Dynamics of networks

Dynamics of networks

Jure LeskovecMachine Learning DepartmentCarnegie Mellon University

2

Cyberspace

The emergence of ‘cyberspace’ and the World Wide Web is like the discovery of a new continent. Jim Gray, 1998 Turing Award address

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Today: Rich data

Large on-line systems have detailed records of human activity On-line communities:

▪ Facebook (64 million users, billion dollar business)▪ MySpace (300 million users)

Communication:▪ Instant Messenger (~1 billion users)

News and Social media:▪ Blogging (250 million blogs world-wide, presidential candidates run

blogs) On-line worlds:

▪ World of Warcraft (internal economy 1 billion USD)▪ Second Life (GDP of 700 million USD in ‘07)

Opportunities for impact in science and industry

4

The networks

b) Internet (AS)c) Social networksa) World wide web

d) Communication e) Citationsf) Protein interactions

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Networks: What do we know? We know lots about the network structure:

Properties: Scale free [Barabasi ’99], 6-degrees of separation [Milgram ’67], Navigation [Adamic-Adar ’03, LibenNowell ’05], Bipartite cores [Kumar et al. ’99], Network motifs [Milo et al. ‘02], Communities [Newman ‘99], Conductance [Mihail-Papadimitriou-Saberi ‘06], Hubs and authorities [Page et al. ’98, Kleinberg ‘99]

Models: Preferential attachment [Barabasi ’99], Small-world [Watts-Strogatz ‘98], Copying model [Kleinberg el al. ’99], Heuristically optimized tradeoffs [Fabrikant et al. ‘02], Congestion [Mihail et al. ‘03], Searchability [Kleinberg ‘00], Bowtie [Broder et al. ‘00], Transit-stub [Zegura ‘97], Jellyfish [Tauro et al. ‘01]

We know much less about processes and

dynamics of networks

6

My research: Network dynamics

Network Dynamics: Network evolution

▪ How network structure changes as the network grows and evolves?

Diffusion and cascading behavior▪ How do rumors and diseases spread over networks?

7

My research: Scale matters We need massive network data for the

patterns to emerge: MSN Messenger network [WWW ’08, Nature ‘08]

(the largest social network ever analyzed)▪ 240M people, 255B messages, 4.5 TB data

Product recommendations [EC ‘06]

▪ 4M people, 16M recommendations Blogosphere [work in progress]

▪ 60M posts, 120M links

8

My research: The structure

Diffusion & Cascades

Network Evolution

Patterns & Observatio

ns

Q1: What do cascades look like?

Q2: How likely are people to get influenced?

Q5: How does network structure change as the network grows/evolves?

Models & Algorithms

Q3: How do we find influential nodes?

Q4: How do we quickly detect epidemics?

Q6:How do we generate realistic looking and evolving networks?

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Network Evolution


ns

Q1: What do cascades look like?

Q2: How likely are people to get influenced?


Models & Algorithms




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Diffusion and Cascades

Behavior that cascades from node to node like an epidemic News, opinions, rumors Word-of-mouth in marketing Infectious diseases

As activations spread through the network they leave a trace – a cascade

Cascade (propagation graph)Network

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Setting 1: Viral marketing

People send and receive product recommendations, purchase products

Data: Large online retailer: 4 million people, 16 million recommendations, 500k products

10% credit 10% off

[w/ Adamic-Huberman, EC ’06]

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Setting 2: Blogosphere

Bloggers write posts and refer (link) to other posts and the information propagates

Data: 10.5 million posts, 16 million links

[w/ Glance-Hurst et al., SDM ’07]

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Viral marketing cascades are more social: Collisions (no summarizers) Richer non-tree structures

Q1) What do cascades look like? Are they stars? Chains? Trees?

Information cascades (blogosphere):

Viral marketing (DVD recommendations):

(ordered by frequency)

pro

pagati

on

[w/ Kleinberg-Singh, PAKDD ’06]

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Q2) Human adoption curves Prob. of adoption depends on the number of friends

who have adopted [Bass ‘69, Granovetter ’78] What is the shape?

Distinction has consequences for models and algorithms

k = number of friends adopting

Pro

b.

of

ad

op

tion

k = number of friends adopting

Pro

b.

of

ad

op

tion

Diminishing returns? Critical mass?

