An Efficient reconciliation algorithm for social networks Silvio - - PowerPoint PPT Presentation

An Efficient reconciliation algorithm for social networks

Silvio Lattanzi (Google Research NY) Joint work with: Nitish Korula (Google Research NY) ICERM Stochastic Graph Models

Outline

Stochastic Graph Models, ICERM

Graph reconciliation Model and theoretical results. Experimental results From theory to practice. Open problems and future directions

Graph reconciliation

Stochastic Graph Models, ICERM

Real world motivations

Stochastic Graph Models, ICERM

Real world motivations

Stochastic Graph Models, ICERM

Intra-language network

Real world motivations

Stochastic Graph Models, ICERM

Intra-language network Inter-language network

Real world motivations

Stochastic Graph Models, ICERM

Can we use intra-language information to improve inter- language graph?

Real world motivations

Stochastic Graph Models, ICERM

Can we use intra-language information to improve inter- language graph?

Real world motivations

Stochastic Graph Models, ICERM

Can we use intra-language information to improve inter- language graph?

?

Real world motivations

Stochastic Graph Models, ICERM

Real world motivations

Stochastic Graph Models, ICERM

Real world motivations

Stochastic Graph Models, ICERM

Real world motivations

Stochastic Graph Models, ICERM

Given two networks, identify as many users as possible across them. Applications:

social networks

• ntology reconciliation

Graph reconciliation problem

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Problem of reconciliation introduced by Novak et al.

Previous work

Stochastic Graph Models, ICERM

Problem of reconciliation introduced by Novak et al. Two main approaches:

• ML on user profile features

(name, location, image)

Previous work

Stochastic Graph Models, ICERM

Problem of reconciliation introduced by Novak et al. Two main approaches:

• ML on user profile features

(name, location, image)

• ML on neighborhood topology

Previous work

Stochastic Graph Models, ICERM

Problem of reconciliation introduced by Novak et al. Two main approaches:

• ML on user profile features

(name, location, image)

• ML on neighborhood topology

Limitations:

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack

Previous work

Stochastic Graph Models, ICERM

Very rich literature in de-anonymization Two relevant works:

• Backstrom et al. propose an active and passive attack
• Narayanan and Shmatikov successful

de-anonymization attack

Previous work

Stochastic Graph Models, ICERM

Ground truth 24000 matching across the two social networks

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

Ground truth 24000 matching across the two social networks

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

80 me-links

Ground truth 24000 matching across the two social networks They could re-identify 30.8% of the mappings.

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

80 me-links

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

?

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

2

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

2 1

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

2 1

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

Algorithm:

Narayanan and Shmatikov experiment

Stochastic Graph Models, ICERM

Why? Is it necessary to have high degree me-links?

Input: two graphs and a set of trusted matching We want to maximize the number of final matches.

Abstraction

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Is the problem tractable?

Stochastic Graph Models, ICERM

Problem is similar to graph isomorphism

Problem is similar to graph isomorphism Problem seems even harder because we want to detect similar structure

Is the problem tractable?

Stochastic Graph Models, ICERM

Problem is similar to graph isomorphism Problem seems even harder because we want to detect similar structure

Is the problem tractable?

Stochastic Graph Models, ICERM

Abstraction

Formalization of the problem:

Underlying social network

Stochastic Graph Models, ICERM

Abstraction

Formalization of the problem:

Underlying social network p1 p2 Delete the edges independently

Stochastic Graph Models, ICERM

Abstraction

Formalization of the problem:

Underlying social network p1 p2 Delete the edges independently Initial matchings

Stochastic Graph Models, ICERM

Questions

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Having a constant fraction of me-links, can we reconcile the entire network? If we have k me-links which fraction of networks can we reconcile?

Without additional assumption on the underling network problem seems still very hard

Underlying social network

Stochastic Graph Models, ICERM

Without additional assumption on the underling network problem seems still very hard We study two different models for social networks:

• G(n,p)
• Preferential attachment

Underlying social network

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Our algorithm

Stochastic Graph Models, ICERM

Algorithm: Narayanan Shmatikov + degree bucketing + acceptance threshold

G(n,p)

Does the technique works if the underlying graph is random?

Stochastic Graph Models, ICERM

p p1 p2

E[NG1(∗) ∩ NG2(∗)] = (n − 2)p2p1p2

G(n,p)

Does the technique works if the underlying graph is random?

