Lecture 27:

Information diffusion

CS 765 Complex Networks

Slides are modified from Lada Adamic, David Kempe, Bill Hackbor

outline

factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections?

studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices

network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage

networks and innovation Lazer and Friedman: innovation

factors influencing diffusion

network structure (unweighted) density degree distribution clustering connected components community structure

strength of ties (weighted) frequency of communication strength of influence

spreading agent attractiveness and specificity of information

Strong tie defined

A strong tie frequent contact affinity many mutual contacts

Less likely to be a bridge (or a local bridge)

“forbidden triad”:

strong ties are likely to “close”

Source: Granovetter, M. (1973). "The Strength of Weak Ties",

edge embeddeness

embeddeness: number of common neighbors the two endpoints have

neighborhood overlap:

school kids and 1st through 8th choices of friends

snowball sampling: will you reach more different kids by asking each kid to name

their 2 best friends, or their 7th & 8th closest friend?

Source: M. van Alstyne, S. Aral. Networks, Information & Social Capital

is it good to be embedded?

What are the advantages of occupying an embedded position in the network?

What are the disadvantages of being embedded?

Advantages of being a broker (spanning structural holes)?

Disadvantages of being a broker?

outline

factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections?

studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices

network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage

networks and innovation Lazer and Friedman: innovation

how does strength of a tie influence diffusion?

M. S. Granovetter: The Strength of Weak Ties, AJS, 1973:

finding a job through a contact that one saw frequently (2+ times/week) 16.7% occasionally (more than once a year but < 2x week) 55.6% rarely 27.8%

but… length of path is short contact directly works for/is the employer or is connected directly to employer

strength of tie: frequency of communication

Kossinets, Watts, Kleinberg, KDD 2008: which paths yield the most up to date info? how many of the edges form the “backbone”?

source: Kossinets et al. “The structure of information pathways in a social communication network”

the strength of intermediate ties

strong ties frequent communication, but ties are redundant due to high

clustering

weak ties reach far across network, but communication is infrequent…

“Structure and tie strengths in mobile communication networks” use nation-wide cellphone call records and simulate diffusion

using actual call timing

source: Onnela J. et.al. Structure and tie strengths in mobile communication networks

Localized strong ties slow infection spread.

how can information diffusion be different from simple contagion (e.g. a virus)?

simple contagion: infected individual infects neighbors with information at some

rate

threshold contagion: individuals must hear information (or observe behavior) from a

number or fraction of friends before adopting

in lab: complex contagion (Centola & Macy, AJS, 2007) how do you pick individuals to “infect” such that your opinion

prevails

http://www.ladamic.com/netlearn/NetLogo4/DiffusionCompetition.html

Framework

The network consists of nodes (individual) and edges (links between nodes)

Each node is in one of two states Susceptible (in other words, healthy) Infected

Susceptible-Infected-Susceptible (SIS) model Cured nodes immediately become susceptible

Susceptible Infected

Infected by neighbor

Cured internally

Framework (Continued)

Homogeneous birth rate β on all edges between infected and susceptible nodes

Homogeneous death rate δ for infected nodes

Infected

Healthy

XN1

N3

N2

Prob. β

Prob. β

Prob. δ

SIR and SIS Models

An SIS model consists of two group Susceptible: Those who may contract the disease Infected: Those infected

An SIR model consists of three group Susceptible: Those who may contract the disease Infected: Those infected Recovered: Those with natural immunity or those that

have died.

Important Parameters

α is the transmission coefficient, which determines the rate at which the disease travels from one population to another.

γ is the recovery rate: (I persons)/(days required to recover)

R0 is the basic reproduction number.

