Transcript here
Influence propagation in
large graphs - theorems and
algorithms
B. Aditya Prakash
http://www.cs.cmu.edu/~badityap
Christos Faloutsos
http://www.cs.cmu.edu/~christos
Carnegie Mellon University
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Networks are everywhere!
Facebook Network [2010]
Gene Regulatory Network
[Decourty 2008]
Human Disease Network
[Barabasi 2007]
The Internet [2005]
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Dynamical Processes over networks
are also everywhere!
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Why do we care?
• Information Diffusion
• Viral Marketing
• Epidemiology and Public Health
• Cyber Security
• Human mobility
• Games and Virtual Worlds
• Ecology
• Social Collaboration
........
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Why do we care? (1:
Epidemiology)
• Dynamical Processes over networks
[AJPH 2007]
Diseases over contact networks
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CDC data: Visualization of
the first 35 tuberculosis
(TB) patients and their
1039 contacts
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Why do we care? (1:
Epidemiology)
• Dynamical Processes over networks
• Each circle is a hospital
• ~3000 hospitals
• More than 30,000 patients
transferred
[US-MEDICARE
NETWORK 2005]
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Problem: Given k units of
disinfectant, whom to immunize?
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Why do we care? (1:
Epidemiology)
~6x
fewer!
CURRENT PRACTICE
[US-MEDICARE
NETWORK 2005]
OUR METHOD
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Hospital-acquired
inf. took
99K+ lives, cost $5B+ (all per year)
Why do we care? (2: Online
Diffusion)
> 800m users, ~$1B
revenue [WSJ 2010]
~100m active users
> 50m users
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Why do we care? (2: Online
Diffusion)
• Dynamical Processes over networks
Buy Versace™!
Followers
Celebrity
Social Media Marketing
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High Impact – Multiple Settings
epidemic out-breaks
Q. How to squash rumors faster?
products/viruses
Q. How do opinions spread?
transmit s/w patches
Q. How to market better?
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Research Theme
ANALYSIS
Understanding
POLICY/
ACTION
DATA
Large real-world
networks & processes
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Managing
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In this talk
Given propagation models:
Q1: Will an epidemic
happen?
ANALYSIS
Understanding
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In this talk
Q2: How to immunize and
control out-breaks better?
POLICY/
ACTION
Managing
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Outline
• Motivation
• Epidemics: what happens? (Theory)
• Action: Who to immunize? (Algorithms)
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A fundamental question
Strong
Virus
Epidemic?
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example (static graph)
Weak Virus
Epidemic?
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Problem Statement
# Infected
above (epidemic)
below (extinction)
Separate the
regimes?
time
Find, a condition under which
– virus will die out exponentially quickly
– regardless of initial infection condition
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Threshold (static version)
Problem Statement
• Given:
–Graph G, and
–Virus specs (attack prob. etc.)
• Find:
–A condition for virus extinction/invasion
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Threshold: Why important?
•
•
•
•
Accelerating simulations
Forecasting (‘What-if’ scenarios)
Design of contagion and/or topology
A great handle to manipulate the spreading
– Immunization
– Maximize collaboration
…..
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Outline
• Motivation
• Epidemics: what happens? (Theory)
– Background
– Result (Static Graphs)
– Proof Ideas (Static Graphs)
– Bonus 1: Dynamic Graphs
– Bonus 2: Competing Viruses
• Action: Who to immunize? (Algorithms)
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“SIR” model: life immunity
(mumps)
• Each node in the graph is in one of three states
– Susceptible (i.e. healthy)
– Infected
– Removed (i.e. can’t get infected again)
Prob. δ
t=1
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t=2
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t=3
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Terminology: continued
• Other virus propagation models (“VPM”)
– SIS : susceptible-infected-susceptible, flu-like
– SIRS : temporary immunity, like pertussis
– SEIR : mumps-like, with virus incubation
(E = Exposed)
….………….
