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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. Networks are everywhere!. Facebook Network [2010].

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Influence propagation in large graphs theorems and algorithms

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


Networks are everywhere
Networks are everywhere! algorithms

Facebook Network [2010]

Gene Regulatory Network [Decourty 2008]

Human Disease Network [Barabasi 2007]

The Internet [2005]

Prakash and Faloutsos 2012


Dynamical Processes algorithmsover networks are also everywhere!

Prakash and Faloutsos 2012


Why do we care
Why do we care? algorithms

  • 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
Why do we care? (1: Epidemiology) algorithms

  • 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 epidemiology1
Why do we care? (1: Epidemiology) algorithms

  • 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 epidemiology2
Why do we care? (1: Epidemiology) algorithms

~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
Why do we care? (2: Online Diffusion) algorithms

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

~100m active users

> 50m users

Prakash and Faloutsos 2012


Why do we care 2 online diffusion1
Why do we care? (2: Online Diffusion) algorithms

  • Dynamical Processes over networks

Buy Versace™!

Followers

Celebrity

Social Media Marketing

Prakash and Faloutsos 2012


High impact multiple settings
High Impact – Multiple Settings algorithms

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
Research Theme algorithms

ANALYSIS

Understanding

POLICY/ ACTION

Managing

DATA

Large real-world networks & processes

Prakash and Faloutsos 2012


In this talk
In this talk algorithms

Given propagation models:

Q1: Will an epidemic happen?

ANALYSIS

Understanding

Prakash and Faloutsos 2012


In this talk1
In this talk algorithms

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

POLICY/ ACTION

Managing

Prakash and Faloutsos 2012


Outline
Outline algorithms

  • Motivation

  • Epidemics: what happens? (Theory)

  • Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012


A fundamental question
A fundamental question algorithms

Strong Virus

Epidemic?

Prakash and Faloutsos 2012


Example static graph
example (static graph) algorithms

Weak Virus

Epidemic?

Prakash and Faloutsos 2012


Problem statement
Problem Statement algorithms

# 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
Threshold (static version) algorithms

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
Threshold: Why important? algorithms

  • 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


Outline1
Outline algorithms

  • 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

Background algorithms

“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

Background algorithms

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

Background algorithms

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


Outline2
Outline algorithms

  • 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
How should the answer look like? algorithms

…..

  • 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
Static Graphs: Our Main Result algorithms

  • 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
Our thresholds for some models algorithms

s = effective strength

s < 1 : below threshold

Prakash and Faloutsos 2012


Our result intuition for
Our result: Intuition for algorithmsλ

“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
Largest Eigenvalue ( algorithmsλ)

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
Examples: Simulations – SIR (mumps) algorithms

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
Examples: Simulations – SIRS (pertusis) algorithms

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


Outline3
Outline algorithms

  • 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 algorithms

Model-based

See paper for full proof

λ * < 1

Graph-based

Topology and stability

Prakash and Faloutsos 2012


Outline4
Outline algorithms

  • 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
Dynamic Graphs: Epidemic? algorithms

Alternating behaviors

DAY

(e.g., work)

adjacency matrix

8

8

Prakash and Faloutsos 2012


Dynamic graphs epidemic1
Dynamic Graphs: Epidemic? algorithms

Alternating behaviors

NIGHT

(e.g., home)

adjacency matrix

8

8

Prakash and Faloutsos 2012


Model description

Prob. algorithmsβ

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
Our result: Dynamic Graphs Threshold algorithms

  • 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
Infection-profile algorithms

log(fraction infected)

MIT Reality Mining

Synthetic

ABOVE

ABOVE

AT

AT

BELOW

BELOW

Time

Prakash and Faloutsos 2012


Take off plots
“Take-off” plots algorithms

Footprint (# infected @ “steady state”)

Synthetic

MIT Reality

EPIDEMIC

Our threshold

Our threshold

EPIDEMIC

NO EPIDEMIC

NO EPIDEMIC

(log scale)

