lecture 27: information diffusion cs 765 complex networks slides are modified from lada adamic,...
TRANSCRIPT
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