Dynamics of networks

Dynamics of networks

Jure LeskovecMachine Learning DepartmentCarnegie Mellon University

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Networks: Rich data Today: 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

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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

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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?

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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

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My research: The structure

Diffusion & Cascades

Network Evolution

Patterns & Observatio

ns

Q1: What do cascades look like? Trees? Stars? Chains?

Q4: How does network structure change as the network grows/evolves?

Models & Algorithms

Q2: How do we find influential nodes?Q3: How do we quickly detect epidemics?

Q5: How do we generate realistic looking and evolving networks?

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Network Evolution


ns

Q1: What do cascades look like? Trees? Stars? Chains?


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)

prop

agat

ion

[w/ Kleinberg-Singh, PAKDD ’06]

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Network Evolution


ns

A1: Cascade shapes. Viral marketing cascades more social.


Models & Algorithms


Q5:How do we generate realistic looking and evolving networks?

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Cascade & outbreak detection Blogs – information epidemics

Which are the influential/infectious blogs?

Viral marketing 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

Problem structure:Submodularity of the reward functions

CELF:algorithm with approximate guarantee

Speed up:Lazy evaluation to speed-up CELF

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


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Reward function are submodular

Theorem: Reward function R is submodular (diminishing returns, think of it as “convexity”)

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

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)

A B

S1

S2

Placement A={S1, S2}

S’

New monitored

node:

Adding S’ helps a lotS2

S4

S1

S3

Placement B={S1, S2, S3, S4}

S’

Adding S’ helps very

little

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Solution: CELF Algorithm

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: If R is submodular then 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 21U Exter 20Bentley systems 19Technion (1) 14Bordeaux 12U Cyprus 11U Guelph 7U Michigan 4Michigan Tech U 3Malcolm 2Proteo 2Technion (2) 1

Number of placed sensors

Population

saved(higher is better)

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

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Network Evolution


ns



Models & Algorithms

A2, A3: CELF algorithm for detecting cascades and outbreaks


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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)

Q4) 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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Q4) 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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Q5) Generating realistic graphs Want to generate realistic networks:

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

sensitive graphs, …

Q: What is a good model that we can fit to the data? A: Next slide. 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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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

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

pij

Edge probability

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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( | ) Kroneckerarg 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

prob

abilit

y

0.99 0.54

0.49 0.13

Path lengths

number of hops

# re

acha

ble

pairs

“Network” values

rankne

twor

k va

lue

[w/ Faloutsos, ICML ’07]

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Diffusion &

CascadesNetwork Evolution

Patterns &

Observations


A4: Densification,

Shrinking diameters

Models & Algorithm

s

algorithm for

detecting cascades and

outbreaks

A5: Kronecker graphs,

Estimating Kronecker initiator

Other topics

Conclusion andFuture work

A2, A3: CELF

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Other work

P1) Query Projections: Predicting search result quality without

page content

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

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P1) Query 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

HyperlinksNon-search resultsconnecting the graph

[w/ Dumais-Horvitz, WWW ’07]

QQuery Results Projection on the web graph

Query projection graph

Generate graphical features

Query connection graph

• -- -- ----• --- --- ----• ------ ---• ----- --- --• ------ -----• ------ -----

Construct case

libraryPredictions

Query projections: Idea

Extracting graph features The idea

Find features that describe the structure of the graph

Then use the features for machine learning

Want features that describe Connectivity of the graph Centrality of projection and

connector nodes Clustering and density of the

core of the graph

vs.

Search results quality Dataset:

Graph: 40 million nodes, 720 million edges 30,000 queries with top 20 results for each

▪ Human assigned relevance. 6-point scale: Perfect to Bad

Task: Predict the highest rating in the set of results

▪ 2-class problem: “Good” (top 3 ratings) vs. “Poor” (bottom 3)

Result: Predict search result quality with 80% accuracy

just from the connection patterns between the results

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Query Projections: Intuition

Predict “Good”Predict “Poor”

Good?

Poor?

[w/ Dumais-Horvitz, WWW ’07]

Search quality: The model

Good result sets have: Search result nodes are hub

nodes in the graph Small connector node degrees Big connected component Few isolated nodes in projection

graph Few connector nodes

Attributes Accuracy

Marginals 0.55RankNet 0.63Query

projection 0.82

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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

46



Network Evolution


ns


A4: 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 Steer the network evolution 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

52

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 ManoheyZoubin 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

Acknowledgments

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Network Evolution


ns


A4: Densification,

Shrinking diameters

Models & Algorithms




Big data matters

Documents

Dynamics of networks