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Influence propagation in large graphs - theorems and algorithms

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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

AUTH’12

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

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

Prakash and Faloutsos 2012

- 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

- 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

~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

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

~100m active users

> 50m users

Prakash and Faloutsos 2012

- Dynamical Processes over networks

Buy Versace™!

Followers

Celebrity

Social Media Marketing

Prakash and Faloutsos 2012

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

ANALYSIS

Understanding

POLICY/ ACTION

Managing

DATA

Large real-world networks & processes

Prakash and Faloutsos 2012

Given propagation models:

Q1: Will an epidemic happen?

ANALYSIS

Understanding

Prakash and Faloutsos 2012

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

POLICY/ ACTION

Managing

Prakash and Faloutsos 2012

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

Prakash and Faloutsos 2012

Strong Virus

Epidemic?

Prakash and Faloutsos 2012

Weak Virus

Epidemic?

Prakash and Faloutsos 2012

# 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

Problem Statement

- Given:
- Graph G, and
- Virus specs (attack prob. etc.)

- Find:
- A condition for virus extinction/invasion

Prakash and Faloutsos 2012

- 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

- 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

Background

- 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

Background

- 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

Background

- 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

- 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

…..

- 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

- 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

s = effective strength

s < 1 : below threshold

Prakash and Faloutsos 2012

“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

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

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

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

- 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

General VPM structure

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

Alternating behaviors

DAY

(e.g., work)

adjacency matrix

8

8

Prakash and Faloutsos 2012

Alternating behaviors

NIGHT

(e.g., home)

adjacency matrix

8

8

Prakash and Faloutsos 2012

Prob. β

Prob. β

- 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

- 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

log(fraction infected)

MIT Reality Mining

Synthetic

ABOVE

ABOVE

AT

AT

BELOW

BELOW

Time

Prakash and Faloutsos 2012

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

iPhone v Android

Blu-ray v HD-DVD

Biological common flu/avian flu, pneumococcal inf etc

Prakash and Faloutsos 2012

Details

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

Virus 2

Virus 1

Prakash and Faloutsos 2012

green: virus 1

red: virus 2

Number of Infections

- Footprint @ Steady State

= ?

ASSUME:

Virus 1 is stronger than Virus 2

Prakash and Faloutsos 2012

- 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

green: virus 1

red: virus 2

Number of Infections

ASSUME:

Virus 1 is stronger than Virus 2

Prakash and Faloutsos 2012

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

[Google Search Trends data]

Reddit v Digg

Blu-Ray v HD-DVD

Prakash and Faloutsos 2012

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

Prakash and Faloutsos 2012

Given: a graph A, virus prop. model and budget k;

Find: k ‘best’ nodes for immunization (removal).

?

?

k = 2

?

?

Prakash and Faloutsos 2012

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

Prakash and Faloutsos 2012

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

λ is the epidemic threshold

“Safe”

“Vulnerable”

“Deadly”

Increasing λ

Increasing vulnerability

Prakash and Faloutsos 2012

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

- 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!

- Example:

Prakash and Faloutsos 2012

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

Log(fraction of

infected

nodes)

PageRank

Betweeness (shortest path)

Degree

Lower is better

Acquaintance

Eigs (=HITS)

NetShield

Time

Prakash and Faloutsos 2012

- 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

Drug-resistant Bacteria (like XDR-TB)

Another Hospital

Hospital

Prakash and Faloutsos 2012

Drug-resistant Bacteria (like XDR-TB)

Another Hospital

Hospital

Prakash and Faloutsos 2012

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

Another Hospital

Hospital

Prakash and Faloutsos 2012

~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

Wall-Clock Time

> 1 week

≈

> 30,000x speed-up!

Lower is better

14 secs

Simulations

SMART-ALLOC

Prakash and Faloutsos 2012

Lower is better

SECOND-LIFE

PENN-NETWORK

~5 x

~2.5 x

K = 200

K = 2000

Prakash and Faloutsos 2012

Funding

Prakash and Faloutsos 2012

- 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

B. Aditya Prakash

Christos Faloutsos

Data

Analysis

Policy/Action

Prakash and Faloutsos 2012