1. Propagation on Large Networks
B. Aditya Prakash
Christos Faloutsos
Carnegie Mellon University
INARC Meeting – May 2nd

2. Preaching to the choir: Networks are everywhere!
Facebook Network [2010]
Gene Regulatory Network [Decourty 2008]
Human Disease Network [Barabasi 2007]
The Internet [2005]
Prakash and Faloutsos 2012

3. Dynamical Processes over networks are also everywhere!
Focus of this talk:
Dynamical Processes over networks are also everywhere!
Prakash and Faloutsos 2012

4. Why do we care? Social collaboration Information Diffusion
Viral Marketing
Epidemiology and Public Health
Cyber Security
Human mobility
Games and Virtual Worlds
Ecology
Prakash and Faloutsos 2012

5. 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
Prakash and Faloutsos 2012

6. 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?
Prakash and Faloutsos 2012

7. Why do we care? (1: Epidemiology)
~6x fewer!
[US-MEDICARE NETWORK 2005]
CURRENT PRACTICE
OUR METHOD
Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)
Prakash and Faloutsos 2012

8. Why do we care? (2: Online Diffusion)
> 800m users, ~$1B revenue [WSJ 2010]
~100m active users
> 50m users
Prakash and Faloutsos 2012

9. Why do we care? (2: Online Diffusion)
Dynamical Processes over networks
Buy Versace™!
Celebrity
Followers
Social Media Marketing
Prakash and Faloutsos 2012

10. Why do we care? (3: To change the world?)
Dynamical Processes over networks
Social networks and Collaborative Action
Prakash and Faloutsos 2012

11. High Impact – Multiple Settings
epidemic out-breaks
Q. How to squash rumors faster?
Q. How do opinions spread?
Q. How to market better?
products/viruses
transmit s/w patches
Prakash and Faloutsos 2012

12. Large real-world networks & processes
Research Theme
ANALYSIS
Understanding
POLICY/ ACTION
Managing
DATA
Large real-world networks & processes
Prakash and Faloutsos 2012

13. Research Theme – Public Health
ANALYSIS
Will an epidemic happen?
POLICY/ ACTION
How to control out-breaks?
DATA
Modeling # patient transfers
Prakash and Faloutsos 2012

14. Research Theme – Social Media
ANALYSIS
# cascades in future?
POLICY/ ACTION
How to market better?
DATA
Modeling Tweets spreading
Prakash and Faloutsos 2012

15. In this talk Given propagation models: Q1: Will an epidemic happen?
ANALYSIS
Understanding
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/ ACTION
Managing
Q2: How to immunize and control out-breaks better?
Prakash and Faloutsos 2012

17. Outline Motivation Epidemics: what happens? (Theory)
Action: Who to immunize? (Algorithms)
Prakash and Faloutsos 2012

18. A fundamental question
Strong Virus
Epidemic?
Prakash and Faloutsos 2012

19. example (static graph)
Weak Virus
Epidemic?
Prakash and Faloutsos 2012

20. Problem Statement Find, a condition under which # Infected
above (epidemic)
below (extinction)
# Infected
time
Find, a condition under which
virus will die out exponentially quickly
regardless of initial infection condition
Separate the regimes?
Prakash and Faloutsos 2012

21. Threshold (static version)
Problem Statement
Given:
Graph G, and
Virus specs (attack prob. etc.)
Find:
A condition for virus extinction/invasion
Prakash and Faloutsos 2012

22. 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)
Background
“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
Prob. β
t = 1
t = 2
t = 3
Prakash and Faloutsos 2012

25. Terminology: continued
Background
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’
Prakash and Faloutsos 2012

26. Background
Related Work
All are about either:
Structured topologies (cliques, block-diagonals, hierarchies, random)
Specific virus propagation models
Static graphs
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/ v2, Aug
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.
………
Prakash and Faloutsos 2012

27. 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

28. 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?
…..
Prakash and Faloutsos 2012

29. 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
on the λ, first eigenvalue of A, and
some constant , determined by the virus propagation model λ
No epidemic if λ * < 1
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, SEIR
s = λ .
s = 1
SIV, SEIV
(H.I.V.)
Prakash and Faloutsos 2012

31. Our result: Intuition for λ
“Official” definition:
“Un-official” Intuition 
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!
λ ~ # paths in the graph u u
≈ .
(i, j) = # of paths i  j of length k
Prakash and Faloutsos 2012

32. Largest Eigenvalue (λ)
better connectivity higher λ
Prakash and Faloutsos 2012

33. Largest Eigenvalue (λ)
better connectivity higher λ
λ ≈ 2
λ = N
λ = N-1
λ ≈ 2
λ= 31.67
λ= 999
N = 1000
N nodes
Prakash and Faloutsos 2012

34. Examples: Simulations – SIR (mumps)
Fraction of Infections
Footprint
(a) Infection profile (b) “Take-off” plot
PORTLAND graph: synthetic population,
31 million links, 6 million nodes
Effective Strength
Time ticks
Prakash and Faloutsos 2012

35. Examples: Simulations – SIRS (pertusis)
Fraction of Infections
Footprint
(a) Infection profile (b) “Take-off” plot
PORTLAND graph: synthetic population,
31 million links, 6 million nodes
Time ticks
Effective Strength
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)
Prakash and Faloutsos 2012

37. Proof Sketch λ * < 1 General VPM structure Model-based Graph-based
λ * < 1
Dimensional arguments…
Full proof 30 pages on arxiv
Graph-based
Topology and stability

38. Models and more models H.I.V. Model Used for SIR Mumps SIS Flu SIRS
Pertussis
SEIR
Chicken-pox
……..
SICR
Tuberculosis
MSIR
Measles
SIV
Sensor Stability
H.I.V.
……….