(Number of new cases arising from one infective) x (Average duration of infection)

If R0 > 1 then ∆I > 0 and an epidemic occurs

/1)(0 oo SSR

SIR and SIS Models

SIS Model:

SIR Model:

IR

ISII

SIS

ISII

ISIS

Diffusion in networks: ER graphs

review: diffusion in ER graphs

http://www.ladamic.com/netlearn/NetLogo501/ERDiffusion.html

Quiz Q:

When the density of the network increases, diffusion in the network is faster slower unaffected

ER graphs: connectivity and density

average degree = 2.5 average degree = 10

nodes infected after 10 steps, infection rate = 0.15

Quiz Q:

When nodes preferentially attach to high degree nodes, the diffusion over the network is faster slower unaffected

Diffusion in “grown networks”

nodes infected after 4 steps, infection rate = 1

http://www.ladamic.com/netlearn/NetLogo501/BADiffusion.html

preferential attachment non-preferential growth

Diffusion in small worlds

What is the role of the long-range links in diffusion over small world topologies?

http://www.ladamic.com/netlearn/NetLogo4/SmallWorldDiffusionSIS.html

diffusion of innovation

surveys: farmers adopting new varieties of hybrid corn by observing what

their neighbors were planting (Ryan and Gross, 1943) doctors prescribing new medication (Coleman et al. 1957) spread of obesity & happiness in social networks (Christakis and

Fowler, 2008)

online behavioral data: Spread of Flickr photos & Digg stories

(Lerman, 2007) joining LiveJournal groups & CS conferences

(Backstrom et al. 2006) + others e.g. Anagnostopoulos et al. 2008

27

Open question: how do we tell influence from correlation?

approaches: time resolved data: if adoption time is shuffled, does it yield the

same patterns? if edges are directed: does reversing the edge direction yield

less predictive power?

poison pills diffused through interlocks geography had little to do with it more likely to be influenced

by tie to firm doing something

similar & having similar centrality

golden parachutes did not diffuse through interlocks geography was a significant factor more likely to follow “central” firms

why did one diffuse through the “network” while the other did not?

Example: adopting new practices

Source: Corporate Elite Networks and Governance Changes in the 1980s.

Social Network and Spread of Influence

Social network plays a fundamental role as a medium for the spread of INFLUENCE among its members Opinions, ideas, information,

innovation…

Direct Marketing takes the “word-of-mouth” effects to significantly increase profits Gmail, Tupperware popularization, Microsoft

Origami …

Problem Setting

Given a limited budget B for initial advertising

e.g. give away free samples of product estimates for influence between individuals

Goal trigger a large cascade of influence

e.g. further adoptions of a product

Question Which set of individuals should B target at?

Application besides product marketing spread an innovation detect stories in blogs

Models of Influence

First mathematical models [Schelling '70/'78, Granovetter '78]

Large body of subsequent work: [Rogers '95, Valente '95, Wasserman/Faust '94]

Two basic classes of diffusion models: threshold and cascade

General operational view: A social network is represented as a directed graph,

with each person (customer) as a node Nodes start either active or inactive An active node may trigger activation of neighboring nodes Monotonicity assumption: active nodes never deactivate

Threshold dynamics

The network:

• aij is the adjacency matrix (N ×N)

• un-weighted

• undirected

The nodes:

• are labelled i , i from 1 to N;

• have a state ;

• and a threshold ri from some distribution.

{0,1}ija

jiij aa

}1,0{)( tvi

The fraction of nodes in state vi=1 is (t):

Threshold dynamics

Updating: 1 if

unchanged otherwisei i

i

rv

Neighbourhood average:

1i ij j

ji

a vk

}1,0{)( tviNode i has state

irand threshold

Example

Inactive Node

Active Node

Threshold

Active neighbors

vw 0.5

0.30.2

0.5

0.10.4

0.3 0.2

0.6

0.2

Stop!

U

X

Independent Cascade Model

When node v becomes active, it has a single chance of activating each currently inactive neighbor w.

The activation attempt succeeds with probability pvw .

Example

vw 0.5

0.3 0.20.5

0.10.4

0.3 0.2

0.6

0.2

Inactive Node

Active Node

Newly active

node

Successful

attempt

Unsuccessful

attempt

Stop!

UX

outline

factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections?

studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices

network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage

networks and innovation Lazer and Friedman: innovation

Burt: structural holes and good ideas

Managers asked to come up with an idea to improve the supply chain

Then asked: whom did you discuss the idea with? whom do you discuss supply-chain issues with in general do those contacts discuss ideas with one another?