• Underlying contact-network – ‘who-can-infectwhom’
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Related Work
R. M. Anderson and R. M. May. Infectious Diseases of Humans. Oxford University Press, 1991.
A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex Networks.
Cambridge University Press, 2010.
F. M. Bass. A new product growth for model consumer durables. Management Science,
15(5):215–227, 1969.
D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in real
networks. ACM TISSEC, 10(4), 2008.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly
Connected World. Cambridge University Press, 2010.
A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of
epidemics. IEEE INFOCOM, 2005.
Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free
networks and the effective immunization. arXiv:cond-at/0305549 v2, Aug. 6 2003.
H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42, 2000.
H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer
Lecture Notes in Biomathematics, 46, 1984.
J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses.
IEEE Computer Society Symposium on Research in Security and Privacy, 1991.
J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE
Computer Society Symposium on Research in Security and Privacy, 1993.
R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical
Review Letters 86, 14, 2001.
………
………
………
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All are about either:
• Structured
topologies (cliques,
block-diagonals,
hierarchies, random)
• Specific virus
propagation models
• Static graphs
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Outline
• Motivation
• Epidemics: what happens? (Theory)
– Background
– Result (Static Graphs)
– Proof Ideas (Static Graphs)
– Bonus 1: Dynamic Graphs
– Bonus 2: Competing Viruses
• Action: Who to immunize? (Algorithms)
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How should the answer look
like?
• Answer should depend on:
– Graph
– Virus Propagation Model (VPM)
• But how??
– Graph – average degree? max. degree? diameter?
– VPM – which parameters?
– How to combine – linear? quadratic? exponential?
2
2
(
d
d
)
/
d
?
d
diameter
? avg
…..
avg
m
ax
avg
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Static Graphs: Our Main Result
• Informally,
For,
any arbitrary topology (adjacency
matrix A)
any virus propagation model (VPM) in
standard literature
•
the
epidemic threshold depends only
1.on the λ, first eigenvalue of A, and
2.some constant CVPM
, determined by the
virus propagation model
w/ Deepay
Chakrabarti
λ
CVPM
No
epidemic if
λ * CVPM< 1
In Prakash+ ICDM 2011 (Selected among best papers).
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Our thresholds for some models
• s = effective strength
• s < 1 : below threshold
Models
SIS, SIR, SIRS, SEIR
SIV, SEIV
Effective Strength
(s)
s=λ.
s=λ.
SI
I
V
V
(H.I.V.) s
1
2
1
2
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1v2 2
= λ . v v
1
2
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Threshold (tipping
point)
s=1
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Our result: Intuition for λ
“Official” definition:
• Let A be the adjacency
matrix. Then λ is the root
with the largest magnitude of
the characteristic polynomial
of A [det(A – xI)].
“Un-official” Intuition
• λ ~ # paths in the graph
k
A
≈
u
k
.u
• Doesn’t give much intuition!
k(i, j) = # of paths i j
A
of length k
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Largest Eigenvalue (λ)
better connectivity
λ≈2
λ≈2
N = 1000
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higher λ
λ= N
λ = N-1
λ= 31.67
λ= 999
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N nodes
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Footprint
Fraction of Infections
Examples: Simulations – SIR
(mumps)
Effective Strength
Time ticks
(a) Infection profile
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(b) “Take-off” plot
PORTLAND graph: synthetic population,
31 million links, 6 million nodes
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Footprint
Fraction of Infections
Examples: Simulations – SIRS
(pertusis)
Time ticks
Effective Strength
(a) Infection profile
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(b) “Take-off” plot
PORTLAND graph: synthetic population,
31 million links, 6 million nodes
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Outline
• Motivation
• Epidemics: what happens? (Theory)
– Background
– Result (Static Graphs)
– Proof Ideas (Static Graphs)
– Bonus 1: Dynamic Graphs
– Bonus 2: Competing Viruses
• Action: Who to immunize? (Algorithms)
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See paper for
full proof
General VPM
structure
Model-based
λ * CVPM< 1
Topology and
stability
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Graph-based
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Outline
• Motivation
• Epidemics: what happens? (Theory)
– Background
– Result (Static Graphs)
– Proof Ideas (Static Graphs)
– Bonus 1: Dynamic Graphs
– Bonus 2: Competing Viruses
• Action: Who to immunize? (Algorithms)
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Dynamic Graphs: Epidemic?
