differently
35/ 160
Distance-Constraint Reliability
Reliable Set Query
Find the probability that the distance from source node S to
a destination node T is less than or equal to a user-defined
threshold d [Jin et. al., VLDB 2011]
Given a source nodes S, find all other nodes that are
reachable from S with probability greater than or equal to a
user-defined threshold η [Khan et. al., EDBT 2014]

41
Recursive Sampling for distance-
constraint Reliability [Jin et. al., VLDB 2011]
36/ 160
   
   }{,)(1
},{)(,
21,
21,21,
eEERep
EeERepEER
d
TS
d
TS
d
TS


Enumeration tree for recursive computation
of distance-constraint reachability
If inclusion set E1 contains a d-path
from S to T, then
  1, 21, EERd
TS
If exclusion set E2 contains a d-cut
for S to T, then
  0, 21, EERd
TS

42
37/ 160
Dynamic Monte-Carlo, Zhu et. al., DASFAA 2011

43
38/ 160

44
Index for Reliable Set Query [Khan et. al., EDBT 2011]
39/ 160
Can we quickly determine the nodes that are certainly not reachable
from S with probability greater than or equal to ɳ
Uncertain Graph
T
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6S
W
U
V
ɳ = 0.5
Indexing (offline) – RQ Tree
Filtering + Verification (Online)
Reliable Set Query: Given a source nodes S, find all other nodes that
are reachable from S with probability greater than or equal to a user-
defined threshold η

45
RQ-Tree Index [Khan et. al., EDBT 2011]
40/ 160
S, U, W, V, T
U V T
WS
RQ-Tree Index
Uout(S, *)=0.8
Uout(S, *)=0.496
Uout(S, *)=0
Uout(S, *)=0.8
ɳ = 0.5
Uncertain Graph
0.5
0.7
0.6
0.5
0.1
0.2
0.3
0.6
S
ɳ = 0.5
U
W
V
T
V,TS, U, W
S, W

46
Pruning Capacity: RQ-Tree Index
41/ 160
# Nodes # Edges Edge Prob: Mean, SD, Quartiles
DBLP 684 911 4 569 982 0.14 ± 0.11, {0.09, 0.09, 0.18}
Flickr 78 322 20 343 018 0.09 ± 0.06, {0.06, 0.07, 0.09}
BioMine 1 008 201 13 445 048 0.27 ± 0.21, {0.12, 0.22, 0.36}
Dataset Characteristics
Precision of RQ-Tree Filtering Phase

47
Shortest Path Query
Shortest Path Distribution
Uncertain and edge-weighted graph G = (V, E, W, p)
Uncertain Edge-Weighted Graph (G)
10, 0.6
S
B
A
C
D
T
E
15, 0.7
5, 0.8
5, 0.4
20, 0.5
20, 0.8
10, 0.9
15, 0.8
25, 0.4
Shortest Path Distribution
Possible World Graph G1
10
S
B
A
C
D
T
E
15
5
20
10 25
Possible World Graph G2
S
B
A
C
D
T
E
15 10 25 

dTSdG
TS
G
Gdp
),(|
, ]Pr[)(

Distance Metric in Uncertain Graphs
43/ 160
Median Distance






 
D
d
TS
D
M dpTSd
0
, 5.0)(maxarg),(
Majority Distance
)(maxarg),( , dpTSd TS
d
J 
Expected Reliable Distance
 

dd ts
ts
E
p
dp
dtsd
| ,
,
)(1
)(
),(

Distance Metric in Uncertain Graphs
44/ 160
Median Distance
Majority Distance
Expected Reliable Distance






 
D
d
TS
D
M dpTSd
0
, 5.0)(maxarg),(
)(maxarg),( , dpTSd TS
d
J 
 

dd ts
ts
E
p
dp
dtsd
| ,
,
)(1
)(
),(

50
Nearest Neighbor Query
45/ 160
Find the top-k nearest neighbors of a given query node based on
distance metrics defined previously
#P-hard
Pruning Techniques: Find top-k nearest neighbors without computing
distances to all nodes from S

51
Pruning Algorithms for
Nearest Neighbor Query
46/ 160






 
D
d
TS
D
M dpTSd
0
, 5.0)(maxarg),(
Median Distance
Distance-based Pruning









 


Ddif
Ddifxp
Ddifdp
dp
Dx
TS
TS
TSD
0
)(
)(
)( ,
,
,,
Initialize D to a small value. Only consider nodes that are within distance
D from query node S
If k nodes found with median distance less than D, terminate
Otherwise increase D and repeat
Pruning
Criteria

52
Variations of Shortest Path Query
47/ 160
52
Threshold-based Shortest Path Query
Top-k Shortest Path Query
Given a source node S, a destination node T, and a probability
threshold η, find a path set {P1, P2, …, Pr} from S to T, such that
each path Pi has a shortest path probability larger than threshold η
[Cheng et. al., DASFAA 2014]
Given a source node S and a destination node T, find a set of k
paths {P1, P2, …, Pr} from S to T, such that their shortest path
probabilities are the largest among all possible shortest paths from
S to T [Zou et. al., WISE 2011]

