Information Diffusion
Kristina Lerman
University of Southern California
CS 599: Social Media Analysis
University of Southern California
1
Information diffusion on Twitter follower graph
Diffusion on networks
• The spread of disease, ideas, behaviors, … on a network can
be described as a contagion process where an active node
(infected/informed/adopted) activates its non-active
neighbors with some probability
– … creates a cascade on a network
• How large do cascades become?
• What determines their growth?
Gangnam style
• "Gangnam Style" became the first YouTube video to reach
one billion views
• As of May 31, 2014, the music video has been viewed over
two billion times  almost 13,000 man-years!
Ebola outbreak
Studying diffusion: data
• Large-scale data about contagion processes is now available
– YouTube views [Crane & Sornette 2008]
– Flickr favorites [Cha, Mislove & Gummadi, 2009]
– Twitter retweets [Ghosh & Lerman, 2011]
– Facebook likes [Dow, Adamic & Frigerri, 2014]
• Challenges
– Volume of the data
• Storing and processing data
– Complexity
• How does the “whole” depend on its “parts”
• Networks add to complexity
Studying diffusion: methods
• Analytic models
– Model cascading behavior, e.g., differential equation
• Solve model under different conditions
• Simulations
– Implement a model to synthetically recreate the process
• Empirical studies
– Does observations of real-world data agree with model
and simulations results?
Cascading Behavior in Complex Socio-technical
Networks (Borge-Holthoefer et al.)
• Research questions
– How can global cascades occur on sparse networks?
– What affects cascade growth?
• Network topology
• How node is activated by an active neighbor
• Properties of the diffusing item?
– How can cascades be characterized?
• Models of diffusion on networks
– Threshold model
– Epidemic models
– Complex contagion
• Empirical data allow testing of models
Threshold model (Watts 2002)
• Each node has some “infection threshold” fi
– Node becomes infected if fraction of infected neighbors is
more than threshold
Exposure response function
f3
f4
1
f2
r
infected
exposed
infection prob.
f1
fiki
number infected neighbors
Threshold model (Watts 2002)
• Under some conditions, global cascades can start from a few
“infected” seeds
– Network topology and individual thresholds interact in
cascading behavior
Epidemic models
• Infected nodes propagate contagion to susceptible neighbors
with probability m (transmissibility or virality of contagion)
Exposure response function
infected
infection prob.
1
exposed
number infected neighbors
Epidemic models
• Epidemic threshold t:
– For m < t, localized cascades (epidemic dies out)
– For m > t, global cascades
• Epidemic threshold depends on topology only: largest
eigenvalue of adjacency matrix of the network
– True for any network
Num.
infected
nodes
N
0
Epidemic
threshold
Transmissibility, m
Complex contagion
• Virus can propagate with a single exposure. Spread of
behaviors requires multiple exposures.
• Non-monotonic exposure response
Exposure response function
infected
infection prob.
1
exposed
number infected neighbors
Characterizing cascades
• Connected tree-like subgraph. Typically star-like
• Size related to centrality
Seeding large outbreaks
• How to select seeds that will initiate large outbreaks?
– Influence maximization
• Are some network positions better at triggering large
outbreaks?
– Being a hub is sufficient but not necessary
• “Million follower fallacy” (Cha et al)
– “hub fire wall” – epidemics die out when reaching a hub
Epidemic Spreading on Real Networks: An
Eigenvalue Viewpoint [Wang et al, 2003]
• Research questions
– How do epidemic cascades on a real network?
– Does an epidemic threshold exist for a given network?
• Contributions
– Model how epidemics propagate on a network
– Propagation depends on network topology
• epidemic threshold is related to the largest eigenvalue of the
adjacency graph describing the network
Homogeneous mixing model
• Homogeneous mixing
– Each node interacts with every other node
• Infection rate m: a node infects neighbor with probability m
• Curing rate d: infected node is cured with probability d
infected
exposed
cured
Homogeneous mixing model
• Homogeneous mixing
– Each node interacts with every other node
• Infection rate m: a node infects neighbor with probability m
• Curing rate d: infected node is cured with probability d
infected
exposed
cured
Homogeneous mixing model
• Homogeneous mixing
– Each node interacts with every other node
• Infection rate m: a node infects neighbor with probability m
• Curing rate d: infected node is cured with probability d
infected
exposed
cured
Homogeneous mixing model
• Homogeneous mixing
– Each node interacts with every other node
• Infection rate m: a node infects neighbor with probability m
• Curing rate d: infected node is cured with probability d
infected
exposed
cured
Homogeneous mixing: epidemic threshold
• Infection rate m: node infects neighbor with probability m
• Curing rate d: node is cured with probability d
– Number of infected nodes: Ninf = (1-d/m<k>)N
– Epidemic threshold: critical value of m/d = t =1/<k>
• beyond which Ninf N, but below Ninf 0
infected
exposed
cured
Epidemics on networks
• Homogeneous mixing model is a good approximation of virus
propagation in a population where contact among individuals
is homogeneous, i.e., each individual is equally likely to
encounter another
– Public spaces: airports, shopping centers, …
– Schools
– Public transportation
• But, social interactions are usually structured
– what role does network structure play in epidemic
spread?