Alternating behaviors
DAY
(e.g., work)
adjacency
matrix
8
8
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Dynamic Graphs: Epidemic?
Alternating behaviors
NIGHT
(e.g., home)
adjacency
matrix
8
8
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Model Description
Healthy
• SIS model
N2
Prob. β
– recovery rate δ
– infection rate β
N1
X
Prob. δ
Infected
N3
• Set of T arbitrary graphs
day
N
night
N
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N
, weekend…..
N
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Our result: Dynamic Graphs
Threshold
• Informally, NO epidemic if
eig (S) =
Single number!
Largest eigenvalue of
The system matrix S
In Prakash+, ECML-PKDD 2010
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<1
S =
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Infection-profile
log(fraction infected)
MIT Reality
Mining
Synthetic
ABOVE
ABOVE
AT
AT
BELOW
BELOW
Time
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Footprint (#
infected @
“steady state”)
“Take-off” plots
Synthetic
MIT Reality
EPIDEMIC
Our
threshold
NO EPIDEMIC
Our
threshold
EPIDEMIC
NO EPIDEMIC
(log scale)
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Outline
• Motivation
• Epidemics: what happens? (Theory)
– Background
– Result (Static Graphs)
– Proof Ideas (Static Graphs)
– Bonus 1: Dynamic Graphs
– Bonus 2: Competing Viruses
• Action: Who to immunize? (Algorithms)
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Competing Contagions
iPhone v Android
Blu-ray v HD-DVD
Biological common flu/avian flu, pneumococcal inf etc
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A simple model
• Modified flu-like
• Mutual Immunity (“pick one of the two”)
• Susceptible-Infected1-Infected2-Susceptible
Virus 2
Virus 1
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Question: What happens in the
end?
Number of
Infections
green: virus 1
red: virus 2
Footprint @ Steady State
Footprint @ Steady State
ASSUME:
Virus 1 is stronger than Virus 2
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= ?
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Question: What happens in the
end? Footprint @ Steady State
Number of
Infections
green: virus 1
red: virus 2
Footprint @ Steady State
??
Strength
Strength
=
2
Strength
Strength
ASSUME:
Virus 1 is stronger than Virus 2
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Answer: Winner-Takes-All
Number of
Infections
green: virus 1
red: virus 2
ASSUME:
Virus 1 is stronger than Virus 2
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Our Result: Winner-Takes-All
Given our model, and any graph, the
weaker virus always dies-out completely
1. The stronger survives only if it is above threshold
2. Virus 1 is stronger than Virus 2, if:
strength(Virus 1) > strength(Virus 2)
3. Strength(Virus) = λ β / δ same as before!
In Prakash+ WWW 2012
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Real Examples
[Google Search Trends data]
Reddit v Digg
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Blu-Ray v HD-DVD
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Outline
• Motivation
• Epidemics: what happens? (Theory)
• Action: Who to immunize? (Algorithms)
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Full Static Immunization
Given: a graph A, virus prop. model and budget k;
Find: k ‘best’ nodes for immunization (removal).
?
?
k=2
?
?
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Outline
• Motivation
• Epidemics: what happens? (Theory)
• Action: Who to immunize? (Algorithms)
– Full Immunization (Static Graphs)
– Fractional Immunization
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Challenges
• Given a graph A, budget k,
Q1 (Metric) How to measure the ‘shieldvalue’ for a set of nodes (S)?
Q2 (Algorithm) How to find a set of k nodes
with highest ‘shield-value’?
