Propagation on Large NetworksB. Aditya Prakash
http://www.cs.cmu.edu/~badityap
Christos Faloutsoshttp://www.cs.cmu.edu/~christos
Carnegie Mellon UniversityINARC Meeting – May 2nd
Prakash and Faloutsos 2012 2
Preaching to the choir:Networks are everywhere!
Human Disease Network [Barabasi 2007]
Gene Regulatory Network [Decourty 2008]
Facebook Network [2010]
The Internet [2005]
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Focus of this talk:Dynamical Processes over
networks are also everywhere!
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Why do we care?• Social collaboration• Information Diffusion• Viral Marketing• Epidemiology and Public Health• Cyber Security• Human mobility • Games and Virtual Worlds • Ecology........
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Why do we care? (1: Epidemiology)
• Dynamical Processes over networks[AJPH 2007]
CDC data: Visualization of the first 35 tuberculosis (TB) patients and their 1039 contacts
Diseases over contact networks
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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]
Problem: Given k units of disinfectant, whom to immunize?
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Why do we care? (1: Epidemiology)
CURRENT PRACTICE OUR METHOD
~6x fewer!
[US-MEDICARE NETWORK 2005]
Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)
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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
Celebrity
Buy Versace™!
Followers
Social Media Marketing
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Why do we care? (3: To change the world?)
• Dynamical Processes over networks
Social networks and Collaborative Action
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High Impact – Multiple Settings
Q. How to squash rumors faster?
Q. How do opinions spread?
Q. How to market better?
epidemic out-breaks
products/viruses
transmit s/w patches
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Research Theme
DATALarge real-world
networks & processes
ANALYSISUnderstanding
POLICY/ ACTIONManaging
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Research Theme – Public Health
DATAModeling # patient
transfers
ANALYSISWill an epidemic
happen?
POLICY/ ACTION
How to control out-breaks?
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Research Theme – Social Media
DATAModeling Tweets
spreading
POLICY/ ACTION
How to market better?
ANALYSIS# cascades in
future?
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In this talk
ANALYSISUnderstanding
Given propagation models:
Q1: Will an epidemic happen?
Prakash and Faloutsos 2012 16
In this talk
Q2: How to immunize and control out-breaks better?
POLICY/ ACTIONManaging
Prakash and Faloutsos 2012 17
Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)
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A fundamental questionStrong Virus
Epidemic?
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example (static graph)Weak Virus
Epidemic?
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Problem Statement
Find, a condition under which– virus will die out exponentially quickly– regardless of initial infection condition
above (epidemic)
below (extinction)
# Infected
time
Separate the regimes?
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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…..
Prakash and Faloutsos 2012 23
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)
Prakash and Faloutsos 2012 24
“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. β Prob. δ
t = 1 t = 2 t = 3
Background
Prakash and Faloutsos 2012 25
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-infect-whom’
Background
Prakash and Faloutsos 2012 26
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.
……… ……… ………
All are about either:
• Structured topologies (cliques, block-diagonals, hierarchies, random)
• Specific virus propagation models
• Static graphs
Background
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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?
?diameterdavg ?/)( max22 ddd avgavg …..
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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 , determined by
the virus propagation model
λVPMC
No epidemic if λ *
< 1
VPMCVPMC
In Prakash+ ICDM 2011 (Selected among best papers).
Prakash and Faloutsos 2012 30
Our thresholds for some models
• s = effective strength• s < 1 : below threshold
Models Effective Strength (s) Threshold (tipping point)
SIS, SIR, SIRS, SEIRs = λ .
s = 1
SIV, SEIV s = λ .
(H.I.V.) s = λ .
12
221
vvv
2121 VVISI
Prakash and Faloutsos 2012 31
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)].
• Doesn’t give much intuition!