53
Pruning Algorithms for Top-K
Shortest Path Query
48/ 160
Top-r shortest paths {P1, P2, P3, …, Pr} from S to T in certain graph
G* by Yen’s algorithm [J. Y. Yen, Management Science 1971]
Probability that Pr is the shortest path from S to T in uncertain
graph G is given by none of the paths {P1, P2, P3, …, Pr-1} exists and
Pr exists.
Upper bound: UB[Pr(Pr = SP(G))] Lower bound: LB[Pr(Pr = SP(G))]
𝞓 = K-th largest lower bound found so far
Terminate if UB[Pr(Pr = SP(G))] < 𝞓
Pruning
Criteria
Zou et. al. [WISE 2011]

54
Pruning Algorithms for Top-K
Shortest Path Query
49/ 160
UB[Pr(Pr = SP(G))] ≤ 1 - LB[Pr(Pr = SP(G))]
      



1
1
Pr)(Pr)(Pr
r
i
rirr PPEPESPP G
First Lower Bound
Second Lower Bound
     
   


















t
i
ir
t
i
irr
SEPE
SEPESPP
1
1
Pr)(Pr
Pr)(Pr)(Pr G
Zou et. al. [WISE 2011]
Si: Edge-set cover for the paths { (Pi – Pr): i ∈ (1, r-1) }
S’i: Pairwise independent set covers

55
Reliability with Edge Colors
50/ 160
Uncertain, edge-colored multi-graph G
Given a source node S and destination node T, find the top-k edge
colors that maximize the reliability from S to T
Barbieri et. al. [ICDM 2012]; Chen er. al. [DASFAA 2014]; Khan et. al. [CIKM 2015]
S
A
B
C
T
0.6
0.2
0.7
0.8
0.4
0.7
0.5
Uncertain, Edge-Colored Multi-Graph:
Select at most K edge-colors

56
51/ 160
Khan et. al. [CIKM 2015]
S
A
B
C
T
0.6
0.2
0.7
0.8
0.4
0.7
0.5
S
A
B
C
T
0.6
0.2
0.7
0.7
0.5
Green and Red
Reliability: R(S,T) = 0
Select at most 2 edge-colors

57
52/ 160
S
A
B
C
T
0.6
0.2
0.7
0.8
0.4
0.7
0.5
Green and Blue
Reliability: R(S,T) = 0.28
S
A
B
C
T
0.6
0.8
0.4
0.7

58
53/ 160
S
A
B
C
T
0.6
0.2
0.7
0.8
0.4
0.7
0.5
Red and Blue
Reliability: R(S,T) = 0.29
S
A
B
C
T
0.2
0.7
0.8
0.4
0.5

59
54/ 160
Top-k enzymes to create pathways
in biological networks
Top-k Advertisement contents for
topic-aware information cascade
Top-k themes to organize a party
among a group of people
Applications
S
A
B
C
T
0.6
0.2
0.7
0.8
0.4
0.7
0.5
Select at most K edge-colors

60
What if Correlated
Probabilities
55/ 160
Potamias et. al. [VLDB 2010]; Cheng et. al. [DASFAA 2014]
S
A
B
D
C
E
T
state(eCT)=1 state(eCT)=0
state(eAC)=1, state(eBC)=1 0.5 0.5
Conditional Probability Table
If DAG, sample each edge of G according to their topological order
If not a DAG, obtaining independent samples is more difficult  Gibbs
sampling
Uncertain Graph (G)

61
Summary: Reliability Queries
56/ 160
Two-terminal reliability computation over uncertain graphs is a #P-
complete problem
Several variations of reliability query – shortest path, nearest
neighbors, reliable set, edge-colored reliability
Application-specific semantics for shortest paths, nearest neighbors,
edge-color and uncertainty
Efficient indexing and sampling techniques, pruning algorithms

62
Tutorial Outline
Open Problems
57/ 160
Nearest Neighbor

Why Uncertain Graphs
Protein-Protein Interaction Networks
False Positive > 45%
In our daily life, uncertainty is ubiquitous!
Protein-Protein Interaction Network
Social Networks
Social Networks
Probabilistic Trust/Influence Model
58/ 160

Why Uncertain Graphs
Uncertain graph has many applications.
In these applications, graph data is usually noisy and
incomplete, which leads to uncertain graphs.
STRING database (http://string-db.org) is a data source that
contains PPIs with uncertain edges provided by biological
experiments.
 Subjective reasons: imprecise physical instrument, network delay,
complex sensing
 Objective reasons: privacy-preserving, information extraction, data
integration
Therefore, it is important to study query
processing on large uncertain graphs.
59/ 160

Our Roadmap …
Efficient Subgraph Search
Efficient Supergraph Search
Efficient Pattern Graph Search
60/ 160
Pattern Matching Queries

66
Probabilistic Subgraph Search
Vertex uncertainty (existence probability)
Edge uncertainty (existence probability given its two endpoints)
Y. Yuan et. al. [VLDB 2011]
Uncertain graph
A (0.6)
A (0.8)
B (0.9)
b
1
2 3a
b
0.9 0.7
0.5
61/ 160

