Dynamics of networks

Dynamics of networksJure LeskovecComputer Science DepartmentCornell University / Stanford University

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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)

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Rich data as 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-Albert ’99, Faloutsos3 ‘99], 6-degrees of separation [Milgram ’67], Small-diameter [Albert-Jeong-Barabasi ‘99], Bipartite cores [Kumar et al. ’99], Network motifs [Milo et al. ‘02], Communities [Newman ‘99], 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], Latent Space Models [Raftery et al. ‘02], Searchability [Kleinberg ‘00], Bowtie [Broder et al. ‘00], Exponential Random Graphs [Frank-Strauss ‘86], Transit-stub [Zegura ‘97], Jellyfish [Tauro et al. ‘01]

We know much less about processes and

dynamics of networks

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This talk: 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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This talk: Scale matters We need massive network data for the

patterns to emerge: MSN Messenger network [WWW ’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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This talk: The structure

Network Evolution

Diffusion &

Cascades

Patterns & Observatio

ns

Q1: How does the network structure change as the network evolves/grows?Q2: What is edge attachment at the microscopic level?

Q4: What do cascades look like?Q5: How likely are people to get influenced?

Models & Algorithms

Q3: How do we generate realistically looking and evolving networks?

Q6: How do we find influential nodes and quickly detect epidemics?

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Network Evolution

Diffusion &

Cascades


ns

Q1: How does the network structure change as the network evolves/grows?Q2: What is edge attachment at the microscopic level?


Models & Algorithms

Q3: How do we generate realistically looking and evolving networks?


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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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Q1) 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]

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Networks are denser over time Densification Power Law:

a … densification exponent (1 ≤ a ≤ 2)

Q1) 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]

Is shrinking diameter just a consequence of

densification?

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Q1) Diameter of a densifying Gnp

diam

eter

size of the graph

Erdos-Renyi random graph

Densification exponent a =1.3

There is more to shrinking diameter

than just densification

Densifying random graph has increasing diameter

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Q1) Diameter of a rewired network Compare diameter

of a: True network (red) Random network with

the same degree distribution (blue)

diam

eter

size of the graphdi

amet

er

Citations

Densification + scale free structure give shrinking diameter

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Q2) Edge attachment We directly observe atomic events of network

evolution (and not only network snapshots)

We observe evolution at finest scale Test individual edge attachment

Directly observe events leading to network properties Compare network models by likelihood

(and not by just summary network statistics)

and so on for

millions…

[w/ Backstrom-Kumar-Tomkins, KDD ’08]

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Setting: Edge-by-edge evolution Network datasets

Full temporal information from the first edge onwards LinkedIn (N=7m, E=30m), Flickr (N=600k, E=3m), Delicious

(N=200k, E=430k), Answers (N=600k, E=2m)

How are edge destinations selected? Where do new edges attach?

Are edges more likely to attach to high degree nodes? Are edges more likely to attach to nodes that are close?


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Edge attachment degree bias

Are edges more likely to connect to higher degree nodes?

kkpe )(Gnp

PA

Flickr

Network

τ

Gnp 0PA 1

Flickr 1Delicio

us1

Answers

0.9

LinkedIn

0.6


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u wv

But, edges also attach locally Just before the edge (u,w) is placed how many

hops is between u and w?

Network

% Δ

Flickr 66%Delicio

us28%

Answers

23%

LinkedIn

50%

Fraction of triad closing

edges

Real edges are local.Most of them close

triangles!


GnpPA

Flickr

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Q3) 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? Q: What is a good model and how do we fit it?

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)

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

pij

Edge probability Edge probability

Kronecker product of graph adjacency matrices(actually, there is also a nice social interpretation of the model)

Given a real graph. How to estimate the initiator G1?

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Kronecker graphs: Interpretation Initiator matrix is a similarity matrix Node u is described with k binary

attributes u1, u2 ,…, uk Probability of link between nodes u,v:

P(u,v) = ∏ G1[ui, vi]

1G a bc d

a b

c d

a bc d

This is exactly Kronecker product

v

u P(u,v) = b∙d∙c∙b

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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

1G a bc d

1GP( | ) Kroneckerarg max

We estimate the model in O(E)

1G

[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

1G0.99 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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Kronecker & Network structure Fitting Epinions we obtained G1 =

What does this tell about the network structure?

0.91 0.54

0.49 0.13

Core0.9

edgesPeriphery0.1 edges

0.5 edges

0.5 edges

Core-periphery (jellyfish, octopus)

[w/ Dasgupta-Lang-Mahoney, WWW ’08]

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Small vs. Large networks Small and large networks are fundamentally

different

Collaboration network (N=4,158, E=13,422)

Scientific collaborations (N=397, E=914)

0.91 0.54

0.49 0.13

0.9 0.1

0.1 0.9G1 = G1 =

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Network Evolution

Diffusion & Cascades


ns

A1: Densification, shrinking diameters A2: Local edge attachment


Models & Algorithms

A3: Kronecker graphs


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

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

Distinction has consequences for models and algorithms

k = number of friends adoptingProb

. of a

dopt

ion

k = number of friends adoptingProb

. of a

dopt

ion

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 receivedAdoption curve follows the

diminishing returns.Can we exploit this?

[w/ Adamic-Huberman, EC ’06]

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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]

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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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Problem structure: Submodularity

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


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

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

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Solution: CELF Algorithm We develop CELF 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’’))


a

b

ca

b

c

d

d

Marginal rewarde

Current solution: {a}Current solution: {}Current solution: {a, c}

Theorem: If R is submodular then CELF is near optimal:

CELF achieves ½(1-1/e) factor approximation

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Blogs: Information epidemics Question: Which blogs should one read to be

most up to date? Idea: Select blogs to cover the blogosphere.

• Each dot is a blog• Proximity is based on the number of common cascades

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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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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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Conclusions and reflections

Why are networks the way they are? Only recently have basic properties been observed

on a large scale Confirms social science intuitions; calls others into

question.

Benefits of working with large data What patterns do we observe in massive networks? What microscopic mechanisms cause them?

Social network of the whole world?

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Planetary look on a small-world

Milgram’s small world experiment

(i.e., hops + 1)

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

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Reflections on Diffusion Predictive modeling of information diffusion

When, where and what information will create a cascade?

Where should one tap the network to get the effect they want? Social Media Marketing

How do news and information spread New ranking and influence measures Sentiment analysis from cascade structure

How to introduce incentives?

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What’s next?

Observations: Data analysis

Models: Predictions

Algorithms: Applications

Actively influencing the network

Documents

Dynamics of networks