Stochastic Graph Models, ICERM

p p1 p2

E[NG1(∗) ∩ NG2(∗)] = (n − 1)pp1p2

Concentration

We assume Two cases:

• , Chernoff bound is enough
• , we never make error

Stochastic Graph Models, ICERM

c log n n ≤ p ≤ 1 6, l, p1, p2 ∈ O(1)

x = (n − 2)p2p1p2

P = " n X

i=1

Bi ≤ 2 # = (1 − x)n + nx(1 − x)n−1 + ✓n 2 ◆ x2(1 − x)n−2 = 1 − n3x3 − o(n3x3)

npp1p2l ≥ 24 log n

npp1p2l ≤ 24 log n

More realistic model

Preferential attachment:

• is a single node with

self-loops

• adding a node to and

edges with probability proportional to the current degrees

Stochastic Graph Models, ICERM

Gm

1

m

Gm

n

Gm

n−1

m

Preferential attachment

A bit harder

• Several nodes of constant degree, we need to have a cascade
• Objective is reconcile a constant fraction of the network

Stochastic Graph Models, ICERM

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results.

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

High degree nodes help to detect intermediate degree nodes that

in turn help to detect small degree nodes.

PA structural lemmas

Stochastic Graph Models, ICERM

High degree nodes are early birds.

Nodes inserted after time , for constant , have degree in

φn φ

• (log2 n)

PA structural lemmas

Stochastic Graph Models, ICERM

High degree nodes are early birds.

Nodes inserted after time , for constant , have degree in

The rich get richer.

For nodes of degree greater than a constant fraction of their neighbors has been inserted after time , for constant

φn φ

• (log2 n)

log2 n

✏n

✏

PA structural lemmas

Stochastic Graph Models, ICERM

High degree nodes are early birds.

Nodes inserted after time , for constant , have degree in

The rich get richer.

For nodes of degree greater than a constant fraction of their neighbors has been inserted after time , for constant

First-mover advantage.

All nodes inserted before time , have degree at least

φn φ

• (log2 n)

log2 n

✏n

✏

n0.3

log3 n

High degree nodes are early birds

Stochastic Graph Models, ICERM

Gm

1

Gm

n

High degree nodes are early birds

Stochastic Graph Models, ICERM

Gm

1

Gm

n

φn

High degree nodes are early birds

Stochastic Graph Models, ICERM

Gm

1

Gm

n

φn

λn

High degree nodes are early birds

Stochastic Graph Models, ICERM

Let be the degree at the beginning of a phase.

The probability that a node increase its degree is dominated by the probability of an head in a coin toss for a biased coin that gives head with probability

Gm

1

Gm

n

φn

λn

di

3di φn

The rich get richer

Stochastic Graph Models, ICERM

If at time , the node has degree less than we are done

✏n

1 2d

Gm

1

Gm

n

✏n

The rich get richer

Stochastic Graph Models, ICERM

If at time , the node has degree less than we are done

The probability that the node increases its degree is dominated by the probability of an head in a coin toss for a biased coin that gives head with probability ✏n

1 2d

Gm

1

Gm

n

✏n

d 2nm

First-mover advantage

Stochastic Graph Models, ICERM

From Cooper and Frieze result on the cover time of PA graphs, Playing a bit with algebra we can get the final result.

Dk = dnm(v1) + dnm(v2) + · · · + dnm(vk) Pr ⇣ |Dk − 2 √ 2kn| ≥ 3 p mn log mn ⌘ ≤ (mn)−2

Pr(dn(vk+1) = d + 1|Dk − 2k = s) ≤ s + d 2N − 2k − s − d

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

High degree nodes help to detect intermediate degree nodes that

in turn help to detect small degree nodes.

Matching high degree nodes

Stochastic Graph Models, ICERM

By Chernoff w.h.p.

E[NG1(∗) ∩ NG2(∗)] = d(v)p1p2l

NG1(∗) ∩ NG2(∗) ≥ 7 8d(v)p1p2l

NG1(∗) ∩ NG2(∗) ≤ ✓2 3 + ✏ ◆ d(v)p1p2l + o(d(v))

Matching high degree nodes

Stochastic Graph Models, ICERM

By Chernoff w.h.p.

E[NG1(∗) ∩ NG2(∗)] = d(v)p1p2l

NG1(∗) ∩ NG2(∗) ≥ 7 8d(v)p1p2l

Gm

1

Gm

n

✏n

Matching high degree nodes

Stochastic Graph Models, ICERM

By Chernoff w.h.p. has degree at most and so the probability of connecting to it is

E[NG1(∗) ∩ NG2(∗)] = d(v)p1p2l

NG1(∗) ∩ NG2(∗) ≥ 7 8d(v)p1p2l

Gm

1

Gm

n

✏n

˜ O(√n)

• (1)

NG1(∗) ∩ NG2(∗) ≤ ✓2 3 + ✏ ◆ d(v)p1p2l + o(d(v))

Matching high degree nodes

Stochastic Graph Models, ICERM

By Chernoff w.h.p. has degree at most and so the probability of connecting to it is

E[NG1(∗) ∩ NG2(∗)] = d(v)p1p2l

NG1(∗) ∩ NG2(∗) ≥ 7 8d(v)p1p2l

Gm

1

Gm

n

✏n

˜ O(√n)

• (1)

NG1(∗) ∩ NG2(∗) ≤ ✓2 3 + ✏ ◆ d(v)p1p2l + o(d(v))

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

High degree nodes help to detect intermediate degree nodes that

in turn help to detect small degree nodes.