– How does the size of cascades depend on network
properties?
Model of epidemic cascades on a network
Simulations on real and synthetic graphs
• Simulate epidemics on
– Real-world networks
– Scale-free graphs (power law degree distribution)
– Random graphs (Poisson degree distribution)
• Results are the same as homogeneous mixing model
• Simulations steps
– Start with a set of randomly chosen infected nodes
– At each time step
• Infected node attempts to infect each neighbor (probability m)
• An infected node is cured (probability d)
– Continue until number of infected nodes no longer
changes
Simulation results on real-world network
• Simulations on 10,900 node Oregon network graph, with
<k>=5.72, m=0.14
Cascade size vs time
m/d=1.75
m/d=0.58
Epidemic threshold
Epidemic threshold and cascade growth
m/d=0.4
m/d=0.2
m/d=0.13
m/d=0.06
m/d=0.1
Epidemic threshold and cascade size
Num. infected nodes
N
Epidemic threshold
0
Effective Transmissibility, m/d
Summary
• A variety of models proposed to explain cascading behavior
on networks
– Some models explain the relationship between properties
of the network and properties of cascades, e.g., epidemic
threshold depends on the eigenvalue of the adjacency
matrix of the graph
– Some models can produce global cascades
• What does data say?
The Structure
of
Online Diffusion
Networks
SHARAD GOEL, Yahoo! Research
DUNCAN J. WATTS, Yahoo! Research
DANIEL G. GOLDSTEIN, Yahoo! Research
“A relatively small number of seeds can trigger a
relatively large number of adoptions via some, usually
multistep, diffusion process”
How often
How much
Is it worth it
Findings
Most cascades small and shallow
Most adoptions lie in such cascades.
Rare for adoptions to result from chains of
referrals
Yahoo! Kindness
one month period in 2010, Yahoo!’s
philanthropic arm launched a website
(kindness.yahoo.com)
7 Different
Sources
59,000 users adopted the campaign
Zync
7 Different
Sources
a plug-in for Yahoo! Messenger, an
instant messaging (IM) application, that
allows pairs of users to watch videos
synchronously while sending instant
messages to one another.
The Secretary Game
7 Different
Sources
Players are encouraged to share the
game’s URL with at least three other
people with an explanation that the
game designers are seeking the world’s
best players.
Twitter News Stories.
80,000 news stories posted on the Twitter
during November 2011, where the original
article was distributed by one of five popular
news sites: The New York Times, CNN, MSNBC,
Yahoo! News, and The Huffington Post.
7 Different
Sources
Tweeted  Adopted
Twitter Videos
540,000 YouTube videos posted on
Twitter during November 2011
7 Different
Sources
Tweeted  Adopted
Friend Sense
7 Different
Sources
third-party Facebook application that
queried respondents about their
political views as well as their beliefs
about their friends’ political views
Yahoo! Voice
paid service launched in 2004 that allows
users to make voice- over-IP calls to phones
through Yahoo! Messenger.
7 Different
Sources
1.8 million users purchased voice credits,
who are defined as adopters
Data Sources
Varied
Cost
Nature of the
network
Incentive
Timescale
• d
“The usual intuition regarding heavy-tailed
distributions, however, is that large events,
although rare, are sufficiently large to
dominate certain key properties of the
corresponding system.”
Authors point of view
Diffusion on online social networks does not
really follow epidemic models.
Researchers should focus on sub-critical process.
Authors point of view
What accounts for sudden popularity of some
YouTube videos or products like Gmail and
Facebook?
Mass Media and traditional advertisement.
Implementation Details
(Time Permitting)
MapReduce parallel computation
framework
Tree Canonicalization
Thanks For
Listening
Questions?!
Comments!