To find the answer we need lots of

data

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Q2) Adoption curve: Validation

Later similar findings were made for group membership [Backstrom-Huttenlocher-Kleinberg ‘06], and probability of communication [Kossinets-Watts ’06]

Prob

abili

ty o

f pur

chas

ing

0 5 10 15 20 25 30 35 400

0.010.020.030.040.050.060.070.080.09

0.1DVD

recommendations(8.2 million

observations)

# recommendations received

Adoption curve follows the diminishing returns.Can we exploit this?

[w/ Adamic-Huberman, EC ’06]

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Network Evolution


ns

A1: Cascade shapes

A2: Human adoption follows diminishing returns


Models & Algorithms




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Cascade & outbreak detection

Blogs – information epidemics Which are the influential/infectious blogs?

Viral marketing [Domingos-Richardson]

Who are the trendsetters? Influential people?

Disease spreading Where to place monitoring stations to detect

epidemics?

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The problem: Detecting cascades

How to quickly detect epidemics as they spread?

c1

c2

c3

[w/ Krause-Guestrin et al., KDD ’07](best student paper)

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Two parts to the problem Cost:

Cost of monitoring is node dependent

Reward: Minimize the number of affected

nodes:▪ If A are the monitored nodes, let R(A)

denote the number of nodes we save

We also consider other rewards: ▪ Minimize time to detection▪ Maximize number of detected outbreaks

R(A)A

( )


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Reward for detecting cascade i

Given: Graph G(V,E), budget M Data on how cascades C1, …, Ci, …,CK spread over time

Select a set of nodes A maximizing the reward

subject to cost(A) ≤ M

Solving the problem exactly is NP-hard Max-cover [Khuller et al. ’99]

Optimization problem

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Detection: Solution outline

Submodularity of the reward functions

CELF: algorithm with approximate guarantee

Lazy evaluation to speed-up CELF

We extend the result of [Kempe-Kleinberg-Tardos ’03]


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Background: Greedy hill-climbing

If objective function exhibits diminishing returns: Submodularity(think of it as “convexity”)

Theorem [Nemhauser et al. ‘78]If R is a function that is monotone and submodular, then k-step greedy produces set A for which R(A) is within (1-1/e) ~63% of optimal.

a

b

c

a

b

c

d

d

Marginal reward

e

GreedyAdd sensor with highest marginal

gain

Slow! O(kN)

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Reward function is submodular

R(A) is monotone: monitoring more nodes never hurts We must show R is submodular: A B

Natural example: Sets A1, A2,…, An

R(A) = size of union of Ai

(size of covered area)

If R1,…,RK are submodular, then ∑pi Ri is submodular

AB

u

Gain of adding a node to a small set

Gain of adding a node to a large set

R(A {u}) – R(A) ≥ R(B {u}) – R(B)

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Reward function is submodular Theorem:

Reward function is submodular

Consider cascade i: Ri(uk) = set of nodes saved from uk

Ri(A) = size of union Ri(uk), ukAÞ Ri is submodular

Global optimization: R(A) = Prob(i) Ri(A)

R is submodular

u1 Ri(u1)

Cascade i

u2

Ri(u2)

(best student paper)[w/ Krause-Guestrin et al., KDD ’07]

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Solution: CELF Algorithm

We extend the setting to variable node cost case

We develop CELF (cost-effective lazy forward-selection) algorithm: Two independent runs of a modified greedy

▪ Solution set A’: ignore cost, greedily optimize reward▪ Solution set A’’: greedily optimize reward/cost ratio

Pick best of the two: arg max(R(A’), R(A’’))

Theorem: CELF is near optimal CELF achieves ½(1-1/e) factor approximation

For the size of our problems naïve CELF is too slow

(best student paper)[w/ Krause-Guestrin et al., KDD ’07]

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Scaling up: Lazy evaluation Observation:

Submodularity guarantees that marginal rewards decrease with the solution size

Idea: Use marginal reward from previous step as a upper

bound on current marginal reward

d

Marginal reward

ab

c

d

e

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CELF: Lazy evaluation

CELF algorithm: Keep an ordered list of

marginal rewards ri from previous step

Re-evaluate ri only for the top node

a

b

c

ab

cd

d

e

e

Marginal reward

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a

ab

c

d

d

b

c

e

e

Marginal reward

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a

c

ab

c

d

d

b

Marginal reward

e

e

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Blogs: Information epidemics

For more info see our website: www.blogcascade.org

Which blogs should one read to catch big stories?