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Proposed vulnerability measure
λ
λ is the epidemic threshold
“Safe”
“Vulnerable”
“Deadly”
Increasing λ
Increasing vulnerability
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A1: “Eigen-Drop”: an ideal shield
value
Eigen-Drop(S)
Δ λ = λ - λs
9
9
11
10
Δ
9
10
1
1
4
4
8
8
2
2
5
5
6
Original Graph
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7
3
7
3
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Without {2, 6}
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(Q2) - Direct Algorithm too
expensive!
• Immunize k nodes which maximize Δ λ
S = argmax Δ λ
• Combinatorial!
• Complexity:
– Example:
• 1,000 nodes, with 10,000 edges
• It takes 0.01 seconds to compute λ
• It takes 2,615 years to find 5-best nodes!
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A2: Our Solution
• Part 1: Shield Value
– Carefully approximate Eigen-drop (Δ λ)
– Matrix perturbation theory
• Part 2: Algorithm
– Greedily pick best node at each step
– Near-optimal due to submodularity
• NetShield (linear complexity)
– O(nk2+m) n = # nodes; m = # edges
In Tong, Prakash+ ICDM 2010
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Experiment: Immunization
quality
Log(fraction of
infected
nodes)
PageRank
Betweeness (shortest path)
Degree
Lower
is
better
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Acquaintance
Eigs (=HITS)
NetShield
Time
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Outline
• Motivation
• Epidemics: what happens? (Theory)
• Action: Who to immunize? (Algorithms)
– Full Immunization (Static Graphs)
– Fractional Immunization
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Fractional Immunization of Networks
B. Aditya Prakash, Lada Adamic, Theodore
Iwashyna (M.D.), Hanghang Tong, Christos
Faloutsos
Under review
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Fractional Asymmetric
Immunization
Drug-resistant Bacteria
(like XDR-TB)
Another
Hospital
Hospital
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Fractional Asymmetric
Immunization
Drug-resistant Bacteria
(like XDR-TB)
Another
Hospital
Hospital
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Fractional Asymmetric
Immunization
Problem: Given k units of disinfectant,
how to distribute them to maximize
hospitals saved?
Another
Hospital
Hospital
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Our Algorithm “SMARTALLOC”
~6x
fewer!
[US-MEDICARE NETWORK 2005]
• Each circle is a hospital, ~3000 hospitals
• More than 30,000 patients transferred
CURRENT PRACTICE
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SMART-ALLOC
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Wall-Clock
Time
Running Time
> 1 week
≈
> 30,000x
speed-up!
Lower
is
better
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14 secs
Simulations
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SMART-ALLOC
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Lower
is
better
Experiments
PENN-NETWORK
SECOND-LIFE
~5 x
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K = 200
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~2.5 x
K = 2000
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Acknowledgements
Funding
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References
1.
2.
3.
4.
5.
6.
7.
Threshold Conditions for Arbitrary Cascade Models on Arbitrary Networks (B. Aditya
Prakash, Deepayan Chakrabarti, Michalis Faloutsos, Nicholas Valler, Christos Faloutsos) In IEEE ICDM 2011, Vancouver (Invited to KAIS Journal Best Papers of ICDM.)
Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms (B.
Aditya Prakash, Hanghang Tong, Nicholas Valler, Michalis Faloutsos and Christos
Faloutsos) – In ECML-PKDD 2010, Barcelona, Spain
Epidemic Spreading on Mobile Ad Hoc Networks: Determining the Tipping Point
(Nicholas Valler, B. Aditya Prakash, Hanghang Tong, Michalis Faloutsos and Christos
Faloutsos) – In IEEE NETWORKING 2011, Valencia, Spain
Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash,
Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon
On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina EliassiRad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia
Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore
Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission
Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko
Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under
Submission
http://www.cs.cmu.edu/~badityap/
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Propagation on Large Networks
B. Aditya Prakash
Christos Faloutsos
Analysis
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Policy/Action
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Data
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