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1 情報伝播研究会’12.3)
複雑ネットワーク上での確率的伝搬とカスケード現象に関する研究動向
生天目 章防衛大学校情報工学科www.nda.ac.jp/~nama
2 情報伝播研究会’12.3)
– Basic Literatures of Diffusion and Cascade
– Agent –based Diffusion in Networks
– Epidemic (Probabilistic )Diffusion Models
– Optimizing of Probabilistic Diffusion
– Cascade Dynamics with Threshold Rules
– Optimizing of Cascade
– Final Remarks
Outline
3 情報伝播研究会’12.3)
<positive diffusion>
• Diffusion of innovation
<negative diffusion>
• Epidemic diffusion
• Failure (risk) diffusion
Ex: i-phoneS4 vs. Xperia
Ex: domino effect
Diffusion in Network
4 情報伝播研究会’12.3)
Aim (1) • Concept of “diffusion” arise quite generally in social sciences
– Diffusion of innovations– Spread of infectious disease
• Concept of “cascade” arises in social sciences and engineering– Cascade phenomena– Cascade failure
• Concept of “systemic risk” arises quite recently – Emergence of collective belief– Transmission of financial distress
We would like to understand in what sense these concepts are the same and how they are different.
5 情報伝播研究会’12.3)
Aim (2)
・Relationship between diffusion and network topology・What types of networks are optimal for maximum or minimum diffusion?
positive type <promotion>
negative type<inhibition>
diffusion
RegularRegular RandomRandom Scale freeScale freeSmall worldSmall world
Other networks?Other
networks?
6 WEHIA‘08 (Taipei)
Penetration of Technological Innovation
6
Color TV
Mobile phone
Personal computer
DVD
VIDEO camera
year
JapanHistorical Data: Diffusion of Innovation
Diffusion curves with different shapesPenetration rate
7 情報伝播研究会’12.3)
Macroscopic Diffusion Model: Bass Model• f(t)=(p+qF(t))[1‐F(t)]: Hazard Model• f(t): the rate of the adoption (growth rate)• F(t): cumulative proportion of adoption• p=coefficient of innovation• q=coefficient of imitation
individuals who already adopted
individuals who are unaware
Special Cases:
q=0: Exponential Distribution
p=0: Logistic Distribution,
f(t) = [p+qF(t)] [(1-F(t)]
external effect (advertising)
Generations of Mainframe Computers (Performance Units) 1974-1992
0
20000
40000
60000
80000
100000
120000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Year
Sale
s
Gen1 Actual Gen1 Fit and Forecast Gen2 Actual Gen2 Fit and Forecast
Gen3 Actual Gen3 Fit and Forecast Gen4 Actual Gen4 Fit and Forecast
8 WEHIA‘08 (Taipei)
Diffusion on Networks<Two-step flow model>
• Describe the flow of information from media to population and the formation of public opinion.
: media → influentials → others influentials
individuals
<Mutual influence model>
D. Watts (2007): Influentials, networks and public opinion formation, Journal of consumer research
Average individuals become much more influent than influentials.
9 情報伝播研究会’12.3)
Diffusion and Contagion
9K: time
Burst diffusionSlow start then rapid spread
Diffusion with critical mass
•We want to characterize macroscopic diffusion patterns from the individual decision level (agent‐based modeling)
•We analyze the network properties that allow or prohibit the wide spread propagation of innovation
Typical Macroscopic Diffusion Patterns
•Question 1: How do we induce individual behaviors behind a specific macroscopic diffusion pattern?
•Question 2: How do we induce the network topology behind a specific macroscopic diffusion pattern?
10 情報伝播研究会’12.3)
Social influenceAGENT
Social network changes an agent’s behaviour
(1) Preference heterogeneity(2) Social influence (3) Social Networks
Agent preference
Preference determines an agent’s behaviour
Social influence changes an agent’s behaviour
An Agent-based Model
Which itemsto buy?