673 managers (455 (68%) completed the survey) ~ 4000 relationships (edges)

results

people whose networks bridge structural holes have higher compensation positive performance evaluations more promotions more good ideas

these brokers are more likely to express ideas less likely to have their ideas dismissed by judges more likely to have their ideas evaluated as valuable

Aral & Alstyne: Study of a head hunter firm

Three firms initially Unusually measurable inputs and outputs

1300 projects over 5 yrs and 125,000 email messages over 10 months (avg 20% of time!) Metrics

(i) Revenues per person and per project, (ii) number of completed projects, (iii) duration of projects, (iv) number of simultaneous projects, (v) compensation per person

Main firm 71 people in executive search (+2 firms partial data) 27 Partners, 29 Consultants, 13 Research, 2 IT staff

Four Data Sets per firm 52 Question Survey (86% response rate) E-Mail Accounting 15 Semi-structured interviews

Email structure matters

Coefficientsa

(Base Model)

Best structural pred.

Ave. E-Mail Size

Colleagues’ Ave.Response Time

B Std. Error

Unstandardized Coefficients

Adj. R2 Sig. F

Dependent Variable: Bookings02a.

Coefficientsa

B Std. Error

Unstandardized Coefficients

Adj. R2 Sig. F

Dependent Variable: Billings02a.

New Contract Revenue Contract Execution Revenue

0.40

12604.0*** 4454.0 0.52 .006

-10.7** 4.9 0.56 .042

-198947.0 168968.0 0.56 .248

0.19

1544.0** 639.0 0.30 .021

-9.3* 4.7 0.34 .095

-368924.0** 157789.0 0.42 .026

Base Model: YRS_EXP, PARTDUM, %_CEO_SRCH, SECTOR(dummies), %_SOLO.b.

N=39. *** p<.01, ** p<.05, * p<.1b.

Sending shorter e-mail helps get contracts and finish them.

Faster response from colleagues helps finish them.

diverse networks drive performance by providing access to novel information

network structure (having high degree) correlates with receiving novel information sooner (as deduced from hashed versions of their email)

getting information sooner correlates with $$ brought in controlling for # of

years worked job level ….

Network Structure Matters

Coefficientsa

(Base Model)

Size Struct. Holes

Betweenness

B Std. Error

Unstandardized Coefficients

Adj. R2 Sig. F

Dependent Variable: Bookings02a.

Coefficientsa

B Std. Error

Unstandardized Coefficients

Adj. R2 Sig. F

Dependent Variable: Billings02a.

New Contract Revenue Contract Execution Revenue

0.40

13770*** 4647 0.52 .006

1297* 773 0.47 .040

0.19

7890* 4656 0.24 .100

1696** 697 0.30 .021

Base Model: YRS_EXP, PARTDUM, %_CEO_SRCH, SECTOR(dummies), %_SOLO.b.

N=39. *** p<.01, ** p<.05, * p<.1b.

Bridging diverse communities is significant.

Being in the thick of information flows is significant.

outline

factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections?

studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices

network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage

networks and innovation Lazer and Friedman: innovation

networked coordination game

choice between two things, A and B e.g. basketball and soccer

if friends choose A, they get payoff a if friends choose B, they get payoff b

if one chooses A while the other chooses B, their payoff is 0

coordinating with one’s friends

Let A = basketball, B = soccer. Which one should you learn to play?

fraction p = 3/5 play basketball fraction p = 2/5 play soccer

which choice has higher payoff?

d neighbors p fraction play basketball (A) (1-p) fraction play soccer (B)

if choose A, get payoff p * d *a

if choose B, get payoff (1-p) * d * b

so should choose A if p d a ≥ (1-p) d b

or p ≥ b / (a + b)

two equilibria

everyone adopts A

everyone adopts B

what happens in between?

