Searching for Influencers via Optimal Percolation: from ...€¦ · The superspreaders are the best...

Searching for Influencers via Optimal Percolation: from Twitter to the Brain

MAIN QUESTION: Where are the superspreaders of information?

Hernán Makse, Physics Department, City College of New York

NetSci2016

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in the kcore

?

hubs?


NetSci2016

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in the kcore

?

hubs?

ANSWER: Optimal Percolation and the Non-Backtracking Matrixuncover the Collective Influencers


NetSci2016

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Longstanding Problem in Network Science

Find the minimum number of influencers/superspreaders to “infect” the entire network

- Viral marketing: $$$- Predicting trends from social media- Immunization epidemics- Essential genes: gene regulation protein networks- Essential species: ecological networks- Brain Networks: neural correlates of consciousness - Financial networks “too big to fail” --> “too weak to fail”?- Predicting stock markets

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3

How to become viral in social media?EASY

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3

How to become viral in social media?EASY

50 million hits

1. Funny baby faces videos

lindosbebitos!!

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4

How to spread information in social media?EASY

50 million hits

2. Funny dog faces videos

FUNNY DOG VIDEOSlindos

perritos!!Friday, June 3, 16

5

3. Most successful viral superspreading event in the history of humankind

Gangnam style videoby Mr Psy

as of today: 2 billion hits(mainly teenagers)

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5

3. Most successful viral superspreading event in the history of humankind

Gangnam style videoby Mr Psy

as of today: 2 billion hits(mainly teenagers)

-‐Can we predict the onset of collec3ve behavior?-‐More is different, Phil Anderson, Nobel ’77

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Network Science: Introduction January 10, 2011

Years

Timescited

1. COMPLEX NETWORK SCIENCE: superspreading of ideas Courtesy of Gene Stanley-‐ Analogous to herding behavior in the stock market

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Network Science: Introduction January 10, 2011

Years

Timescited “IrraMonal exuberance”

-‐ Chairman Greenspan (FED) on the dot-‐com bubble-‐ Keynes: “animal spirits” of excessive opMmism

1. COMPLEX NETWORK SCIENCE: superspreading of ideas Courtesy of Gene Stanley-‐ Analogous to herding behavior in the stock market

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THE ANSWER:Understanding BIG DATA

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What is the Problem with Big Data?

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What is the Problem with Big Data?

Eric Schmidt (CEO Google): “Every two years we create as much information (5 exabytes) as we did from the dawn of civilization until 2003”

Age of Big Data

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THE QUESTION ARISES:What happened in the last two years?

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Did we all get clever all of a sudden?

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Did we all get clever all of a sudden?Most probably not

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Evidence #1: The USA Presidential Primary Election Debates

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Evidence #1: The USA Presidential Primary Election Debates

Evidence #2: Peter Thiel (founder PayPal)

“The rate of technological innovation is actually slowing”

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WHY IS THAT?

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WHY IS THAT?

THE AGE OF BIG DATA

OR

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WHY IS THAT?

THE AGE OF BIG DATA

OR

THE AGE OF BIG DATA JUNK?

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THE SOLUTION: REDUCE BIG DATA TO A FEW INFLUENCERS

MINIMAL INFLUENCERS

NP-hard Maximization

of Influence Problem

Kempe, Kleinberg,

Tardos (2003)

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How to predict the most influentials in social networks

Typical approach: Brute-Force (heuristics)

Twitter“Attacking” the hubs (high degree)

Pres Obama: 55M followers Lady Gaga: 45M followers (very few: 1 percenters....)

Other heuristics: ranking the nodes byPageRank (Google), betweeness, eigenvector, closeness centralities, kcore, equal graph partitioning, etc...Problems: heuristics do not maximize any function of influence Friday, June 3, 16

Our approach

Inspired by French mime Marcel Marceau:“Making visible the invisible”

Collective optimization theory

unravels the strength of “weak nodes”

Granovetter, 1973: “The strength of weak ties”

Greatest mime of all time

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Mapping to Optimal Percolation to find the minimal set of “influencers” to fragment the network

Best spreaders = minimize the inactive nodes = maximize giant

component

Best immunizators =minimize the giant connected component

Optimal Percolation =Best immunizators = best influencers

G1G1

Linear-threshold model with

k-1 thresholdSIR model

Morone, Makse,

Nature (2015)

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G1



component



G1G1



Morone, Makse,

Nature (2015)

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G1

G1



component



G1G1



Morone, Makse,

Nature (2015)

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qrand =� 2

� 1

qrand

Optimal Percolation = minimize : minimal fraction of influencers

to destroy the network

qqc

G1

qcqc

Minimize qc = find the minimal influencers until the solution

G1 = 0 ! {⌫i!j = 0} becomes unstable

Transform the problem

into a stability problem

Random percolation

Strategy = minimize the giant componentor

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⌫i!j = ni

2

41 �Y

k2@i\j

(1 � ⌫k!i)

3

5

ni = 0

ni = 1

How to calculate the stability matrix

ij

k

G1Order parameter = Prob to belong to giant component:

G1

removed node

other nodes

Message passing equation in a locally tree-like network

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@⌫i!j

@⌫k!`

��⌫i!j=0

⌘ nkBk!`,i!j

�max

(n, q) 1

minn

�max

(n,qc) = 1

Stability of is given by largest eigenvalue of non-backtracking matrix

G1 = 0

Non-Backtracking Matrix

Solution is stable when the largest eigenvalue of the modified NB matrix:

Finding the superspreaders = finding the nodes that minimizes the largest eigenvalue of the modified NB matrix:

2M x 2M matrix

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The superspreaders are the best non-backtracking walkers