Prakash and Faloutsos 2012


Outline5
Outline algorithms

  • 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
Competing Contagions algorithms

iPhone v Android

Blu-ray v HD-DVD

Biological common flu/avian flu, pneumococcal inf etc

Prakash and Faloutsos 2012


A simple model

Details algorithms

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
Question: What happens in the end? algorithms

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 end1
    Question: What happens in the end? algorithms

    • 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
    Answer: Winner-Takes-All algorithms

    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
    Our Result: Winner-Takes-All algorithms

    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


    Real examples
    Real Examples algorithms

    [Google Search Trends data]

    Reddit v Digg

    Blu-Ray v HD-DVD

    Prakash and Faloutsos 2012


    Outline6
    Outline algorithms

    • Motivation

    • Epidemics: what happens? (Theory)

    • Action: Who to immunize? (Algorithms)

    Prakash and Faloutsos 2012


    Full static immunization
    Full Static Immunization algorithms

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

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

    ?

    ?

    k = 2

    ?

    ?

    Prakash and Faloutsos 2012


    Outline7
    Outline algorithms

    • Motivation

    • Epidemics: what happens? (Theory)

    • Action: Who to immunize? (Algorithms)

      • Full Immunization (Static Graphs)

      • Fractional Immunization

    Prakash and Faloutsos 2012


    Challenges
    Challenges algorithms

    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
    Proposed vulnerability measure algorithmsλ

    λ is the epidemic threshold

    “Safe”

    “Vulnerable”

    “Deadly”

    Increasing λ

    Increasing vulnerability

    Prakash and Faloutsos 2012


    A1 eigen drop an ideal shield value
    A1 algorithms: “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
    (Q2) - Direct Algorithm too expensive! algorithms

    • 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
    A2: algorithms 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
    Experiment: Immunization quality algorithms

    Log(fraction of

    infected

    nodes)

    PageRank

    Betweeness (shortest path)

    Degree

    Lower is better

    Acquaintance

    Eigs (=HITS)

    NetShield

    Time

    Prakash and Faloutsos 2012


    Outline8
    Outline algorithms

    • Motivation

    • Epidemics: what happens? (Theory)

    • Action: Who to immunize? (Algorithms)

      • Full Immunization (Static Graphs)

      • Fractional Immunization

    Prakash and Faloutsos 2012


    Fractional Immunization of Networks algorithms

    B. Aditya Prakash, Lada Adamic, Theodore

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

    Faloutsos

    Under review

    Prakash and Faloutsos 2012


    Fractional asymmetric immunization
    Fractional Asymmetric Immunization algorithms

    Drug-resistant Bacteria (like XDR-TB)

    Another Hospital

    Hospital

    Prakash and Faloutsos 2012


    Fractional asymmetric immunization1
    Fractional Asymmetric Immunization algorithms

    Drug-resistant Bacteria (like XDR-TB)

    Another Hospital

    Hospital

    Prakash and Faloutsos 2012


    Fractional asymmetric immunization2
    Fractional Asymmetric Immunization algorithms

    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
    Our Algorithm “SMART-ALLOC” algorithms

    ~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
    Running Time algorithms

    Wall-Clock Time

    > 1 week

    > 30,000x speed-up!

    Lower is better

    14 secs

    Simulations

    SMART-ALLOC

    Prakash and Faloutsos 2012


    Experiments
    Experiments algorithms

    Lower is better

    SECOND-LIFE

    PENN-NETWORK

    ~5 x

    ~2.5 x

    K = 200

    K = 2000

    Prakash and Faloutsos 2012


    Acknowledgements
    Acknowledgements algorithms

    Funding

    Prakash and Faloutsos 2012


    References
    References algorithms

    • 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
    Propagation on Large Networks algorithms

    B. Aditya Prakash

    Christos Faloutsos

    Data

    Analysis

    Policy/Action

    Prakash and Faloutsos 2012


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