CELF

In-links

Random

# postsOut-links

Number of selected blogs (sensors)

Reward

(higher is

better)

(used by Technorati)

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CELF: Scalability

CELF

GreedyExhaustive search

CELF runs 700x faster than simple greedy algorithm

Number of selected blogs (sensors)

Run time

(seconds)

(lower is better)

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Same problem: Water Network Given:

a real city water distribution network

data on how contaminants spread over time

Place sensors (to save lives)

Problem posed by the US Environmental Protection Agency

SS

c1

c2

[w/ Krause et al., J. of Water Resource Planning]

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Water network: Results

Our approach performed best at the Battle of Water Sensor Networks competition

CELF

PopulationRandom

Flow

Degree

Author Score

CMU (CELF) 26

Sandia 21

U Exter 20

Bentley systems 19

Technion (1) 14

Bordeaux 12

U Cyprus 11

U Guelph 7

U Michigan 4

Michigan Tech U 3

Malcolm 2

Proteo 2

Technion (2) 1

Number of placed sensors

Population

saved(higher is better)

[w/ Ostfeld et al., J. of Water Resource Planning]

34



Network Evolution


ns

A1: Cascade shapes

A2: Human adoption follows diminishing returns


Models & Algorithms

A3, A4: CELF algorithm for detecting cascades and outbreaks

Q6: How do we generate realistic looking and evolving networks?

35

Background: Network models

Empirical findings on real graphs led to new network models

Such models make assumptions/predictions about other network properties

What about network evolution?

log degree

log pro

b. Model

Power-law degree distribution

Preferential attachment

Explains

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Networks are denser over time Densification Power Law:

a … densification exponent (1 ≤ a ≤ 2)

Q5) Network evolution

What is the relation between the number of nodes and the edges over time?

Prior work assumes: constant average degree over time

Internet

Citations

a=1.2

a=1.6

N(t)

E(t

)

N(t)

E(t

)

[w/ Kleinberg-Faloutsos, KDD ’05](best research paper)

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Q5) Network evolution

Prior models and intuition say that the network diameter slowly grows (like log N, log log N)

time

diam

eter

diam

eter

size of the graph

Internet

Citations

Diameter shrinks over time as the network grows the

distances between the nodes slowly decrease

[w/ Kleinberg-Faloutsos, KDD ’05](best research paper)

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Q6) Need a new network model

None of the existing models generates graphs with these properties

We need a new model Idea: Generate graphs recursively

Recursion (fractals)

Skewed distributions (power-laws)

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The model: Kronecker graphs

We prove Kronecker graphs mimic real graphs: Power-law degree distribution, Densification,

Shrinking/stabilizing diameter, Spectral properties

Initiator

(9x9)(3x3)

(27x27)

Kronecker product of graph adjacency matrices

(actually, there is also a nice social interpretation of the model)

[w/ Chakrabarti-Kleinberg-Faloutsos, PKDD ’05]

pij

Edge probability Edge probability

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Q6) Generating realistic graphs Want to generate realistic networks:

Why synthetic graphs? Anomaly detection, Simulations, Predictions, Null-model,

Sharing privacy sensitive graphs, …

Q: Which network properties do we care about? A: Don’t commit, let’s match adjacency matrices

Compare graphs properties, e.g., degree

distribution

Given a real network

Generate a synthetic network

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Maximum likelihood estimation

Kronecker graphs: Estimation

Naïve estimation takes O(N!N2): N! for different node labelings:

▪ Our solution: Metropolis sampling: N! (big) const N2 for traversing graph adjacency matrix

▪ Our solution: Kronecker product (E << N2): N2 E Do stochastic gradient descent

a bc d

P( | ) Kronecker

arg max

We estimate the model in O(E)

[w/ Faloutsos, ICML ’07]

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Estimation: Epinions (N=76k, E=510k)

We search the space of ~101,000,000

permutations Fitting takes 2 hours Real and Kronecker are very close

Degree distribution

node degree

pro

babili

ty

0.99 0.54

0.49 0.13

Path lengths

number of hops

# r

each

able

pair

s

“Network” values

ranknetw

ork

valu

e

[w/ Faloutsos, ICML ’07]

