On Social-Temporal Group Query with Acquaintance Constraint

ON SOCIAL-TEMPORAL GROUP

QUERY WITH ACQUAINTANCE

CONSTRAINT D.N Yang

Y.L Chen

W.C Lee

M.S Chen

Sep 2011

Presented by

Roi Fridburg

1

AGENDA

Activity Planning

Social Graphs

Proposed Algorithms

SGSelect

SGTSelect

Experimental Results

Efficiency and Reliability

Conclusions

2

ACTIVITY PLANNING

How do we plan an activity?

Who do we invite?

3

Familiarity with the initiator

Social relations with the

members

Available time

ACTIVITY PLANNING – CONT.

PROPOSAL: • SOCIAL-TEMPORAL GROUP QUERY (STGQ)

TO FIND ACTIVITY TIME AND ATTENDEES WITH THE MINIMUM TOTAL SOCIAL DISTANCE TO THE INITIATOR INCORPORATING AN ACQUAINTANCE CONSTRAINT

4

Definitions SGQ STGQ Results Conclusions

ILLUSTRATING EXAMPLE

SOCIAL GRAPH OF CASEY AFFLECK

q = Casey Affleck

5



SOCIAL GRAPH OF MATT DAMON

q = Matt Damon

k=0 each attendee knows everybody else F: {M.Damon, K.Costner, R.De Niro}

F: {M.Damon, R.De Niro, A.Jolie}

F: {M.Damon, N.Cage, J.Roberts}

F: {M.Damon, R.De Niro, J.Roberts}

Which one should we choose?

Answer: The one that has the minimum total distance

between q and every vertex v. → min dv,qv∈F

6

d𝑣𝑖,𝑣𝑗 (𝑖≠𝑗) is not counted

ILLUSTRATING EXAMPLE – CONT.

SOCIAL GRAPH OF MATT DAMON

q = Matt Damon

k=1 each attendee has at most 1

unfamiliar attendee

For example:

F: {M.Damon, K.Costner, R.De Niro, A.Jolie}

There is no solution for k=0 in this case! 7

BASIC TERMS AND PARAMETERS

q – activity initiator

G – social graph of q

p – activity size number of expected attendees

s – social radius Scope of candidate attendees

All acquainted attendees are located no more than s edges away from the initiator on his social graph

k – acquaintance constraint Social relationship between attendees

Each attendee can have at most k other unacquainted attendees

m – activity length of time (slots)

d – social distance Distance between vertices in the group

How “close” the vertices are to each other 8

ACTIVITY EXAMPLES

A person has a given number of tickets for a movie

and would like to invite some friends

Set of mutually close friends and the time that all of them

are available

small k all attendees know each other very well

s=1 all attendees are acquainted with him

9

ACTIVITY EXAMPLES – CONT.

A person would like to arrange a party and invite

some friends

large s friends of friends can be also invited

large k more diverse event

10

WHERE IS ALL THE DATA TAKEN FROM?

Social network websites – social relations and

contact information

Facebook

LinkedIn

Web collaboration tools – shared available time

Google calendar

Enterprise tools

MS Outlook

11


WITH TIME CONSTRAINT

q = Casey Affleck

12


WITH TIME CONSTRAINT

{v7}

{v7,v2}

{v7,v2,v3

}

{v7,v2,v3,v4}

{v7,v2,v3,v6}

{v7,v2,v3,v8}

{v7,v2,v4

}

{v7,v2,v4,v6}

{v7,v2,v4,v8}

{v7,v2,v6

} {v7,v2,v6,v8}

{v7,v3}

{v7,v3,v4

}

{v7,v3,v4,v6}

{v7,v3,v4,v8}

{v7,v3,v6

} {v7,v3,v6,v8}

{v7,v4} {v7,v4,v6

} {v7,v3,v6,v8}

C53= 10

d = 64

d = 65

t1

t2

t3

t4

t5

t6

v1

o o o o

v2

o o o o

v3

o o o o o

v4

o o o o o o

v5

o o o o

v6

o o o o

v7

o o o o o

v8

o o o o o 13

COMPLEXITY AND CANDIDATE GROUPS

Simple approach: consider every possible p

attendees and find the total social distance

Needs to evaluate Cf − 1p − 1

candidate groups

f is the number of candidate attendees

If an initiator would like to invite 10 attendees out of

his 100 friends, the number of candidate groups is

in the order of 𝟏𝟎𝟏𝟑

Scenario considers only the friends of the user

This problem is NP-hard

14

SOCIAL GROUP QUERY (SGQ)

PROBLEM DEFINITION

Given an activity initiator q and his social graph G=(V,E)

v is a candidate attendee

eu,v is the distance between u and v representing their

social closeness

Need to find SGQ(p,s,k) which satisfies all the conditions

15


SOCIAL GROUP QUERY (SGQ)

ALGORITHM DESIGN

Proposed algorithm: SGSelect

Step 1: derive feasible graph GF=(VF, EF) from G which

satisfies:

Social radius constraint

dv,q is the minimal social distance between v and q

q = Ryan Giggs

dv3,q =? dv3,q = 8

dv6,q =? dv3,q = 16

16

RADIUS GRAPH EXTRACTION

How to create the feasible graph?