67
Possible worlds: combination of all uncertain edges and vertices
Uncertain graph
A (0.6)
A (0.8)
B (0.9)
b
1
2 3a
b
0.9 0.7
0.5

(1)
1
(2) (3) (4) (5) (6)
2 3
0.008 0.032 0.012 0.072
0.0432 0.2016
1
2
1
3
1
2 3
(7)
2
3
0.054
(8)
0.0048
1
2
(9)
0.0864
1
3
(10)
0.054
2
3
1
2 3
(11)
0.00648
(15)
0.13608
1
2 3
(12)
0.05832
1
2 3
(13)
0.01512
1
2 3
(14)
0.00648
1
2 3
(16)
0.13608
1
2 3
(17)
0.05832
1
2 3
(18)
0.01512
62/ 160

68
Given: an uncertain graph database G={g1, g2,…, gn}, a query graph q and
probability threshold τ
Query: find all gi ∈G, such that the subgraph isomorphic probability is
not smaller than τ.
Subgraph isomorphic probability (SIP): The SIP between q and gi = the
sum of gi’s possible worlds to which q is subgraph isomorphic
Problem Definition
63/ 160

69
Subgraph isomorphic probability (SIP)
Problem Definition
A (0.6)
A (0.8)
B (0.9)
b
1
2 3a
b
0.9 0.7
0.5
a
A B
g q
1
2 3
(14)
0.00648
(7)
2
3
0.054
1
2 3
(15)
0.13608
1
2 3
(17)
0.05832
1
2 3
(18)
0.01512
+ + + + = 0.27
It is #P-complete to calculate SIP
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70
Probabilistic Subgraph Query Processing Framework
Naïve method：sequence scan D, and decide if the SIP between q and gi
is not smaller than threshold τ.
g1 graph isomorphic to g2 : NP-hard?
g1 subgraph isomorphic to g2 : NP-Complete
Calculating SIP: #P-Complete
Naïve method: very costly, infeasible！
65/ 160

71
A Filtering-and-Verification Query Processing Framework
Filtering
Verification
Candidates
Answers
{g1,g2,..,gn} {g’1,g’2,..,g’m}
{g”1,g”2,..,g”k}Query q
66/ 160

72
Filtering: Structural Pruning
Principle: if we remove all the uncertainty from g, and the resulting
graph still does not contain q, then the original uncertain graph cannot
contain q.
Theorem: if qgc，then Pr(qg)=0
A (0.6)
A (0.8)
B (0.9)
b
1
2 3a
b
0.9 0.7
0.5
g
a
A B
q
67/ 160

73
Filtering: Probabilistic Pruning
Let f be a feature of gc i.e., fgc
Rule 1:
if f  q , UpperB(Pr(f  g))<，then g is pruned.
∵ f  q, ∴ Pr(q  g)Pr(f  g)<
Uncertain Graph Feature Query & 
1
2
3 4
6
5
A (0.5)
A (1)
B (0.3)
A (0.6)
A (0.7)
B (0.4)
b b
b
a
a
a
c0.6
0.8
0.9
0.5 1
0.9
0.2
A
A Ba
c a
c
b
A
B A
, 0.6)(
A
68/ 160

74
Rule 2：
if q  f, LowerB(Pr(f  g))，then g is an answer.
∵ q  f, ∴ Pr(q  g)Pr(f  g)
Uncertain Graph Feature
Query & 
1
2
3 4
6
5
A (0.5)
A (1)
B (0.3)
A (0.6)
A (0.7)
B (0.4)
b b
b
a
a
a
c0.6
0.8
0.9
0.5 1
0.9
0.2
A
A Ba
c a
BA , 0.2 )(
Two main issues for probabilistic pruning
 How to derive lower and upper bounds of SIP?
 How to select features with great pruning power?
69/ 160

75
Technique 1: calculation of lower and upper bounds
Lemma: Let Bf1,…,Bf|Ef|be all embeddings of f in gc, then
Pr(fg)=Pr(Bf1…Bf|Ef|).
UpperB(Pr(fg)):
     EfEf
BfBfBfBfgf   11 1 PrPrPr
   

Ef
i
iEf
BfBfBf
1
1 PrPr 
  )())Pr(1(1)Pr(1Pr
||
1
||
1
fUpperBBfBfgf
Ef
i
i
Ef
i
i   
70/ 160

76
LowerB(Pr(fg)):
Tightest LowerB(f)
          
 
IN
j
ij
IN
ji
Ef
i fLowerBBfBfBfgf
1
11 Pr11PrPrPr
1
2
3 4
6
5
A (0.5)
B (0.3)
A (0.6)
B (0.4)
b b
b
a
a
a
c0.6
0.8
0.9
0.5
1
0.9
0.2
(002) (f2)
A
a
b
A
B
1
2 3
4
5 6
(EM1) (EM3)
1
2 3
(EM2)
EM1
EM2 EM3
Embeddings of f2 in 002 Graph bG of embeddings
Converting into computing the
maximum clique of graph bG
71/ 160