Non-backtracking walk of length l: a walker that cannot

immediately come back

- Second largest eigenvalue of NB is also optimal for community detection. Krzakala PNAS (2013), Newman PRE (2013)- In contrast, most heuristics are based on the adjacency matrix which measures only regular random walks: PageRank largest eigenvalue Aij

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` = 1ki � 1 kj � 1

` = 3

` = 5

�max

(n) ⇡X

i

(ki � 1)X

j2@Ball(`)

nk(kj � 1)

Many-body problem: Power method “Feynman-like” diagrams for

Non-backtracking walk

�max

(n⇤, qc)

l-level = steps

1 NB step2-body interaction

3 NB steps4-body interaction

5 NB steps6-body interaction

Diagramatic expansion contains all physical interactions. Exact solution for random graphs (not for real graphs)

ki � 1 kj � 1

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O(N) ⇠ N logN

Highly Scalable Approximate Solution: Collective Influence Algorithm - CI Algorithm

For a given level l1. Start with all nodes occupied: ni=1

2. Assigned to every node the collective influence index:

CI` = (ki � 1)X

j2@Ball(`)

(kj � 1)

3. Remove (attack) the highest CI node: ni*=0

4. Recalculate CI and repeat until giant component is zero. CI scales as:

ki

kj

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Test: Exact optimal solution and best approximation with CI in Erdos-Renyi network

Others: hub removal (HD, high degree and adaptive HDA), PageRank, Closeness Centrality, Eigenvector Centrality, k-core, EGP

CI outperforms heuristic centralities and approximates well the exact optimal solution

The best possible attack

is to remove the loops to get

a tree at qc

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Why CI outperforms other heuristics?

Collective Influence identifies a new class of influencer neglected

by previous rankings

Weak node: a low degree node surrounded by hierarchical coronas of hubs at level l

Related to Granovetter theory “Strength of weak ties” (1973). Weak node

CI` = (ki � 1)X

j2@Ball(`)

(kj � 1)

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Validation of CI in Big Data: Twitter

A broker node

- CI identifies 40% less influencers than high degree

CI is valid in locally tree-like networks: what about real networks?

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Next, we address the question that will doubtless arise:

Friday, June 3, 16

Next, we address the question that will doubtless arise:

Is all this mathematical gibberish of any real use?

Three applications:1. Marketing campaign in Mexico

2. Predicting trends in Twitter: Trump vs Cruz3. Finding superspreaders in the brain

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1. Validating CI in a Real Marketing Campaign using Mobile Phone Networks

Mexico:110 million users

calling over 3 months

+banking data

Team up with GranData.com(an Argentinian

Corporation)

Market campaign targeting the high CI people, AKA influencers

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Communication patterns of rich and poor

i

E

B

A

Top 10%

Bottom 10%

ED

C

Bottom 20%

Top 1%

F

Top 1% richest people

Bottom 20% poorest people

HIGH CI

LOW CI

(same degree)

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Improve response rate = $$$ by five-fold between low CI and high CI targeting

Low CI: poorest people

High CI: wealthy people

Targeting 60,000 people according to their Collective Influence in the network of entire Mexico

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2. Predicting Trump Sentiment from Twitter

Trumpcamp

Cruzcamp

ALL TWEETS

INFLUENCERS

Trumpwins

Cruzwins

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Awake brain surgery to dissect tumor with no clear boundaries

Neurosurgeon sMmulates areas around the tumor with electrodesto locate the essenMal funcMonal areas.

FuncMonal areas (eg, language, motor)are located by asking paMent to talk, move, etc. Remove as much tumor as possible avoiding the essenMal areas.

GOAL: predict the essenMal areas of the brain with NoN theory

CollaboraMon with Andrei Holodny, MSKCC

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3. Finally: Influencers in a Brain Network of Networks

Finding the essential minimal nodes in the brain- neural correlates of consciousness -

Important for any biological system: genetic networks, proteome, metabolome, etc

We extract the Network from fMRI/DTI experiments on dual tasking: auditory and visual

controllinks

controllinks

innerlinks

Gallos, Makse, Sigman, PNAS (2012)Reis, Andrade, Sigman, Canals, Makse, Nat Phys (2014)Friday, June 3, 16

Influencers in a Network of NetworksInfluencers are the best non-backtracking walkers walking along two types of links. Information flows via two type of messages:

intra-modular and inter-modular

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Collective Influence Map of the Human Brain:reducing the brain to its essential nodes

High CI

CI-map

a d

b

e

Anterior cingulate (AC)Posterior parietalcortex (PPC)

Posterior occipitalcortex (V1/V2)

ACPPC

V1/V2

ACPPC

V1/V2

Complexity Reduction

Inter-modularlinks

Intra-modularlinks

High CI

G(q)

q (Frac. of zero inputs)

Gia

nt a

ctiv

e co

mpo

nent

()

c

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

CI - Rare inputs Random - Typical inputs

High CI

CI-map

a d

b

e

Anterior cingulate (AC)Posterior parietalcortex (PPC)

Posterior occipitalcortex (V1/V2)

ACPPC

V1/V2

ACPPC

V1/V2

Complexity Reduction

Inter-modularlinks

Intra-modularlinks

High CI

G(q)

q (Frac. of zero inputs)

Gia

nt a

ctiv

e co

mpo

nent

()

c

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

CI - Rare inputs Random - Typical inputs

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Data Analytics at the Cutting-Edge for Free!kcore-analytics.com

Search Engine for Influencers

Kcore DREAMTEAM:

Flaviano Morone

Francesca LuciniAlex Bovet Kevin RothGeorge Furbish

Hernan Makse

Andrea Morone

Friday, June 3, 16

Searching for Influencers via Optimal Percolation: from ...€¦ · The superspreaders are the best...

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