Define di v, q

the i-edge minimum distance between the

vertex v and the vertex q

1 ≤ i ≤ s

17

RADIUS GRAPH EXTRACTION – CONT.

Target

G V, E → GF VF, EF

dv,q satisfying s constraint

1. 𝑑𝑞,𝑞0 = 0, 𝑑𝑢,𝑞

0 = ∞ 𝑓𝑜𝑟 𝑢 ≠ 𝑞;

2. 𝑓𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑠 𝑑𝑜

3. 𝑑𝑞,𝑞𝑖 = 0;

4. 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑒𝑟𝑡𝑒𝑥 𝑢 ≠ 𝑞 𝑖𝑛 𝑉 𝑑𝑜

5. 𝑑𝑢,𝑞𝑖 = 𝑑𝑢,𝑞

𝑖−1;

6. 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑒𝑟𝑡𝑒𝑥 𝑣 𝑖𝑛 𝑁𝑢 𝑑𝑜

7. 𝑖𝑓 𝑑𝑣,𝑞𝑖−1 + 𝑒𝑢,𝑣 < 𝑑𝑢,𝑞

𝑖 𝑡ℎ𝑒𝑛

8. 𝑑𝑢,𝑞𝑖 = 𝑑𝑣,𝑞

𝑖−1 + 𝑒𝑢,𝑣;

9. 𝐸𝑥𝑡𝑟𝑎𝑐𝑡 𝑎𝑙𝑙 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑤 𝑖𝑛 𝑉 𝑤𝑖𝑡ℎ 𝑑𝑤,𝑞

𝑠 < ∞ 𝑎𝑛𝑑 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑉𝑓

q = Ryan Giggs

𝑑𝑞,𝑞0 = 0

𝑑𝑢2,𝑞0 = ∞

𝑑𝑢3,𝑞0 = ∞

𝑑𝑢4,𝑞0 = ∞

𝑑𝑢5,𝑞0 = ∞

𝑑𝑢6,𝑞0 = ∞

𝑑𝑢7,𝑞0 = ∞

𝑑𝑞,𝑞1 = 0

𝑑𝑢2,𝑞1 = 3

𝑑𝑢3,𝑞1 = 10

𝑑𝑢4,𝑞1 = 4

𝑑𝑢5,𝑞1 = ∞

𝑑𝑢6,𝑞1 = ∞

𝑑𝑢7,𝑞0 = ∞

𝑑𝑞,𝑞2 = 0

𝑑𝑢2,𝑞2 = 3

𝑑𝑢3,𝑞2 = 8

𝑑𝑢4,𝑞2 = 4

𝑑𝑢5,𝑞2 = 10

𝑑𝑢6,𝑞2 = 16

𝑑𝑢7,𝑞0 = ∞

18

ACCESS ORDERING

The selection of a vertex at each iteration is critical

Avoid selecting a vertex v that

Increases the total social distance

Leads to violation of the acquaintance constraint

It is preferable to first include

Well-connected vertex

Early pruning of unqualified solutions is a key factor to

the overall performance

Terminology

GF VF, EF : feasible social graph of q

VS : intermediate solution obtained so far

V𝐴 : remaining vertices in V𝐹 (V𝐴 = VF − VS)

19

ACCESS ORDERING – CONT.

Step1: construct the feasible graph GF

Step2: explore GF to find the optimal solution

Initially VS includes only q

The remaining vertex set VA is VF − {q}

At each iteration moving a vertex from VA to VS

VS represents a feasible solution when VS =p

20 𝐕𝐒 𝐕𝐀 q

v1 v3

v2

v4

vi

ACCESS ORDERING – CONT.

Two conditions should be satisfied to exploit the

acquaintance constraint:

Interior unfamiliarity

The maximum number of unacquainted vertices that a vertex v in VS has

Exterior expansibility

The maximum number of vertices that VS can be extended

from

21 𝐕𝐒 𝐕𝐀 q

v6 v3

v2

v4

v8

INTERIOR UNFAMILIARITY

Interior unfamiliarity (U) – definition:

U(VS) = maxv∈VS VS − v − Nv

Nv is a set of neighboring vertices of v in GF

VS − v − Nv is a set of non-neighboring vertices of v in VS

In other words:

The maximum number of unacquainted vertices that a vertex v in VS has

Small value of U indicates that every vertex v ∈ VS has plenty of neighboring vertices in VS

It is preferable first to include a well-connected

vertex (low U) since it will make selections of other

vertices in the later iterations easier 22

INTERIOR UNFAMILIARITY – CONT.