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

Bound the mismatch score

Bound the mismatch score

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

n

4 3 0.3

n( 4

3) 20.3

n( 4

3) 30.3

na = n0.3, nb = n

4 3 0.3

n( 3

2 −✏)0.3

n( 3

2 −✏) 20.3

n( 3

2 −✏) 30.3

The probability that 3 nodes coming between and point to and

Bound the mismatch score

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

n

4 3 0.3

n( 4

3) 20.3

n( 4

3) 30.3

na = n0.3, nb = n

4 3 0.3

na

nb

• nb2

nb

X

i=na nb

X

j=na nb

X

k=na

✓ log3 n (i − 1) ◆2 ✓ log3 n (j − 1) ◆2 ✓ log3 n (k − 1) ◆2 ≈ n2b−3a ∈ o(1)

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

High degree nodes help to detect intermediate degree nodes

that in turn help to detect small degree nodes.

Cascade

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

Cascade

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

n0.25

After one phase

Gm

1

Gm

n

Cascade

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

n0.25

After one phase in each phase we do not identify a small fraction, in total we loose a small constant

Gm

1

Gm

n

Gm

1

Gm

n

Cascade

Stochastic Graph Models, ICERM

Gm

1

Gm

n

n0.3

n0.25

After one phase in each phase we do not identify a small fraction, in total we loose a small constant

Gm

1

Gm

n

Gm

1

Gm

n

Sketch of the proof

Stochastic Graph Models, ICERM

For high degree node we can use concentration results. Different nodes of intermediate degree do not share many

neighbors.

High degree nodes help to detect intermediate degree nodes that

in turn help to detect small degree nodes.

Results

Stochastic Graph Models, ICERM

Theorem 1

If the underlying network is a G(n,p) graph it is possible to reconcile it completely

Theorem 2

If the underlying network is a PA graph it is possible to reconcile it a large fraction

• f it.

Experimental results

Stochastic Graph Models, ICERM

Experiments

Stochastic Graph Models, ICERM

Experiments on different graphs:

PA experiment

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Are our theoretical results robust?

Scalability

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How does the algorithm scale with the size of the graph?

Facebook experiment

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How does the algorithm perform if the underlying graph is a social network?

Facebook experiment

Stochastic Graph Models, ICERM

How does the algorithm perform if the underlying graph is a social network? 80% recall!! Can we explain it in theory?

Facebook cascade experiment

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What does happen if we generate the underlying network using a cascade process? Recover almost all the graph in the intersection. Can we explain it in theory?

Affiliation network model

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What does happen if we delete all the edges inside a subset of the communities? More than 80% recall. Can we explain it in theory?

Reconcile different graphs

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DBLP: we generate two co-authorship graphs. One considering only publications in even years and the other publication only in

• dd years.

Reconcile different graphs

Stochastic Graph Models, ICERM

DBLP: we generate two co-authorship graphs. One considering only publications in even years and the other publication only in

• dd years.

Gowalla: we generate two co-checkin graphs. One considering only checkins in even years and the other checkins only in

• dd years.

Reconcile different graphs

Stochastic Graph Models, ICERM

DBLP: we generate two co-authorship graphs. One considering only publications in even years and the other publication only in

• dd years.

Gowalla: we generate two co-checkin graphs. One considering only checkins in even years and the other checkins only in

• dd years.

German/French Wikipedia: we crawl the inter-languange links, we use few of them as seed and we check how many links we could recover.

Reconcile different graphs

Stochastic Graph Models, ICERM

Recall for Wikipedia ~30%

Reconcile different graphs

Stochastic Graph Models, ICERM

We have really good performance for high degree nodes

Open problems and future directions

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Extensions

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Other model of underlying graphs Other model of generation of networks Adversarial underlying network, error in seed links

Limitation of the current model

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Users’ degree depend varies in different social networks

How can we model this more general setting?

Better algorithm

Stochastic Graph Models, ICERM

Currently exploring only direct neighborhood Can we design better algorithms?

Thanks!

Stochastic Graph Models, ICERM