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Diffusion &

Cascades

Network Evolution

Patterns &

Observations

A1: Cascade shapes

A2: Human adoption follows

diminishing returns

A5: Densification,

Shrinking diameters

Models & Algorithm

s

algorithm for

detecting cascades and

outbreaks

A6: Kronecker graphs,

Estimating Kronecker initiator

A3, A4: CELF

Two quick applicatio

ns

Conclusion andFuture work

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Two applications

P1) Web Projections: Predicting search result quality without

page content

P2) MSN Instant Messenger: Social network of the whole world

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P1) Web Projections

User types in a query to a search engine Search engine returns results:

Is this a good set of search results?

Result returned by the search engine

Hyperlinks

Non-search resultsconnecting the graph

[w/ Dumais-Horvitz, WWW ’07]

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P1) Web Projections: Results We can predict search result quality with

80% accuracy just from the connection patterns between the results

Predict “Good”Predict “Poor”

Good?

Poor?

[w/ Dumais-Horvitz, WWW ’07]

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P2) Planetary look on a small-world

Milgram’s small world experiment

(i.e., hops + 1)

Lots of engineering effort and good hardware: 4 dual-core Opteron

server 48GB memory 6.8TB of fast SCSI disks

How to learn short paths?

Small-world experiment [Milgram ‘67] People send letters from Nebraska to Boston

How many steps does it take? Messenger social network – largest network analyzed

240M people, 30B conversations, 4.5TB data

[w/ Horvitz, WWW ’08, Nature ‘08]

MSN Messenger network

Number of steps

between pairs of people

48



Network Evolution


ns

A1: Cascade shapes


diminishing returns

A5: Densification,

Shrinking diameters

Models & Algorithms




Big data

matters

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Future direction 1: Diffusion

How do news and information spread New ranking and influence measures for blogs Sentiment analysis from cascade structure Crawling 2 million blogs/day (with Spinn3r)

Obscure technology story

Small

tech blog

WiredSlashd

otNew Scientis

t New York

TimesCNN

BBC

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Future direction 1: Diffusion

Models of information diffusion When, where and what post will create a cascade? Where should one tap the network to get the

effect they want? Social Media Marketing

How to handle richer classes of graphs Interaction of network structure and

node/edge attributes

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Future direction 2: Evolution

Why are networks the way they are? Continue work on fundamental properties of networks Build models and understanding Network community structure

Health of a social network How to better design and steer networks Adoption of social networking services (with Facebook)

Predictive modeling of large communities Online multi-player games are closed worlds with detailed

traces of human activity Predict success of guilds, battles, elections (with EVE online)

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Future direction 3: Scaling up

Map-reduce for massive graphs Algorithms and techniques for massive graphs With Yahoo! Research:

4000 node Hadoop cluster ▪ world’s 50th fastest supercomputer▪ 3 TB memory▪ 1.5 PB storage

We already have new ways to peek into the structure of massive networks

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Networks & Computer Science

Computer

systems

Theory and

algorithms

Machine learning /

Data mining

(complex) networks

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Networks & Science

Computer

systems

Theory and

algorithms

Machine learning /

Data mining

Social Science

s

Biology

Physics

Applications &

Industry

Statistics

(complex) networks

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Christos FaloutsosJon Kleinberg

Andreas KrauseCarlos Guestrin

Eric HorvitzSusan Dumais

Andrew TomkinsRavi Kumar

Lada AdamicBernardo Huberman

Kevin LangMichael Manohey

Zoubin GharamaniAnirban Dasgupta

Tom MitchellJohn Lafferty

Larry WassermanAvrim Blum

Samuel MaddenMichalis Faloutsos

Ajit SinghJohn Shawe-Taylor

Lars BackstromMarko Grobelnik

Natasa Milic-FraylingDunja Mladenic

Jeff HammerbacherMatt Hurst

Deepay ChakrabartiNatalie Glance

Jeanne VanBriesen

Microsoft ResearchYahoo Research

FacebookEve online

Spinn3rLinkedIn

Nielsen BuzzMetricsMicrosoft Live Labs

Google

Thanks!

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Network Evolution


ns

A1: Cascade shapes


diminishing returns

A5: Densification,

Shrinking diameters

Models & Algorithms




Big data matters

Dynamics of networks

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