77
Exact value V.S. Upper and lower bound
0
0.2
0.4
0.6
0.8
1
50 100 150 200 250
Database size
Probability
UpperBound Exact LowerBound
0.1
1
10
100
1000
50 100 150 200 250
Database size
Caculationtime(second)
UpperBound Exact LowerBound
Value Computing Time
72/ 160

78
Technique2: Optimal Feature Selection
If we index all features, we will have the most pruning power index. But
it is also very costly to query such index. Thus we would like a small
number of features but with the greatest pruning power.
Cost model:
Max gain = sequence scan cost– query index cost
Integer programmingmaximum set coverage: NP-complete.
Use the greedy algorithm to approximate it.
73/ 160

79
Technique2: Optimal Feature Selection
Integer programming：greedy algorithm
001 002
f1 (0.19,0.19) (0.27,0.49)
f2 (0.27,0.27) (0.4,0.49)
f3 0 (0.01,0.11)
(0.19,0.19) (0.27,0.49)
(0.27,0.27) (0.4,0.49)
0 0
0 (0.27,0.49)
(0.27,0.27) (0.4,0.49)
0 0
0 0
(0.27,0.27) (0.4,0.49)
0 (0.01,0.11)
f1
f2
f3
001 002 001 002 001 002
a
a
b
A
BA
, 0.5q1 )( a
BA , 0.2q2 )( a
c
b
A
B A
, 0.6q3 )(
A
Feature Matrix
Probabilistic Index
Approximate optimal
index within 1-1/e
74/ 160

80
Probabilistic Index
Construct a string for each feature
Construct a prefix tree for all feature strings
Construct an invert list for all leaf nodes
Root
fa
ID-list: {<g1, 0.2, 0.6>, <g2, 0.4, 0.7>, ….}
fb
ID-list: {….}
fc
ID-list: {….}
fd
ID-list: {<g2, 0.3, 0.8>, <g4, 0.4, 0.6>, ….}
75/ 160

81
Verification: Iterative bound pruning
Lemma： Pr(qg)=Pr(Bq1…Bq|Eq|)
Unfolding: 
Let
Based on Inclusion-Exclusion Principle
     
 






iJEJ
qj
J
j
E
i
i
q
q
Bgq
,,,1
1
1
1
Pr1Pr

   qj
J
ji BS 1Pr
 
 
 












evenisiifS
oddisiifS
gq i
w w
i
i
w w
i
1
1
Pr
1
1
1
1
Iterative Bound Pruning
76/ 160

Our Roadmap …
77/ 160

83
Probabilistic Supergraph Search
Y. Tong et. al. [CIKM 2014]
Back to our example of the uncertain graph database
Figure 1: An Uncertain Graph Database
The existing probability of the
specific vertex A.
The conditional probability of the edge
B-C appears when the nodes B and C
already exist.
78/ 160

84
We derive 18
possible world
graphs
Pr(PW6)=0.9*0.8*0.8*(1-0.9)=0.0576
The condition probabilities of A-C and B-C are not
considered since the node C does not exist.

85
SIP(q, ug2)=0.419904+0.046656= 0.46656

86
Given an uncertain graph ug and a query graph q, the SCP between q
and ug is equal to the sum of the probabilities of ug’s possible worlds
where ug is subgraph of q
Supergraph Containment Probability (SCP)
Given an uncertain graph database G={g1,g2,…,gn}, a query graph q and
probability threshold τ.
Query: find all gi ∈G, such that such that the supergraph containment
probability is not smaller than τ.
Probabilistic Supergraph Containment Search
81/ 160

87
Supergraph Containment Probability (SCP)
SCP(q, ug2)=0.002+0.018+…+0.001296+0.005184
=0.352

88
Whether the existing approach of probabilistic subgraph search can be
extended to solve the issue of probabilistic supergraph?
Dq
UGDq
UGDq
Dq
Subgraph Search Supergraph Search
The answer set of q in the
corresponding deterministic
graph database
The final answer set
of q in the uncertain
graph database
The answer set of q in
the corresponding
deterministic graph
database
The final answer set
of q in the uncertain
graph database
The framework of probabilistic subgraph search is not suitable for the
problem of probabilistic supergraph search!

89
However, we prove that it is #P-hard to calculate the supergraph
containment probability (SCP) of a given uncertain graph and a query
graph.
How to compute this hard problem?
Complexity Analysis
84/ 160

90
Offline Index Construction (Using Existing Work)
 Mining probabilistic frequent subgraphs, which are considered as
feature set to build index
Filtering Phase
 Probabilistic-supergraph-filtering-logic-based pruning
Verification Phase
 Sampling-based algorithm (Unequal-Probability Sampling)
85/ 160

91
Principle: If a feature graph and ,
then
Theorem: If a feature graph and , where τ
is the probabilistic threshold, then ug can be pruned safely!
f q Pr( )f ug p 
Pr( ) 1ug q p  
f q Pr( ) 1f ug   
86/ 160

92
The Example of Probabilistic Pruning
0.7 
SIP(f, ug2)=0.4199+0.0466=0.46656>1-0.7=0.3, SCP(q, ug2)
must be lower than the given threshold. Thus, ug2 can be
pruned safely.

