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Influence propagation in large graphs - theorems and algorithms

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### Influence propagation in large graphs - theorems and algorithms

OutlineOutline

B. Aditya Prakash

http://www.cs.cmu.edu/~badityap

Christos Faloutsos

http://www.cs.cmu.edu/~christos

Carnegie Mellon University

AUTH’12

Networks are everywhere!

Facebook Network [2010]

Gene Regulatory Network [Decourty 2008]

Human Disease Network [Barabasi 2007]

The Internet [2005]

Prakash and Faloutsos 2012

Dynamical Processes over networks are also everywhere!

Prakash and Faloutsos 2012

Why do we care?

- Information Diffusion
- Viral Marketing
- Epidemiology and Public Health
- Cyber Security
- Human mobility
- Games and Virtual Worlds
- Ecology
- Social Collaboration

........

Prakash and Faloutsos 2012

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

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

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

Why do we care? (2: Online Diffusion)

> 800m users, ~$1B revenue [WSJ 2010]

~100m active users

> 50m users

Prakash and Faloutsos 2012

Why do we care? (2: Online Diffusion)

- Dynamical Processes over networks

Buy Versace™!

Followers

Celebrity

Social Media Marketing

Prakash and Faloutsos 2012

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

Research Theme

ANALYSIS

Understanding

POLICY/ ACTION

Managing

DATA

Large real-world networks & processes

Prakash and Faloutsos 2012

In this talk

Given propagation models:

Q1: Will an epidemic happen?

ANALYSIS

Understanding

Prakash and Faloutsos 2012

In this talk

Q2: How to immunize and control out-breaks better?

POLICY/ ACTION

Managing

Prakash and Faloutsos 2012

Outline

- Motivation
- Epidemics: what happens? (Theory)
- Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012

Problem Statement

# Infected

above (epidemic)

below (extinction)

time

Separate the regimes?

Find, a condition under which

- virus will die out exponentially quickly
- regardless of initial infection condition

Prakash and Faloutsos 2012

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

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

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

“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

Prakash and Faloutsos 2012

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

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/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.
- ………
- ………
- ………

Prakash and Faloutsos 2012

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

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

Static Graphs: Our Main Result

- Informally,

w/ Deepay

Chakrabarti

- 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

Our thresholds for some models

s = effective strength

s < 1 : below threshold

Prakash and Faloutsos 2012

Our result: Intuition for λ

“Official” definition:

“Un-official” Intuition

λ ~ # paths in the graph

- 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!

u

u

≈ .

(i, j) = # of paths i j of length k

Prakash and Faloutsos 2012

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

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

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

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

Model-based

See paper for full proof

λ * < 1

Graph-based

Topology and stability

Prakash and Faloutsos 2012

- 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

Dynamic Graphs: Epidemic?

Alternating behaviors

DAY

(e.g., work)

adjacency matrix

8

8

Prakash and Faloutsos 2012

Dynamic Graphs: Epidemic?

Alternating behaviors

NIGHT

(e.g., home)

adjacency matrix

8

8

Prakash and Faloutsos 2012

Prob. β

Model Description- SIS model
- recovery rate δ
- infection rate β
- Set of T arbitrary graphs

day

night

N

N

N

N

Healthy

N2

, weekend…..

N1

X

Prob. δ

Infected

N3

Prakash and Faloutsos 2012

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

Infection-profile

log(fraction infected)

MIT Reality Mining

Synthetic

ABOVE

ABOVE

AT

AT

BELOW

BELOW

Time

Prakash and Faloutsos 2012

“Take-off” plots

Footprint (# infected @ “steady state”)

Synthetic

MIT Reality

EPIDEMIC

Our threshold

Our threshold

EPIDEMIC

NO EPIDEMIC

NO EPIDEMIC

(log scale)

Prakash and Faloutsos 2012

- 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

Competing Contagions

iPhone v Android

Blu-ray v HD-DVD

Biological common flu/avian flu, pneumococcal inf etc

Prakash and Faloutsos 2012

A simple model

- Modified flu-like
- Mutual Immunity (“pick one of the two”)
- Susceptible-Infected1-Infected2-Susceptible

Virus 2

Virus 1

Prakash and Faloutsos 2012

Question: What happens in the end?

green: virus 1

red: virus 2

Number of Infections

- Footprint @ Steady State
- Footprint @ Steady State

= ?

