sigmod11 outsource shortest path

Neighborhood-Privacy Protected Shortest Distance Computing in

Cloud

Jun Gao, Jeffrey Yu Xu, Ruoming Jin,

Jiashuai Zhou, Tengjiao Wang, Dongqing Yang

14 Jun, 2011, Greece, SIGMOD 2012

Outline

Motivation

Related work

Our solution

• 1-neighborhood-d-radius graph

• Graph transformation with exact answer

• Graph transformation with approximate answer

Experiment

Conclusion & Future work

2

Graph data management in cloud

3Coauthor Network , from manyeyes.alphaworks.ibm.com

Graph data applications

• Social network, knowledge network...

Time consuming graph operations

• The shortest distance computing takes O(n2)

• The breadth-first-search requires O(n+m)

• ......

Cloud Computing

Advantage of cloud computing

• High computational power

• Easy maintenance

• Easy re-provisioning of resources

• ……

Can we use the cloud serve to manage graph data, such as to answer shortest distance?

Security issues in graph outsourcingAttacks on outsourced graph

• Structural Pattern Attack

- Use sub-graph to re-identify the target part

• Reconstruction Attack

- Recover the original graph from outsourced one.

Security leakage

• Regulation of sensitive data violated

• Untrusted answers produced by cloud server

4

We have to strike a balance between the security and the computational cost saving using cloud

server

Client SideClient Side

Framework of graph outsourcing

5

Original Original GraphGraph

Original Original GraphGraph

Graph Graph TransformationTransformationGraph Graph TransformationTransformation

Link Link graphgraph

Link Link graphgraph

ResultsResultsResultsResults ResultResultCombinationCombinationResultResultCombinationCombination

Cloud ServerCloud Server

OutsourcedOutsourcedGraphGraph

OutsourcedOutsourcedGraphGraph

QueryQueryEvaluationEvaluationQueryQueryEvaluationEvaluation

Query Query RewritingRewritingQuery Query RewritingRewriting

QueryQueryQueryQuery

(1) A reasonable security model on outsourced graph

(2) An efficient method to transform the original graph into the outsourced graph

(3) An approach to rewrite the query and combine the results

(2)

(1)

(3)

Outline

Motivation

Related work

Our solution




Experiment


6

Structural Anonymization

• Structural anonymization in publishing

- 1-neighborhood [icde 08], k-degree [sigmod08], k-automorphism [vldb 09], k-isomorphism [sigmod10], etc

- Using the least amount of modifications of the original graph

7

Original graph 4-isomorphism Attacker’s query

find 4 sub-graphs

No shortest distance preservationNo consideration of edge weight

1

6 2

34 2

362

1

63

1

7

9 6

4

8

3

8

2

4

5

a b c

d gfe

h

p

v

i

k lj

uo 9

7

3

a b c

d gf

e

h

p

v

i

k lj

u o

z

yw x

Feature preservation graph transformation

Eigenvalue preservation [sdm 08]

• Random add/remove/switch edges

• Theoretically prove that the eigenvalue can be preserved.

Shortest path preservation [icde 10]

• Express the shortest path preservation by inequality rules

• Use line programming to find a solution to such rules

• Requires O(dn2) rules in all shortest path preservation

8

No support of exact distance computingNo explicit security guarantee

Shortest distance indexMultiple-level index [tkde98]

• Select nodes to build a higher level graph

• Exploit the shortest paths at a higher level graph to guide the path searching at a lower level

Landmark index [cikm 09, jacm 09]

• Select landmark nodes and build the shortest path

• Exploit the triangle inequality rules to estimate the distance

2-HOP index [soda 02]

• Annotate incoming and outgoing labels on each node

• Compute the distance between two nodes with the intersection

9

No security consideration

Outline

Motivation

Related work

Our solution




Experiment


10

1- Neighborhood-d-Radius GraphIntuition

• Protect the neighborhood information and the close relationship between nodes.

Privacy protection

• Find empty meaningful results for any query pattern11

(1-neighborhood): for any node pair u and v ∈ Vo, (u, v) ∉ E(d-radius): for any node pair u and v ∈ Vo, δG(u, v) >= d.

6 2

34 2

36

2

1

63

7

9 6

4

8

3

8

2

4

5

a b c

d gfe

h

p

v

i

k lj

uo 9

7

3

3

5

5 1116

16

9

10

105

b f

v

io

h

10 z

1

3 6yw 4 x

Original graph Attacker’s query 2-radius graph

1-Neighborhood-d-Radius Graph too strong?

Can we hide the neighbors and relationship with distance less than d, and add direct edges among others?

No, using triangle inequality rules will find the “hidden” edges

• Reconstruction Attack

12

1

6 2

34 2

362

1

63

1

7

9 6

4

8

3

8

2

4

5

a b c

d gfe

h

p

v

i

k lj

uo 9

7

3

ab c

d gfe

p

vi

kl

j

u

o

h

9

1310

11

6

108

1215

16

10

9

6

7

6

4 11

12

14

Original graph non-2-radius graph

Utilization: Shortest Distance Computation

13

1

6 2

3

42

36

2

1

63

1

79

6

4

8

3

8

2

4

5

9

7

3

55 11

16

169

10

105

10

f

a c

d ge

p

k lj

u

b

vi

o

h

b f

v

ioh

Outsourcedgraph

Link Graph

Originalgraph

),(,),(min, vywyxxuwvuoGG

Given a node pair u and v, the shortest distance can be discovered with

…… u v

Graph Transformation ProblemGiven a graph G = (V,E) and d, the graph transformation produces outsourced graphs Go = {G1, ...Gj}, and a local link graph Gl, which achieves the following objectives:

• Security

- Each outsourced graph is a 1-neighborhood-d-radius graph;

• Utility

- The union of Go and Gl can answer the shortest distance in the original graph;

• Local computational cost

- The space cost of Gl and the cost of the shortest distance computation on the client side are minimized.

