Dynamics of networks
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Transcript of 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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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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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?
[w/ Backstrom-Kumar-Tomkins, KDD ’08]
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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
0.6
[w/ Backstrom-Kumar-Tomkins, KDD ’08]
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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%
50%
Fraction of triad closing
edges
Real edges are local.Most of them close
triangles!
[w/ Backstrom-Kumar-Tomkins, KDD ’08]
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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This talk: The structure
Network Evolution
Diffusion & Cascades
Patterns & Observatio
ns
A1: Densification, shrinking diameters A2: Local edge attachment
Q4: What do cascades look like?Q5: How likely are people to get influenced?
Models & Algorithms
A3: Kronecker graphs
Q6: How do we find influential nodes and quickly detect epidemics?
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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
( )
[w/ Krause-Guestrin et al., KDD ’07]
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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”)
[w/ Krause-Guestrin et al., KDD ’07]
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’’))
[w/ Krause-Guestrin et al., KDD ’07]
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)
[w/ Krause-Guestrin et al., KDD ’07]
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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