Interior unfamiliarity (U) – condition:

U(VS ∪ {v}) ≤ kVS∪{v}

p

θ

θ > 0

VS∪{v}

p is the portion attendees that have been considered

With θ = 0 the inequality reaches its maximum (k)

Large θ allows choosing a vertex from VA that connects

to more vertices in VS

At early stages (VS∪{v}

p≪ 1) it is preferable to add well-

connected vertices

Algorithm SGSelect reduces θ if there exists no vertex

in VA that can satisfy the above condition 23

EXTERIOR EXPANSIBILITY

Exterior expansibility (A) – definition:

A(VS) = minv∈VS VA ∩ Nv + k − VS − v − Nv

VA ∩ Nv contains the neighboring vertices of v in V𝐴

VS − v − Nv is a set of non-neighboring vertices of v in VS

In other words:

The maximum number of vertices that VS can be extended from

We can select at most k − VS − v − Nv extra non-

neighboring vertices v in V𝐴

24

EXTERIOR EXPANSIBILITY – CONT.

Exterior expansibility (A) – condition:

A(VS ∪ {v}) ≥ p − VS ∪ {v} p − VS ∪ {v} is the number of attendees that have not been

considered

If the inequality doesn’t hold, the new intermediate

solution set obtained by adding v is not expansible

There is no degree of freedom in this case

25

EXAMPLE


VS = v7 , VA = {v2, v3, v4, v6, v8} Select v2 (since it has smallest social distance)

Calculate exterior expansibility A(VS ∪ {v2})

VA ∩ NV2 + k − VS − v2 − NV2 = 2 + 1 − 0 = 3

VA ∩ NV7 + k − VS − v7 − NV7 = 4 + 1 − 0 = 5

Choose the smallest one -> A(VS ∪ {v2}) = 𝟑

Validate exterior expansibility condition

p − VS ∪ v2 = 4 − 2 = 𝟐

Exterior expansibility holds for v2

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

• GF

• q=v7 • p=4 • s=1 • k=1

26

EXAMPLE – CONT.


VS = v7 , VA = {v2, v3, v4, v6, v8} Calculate interior unfamiliarity U(VS ∪ {v2})

VS − v2 − NV2 = ∅ = 0

VS − v7 − NV7 = ∅ = 0

Choose the largest one -> U(VS ∪ {v2}) = 𝟎

Validate interior unfamiliarity condition

kVS∪{v2}

p

θ= 1 ×

2

4

2=𝟏

𝟒 (assuming that θ = 2)

Interior unfamiliarity holds for v2

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

• GF

• q=v7 • p=4 • s=1 • k=1

27

EXAMPLE – CONT.


VS = v7, v2 , VA = {v3, v4, v6, v8} Select v3

Calculate exterior expansibility A(VS ∪ {v3}) = 1

Validate exterior expansibility condition p − VS ∪ v3 = 1

Exterior expansibility holds for v3

Calculate interior unfamiliarity U(VS ∪ {v3}) = 1

Validate interior unfamiliarity condition kVS∪{v3}

p

θ=

9

16

Interior unfamiliarity is violated for v3

In this stage, no reduction of θ is done, since there are still more

vertices in VA

Put v3 in parentheses and temporarily skip it

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

28

EXAMPLE – CONT.


VS = v7, v2 , VA = { v3 , v4, v6, v8} Select v6

Both conditions hold

VS = v7, v2, v6 , VA = {v3, v4, v8} Select v3

Interior unfamiliarity is violated for v3

Put v3 in parentheses and temporarily skip it

VS = v7, v2, v6 , VA = { v3 , v4, v8} Select v8

Exterior expansibility is violated for v8

Remove v8 from VA

VS = v7, v2, v6 , VA = { v3 , v4} Select v4

Both conditions hold – p=4

Obtain first feasible solution 𝐕𝐒 = 𝐯𝟕, 𝐯𝟐, 𝐯𝟔, 𝐯𝟒

Total social distance = 67

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

29

EXAMPLE – CONT.


𝐕𝐒 = 𝐯𝟕, 𝐯𝟐, 𝐯𝟔, 𝐯𝟒 (Total social distance = 67)

Is this the optimal solution?

Does it have the lowest social distance?