14

Naive Method

Steps

• Enumerate different forms of the candidate solutions

- One local link graph and outsourced graphs.

• Find the one with the minimal space cost of local graph.

Searching space

• The nodes in a outsourced graph are a sub-set of the these

in original graph, and the different forms of outsourced

graph can be O(2n)

• The brute force strategy will lead to exponential time cost

15

Greedy MethodBasic idea

• Generate more “expressive” outsourced graph which can answer more shortest paths.

- Edges in link graph can be reused so that the space cost of link graph is reduced

Challenges

• How to find “expressive” outsourced nodes?

• How to build d-radius graph from the select nodes?

Steps

1. Enumerate all shortest paths, find possible candidate outsourced nodes, and assign benefit on nodes

2. Generate outsourced graphs according to node benefit

16

Step 1: Enumerate shortest path and benefit assignment

Candidate outsourced node pair

• node pair (x,y) can be used to answer

shortest distance between (u,v)

• (x,y) should meet d-radius.

• x is close to u, y is close to v

Benefit function

• Record the frequency of a node (or node

pair) which can be outsourced

17

1 1a b 2c 3f g

2 1d b 2c 7f e

1 2b c

3f

7g v

2 1d b

2c

3f g

1a 3 g

2d 7f

e

1b c v

b

(a) Sampling Shortest Paths

(b) outsourced node pairs

1 1a b 2c 3f g

Step 2: Generate one outsourced graph

Node selection

• The node which is with the next maximal benefit and is not in any cluster, can be selected

• Build a d-radius cluster for the selected node

Edge building

• The edge weight is the shortest distance between cluster centers

18

3b f

(c) outsourced graphs(part)

1a 3 g

2d 7f

e1

b c v

b

(a) outsourced node pairs

6 2

34 2

36

2

1

63

7

9 6

4

8

3

8

2

4

5

a b c

d gfe

h

p

v

i

k lj

uo 9

7

3

(b) d-radius cluster

Graph transformation with approximate answer

Graph transformation with exact answer at least requires enumeration of all shortest paths.

Approximate distance can be acceptable in many domains

Approximate distance can be measured by

Basic idea

• Transform graph to achieve α = 1 and a given average additive error β?

Main steps

• Construct outsourced graph in a relaxed way

• Estimate the average additive error

19

vuG ,

Q

vuplenQq Gq

,

Relaxed outsourced graph construction

Select outsourced nodes randomly.

Relax edge weight assignment

• Build k shortest path trees

• In each tree, link the outsourced node with its lowest ancestor as the edge.

20

6 2

34 2

362

1

63

7

9 6

4

8

3

8

2

4

5

a b c

d gfe

h

p

v

i

k lj

uo 9

7

3

1 1

2

3

2 3

2

4 8

3

84

5

a

bcd

g

f

e

h

p

v

i

k

l

j

u o7

3

11 5

5

10

11

b

fh

v

i o

3

Estimation of average additive error The error for distance query (u,v) varies according to whether u and v have been outsourced

β can be computed as follows:

• We estimate the percentage of each category with the random node selection assumption

• The average additive error can be estimated by sampling

21

221100 lQpctlQpctlQpct

u x

(a) Q0 Query (b) Q1 Query

x yvu v

dd d d

(c) Q2 Query

u

d d

v

Heuristic outsourced node selection

Single outsourced graph

• Degree based construction

- First select the node with the higher degree

• Cluster size based construction

- First select the node with more nodes in its cluster

Multiple outsourced graphs

• Avoid outsourcing the same graph.

22

Outline

Motivation

Related work

Our solution




Experiment


23

ExperimentMeasures:

• transformation time cost

• space cost of link graph

• average additive error

• local overhead ratio=

Competitor

• LP-based Edge weight anonymization in ICDE 2010

Datasets:

24

Time cost with cloud server

Time cost without cloud server

Results related with exact answers

Scalability

• Better than LP based method

Impact of increase of d

• Strengthen security of outsourced graphs

• Increase the transformation time cost, the space cost of the link graph

25

Results related with exact answers (cont.)

Benefit function

• Vertex pair based method works better

Local overhead ratio

• Very low

• Goes down with the increase of graph size

26

Results related with approximate answers

Scalability

• Support large graph

Impact of increase of error bound

• Decrease of space cost and time cost in outsourcing

27

Results related with approximate answers(cont.)

Additive error bound

• Achieves the given additive error quite well

Local overhead ratio

• Declines with the increase of nodes

28

Outline

Motivation

Related work

Our solution




Experiment


29


Conclusion:

• A 1-neighbourhood-d- radius security model

• A greedy method to transform graph with exact answer

• A method to transform graph with approximate answer

• Extensive experimental results on real and synthetic data

Future work:

• More graph operations.

• Stronger security model

• Incremental graph outsourcing over dynamic graphs

30

31

[email protected]

sigmod11 outsource shortest path

Technology