No purchase
11 情報伝播研究会’12.3)
A Binary Decision Model with Social Influence
Individual preference μ
Social trendF(t)Social influence α
)()1()1( tFtp αμα +−=+
Probability of choosing A at time t
11
α∈[0,1] : social influence factor
A
preference social trendF(t): the proportion of the agents to choose A
)}()(/{)()( tBtAtAtF
B
+=
12 情報伝播研究会’12.3)
Experiments:
12
α = ｛0.1, 0.5, 1｝, μ=1,
Rate of penetration:・α=0.1 (very weak social influence)> Monotonically increasing・ α=0.5 (mild social influence)> S-shape function・ α=1 (very strong social influence)> step function
)()1()1( tFtp αμα +−=+
)10( << α
Speed of penetration:・α=0.1 (very weak social influence)> Monotonically decreasing・ α=0.5 (mild social influence)> steady-decreasing・ α=1 (very strong social influence)> increasing-decreasing
α=1
α=1
α=0.1
α=0.1α=0.5
α=0.5
13 情報伝播研究会’12.3)
Empirical Study(1): Household Products
car
Fan heater
Color TV
PC
(%)(%)
14 情報伝播研究会’12.3)
Empirical Study: Household Products (cont.)weak social influence
Fan heater PC
Car Color TV
strong social influence
15 情報伝播研究会’12.3)
1st rank
2nd
rank:3000
rank:4000
rank: 3000
rank:4000
x104
best seller
second best seller
Empirical Study(2): Magazine Sales
weak social influence
strong social influence
Top sales low sales
16 情報伝播研究会’12.3)
Summary:
K: time
S-shape: Slow start then rapid spread Diffusion with critical massBurst diffusion
social influence: very low social influence: mild social influence: very strong
)()1()1( tFtp αμα +−=+
Macroscopic phenomena
Individual characteristic: α
1.0=α 5.0=α 1=α
Social influence level determines the shape of diffusion patterns
17 情報伝播研究会’12.3)
Probabilistic Diffusion
Epidemic diffusion model: <SIS model>
• Consider a fixed population of size N• Each individual is in one of three states:susceptible (S), infected (I), recover & susceptible (S)
S I Sβ j
i
δ
18 情報伝播研究会’12.3)
An Epidemic Diffusion Model
18
α
Awareness=0Preference=0Adoption=0
Awareness=1Preference=0Adoption=0
Awareness=1Preference=1Adoption=0
Awareness=1Preference=1Adoption=1
Mizuno (2008) β
Each individual is in one of three cognitive states:Susceptible (S) (unaware, also inactive, non-adopter)Infected (I) (aware, also active, informed, adopter)Removed (R) (lose interest or forget)
aware
lose interest or forget
Agent State Transitions
19 情報伝播研究会’12.3)
Epidemic Dynamics• The expected state of the system at time t is given by
• As t ∞
• the probability that all copies die converges to 1
• the probability that all copies die converges to 1
• the probability that all copies die converges to a constant < 1
( )( ) 1tt −−+= vIAv δβ 1
( )( ) ( ) 0 then ,λ11λ if t11 →<⇔<−+ vAIA βδδβ
( )( ) ( ) cvAIA →=⇔=−+ t11 then ,λ11λ if βδδβ
( )( ) ( ) ∞→=⇔>−+ t11 then ,λ11λ if vAIA βδδβ
( )A1λ The largest eigenvalue of the adjacent matrix A
vi :the probability of being infected of node iv=(v1,v2,…,vN)
20 情報伝播研究会’12.3)
Threshold of Epidemic Spread
• The virus spreads if λ>1/λ1(A)
• The virus dies out if λ<1/λ1(A)
)(1
1 Ac λλ =
１
２
５ ４
３
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
0100110001000010000111110
Adjacency matrix
)(1 Aλ
Network
Largest eigenvalue
δβλ /=Relative propagation probability:
threshold
21 情報伝播研究会’12.3)
Largest Eigenvalues
: the largest eigenvalue of the adjacency matrix Amaxλstar
1λ max −= N
complete
1λ max −= N
random
>=<= kpNmaxλ
Multi Hub
Scale free
4/1maxλ N≅