93
Simple-Random-Sampling-based Approach
Analysis of Simple-Random-Sampling-based Approach
 This method is unbiased.
 However, its variance is , which is larger.
Verification Solutions
88/ 160

94
Simple-Random-Sampling-based Approach
Analysis of Simple-Random-Sampling-based Approach
 This method is unbiased.
 However, its variance is , which is larger.
Verification Solutions: Simple-Random-Sampling-based Approach
89/ 160

Verification Solutions: Unequal-Probability-Sampling-based Approach
Simple-Random-Sampling Unequal-Probability Sampling
Early Pruning
The stopping condition 1 means that all
subsequent sampled possible world graphs
must be contained by the given query graph
The stopping condition 2 means that all
subsequent sampled possible world graphs must
NOT be contained by the given query graph

Our Roadmap …
91/ 160

97
Given a graph G and a query q with distance constraint γ
 Vertex labeled G and q
An answer m is a set of vertices in G：
 A vertex in m has the same label as a vertex in G
 Any pair of vertices has a shortest path distance ≤ γ
Y. Yuan et. al. [CIKM 2014]
Deterministic Graph Pattern Matching
Probabilistic Pattern Graph Matching
92/ 160

98
Distance constraint γ=3
 Correct answer: {2, 5, 7}, {5, 6, 7}
 Incorrect answer: {1, 5, 7}: distance between 1 and 7=4> γ
Deterministic Graph Pattern Matching
93/ 160

99
Distance constraint γ=3
 Vertex is deterministic
 Edge uncertainty (existence probability)
Probabilistic Graph Pattern Matching
94/ 160

100
Possible worlds: combination of all uncertain edges
Probabilistic Graph Pattern Matching
......
Uncertain Graph
29 =512 possible worldsY. Yuan et. al. [CIKM 2014]
95/ 160

101
Given: an uncertain graph G, a query graph q and a probability
threshold 
Query: find all matches {m} in G, such that the pattern matching
probability is not smaller than .
Pattern matching probability (PMP): The PMP of m in G = the sum of
G’s possible worlds in which m is a valid match.
For example, m={2, 5, 7} : PMP of m in G= 0.01248+0.009126+...=0.65.
Problem Definitions
It is #P-complete to calculate PMP
96/ 160

102Y. Yuan et. al. [CIKM 2014]
Probabilistic Pattern Graph Matching Framework
Naïve method：in G enumerate all vertex sets {m} with size of V(q), and
decide if the PMP of m in G is not smaller than threshold .
Number of {m}= Comb(|G |, |V(q)|)
Calculating PMP: #P-Complete
97/ 160

103Y. Yuan et. al. [CIKM 2014]
Filtering
Verification
Candidates
Answers
G: {m1,m2,..,ma} {m’1,m’2,..,m’b}
{m”1,m”2,..,m”c}Query q
98/ 160

104
We remove all the uncertainty from G, and obtain the resulting vertex
sets {m} after certain pattern matching on G, then the vertex sets {m}
is input for the uncertain filtering.
{2, 5, 7}, {5, 6, 7}, {1, 2, 4}, …
99/ 160

105
Edge cut: a set of edges whose removing results in a partition of G
Probabilistic Index
Edge cut: {e1, e2,…,ef}
Connected probability:
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106
Structure: PI is a tree structure. Each node of PI is a vertex of G, and
each edge of PI indexes a edge cut. In PI, suppose a path (s, t) has an
edge, then the indexed edge cut is a cut of (s, t) in G.
Probabilistic Index
G
Index
101/ 160

107
Lemma: Let Bc1,…,Bc|Mc| be the cuts of m in Gc, and Bc1,…,Bc|IN| be
the disjoint cuts, then
Many groups of disjoint cuts Many upper bounds  Best upper
bound  Maximum packing set problem.
102/ 160

108
One-by-one algorithm: scan the candidate match set {m1, m2,…,mk},
and for mi, if UpperB(mi) ≤ γ, mi can be pruned.
Collective algorithm:
103/ 160

109
Tutorial Outline
Open Problems
Nearest Neighbor
104/ 160

110
Uncertain graph：
 Vertices are deterministic
 Edge uncertainty: neighbor edges are corrected
Probabilistic Subgraph Similarity Search
e1
e2 e3
e4
e5
a
a
b
b
c
e1 e2 e3 Prob
1 1 1 0.3
0 1 1 0.3
-- -- -- --
e3 e4 e5 Prob
1 1 0 0.25
1 1 1 0.15
JPT2
JPT1
-- -- -- --
Road Network
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111
Possible worlds: combination of all uncertain edges
e1
e2 e3
e4a
a
b
b
0.075
(1)
e1
e2 e3
e4
e5
a
a
b
b
c
0.045
(2)
e2 e3
e4a
b
b
0.075
(3)
e2 e3
e4
e5
a
b
b
c
0.045
(4)
e1
e2 e3
e4
e5
a
a
b
b
c
e1 e2 e3 Prob
1 1 1 0.3
0 1 1 0.3
-- -- -- --
e3 e4 e5 Prob
1 1 0 0.25
1 1 1 0.15
JPT2
JPT1
-- -- -- --
106/ 160