What if two nodes switch at random? Will a cascade occur?

example: a = 3, b = 2

payoff for nodes interaction using behavior A is 3/2 as large as what they get if they both choose B

nodes will switch from B to A if at least q = 2/(3+2) = 2/5 of their neighbors are using A

how does a cascade occur

suppose 2 nodes start playing basketball due to external factors e.g. they are bribed with a free pair of shoes by some

devious corporation

Quiz Q:

Which node(s) will switch to playing basketball next?

the complete cascade

you pick the initial 2 nodes

A larger example (Easley/Kleinberg Ch. 19) does the cascade spread throughout the network?

http://www.ladamic.com/netlearn/NetLogo412/CascadeModel.html

implications for viral marketing

if you could pay a small number of individuals to use your product, which individuals would you pick?

Quiz question:

What is the role of communities in complex contagion enabling ideas to spread in the presence of thresholds creating isolated pockets impervious to outside ideas allowing different opinions to take hold in different parts of the

network

bilingual nodes

so far nodes could only choose between A and B

what if you can play both A and B, but pay an additional cost c?

try it on a line

Increase the cost of being bilingual so that no node chooses to do so. Let the cascade run

Now lower the cost. What happens?

Quiz Q:

The presence of bilingual nodes helps the superior solution to spread throughout the network helps inferior options to persist in the network causes everyone in the network to become bilingual

knowledge, thresholds, and collective action

nodes need to coordinate across a network, but have limited horizons

can individuals coordinate?

each node will act if at least x people (including itself) mobilize

nodes will not mobilize

mobilization

there will be some turnout

innovation in networks

network topology influences who talks to whom who talks to whom has important implications for

innovation and learning

better to innovate or imitate?

brainstorming:more minds together,but also danger of groupthink

working in isolation:more independenceslower progress

in a network context

modeling the problem space

Kauffman’s NK model

N dimensional problem space N bits, each can be 0 or 1

K describes the smoothness of the fitness landscape how similar is the fitness of sequences with only 1-2 bits flipped

(K = 0, no similarity, K large, smooth fitness)

Kauffman’s NK model

distance

fitn

ess

K large K medium K small

Update rules

As a node, you start out with a random bit string

At each iteration If one of your neighbors has a solution that is more fit than yours,

imitate (copy their solution) Otherwise innovate by flipping one of your bits

Quiz Q:

Relative to the regular lattice, the network with many additional, random connections has on average: slower convergence to a local optimum smaller improvement in the best solution relative to the initial

maximum more oscillations between solutions

networks and innovation:is more information diffusion always better?

Nodes can innovate on their own (slowly) or adopt their neighbor’s solution

Best solutions propagate through the network

source: Lazer and Friedman, The Parable of the Hare and the Tortoise: Small Worlds, Diversity, and System Performance

linear network fully connected network

networks and innovation

fully connected network converges more quickly on a solution, but if there are lots of local maxima in the solution space, it may get stuck without finding optimum.

linear network (fewer edges) arrives at better solution eventually because individuals innovate longer

Coordination: graph coloring

Application: coloring a map: limited set of colors, no two adjacent countries should have the same color

graph coloring on a network

Each node is a human subject. Different experimental conditions: knowledge of neighbors’ color knowledge of entire network

Compare: regular ring lattice small-world topology scale-free networks

Kearns et al., ‘An Experimental Study of the Coloring Problem on Human Subject Networks’, Science, 313(5788), pp. 824-827, 2006

simulation

Kearns et al. “An Experimental Study of the Coloring Problem on Human Subject Networks” network structure affects convergence in coordination games,

e.g. graph coloring

http://www.ladamic.com/netlearn/NetLogo4/GraphColoring.html

Quiz Q:

As the rewiring probability is increased from 0 to 1 the following happens: the solution time decreases the solution time increases the solution time initially decreases then increases again

to sum up network topology influences processes occurring on networks

what state the nodes converge to how quickly they get there

process mechanism matters: simple vs. complex contagion coordination learning

diffusion can be simple (person to person) or complex (individuals having thresholds)

people in special network positions (the brokers) have an advantage in receiving novel info & coming up with “novel” ideas

in some scenarios, information diffusion may hinder innovation