39. Ingredient 1: Our generalized model
Exogenous Transitions
Endogenous Transitions
Susceptible
Infected
Vigilant
Endogenous Transitions
Susceptible
Infected
Vigilant
(S*I*V*?)

40. Special case
Susceptible
Infected
(S*I*V*?)
Vigilant

41. Special case: H.I.V. Multiple Infectious, Vigilant states
“Non-terminal”
“Terminal”
Multiple Infectious, Vigilant states

42. Ingredient 2: NLDS+Stability
Details
Ingredient 2: NLDS+Stability
View as a NLDS
discrete time
non-linear dynamical system (NLDS)
size
mN x 1 .
size N (number of nodes in the graph) S
Probability vector Specifies the state of the system at time t
Curly braces…say what the blue yellow are… I V

43. Ingredient 2: NLDS + Stability
Details
Ingredient 2: NLDS + Stability
View as a NLDS
discrete time
non-linear dynamical system (NLDS)
size
mN x 1 .
Non-linear function
Explicitly gives the evolution of system

44. Ingredient 2: NLDS + Stability
View as a NLDS
discrete time
non-linear dynamical system (NLDS)
Threshold  Stability of NLDS

45. Special case: SIR Details NLDS size 3N x 1 = probability that node
Apply g on p_T, you get p_t+1
= probability that node
i is not attacked by
any of its infectious
neighbors
NLDS

46. Fixed Point Details State when no node is infected Q: Is it stable? 1.
State when no node is infected
Q: Is it stable?

47. Stability for SIR
Stable
under threshold
Unstable
above threshold

48. λ * < 1 See paper for full proof General VPM structure Model-based
λ * < 1
Graph-based
Topology and stability
Prakash and Faloutsos 2012

49. 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

50. Dynamic Graphs: Epidemic?
Alternating behaviors
DAY
(e.g., work)
adjacency matrix 8
Prakash and Faloutsos 2012

51. Dynamic Graphs: Epidemic?
Alternating behaviors
NIGHT
(e.g., home)
adjacency matrix 8
Prakash and Faloutsos 2012

52. Model Description SIS model Set of T arbitrary graphs recovery rate δ
Infected
Healthy X N1 N3 N2
Prob. β
Prob. δ
SIS model
recovery rate δ
infection rate β
Set of T arbitrary graphs
day N
night N
, weekend…..
Prakash and Faloutsos 2012

53. Our result: Dynamic Graphs Threshold
Informally, NO epidemic if
eig (S) = < 1
Single number!
Largest eigenvalue of
The system matrix S
Details
S =
In Prakash+, ECML-PKDD 2010
Prakash and Faloutsos 2012

54. Infection-profile Synthetic MIT Reality Mining log(fraction infected)
ABOVE
ABOVE AT AT
BELOW
BELOW
Time
Prakash and Faloutsos 2012

55. “Take-off” plots Synthetic MIT Reality Our threshold Our threshold
Footprint (# “steady state”)
Synthetic
MIT Reality
EPIDEMIC
Our threshold
Our threshold
EPIDEMIC
NO EPIDEMIC
NO EPIDEMIC
(log scale)
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)
Prakash and Faloutsos 2012

57. Competing Contagions iPhone v Android Blu-ray v HD-DVD v Attack
Retreat
Biological common flu/avian flu, pneumococcal inf etc

58. A simple model Virus 2 Virus 1 Details Modified flu-like
Mutual Immunity (“pick one of the two”)
Susceptible-Infected1-Infected2-Susceptible
Virus 1
Virus 2
Prakash and Faloutsos 2012

59. Question: What happens in the end?
green: virus 1
red: virus 2
Number of Infections
Steady State
= ?
ASSUME:
Virus 1 is stronger than Virus 2
Prakash and Faloutsos 2012

60. Question: What happens in the end?
Steady State
green: virus 1
red: virus 2
Number of Infections
Strength ?? = 2
Strength
ASSUME:
Virus 1 is stronger than Virus 2

61. Answer: Winner-Takes-All
green: virus 1
red: virus 2
Number of Infections
ASSUME:
Virus 1 is stronger than Virus 2

62. Our Result: Winner-Takes-All
Given our model, and any graph, the weaker virus always dies-out completely
Details
The stronger survives only if it is above threshold
Virus 1 is stronger than Virus 2, if:
strength(Virus 1) > strength(Virus 2)
Strength(Virus) = λ β / δ  same as before!
In Prakash+ WWW 2012