“Un-official” Intuition • λ ~ # paths in the
graph
uu≈ .k
kA
(i, j) = # of paths i j of length k
kA
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better connectivity higher λ
Largest Eigenvalue (λ)
Prakash and Faloutsos 2012 33N nodes
Largest Eigenvalue (λ)
λ ≈ 2 λ = N λ = N-1
N = 1000λ ≈ 2 λ= 31.67 λ= 999
better connectivity higher λ
Prakash and Faloutsos 2012 34
Examples: Simulations – SIR (mumps)
(a) Infection profile (b) “Take-off” plot
PORTLAND graph: synthetic population, 31 million links, 6 million nodes
Frac
tion
of In
fecti
ons
Foot
prin
tEffective StrengthTime ticks
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Examples: Simulations – SIRS (pertusis)
Frac
tion
of In
fecti
ons
Foot
prin
tEffective StrengthTime ticks
(a) Infection profile (b) “Take-off” plot
PORTLAND graph: synthetic population, 31 million links, 6 million nodes
Prakash and Faloutsos 2012 36
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)
37
λ * < 1VPMC
Graph-based
Model-based
Proof Sketch
General VPM structure
Topology and stability
38
Models and more modelsModel Used for
SIR Mumps
SIS Flu
SIRS Pertussis
SEIR Chicken-pox
……..
SICR Tuberculosis
MSIR Measles
SIV Sensor Stability
H.I.V.……….
2121 VVISI
39
Ingredient 1: Our generalized model
Endogenous Transitions
Susceptible Infected
Vigilant
Exogenous Transitions
Endogenous Transitions
Endogenous Transitions
Susceptible Infected
Vigilant
40
Special case
Susceptible Infected
Vigilant
41
Special case: H.I.V.
2121 VVISI
Multiple Infectious, Vigilant states
“Terminal”
“Non-terminal”
42
Ingredient 2: NLDS+Stability
• View as a NLDS– discrete time – non-linear dynamical system (NLDS)
Probability vector Specifies the state of the system at time t
Details
size mN x 1
.
.
.
.
.
size N (number of nodes in the graph)
.
.
.
S
I
V
43
Ingredient 2: NLDS + Stability
• View as a NLDS– discrete time – non-linear dynamical system (NLDS)
Non-linear functionExplicitly gives the evolution of system
Details
size mN x 1
.
.
.
.
.
.
.
.
44
Ingredient 2: NLDS + Stability
• View as a NLDS– discrete time – non-linear dynamical system (NLDS)
• Threshold Stability of NLDS
45
= probability that node i is not attacked by any of its infectious neighbors
Special case: SIR
size 3N x 1 I
R
S
NLDS
I
R
S
Details
46
Fixed Point
11.
00.
00.
State when no node is infected
Q: Is it stable?
Details
47
Stability for SIR
Stableunder threshold
Unstableabove threshold
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λ * < 1VPMC
Graph-based
Model-basedGeneral VPM structure
Topology and stability
See paper for full proof
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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?
adjacency matrix
8
8
Alternating behaviorsDAY (e.g., work)
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adjacency matrix
8
8
Dynamic Graphs: Epidemic?Alternating behaviorsNIGHT
(e.g., home)
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• SIS model– recovery rate δ– infection rate β
• Set of T arbitrary graphs
Model Description
day
N
N night
N
N , weekend…..
Infected
Healthy
XN1
N3
N2
Prob. βProb. β Prob. δ
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• Informally, NO epidemic if
eig (S) = < 1
Our result: Dynamic Graphs Threshold
Single number! Largest eigenvalue of The system matrix S
In Prakash+, ECML-PKDD 2010
S =
Details
Prakash and Faloutsos 2012 54
Synthetic MIT Reality Mining
log(fraction infected)
Time
BELOW
AT
ABOVE ABOVE
AT
BELOW
Infection-profile
Prakash and Faloutsos 2012 55
“Take-off” plotsFootprint (# infected @ “steady state”)
Our threshold
Our threshold
(log scale)
NO EPIDEMIC
EPIDEMIC
EPIDEMIC
NO EPIDEMIC
Synthetic MIT Reality
Prakash and Faloutsos 2012 56
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)
57
Competing ContagionsiPhone v Android
Blu-ray v HD-DVD
Biological common flu/avian flu, pneumococcal inf etc
Attack Retreatv
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A simple model• Modified flu-like • Mutual Immunity (“pick one of the two”)• Susceptible-Infected1-Infected2-Susceptible
Virus 1 Virus 2
Details
Prakash and Faloutsos 2012 59
Question: What happens in the end?
green: virus 1red: virus 2
Footprint @ Steady State Footprint @ Steady State = ?