ASSUME:

Virus 1 is stronger than Virus 2

Prakash and Faloutsos 2012

Question: What happens in the end?

- Footprint @ Steady State
- Footprint @ Steady State

green: virus 1

red: virus 2

Number of Infections

Strength

Strength

??

=

2

Strength

Strength

ASSUME:

Virus 1 is stronger than Virus 2

Prakash and Faloutsos 2012

Answer: Winner-Takes-All

green: virus 1

red: virus 2

Number of Infections

ASSUME:

Virus 1 is stronger than Virus 2

Prakash and Faloutsos 2012

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) = λβ / δ same as before!

In Prakash+ WWW 2012

Prakash and Faloutsos 2012

Outline

- Motivation
- Epidemics: what happens? (Theory)
- Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012

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

Outline

- Motivation
- Epidemics: what happens? (Theory)
- Action: Who to immunize? (Algorithms)
- Full Immunization (Static Graphs)
- Fractional Immunization

Prakash and Faloutsos 2012

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

Proposed vulnerability measure λ

λ is the epidemic threshold

“Safe”

“Vulnerable”

“Deadly”

Increasing λ

Increasing vulnerability

Prakash and Faloutsos 2012

A1: “Eigen-Drop”: an ideal shield value

Eigen-Drop(S)

Δ λ = λ - λs

9

Δ

9

9

11

10

10

2

1

1

4

4

8

8

6

2

7

3

7

3

5

5

6

Original Graph

Without {2, 6}

Prakash and Faloutsos 2012

(Q2) - Direct Algorithm too expensive!

- Immunize k nodes which maximize Δλ

S = argmax Δλ

- Combinatorial!
- Complexity:
- Example:
- 1,000 nodes, with 10,000 edges
- It takes 0.01 seconds to compute λ
- It takes2,615 yearsto find 5-best nodes!

Prakash and Faloutsos 2012

A2: Our Solution

In Tong, Prakash+ ICDM 2010

- Part 1: Shield Value
- Carefully approximate Eigen-drop (Δλ)
- 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

Prakash and Faloutsos 2012

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

Outline

- Motivation
- Epidemics: what happens? (Theory)
- Action: Who to immunize? (Algorithms)
- Full Immunization (Static Graphs)
- Fractional Immunization

Prakash and Faloutsos 2012

Fractional Immunization of Networks

B. Aditya Prakash, Lada Adamic, Theodore

Iwashyna (M.D.), Hanghang Tong, Christos

Faloutsos

Under review

Prakash and Faloutsos 2012

Fractional Asymmetric Immunization

Drug-resistant Bacteria (like XDR-TB)

Another Hospital

Hospital

Prakash and Faloutsos 2012

Fractional Asymmetric Immunization

Drug-resistant Bacteria (like XDR-TB)

Another Hospital

Hospital

Prakash and Faloutsos 2012

Fractional Asymmetric Immunization

Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved?

Another Hospital

Hospital

Prakash and Faloutsos 2012

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

Running Time

Wall-Clock Time

> 1 week

≈

> 30,000x speed-up!

Lower is better

14 secs

Simulations

SMART-ALLOC

Prakash and Faloutsos 2012

Experiments

Lower is better

SECOND-LIFE

PENN-NETWORK

~5 x

~2.5 x

K = 200

K = 2000

Prakash and Faloutsos 2012

References

- 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 Eliassi-Rad 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/

Prakash and Faloutsos 2012

Propagation on Large Networks

B. Aditya Prakash

Christos Faloutsos

Data

Analysis

Policy/Action

Prakash and Faloutsos 2012

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