112
Given: an uncertain graph database G={g1,g2,…,gn}, a query graph q and
probability threshold ε
Query: find all gi ∈G, such that the subgraph similarity probability is
not smaller than ε.
Subgraph similarity probability (SSP):
 The SSP between q and gi = the sum of gi’s possible worlds g’ to
which q is subgraph similar
 q is subgraph similar to g’: the distance between g’ and q is not
larger than a distance threshold 
 Subgraph distance between q and g’= |q|-|MCS(q,g)| where
MCS(q,g) is the maximum common subgraph of q and g’.
Problem Definitions
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113
Subgraph similar probability (SSP)
Problem Definitions
g q
+ + + = 0.45
It is #P-complete to calculate SSP
a
b
c
e1
e2 e3
e4
e5
a
a
b
b
c
e1 e2 e3 Prob
1 1 1 0.3
0 1 1 0.3
-- -- -- --
e3 e4 e5 Prob
1 1 0 0.25
1 1 1 0.15
-- -- -- --
e1
e2 e3
e4a
a
b
b
0.075
e1
e2 e3
e4
e5
a
a
b
b
c
0.045
e2 e3
e4a
b
b
0.075
……

114Y. Yuan et. al. [VLDB 2012]
Probabilistic Subgraph Similarity Query Processing Framework
Naïve method：sequence scan D, and decide if the SSP between q and
gi is not smaller than threshold ε.
g1 subgraph isomorphic to g2 : NP-Complete
the distance between g1 and g2 : NP-Complete
Calculating SSP: #P-Complete
109/ 160

115Y. Yuan et. al. [VLDB 2012]
Structure
pruning
Verification
Candidates
Answers
{g1,g2,..,gn} {g’’1,g’’2,..,g’’m}
{g”’1,g”’2,..,g’”k}
Query q
Prob. pruning
(two rules)
{g’1,g’2,..,g’l}
110/ 160

116
Principle: if we remove all the uncertainty from g, and the resulting
graph is still not subgraph similar to q, then the original uncertain
graph cannot approximately contain q.
Theorem: if qsimgc，then Pr(qsimg)=0
g q
a
b
c
e1
e2 e3
e4
e5
a
a
b
b
c
e1 e2 e3 Prob
1 1 1 0.3
0 1 1 0.3
-- -- -- --
e3 e4 e5 Prob
1 1 0 0.25
1 1 1 0.15
-- -- -- --
111/ 160

117
Probabilistic index: Each column of the matrix corresponds to an
uncertain graph, and each row corresponds to an indexed feature. The
entry gives the upper and lower bounds of the subgraph isomorphism
probability (SIP) of feature f to g.
002
(0.42, 0.5)
(0.26, 0.58)
(0.08, 0.15)
001
(0.55, 0.64)
(0.3, 0.48)
0
f1
f2
f3
graph
feature
a bb
a c
b
f1 f2 f3
PMI
features
e1
e2
e3
b
d
e1
e2 e3
e4
e5
a
a
b
b
c
001 002
112/ 160

118
let U={rq1,…,rqa} be a graph set after q relaxing  edges. For each rqi, in
the index, we find a graph feature fi
1 such that fi
1rqi.
Rule 1：If Usim=UpperB(Pr(q sim g))=UpperB(fi
1) +…+ UpperB(fa
1) < ε，
then g is pruned.
b
a a c
b
crq1 rq2 rq3
f1
a rq1 UpperB(f1)=0.4
f2
c
rq2， UpperB(f2)=0.1rq3
a
b
e1
e2 e3
e4
e5
a
a
b
b
c
c
g q
Usim =0.4+0.1=0.5
113/ 160

let U={rq1,…,rqa} be a graph set after q relaxing  edges. For each rqi,
we find two graph features (fi
1, fi
2) such that fi
1 rqi and rqi  fi
2
Rule 2：If Lsim=LowerB(Pr(q sim g))=Σ1
aLowerB(fi
2)–Σ1≤i,j≤a UpperB(fi
2)
UpperB(fj
2) >ε，then g is an answer.
Lsim=0.28+0.09-0.36*0.15=0.31
b
a a c
b
crq1 rq2 rq3
f1
a
S1:{rq1} LowerB(f1)=0.28 , UpperB(f1)=0.36a
b
f2
a S2:{rq1, rq2, rq3}
c
b
LowerB(f1)=0.09 , UpperB(f1)=0.15
114/ 160

120
If there are 10 features and 10 graphs after relaxation, we get 1010 Usim
Solution: converting it into the set cover problem
Tightest Upper Bound of SSP
Usim =(0.4+0.1=0.5) or (0.1+0.5=0.6) or (0.4+0.5=0.9)
b
a a c
b
crq1 rq2 rq3
f1
a S1:{rq1,rq2} UpperB(f1)=0.4
f2
c S2:{rq2,rq3} UpperB(f2)=0.1
f3
b S3:{rq1,rq3} UpperB(f3)=0.5
115/ 160

