群体智慧--复杂网络上的最佳共识形成

Xiaofan Wang
xfwang@
SOCIAL LEARNING IN COMPLEX NETWORKS
What is social learning?
(Consensus on the true state)
Our recent researches
(Pinning, Similarity ‐based, Chaos)
Social learning algorithms
(Bayesian +
Consensus)
我的在线历程20002005200620102011
汪小帆老师
90
关注4761粉丝
¾极度不均匀：
¾并非很社会：¾高度同质化：¾两步信息流：
¾不同生命期：
人肉搜索为何百发百中？
教育医疗等却为何如此难以形成最佳共识？
围脖上的意见领袖对粉丝的影响力有多大？
热门话题、名人堂。

群体智慧：大众比精英更聪明！Galton, Nature, 1907
¾Private signals
¾Network structure
¾Update rules Observing
Communicating
Updating Beliefs
Social Learning Process
1
2
3
一致性：能否形成共识？
最优性：是否最佳共识？
可控性：能否引导共识？
Case study: Who is singing?
State space True state Private signal
Likelihood function
Network structure ,,0()1,()1
i t k i t k k
μθμθ≤≤=∑,(*)1
i t μθ→Belief
{}
12,,,n θθθΘ=L *θ∈Θ
i t
s
(|)1
i s
l s θ=∑(,)
G V E =
Bayesian Learning ：A Single Agent Case
()
t →∞*
()1P
t μθ→
11
111(
|)()
()(|)()
t t t t t t l s s m s θμθμθμθ+++++=
Observationally equivalent states
Agent i is called an indiscriminative agent
*:,(|)(|)for all i i i
i i i
l s l s s S θθθθθ∗∃∈Θ≠=∈
¾Each agent should know the global structure of the network ¾Each agent tries to deduce the information of every other agent
Bayesian Social Learning: Network Case
Computation burden + high complexity (|)()
(|)()
l s s m s θμθμθ=
人肉搜索有效克服了这两个困难！
Consensus
,1,,()()i
i t ii i t ij j t j N a a μθμθμθ+∈=+∑（）,,()i t j t μθμθ−→（）
0,()i t μθ→0,(*)1
i t μθ→θ∀∈Θ
θ∀∈Θ*
θθ∀≠Social Learning = Best Consensus
Consensus Algorithm
Consensus
,,01
()i t j j
N μθμθ→∑（）
Consensus
,1,,()()i
i t ii i t ij j t j N a a μθμθμθ+∈=+∑（）,,()i t j t μθμθ−→（）
0,()i t μθ→0,(*)1
i t μθ→θ∀∈Θ
θ∀∈Θ*
θθ∀≠Social Learning = Best Consensus
Consensus Algorithm
Consensus
,,01
()i t j j
N μθμθ→∑（）
Consensus
,1,,()()i
i t ii i t ij j t j N a a μθμθμθ+∈=+∑（）,,()i t j t μθμθ−→（）
0,()i t μθ→0,(*)1
i t μθ→θ∀∈Θ
θ∀∈Θ*
θθ∀≠Social Learning = Best Consensus
Consensus Algorithm
Consensus
,,01
()i t j j
N μθμθ→∑（）
Consensus
,1,,()()i
i t ii i t ij j t j N a a μθμθμθ+∈=+∑（）,,()i t j t μθμθ−→（）
0,()i t μθ→0,(*)1
i t μθ→θ∀∈Θ
θ∀∈Θ*
θθ∀≠Social Learning = Best Consensus
Consensus Algorithm
Consensus
,,01
()i t j j
N μθμθ→∑（）θ∀∈Θ
Bayesian vs. Consensus
Consensus/Synchronization
Bayesian Learning
N e t w o r k C o m p l e x i t y
Just average, but not necessarily true!
Demand strict!
Boundedly Rational
Learning W
e s e e k
,i
j t j N μθ∈∑（）,1()i t μθ+=θ
∀∈Θ
,11,()(|)()
i i i t t i t i t m s l s θ
θμθ++∈Θ
=
∑Bayesian
Consensus
1
,,1(|)()
()
i i t i t i
i t t l s
m s θμθ++ii
a ij
a
+Bayesian+ Consensus
(e) There is no other state that is observationally equivalent to the true state from the point of all agents in the network. The Wisdom of Crowds
(a) The social network is strongly connected ;
(b) All agents have strictly positive self-reliances ;
(c) There exists an agent with positive prior belief on the true state;*,()1i t μθ→1,1,,,1(|)()()()i
i i t i t ii i t ij j t
i j N i t t l s a a m s θμθμθμθ++∈+=+∑（）
θ∀∈Θ
Our recent works
1.Uninformed agents:those who can’t observe their
private signals;
2.Similarity-based communication:two agents
are neighbors only if they have similar beliefs;
3.Chaos in social learning with multiple true
states
Uninformed agents Remark:Jadbabaie’s model Consensus
l n
=0l
=Informed agents 1,1,,,1(|)()(),1,2,,()
i i i t i t ii i t ij j t i j N i t t l s a a i l m s θμθμθμθ++∈+=+=∑L （）
,1,,()(),1,,i i t ii i t ij j t j N a a i l n μθμθμθ+∈=+
=+∑L （）
(e) There is no state that is observationally equivalent to the true state from the point of all informed agents
in the network.
Suppose that
(a) The social network is strongly connected ;
(b) There exists at least one informed agent and all self-reliance of informed agents are strictly positive;
(c) There exists at least one agent with positive prior belief on the true state;,(*)1
i t μθ→
In a power-law network with tunable exponent
2.1γ=10γ=Heterogeneous Homogeneous
N=1000, two states, two signals {H, T}
Prior beliefs: uniform distribution in [0,1], a ii =0.5
Similarity Breeds Connection: Homophily Principle in Sociology
{}
,,():i j t i t N t j V r μμ=∈−≤()()()()()1,1,,(),1|1()i i t i t i t j t i j N t i i t t p s N t m s θμθμθμθ++∈+⎛⎞=+⎜⎟⎜⎟⎝⎠
∑Neighbors of agent i
Update rule
Social learning with similarity-based communication Confidence radius
r = 0.3r = 0.1r = 0.02容忍差异才能一致
N=100, 20 discriminative agents, 80 indiscriminative agents
Two states, two signals {H, T}
Prior beliefs: random distribution in [0,1]
Originally connect
Tight and Loose Cultures:
A 33-Nation Study
If one were to order all mankind to choose the best set of rules in the world, each group would, after due consideration, choose its own customs; each group regards its own as being the best by far. ---Herodotus
ScMichele J. Gelfand, et al.ience27 May 2011:
Vol. 332 no. 6033 pp. 1100-1104
The Wisdom of Crowds
The Wisdom of Crowds
Indiscriminative agents
2
r 2
r Feasible region
Belief update without losing old neighbors Connectivity preserve strategy
Connectivity preserve
Different communities might have different underlying true states
Three states Two signals {H,T}
Prior beliefs: 1/3{}
123
,,θθθ2(|)
l s θ1(|)
l s θ
Belief evolution of agents in group A 1Belief evolution of agents in group A 1
Belief evolution of agents in Group A 2
Belief evolution of agents in Group A 2
0.3350
0.3338
0.1481
Group 2
0.19370.30190.1999Group 1 State 3State 2State 1
0.92360.73200.7251Group
What’s next?
•New adaptive social learning model •Methods for misinformation dynamics •Models for belief manipulation
……
Xiaofan Wang Shanghai Jiao Tong University
xfwang@。