By backtracking to steps where interior unfamiliarity constraint was violated and reducing θ to zero, it is possible to get a better solution

𝐕𝐒 = 𝐯𝟕, 𝐯𝟐, 𝐯𝟑, 𝐯𝟒 (Total social distance = 62)

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

30

SOCIAL-TEMPORAL GROUP QUERY (STGQ)

Extension of SGQ by exploring the temporal

dimension (m)

STGQ is more complex than SGQ

Intuitive approach

Find SGQ solution for each individual activity period and

then select the one with the minimum total social

distance

Computationally expensive

31


SOCIAL-TEMPORAL GROUP QUERY (STGQ)

ALGORITHM DESIGN

Proposed algorithm: STGSelect

STGselect constructs the VS with vertices which

have more available times lots in common

Pivot time slot is chosen

It is better to choose time slot which has plenty vertices

It is mandatory to choose a time slot which has more

than p vertices

32

TEMPORAL EXTENSIBILITY

Temporal extensibility (X) – definition:

X(VS) = Ts −m Ts set of the consecutive time slots available for all vertices in VS

In other words:

How many extra mutual time slots are available

Temporal extensibility(X) – condition:

X(VS ∪ {v}) ≥ m − 1p− VS∪{v}

p

φ

φ ≥ 1

p− VS∪{v}

p is the portion of attendees that have not been

considered

33

EXAMPLE (ONCE AGAIN, BUT WITH TEMPORAL CONSTRAIN)


Choose t3 as pivot time slot

VS = v7 , VA = {v2, v3, v4, v6, v8} Select v2 (since it has smallest social distance)

Both exterior expansibility and interior unfamiliarity conditions hold

Calculate temporal extensibility X(VS ∪ {v2})

Ts = t1, t2, t3, t4, t5 → Ts = 5 → X(VS ∪ {v2}) = 5 − 3 = 𝟐

Validate temporal extensibility condition

m− 1p− VS∪{v}

p

φ= 2 ×

2

4

2=𝟏

𝟐 -> condition holds

vi di,7

v2 17

V3 18

V4 27

v6 23

v8 25

• GF

• q=v7 • p=4 • s=1 • k=1 • m=3

t1 t2 t3 t4 t5 t6 t7

v2 o o o o o o o

v3 o o o o

v4 o o o o o o

v6 o o o o o o

v7 o o o o o o

v8 o o o o

34

EXPERIMENTAL RESULTS

Experiment setup – need to find G

Scheduling 194 people from various communities (schools, government,

business, industry)

Google calendar is utilized for scheduling

Social graph Synthetic dataset with 12800 people is generated from a social

network website

The schedule of each person in each day is randomly assigned from the above 194-people real dataset

Existing algorithm - PCArrange Imitating the behavior of manual coordination via phone calls,

where the initiator q sequentially invites close friends first then finds out the common available time slots

Doesn’t include acquaintance constraint

They obtain k from each activity manually

35


RESULT ANALYSIS

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

1.00E+09

1.00E+10

2 3 4 5 6 7 8 9 10 11 12

Exec

uti

on

tim

e [n

s]

p (k=2, s=1)

Running time with different p

SGSelect

Baseline

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

1.00E+09

1.00E+10

1.00E+11

0 2 4 6

Exec

uti

on

tim

e [n

s]

s (k=2, p=4)

Running time with different s

SGSelect

Baseline

1.00E+05

1.00E+06

1.00E+07

1.00E+08

0 1 2 3 4 5 6 7

Exec

uti

on

tim

e [n

s]

k (p=5, s=2)

Running time with different k

SGSelect

Baseline

5.00E+04

5.00E+05

5.00E+06

5.00E+07

5.00E+08

150 1500 15000

Exec

uti

on

tim

e [n

s]

Network Size (p=5, k=3, s=1)

Running time with different

network sizes

SGSelect

Baseline

36

RESULT ANALYSIS – CONT.

0

2

4

6

8

2 3 4 5 6 7 8 9 10 11 12

k

p

Relation of p and k

SGSelect

Baseline

0

50

100

150

2 3 4 5 6 7 8 9 10 11 12Tota

l so

cial

dis

tan

ce

p

Relation of p and the total social distance

SGSelect

Baseline

37

ARE WE ALWAYS GETTING WHAT WE WANT?

Time constraint can sometimes lead to undesirable

solution. E.g. Arrange a party at 09:00 – 14:00

Possible solution:

Define also time interval, e.g. m=3 & 18:00<T<02:00

38


Event may be too diverse for the initiator

p=8

s=2

k=6

Possible solution

Give more “weight” to s=1 – include more direct friends of the initiator

q

10

11

8

3

14

7

39


Data is not complete, currently there are no social

graphs which includes also timetable

Reliability – “inactive” users wrong social distance

calculation

Social graph may not contain actual friends

40

CONCLUSIONS

Novel system for automatic activity planning based

on social and temporal relationship

Two algorithms proposed:

SGSelect

STGSelect

Algorithm proposed, significantly outperforms the

existing algorithms

Research result can be adopted in social network

websites

41


THANK YOU

42

Documents

On Social-Temporal Group Query with Acquaintance Constraint