121
Solution: Converting it into the quadratic programming
Tightest Lower Bound of SSP
b
a a c
b
crq1 rq2 rq3
f1
a
S1:{rq1} LowerB(f1)=0.28 , UpperB(f1)=0.36a
b
f2
a S2:{rq1, rq2, rq3}
c
b
LowerB(f1)=0.09 , UpperB(f1)=0.15
116/ 160

122
Tutorial Outline
Open Problems
117/ 160
Nearest Neighbor

Information Diffusion in
Social Networks
2008 U.S. Presidential Election
Emergencies such as Hurricanes Ike
and Gustav in 2008
Demonstration in Egypt, 2011
Death of Michael Jackson in 2009
118/ 160
0.2
0.3
0.7

Influence Maximization
in Social Networks
Find a small subset of influential individuals in a social network, such
that they can influence the largest number of people in the network
0.7
0.8
0.6
0.8
0.9
0.4
Viral Marketing
119/ 160

in Social Networks
0.7
0.8
0.6
0.8
0.9
0.4
Viral Marketing
120/ 160

in Social Networks
0.7
0.8
0.6
0.8
0.9
0.4
Viral Marketing
121/ 160

in Social Networks
0.7
0.8
0.6
0.8
0.9
0.4
Viral Marketing
122/ 160

Related Tutorials
Information and Influence Spread in Social Networks –
Motivation, Applications, Challenges, Data, and Tools for Information
diffusion and Influence Maximization [Castillo et. al., KDD 2012]
Information Diffusion In Social Networks: Observing and
Affecting What The Society Cares About – Effect on Network
Structure on Information Diffusion [Agrawal et. al., CIKM 2011]
Information Diffusion In Social Networks: Observing and
Influencing Societal Interests – Various Information Diffusion
Models [Agrawal et. al., VLDB 2011]
123/ 160

Our Roadmap …
Influence Maximization Problem
Targeted Influence Maximization
Maximizing Product Adoption
Topic-Aware Influence Maximization
Preventing the Spread of an Existing Negative Campaign
Competitive Influence Maximization
Influence Maximization by Social Network Host
Complementary Influence Maximization
and its Variations
124/ 160

The first influence maximization problem: Markov random
fields formulation [Domingos et. al., KDD 2001]
[Kempe et. al., KDD 2003]
Social network G = (V, E, p)
Seed set 𝑺: initial set of nodes influenced directly by the campaigner
Influence cascade: Nodes are influenced starting from the seed
nodes, in discrete steps and following certain probabilistic influence
cascading model
Influence spread: Number of influenced nodes when the cascading
process starting from the seed set 𝑆 ends
The Problem: Given a user-defined budget K, find the top-K seed
nodes that maximize the expected influence spread
Influence Maximization with
Discrete Diffusion Model
125/ 160

Influence Cascading Models
Independent cascade (IC) model, Linear threshold (LT) model
IC Model
0.6
0.8
1.0
0.4
0.2 0.7
0.2
0.7
0.6
0.3
0.5
126/ 160

IC Model
127/ 160
0.6
0.8
1.0
0.4
0.2 0.7
0.2
0.7
0.6
0.3
0.5

IC Model
128/ 160
0.6
0.8
1.0
0.4
0.2 0.7
0.2
0.7
0.6
0.3
0.5

IC Model
129/ 160
0.6
0.8
1.0
0.4
0.2 0.7
0.2
0.7
0.6
0.3
0.5

LT Model
130/ 160
0.1
0.4
0.1
0.2
0.3 0.2
0.3
0.4
0.1
0.7
0.5
0.2
0.1
0.7
0.3 0.9
0.5

LT Model
131/ 160
0.1
0.4
0.1
0.2
0.3 0.2
0.3
0.4
0.1
0.7
0.5
0.2
0.1
0.7
0.3 0.9
0.5

LT Model
132/ 160
0.1
0.4
0.1
0.2
0.3 0.2
0.3
0.4
0.1
0.7
0.5
0.2
0.1
0.7
0.3 0.9
0.5

LT Model
133/ 160
0.1
0.4
0.1
0.2
0.3 0.2
0.3
0.4
0.1
0.7
0.5
0.2
0.1
0.7
0.3 0.9
0.5

LT Model
134/ 160
0.1
0.4
0.1
0.2
0.3 0.2
0.3
0.4
0.1
0.7
0.5
0.2
0.1
0.7
0.3 0.9
0.5

Influence maximization under both IC and LT models is NP-hard
Expected influence spread is sub-modular and increases
monotonically with inclusion of seed nodes
135/ 160
Influence Maximization: Complexity and
Approximation Algorithm
Iterative hill-climbing algorithm produces solution with
approximation guarantee:
e
1
1
Iterative hill-climbing algorithm:
    SvSv
SVv
 

