1 Graph Analysis with - PowerPoint Presentation

Slide1

1

Graph Analysis with Node Differential Privacy

Sofya RaskhodnikovaPenn State University

Joint work with

Shiva Kasiviswanathan (GE Research), Kobbi Nissim (Ben-Gurion U. and Harvard U.), Adam Smith (Penn State)Slide2

Publishing information about graphs

Many datasets can be represented as graphs

“Friendships” in online social networkFinancial transactionsEmail communication

Romantic relationships2

American J. Sociology

,

 Bearman, Moody, Stovelimage source http://community.expressor-software.com/blogs/mtarallo/36-extracting-data-facebook-social-graph-expressor-tutorial.html

Privacy is a big issue!Slide3

Differential privacy for graph data

Differential privacy

[Dwork McSherry Nissim Smith 06]

An algorithm A is

-differentially private if for all pairs of neighbors

and all sets of answers S: 

Graph

G

A

queries

answers

)

(

Government,

researchers,

businesses

(or)

m

alicious

adversary

3

image

source

http://www.queticointernetmarketing.com/new-amazing-facebook-photo-mapper/

Trusted

curator

UsersSlide4

Two variants of differential privacy for graphs

Edge differential privacy

Two graphs are neighbors if they differ in one edge.Node differential privacyTwo graphs are neighbors if one can be obtained from the other by deleting a node and its adjacent edges.

4

G:

G

:

G:

G

:

 Slide5

Node differentially private analysis of graphs

5

image source http://www.queticointernetmarketing.com/new-amazing-facebook-photo-mapper/

Two conflicting goals:

utility and privacy

Impossible to get both in the worst casePreviously: no node differentially private algorithms that are accurate on realistic graphs

Graph

G

A

queries

answers

)

(

Government,

researchers,

businesses

(or)

m

alicious

adversary

Trusted

curator

UsersSlide6

Our contributions

First node differentially private algorithms that are accurate for sparse graphsnode differentially private for all graphs

accurate for a subclass of graphs, which includes graphs with sublinear (not necessarily constant) degree boundgraphs where the tail of the degree distribution is not too heavydense graphsTechniques for node differentially private algorithmsMethodology for analyzing the accuracy of such algorithms on realistic networks

Concurrent work on node privacy [Blocki

Blum Datta Sheffet

13]

6Slide7

Node differentially private algorithms for releasing

number of edgescounts of small subgraphs (e.g., triangles

, -triangles, -stars)degree distributionAccuracy analysis of our algorithms for graphs with not-too-heavy-tailed degree distribution: with

-decay for constant

Notation:

fraction of nodes in G of degree

Every graph satisfies -decayNatural graphs (e.g., “scale-free” graphs, Erdos-Renyi) satisfy  Our contributions: algorithms

7

A graph G satisfies

-decay

if

for all

…

…

Frequency

Degrees

…

…

…Slide8

Our contributions: accuracy analysis

Node differentially private algorithms for releasing number of edgescounts of small subgraphs

(e.g., triangles, -triangles, -stars)degree distributionAccuracy analysis of our algorithms for graphs with not-too-heavy-tailed degree distribution: with

-decay for constant

number of edges

counts of small subgraphs (e.g., triangles,

-triangles, -stars)degree distribution 8

A graph G satisfies

-decay

if for all

(1+o(1))-approximation

}

…

…Slide9

Previous work on

differentially private computations on graphsEdge differentially private algorithms

number of triangles, MST cost [Nissim Raskhodnikova Smith 07]degree distribution [Hay

Rastogi Miklau

Suciu 09, Hay Li Miklau

Jensen 09, Karwa Slavkovic

12]small subgraph counts [Karwa Raskhodnikova Smith Yaroslavtsev 11]cuts [Blocki Blum Datta Sheffet 12]Edge private against Bayesian adversary (weaker privacy)small

subgraph

counts

[

Rastogi

Hay

Miklau

Suciu

09]

Node zero-knowledge private (

stronger

privacy)average degree, distances to nearest connected, E

ulerian, cycle-free graphs for dense graphs [Gehrke

Lui Pass 12]

9Slide10

Challenge for node privacy: high local sensitivity

Local sensitivity

[NRS’07]:Global sensitivity [DMNS’06]:For many functions of the data, node

is high.

Consider adding a node connected to all other nodes.Examples:

(G)= |E(G)|.