Number of Infections
ASSUME: Virus 1 is stronger than Virus 2
60
Question: What happens in the end?
green: virus 1red: virus 2
Number of Infections
ASSUME: Virus 1 is stronger than Virus 2
Strength Strength
??= Strength Strength
2
Footprint @ Steady State Footprint @ Steady State
61
Answer: Winner-Takes-Allgreen: virus 1red: virus 2
ASSUME: Virus 1 is stronger than Virus 2
Number of Infections
62
Our Result: Winner-Takes-All
In Prakash+ WWW 2012
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!
Details
63
Real Examples
Reddit v Digg Blu-Ray v HD-DVD
[Google Search Trends data]
Prakash and Faloutsos 2012 64
Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)
Prakash and Faloutsos 2012 65
?
?
Given: a graph A, virus prop. model and budget k; Find: k ‘best’ nodes for immunization (removal).
k = 2
??
Full Static Immunization
Prakash and Faloutsos 2012 66
Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)– Full Immunization (Static Graphs)– Fractional Immunization
Prakash and Faloutsos 2012 67
Challenges• Given a graph A, budget k, Q1 (Metric) How to measure the ‘shield-
value’ 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
Increasing λ Increasing vulnerability
λ is the epidemic threshold
“Safe” “Vulnerable” “Deadly”
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1
9
10
3
4
5
7
8
6
2
9
1
11
10
3
4
56
7
8
2
9
Original Graph Without {2, 6}
Eigen-Drop(S) Δ λ = λ - λs
Δ
A1: “Eigen-Drop”: an ideal shield value
Prakash and Faloutsos 2012 70
(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
72
Our Solution: Part 1• Approximate Eigen-drop (Δ λ)
• Δ λ ≈ SV(S) =
– Result using Matrix perturbation theory–u(i) == ‘eigenscore’ ~~ pagerank(i) A u = λ . u
u(i)
Details
73
P1: node importance P2: set diversity
Original Graph Select by P1 Select by P1+P2
Details
74
Our Solution: Part 2: NetShield
• We prove that: SV(S) is sub-modular (& monotone non-decreasing)
• NetShield: Greedily add best node at each step
Corollary: Greedy algorithm works 1. NetShield is near-optimal (w.r.t. max SV(S)) 2. NetShield is O(nk2+m)
Footnote: near-optimal means SV(S NetShield) >= (1-1/e) SV(S Opt)
Details
Prakash and Faloutsos 2012 75
Experiment: Immunization qualityLog(fraction of infected nodes)
NetShield
Degree
PageRank
Eigs (=HITS)Acquaintance
Betweeness (shortest path)
Lower is
better Time
Prakash and Faloutsos 2012 76
Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)– Full Immunization (Static Graphs)– Fractional Immunization
Prakash and Faloutsos 2012 77
Fractional Immunization of NetworksB. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos
Submitted to Science
Prakash and Faloutsos 2012 78
Fractional Asymmetric Immunization
Hospital Another Hospital
Drug-resistant Bacteria (like XDR-TB)
Prakash and Faloutsos 2012 79
Fractional Asymmetric Immunization
Hospital Another Hospital
Drug-resistant Bacteria (like XDR-TB)
= f
Prakash and Faloutsos 2012 80
Fractional Asymmetric Immunization
Hospital Another Hospital
Problem: Given k units of disinfectant, how to distribute them to maximize
hospitals saved?
Prakash and Faloutsos 2012 81
Our Algorithm “SMART-ALLOC”
CURRENT PRACTICE SMART-ALLOC
[US-MEDICARE NETWORK 2005]• Each circle is a hospital, ~3000 hospitals• More than 30,000 patients transferred
~6x fewer!
Prakash and Faloutsos 2012 82
Running Time
≈
Simulations SMART-ALLOC
> 1 week
14 secs
> 30,000x speed-up!
Wall-Clock Time
Lower is better
* = ones which I talked aboutPublications1. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld,
Christos Faloutsos) – In WWW 2012, Lyon2. 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.)