}{maxarg*

Time Complexity:
 )( enKnrO 
Kempe et. al. [KDD 2003]

136/ 160
More on Influence Maximization
Exact Methods (CELF, CELF++, TIM, …)
Scalable Influence Maximization
Heuristic Methods (MIA, Community-based approach,
Sparsification, Degree Discount IC, …)
General Threshold Model
Other Information Diffusion Models
Susceptible-Infected-Removed Model
Continuous-Time Diffusion
………
[Castillo et. al., KDD 2012]
[Agrawal et. al., VLDB 2011]

Targeted Influence Maximization
A campaigner often promotes
her product with a group of
target customers in mind
Target marketing by maximizing
the influence over a region of
the social network
k-effectors — identify k seed nodes
such that a given activation pattern
can be established
137/ 160
[Aggarwal et. al., SDM 2011, Li et. al., SocialCom 2011]
[Lappas. al., KDD 2010]

Maximizing Product Adoption
Influence ≠ Adoption
Conformity-Aware Influence
Maximization
[Li et. al., VLDB J. 2015]
U
V T
If both U and V adopted, the probability
that T will also adopt is:
+
-
Signed Network:
Each User has a
Influence index and
a Conformity Index
         TCVITCUI  111
LT-C Model
[Bhagat et. al., WSDM 2012]

Topic-Aware Influence Maximization
Topic-aware Influence Maximization
[Chen et. al., VLDB 2015]
139/ 160
Topic-aware Social Influence
Propagation Models [Barbieri
et. al., ICDM 2012]
Online Topic-aware Influence
Maximization Queries [Aslay et.
al., EDBT 2014]
Online Topic-Aware Influence
Maximization [Chen et. al.,
VLDB 2015]

Competitive and Complementary
140/ 160
Competitive Influence
Maximization
[Bharathi et. al., WINE 2007]
Complementary Influence
Maximization
Preventing the spread of an existing negative campaign
Non-cooperative campaigns who select seeds alternatively
Competing campaigners promote their products at the same
time (e.g., Nintendo’s Wii vs. Sony’s Playstation vs. Microsoft’s X-
Box)
[Borodin et. al., WINE 2007] [Budak et. al., WWW 2011]
[Fazeli et. al., CDC 2012] [Tzoumas et. al., WINE 2012]
[Li et. al., SIGMOD 2015]
iPhone 6 and Apple Watch are complementary products
[Lu et. al., VLDB 2016]

Influence Maximization as a Service:
Social Network Host’s Perspective
141/ 160
Social Network graph is hidden by the host of the social network
(e.g., Facebook, Twitter, LinkedIn)
A campaigner (e.g., AT&T, Sony, Microsoft, Samsung) is unable to
identify the top-k seed sets for maximizing her campaign
Challenges for Campaigners
Social network host sells influence maximization service to its client
campaigners
Challenges for Campaigners
How does the host select the seed nodes for each of its client
campaigners so that the spread of each campaign remains balanced?
Lu et. al. [KDD 2013]

Open Problems
Finding one good possible world instead of sampling
Trade-off between accuracy vs. efficiency
System design issues for uncertain graphs processing
Availability of benchmark datasets, ground-truths, and query
results
Semantics of classical graph queries over uncertain graphs,
e.g., centrality, partitioning, summarization, visualization
142/ 160

Open Problem: One Good
Possible World
143/ 160
Find one deterministic representative instance that maintains
the underlying graph properties
Parchas et. al. [SIGMOD 2013]
S
Representative instance for more complex graph properties –
Reachability, Subgraph containment ?
W
U V
0.51
0.52 0.50
S W
U V
+ 0.97
+ 0.48
- 0.01
- 0.50
Uncertain Graph One Possible Graph
(Discrepancy in Degree Distribution)

Open Problem: Accuracy vs.
Efficiency
144/ 160
Parameters controlling accuracy vs. efficiency, false positive vs.
false negative rates
Reliable Set Computation
Khan et. al. [EDBT 2014]
Most probable path provides a
lower bound of reliability
No false positive; but can have
false negatives
S W
U T
0.7
0.6 0.7
Actual Reliable Set
of S with threshold
0.5 = {W,U,T}
Reliable Set via
Most Probable
Path = {W,U}
0.8

Open Problem: Semantics of Classical
145/ 160
Centrality over uncertain graphs – influential nodes are one
type of central nodes
Partition an uncertain graph
Uncertain graph summarization
Uncertain graph visualization
[Pfeiffer et. al., Purdue Tech. Report 2011]
[Hassanlou et. al., WAIM 2011]
[Cesario et. al., SPIE 2011]

Open Problem: System Issues
146/ 160
Are uncertain databases (DeepDive, BayesStore, PrDB) good for
processing uncertain graphs?
Should graph databases (Neo4J, OrientDB) support uncertainty?

Open Problem: Benchmark
Datasets, Ground-Truths
147/ 160
Benchmark datasets
Open-source software
Ground-truths – how to measure the effectiveness of influence
maximization algorithms in real-world? [Castillo et. al., KDD 2012]

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