(G)= # of s in G.  10

Edge

is 1;

node

is

for all G.

Edge

is

;

node

is |E(G)|.

 Slide11

“Projections” on graphs of small degree

Let

= family of all graphs, = family of graphs of degree .Notation.

=

node

over

. = node over .Observation. is low for many useful

.

Examples

:

=

(compare to

=

)

=

(

compare to

=

)

Idea:

``Project’’ on graphs in

for a carefully chosen d << n.

11

Goal:

privacy for all graphsSlide12

Method 1: Lipschitz extensions

Release

via GS framework [DMNS’06]Requires designing Lipschitz extension for each function

we base ours on maximum flow and linear and convex programs

12

low

high

=

A function

is a

Lipschitz

extension

of

from

to

if

agrees with

on

and

=

 Slide13

Lipschitz

extension of

: flow graph

For a graph G=(V, E), define

flow graph of G:Add edge

iff .(G) is the value of the maximum flow in this graph.Lemma. (G

)/2

is a

Lipschitz

extension of

.

13

s

1

3

5

1'

3'

5'

t

2

4

2'

4'

1

 Slide14

Lipschitz

extension of

: flow graph

For a graph G=(V, E), define

flow graph of G:Add edge

iff .(G) is the value of the maximum flow in this graph.Lemma. (G

)/2

is a

Lipschitz

extension of

.

Proof: (1)

(G

) =

for all

G

(2)

= 2⋅

14

s

1

3

5

1'

3'

5'

t

2

4

2'

4'

1

1/Slide15

Lipschitz

extension of

: flow graph

For a graph G=(V, E), define

flow graph of G:

(G) is the value of the maximum flow in this graph.Lemma. (G)/2 is a Lipschitz extension of .Proof: (1)

(G

) =

for all

G

(2)

= 2⋅

=

2

15

s

1

3

5

1'

3'

5'

t

2

4

2'

4'

1

6

'

6Slide16

For a graph G=([n], E), define

LP with variables

for all triangles :

(G) is the value of LP.

Lemma.

(G)

is a Lipschitz extension of .Can be generalized to other counting queriesOther queries use convex programs Lipschitz extensions via linear/convex programs16

Maximize

for all triangles

for all nodes

 Slide17

Method 2: Generic reduction to privacy over

Time(A) = Time(B) + O(m+n

)Reduction works for all functions

How it works: Truncation T(G) outputs G with nodes of degree

removed.Answer queries on T(G) instead of G

 17

low

high

Input:

Algorithm

B that is node-DP over

Output:

Algorithm

A that is node-DP over

,

has

accuracy similar to B on “nice”

graphs

via Smooth Sensitivity framework

[

NRS’07]

via

finding

a

DP upper bound

on

[

Dwork

Lei

09, KRSY’11

]

and

running any algorithm

that is

-

node-DP

over

T

query

f

+ noise

T(G)

G

S

(G)

ASlide18

Generic Reduction via Truncation

Truncation T(G) removes nodes of degree

.On query , answer

How much noise?

Local sensitivity of

as a map

Lemma

.

,

where

=

nodes of degree

.

Global

sensitivity

is

too

large.

18

…

…

d

Frequency

Degrees

Nodes that determine

 Slide19

Smooth Sensitivity of Truncation

Lemma

.

is a smooth bound

for

, computable in time

“

Chain rule

”:

is a smooth bound

for

19

Smooth Sensitivity Framework

[NRS ‘07]

is a

smooth bound on local sensitivity

of

if

for all neighbors

T

query

f

+ noise

T(G)

G

S

(G)

ASlide20

Utility of the Truncation Mechanism

Lemma.

If we truncate to a random degree in ,

Application to releasing

the degree

distribution:

an -node differentially private algorithm

such that

with

probability

at least

if

satisfies

-decay for

20

Utility:

If G is

-bounded,

expected noise magnitude is

.

T

query

f

+ noise

T(G)

G

S

(G)

ASlide21

Techniques used to obtain our results

Node differentially private algorithms for releasing number of edgescounts of small subgraphs

(e.g., triangles, -triangles, -stars)degree distribution

21

via Lipschitz extensions

} via generic reductionSlide22

Conclusions

It is possible to design node differentially private algorithms with good utility on sparse graphsOne can first test whether the graph is sparse privatelyDirections for future workNode-private algorithm for releasing cutsNode-private synthetic graphs

What are the right notions of privacy for graph data?22