63. Real Examples [Google Search Trends data] Reddit v Digg
Blu-Ray v HD-DVD

64. Outline Motivation Epidemics: what happens? (Theory)
Action: Who to immunize? (Algorithms)
Prakash and Faloutsos 2012

65. Full Static Immunization
Given: a graph A, virus prop. model and budget k;
Find: k ‘best’ nodes for immunization (removal). ? ?
k = 2 ? ?
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’?
Prakash and Faloutsos 2012

68. Proposed vulnerability measure λ
λ is the epidemic threshold
“Safe”
“Vulnerable”
“Deadly”
Increasing λ
Increasing vulnerability
Prakash and Faloutsos 2012

69. A1: “Eigen-Drop”: an ideal shield value
Eigen-Drop(S)
Δ λ = λ - λs 9 9 Δ 9 11 10 10 1 1 6 2 4 4 8 8 2 3 7 3 7 5 5 6
Original Graph
Without {2, 6}
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!
Prakash and Faloutsos 2012

71. A2: Our Solution Part 1: Shield Value Part 2: Algorithm
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
Prakash and Faloutsos 2012

72. Our Solution: Part 1 Details Approximate Eigen-drop (Δ λ) A u u = λ .
Δ λ ≈ SV(S) =
Result using Matrix perturbation theory
u(i) == ‘eigenscore’
~~ pagerank(i)
u(i) A u u
= λ .

73. Details P1: node importance P2: set diversity Select by P1
Select by P1+P2
Original Graph

74. Our Solution: Part 2: NetShield
Details
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)
tell jon’s paper (influence propagation..)
Footnote: near-optimal means SV(S NetShield) >= (1-1/e) SV(S Opt)

75. Experiment: Immunization quality
Log(fraction of
infected
nodes)
PageRank
Betweeness (shortest path)
Degree
Lower is better
Acquaintance
Eigs (=HITS)
NetShield
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 Networks
B. Aditya Prakash, Lada Adamic, Theodore
Iwashyna (M.D.), Hanghang Tong, Christos
Faloutsos
Submitted to Science
Prakash and Faloutsos 2012

78. Fractional Asymmetric Immunization
Drug-resistant Bacteria (like XDR-TB)
Hospital
Another Hospital
Prakash and Faloutsos 2012

79. Fractional Asymmetric Immunization
Drug-resistant Bacteria (like XDR-TB)
Hospital
Another Hospital
Prakash and Faloutsos 2012

80. Fractional Asymmetric Immunization
Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved?
Hospital
Another Hospital
Prakash and Faloutsos 2012

81. Our Algorithm “SMART-ALLOC”
~6x fewer!
[US-MEDICARE NETWORK 2005]
Each circle is a hospital, ~3000 hospitals
More than 30,000 patients transferred
CURRENT PRACTICE
SMART-ALLOC
Prakash and Faloutsos 2012

82. ≈ Running Time Wall-Clock Time > 30,000x speed-up! Lower is better
> 1 week ≈
> 30,000x speed-up!
Lower is better
14 secs
Simulations
SMART-ALLOC
Prakash and Faloutsos 2012

83. * = ones which I talked about
Publications
Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon
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.)
Times Series Clustering: Complex is Simpler! (Lei Li, B. Aditya Prakash) - In ICML 2011, Bellevue
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
Formalizing the BGP stability problem: patterns and a chaotic model (B. Aditya Prakash, Michalis Faloutsos and Christos Faloutsos) – In IEEE INFOCOM NetSciCom Workshop, 2011.
On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia
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
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.
Parsimonious Linear Fingerprinting for Time Series (Lei Li, B. Aditya Prakash and Christos Faloutsos) - In VLDB 2010, Singapore
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
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.
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
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.
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. Submitted
Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos)
How much of Twitter is Influence? (B. Aditya Prakash, Deepayan Chakrabarti, Kunal Punera)
Who is to blame? Finding Culprits in Epidemics (B. Aditya Prakash, Jilles Vreeken, Christos Faloutsos)
Competing Viruses on Composite Networks: Who wins? (Xuetao Wei, Nicholas Valler, B. Aditya Prakash, Iulian Neamtiu, Michalis Faloutsos and Christos Faloutsos)
Gelling, and Melting, Large Graphs through Edge Manipulation (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad, Michalis Faloutsos, Christos Faloutsos)
Worst-case Footprints in the SIS model (B. Aditya Prakash, Varun Gupta and Christos Faloutsos)
Patents
Determining User Communities in Communication Networks (Ashwin Sridharan, Mukund Seshadri, James Schneider, B. Aditya Prakash, Christos Faloutsos) Sprint Inc., filed March 2010
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 *

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. Propagation on Large Networks
B. Aditya Prakash
Christos Faloutsos
Analysis
Policy/Action
Data
Prakash and Faloutsos 2012