p: connection probability<k>: average degree
average max maxd dλ≤ ≤
max 2maxTx Axx
λ =
max max maxd dλ≤ ≤If G is regular of degree d, then max .dλ =
22 情報伝播研究会’12.3)
Network Optimization (1)
• Objective function: Maximization of diffusion– Find network which have lowest number of E by genetic algorithm
– First term maximize largest eigenvalue
– Second term minimize average degree
1)1(
)(1
1 −><
−+=n
kA
E ωλ
ω
23 情報伝播研究会’12.3)
Network Optimization (2)
• Objective function: Minimization of diffusion– Find network which have lowest number of E by genetic algorithm
– First term minimize largest eigenvalue
– Second term maximize average degree
><−+= kAE /)1()(1 ωωλ
24 情報伝播研究会’12.3)
Stochastic Optimization (1)
• Simulated annealing – Probabilistic algorithm for the optimization problem– Rewiring trials - Rewiring a randomly selected link– Fitness function to be optimized: Q
– if δQ = Qfinal − Qinitial < 0accept rewiring
N = 50, and <k> = 4
Optimized network
25 情報伝播研究会’12.3)
Stochastic Optimization (2) Generic Algorithm
Crossover rate ：0.7Mutation rate : 2/nC2Number of Individuals ：10
(networks)
Generic code representation
Networks and adjacency matrix
26 情報伝播研究会’12.3)
Some Evolved Networks • The largest eigenvalue λ1
• Object function
(to be minimized)
Hub network All connected
A Kind of scale-free (KN network) is optimal for maximum diffusion
ω=0.1 ω=0.5 ω=0.9
D=46α=1.98
D=50α=1.98
λ1=8.43λn=-3.41
D=48α=1.98
ω=1
λ1=100.0λn=-1.98
D=2α=98.988
ω=0.92
λ1=8.18λn=-2.96
D=14α=2.52
ω=0.95
λ1=10.27λn=-2.98
D=11α=2.88
ω=0.98
λ1=14.2λn=-3.21
D=9α=3.
Core-dense network
><−+= kF )1(/ 1 ωλω
27 情報伝播研究会’12.3)
Comparative Study
RandomRandom BABA CPACPARandom RegularRandom RegularTorusTorus
・Relationship between diffusion and network topology・What types of networks are optimal to diffuse?
(maximization vs. minimization of diffusion)
41 =λ 41 =λ 2.51 =λ 121 =λ 331 =λ
The largest eigenvalue of the adjacency matrix A
minimum diffusionmaximum diffusion
Power lawi) BA(Barabasi-Albert) λ =3
ii) Scale Free
λ−= Ckkp )(
32 <≤ λ
28 情報伝播研究会’12.3)
Cascade in Socio-economic Networks
<Many social-economic networks show cascading effects>
initial disturbance in some area
Largest blackout
Internet congestion collapse
Drop in speed of a factor 100
Power gridsInternet
29 情報伝播研究会’12.3)
Cascade in Finance
29
ＮＹ Stock Prices
ＮＹ Oil Prices
Cascade
30 情報伝播研究会’12.3)
Cascade: Blackout• Concepts of “cascade” arise quite generally in computer
science, social sciencesOne application component can trigger failures in others.
The down of some sub‐systems can make the entire system down.
Blackout Hits U.S. and Canadian Cities Aug. 14, 2003.
Before After
Failure makes another failure How about global cascade ? Initial failure break the system completely.
31 情報伝播研究会’12.3)
A Cascade Model with Threshold RuleEach node with a binary state shifts depending on the
states of neighboring nodes
: A binary decision rule of a consumer
phone4s)-(i 1 x to(Xperia) 0 xfromshift /)1(#
ii ==≥= φii kxofif : threshold of consumer i
Ex: i‐phone4s vs. XperiaGOGO! Docomo!
φ
32 情報伝播研究会’12.3)
Cascade Window– Cascade occurs on limited sizes of the average degree
– Robust‐yet‐Fragile property
The frequency of global cascades
Largest connected components (cluster size) of the network
zSeS −−=1
z [Average degree]
1000Cascade of #
=
Cascade size# of Node: N=500,