3. Times Series Clustering: Complex is Simpler! (Lei Li, B. Aditya Prakash) - In ICML 2011, Bellevue4. 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, Spain5. Formalizing the BGP stability problem: patterns and a chaotic model (B. Aditya Prakash, Michalis Faloutsos and
Christos Faloutsos) – In IEEE INFOCOM NetSciCom Workshop, 2011.6. On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad and Christos Faloutsos) –
In IEEE ICDM 2010, Sydney, Australia7. 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, Spain8. MetricForensics: A Multi-Level Approach for Mining Volatile Graphs (Keith Henderson, Tina Eliassi-Rad, Christos
Faloutsos, Leman Akoglu, Lei Li, Koji Maruhashi, B. Aditya Prakash and Hanghang Tong) - In SIGKDD 2010, Washington D.C.
9. Parsimonious Linear Fingerprinting for Time Series (Lei Li, B. Aditya Prakash and Christos Faloutsos) - In VLDB 2010, Singapore
10. EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs (B. Aditya Prakash, Ashwin Sridharan, Mukund Seshadri, Sridhar Machiraju and Christos Faloutsos) – In PAKDD 2010, Hyderabad, India
11. BGP-lens: Patterns and Anomalies in Internet-Routing Updates (B. Aditya Prakash, Nicholas Valler, David Andersen, Michalis Faloutsos and Christos Faloutsos) – In ACM SIGKDD 2009, Paris, France.
12. Surprising Patterns and Scalable Community Detection in Large Graphs (B. Aditya Prakash, Ashwin Sridharan, Mukund Seshadri, Sridhar Machiraju and Christos Faloutsos) – In IEEE ICDM Large Data Workshop 2009, Miami
13. FRAPP: A Framework for high-Accuracy Privacy-Preserving Mining (Shipra Agarwal, Jayant R. Haritsa and B. Aditya Prakash) – In Intl. Journal on Data Mining and Knowledge Discovery (DKMD), Springer, vol. 18, no. 1, February 2009, Ed: Johannes Gehrke.
14. Complex Group-By Queries For XML (C. Gokhale, N. Gupta, P. Kumar, L. V. S. Lakshmanan, R. Ng and B. Aditya Prakash) – In IEEE ICDE 2007, Istanbul, Turkey.
**
**
84
Submitted1. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna,
Hanghang Tong, Christos Faloutsos)2. How much of Twitter is Influence? (B. Aditya Prakash, Deepayan Chakrabarti, Kunal Punera)3. Who is to blame? Finding Culprits in Epidemics (B. Aditya Prakash, Jilles Vreeken, Christos
Faloutsos)4. Competing Viruses on Composite Networks: Who wins? (Xuetao Wei, Nicholas Valler, B. Aditya
Prakash, Iulian Neamtiu, Michalis Faloutsos and Christos Faloutsos)5. Gelling, and Melting, Large Graphs through Edge Manipulation (Hanghang Tong, B. Aditya
Prakash, Tina Eliassi-Rad, Michalis Faloutsos, Christos Faloutsos) 6. Worst-case Footprints in the SIS model (B. Aditya Prakash, Varun Gupta and Christos
Faloutsos)
Patents7. Determining User Communities in Communication Networks (Ashwin Sridharan, Mukund
Seshadri, James Schneider, B. Aditya Prakash, Christos Faloutsos) Sprint Inc., filed March 2010
8. Analysis of Computer Network Activity by Successively Removing Accepted Types of Access Events (B. Aditya Prakash, Alice Zheng, Jack Stokes, Eric Fitzgerald, Theodore Hardy) Microsoft Research, filed April 2010
*
Prakash and Faloutsos 2012 85
Acknowledgements
Collaborators Christos Faloutsos Roni Rosenfeld, Michalis Faloutsos, Lada Adamic, Theodore Iwashyna (M.D.), Dave Andersen, Tina Eliassi-Rad, Iulian Neamtiu,
Varun Gupta, Jilles Vreeken,
Deepayan Chakrabarti, Hanghang Tong, Kunal Punera, Ashwin Sridharan, Sridhar Machiraju, Mukund Seshadri, Alice Zheng, Lei Li, Polo Chau, Nicholas Valler, Alex Beutel, Xuetao Wei
Prakash and Faloutsos 2012 86
Acknowledgements
Funding
Prakash and Faloutsos 2012 87
Analysis Policy/Action Data
Propagation on Large Networks
B. Aditya Prakash Christos Faloutsos