na ve learning in social networks and the wisdom of crowds...na ve learning in social networks and...

Naıve Learning in Social Networksand the Wisdom of CrowdsB. Golub and M.O. Jackson (2010)

Matteo CamboniLeonardo Nini

Spring 2014 IGIER Visiting Students presentations

March 28, 2014

M. Camboni L. Nini (IGIER VSI) Naıve Learning in Social Networks March 28, 2014 1 / 32

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What is a network?



Learning in social networks

The model investigates the ability of a population to aggregateinformation in an efficient way through social interaction.

Research question

Which are the social network structures that allow a society made up ofindividuals who communicate and update their beliefs naıvely to aggregatedecentralized information completely and correctly?



Main findings

1 Individuals’ beliefs converge to a consensus, provided that thenetwork is strongly connected and satisfies a weak aperiodicitycondition.

2 Even with a naıve updating process, growing societies will eventuallycome close to the truth, as long as no agent holds a nonvanishinginfluence as the population grows.A society which converges to the truth is called “wise”.



The DeGroot Model (1974)

1 The model, introduced by Morris De Groot in 1974, serves as astarting point for the authors’ analysis.

2 There are n agents, indexed by the set N = {1, ..., n} who interact ina network Γ(n).

3 Each individual has to estimate an uknown parameter µ ∈ R.

4 Time is assumed to be discrete, i.e .t = 0, 1, 2 . . . we can think ofperiods as days.

5 Each agent receives at t=0 a noisy signal about the true value of the

parameter p(0)i = µ+ εi , where εi is an randomly distributed error

term with expected value of 0.



Behavioral Assumption

Attention

Agents are characterized by bounded rationality: they continue to use thesame updating rule throughout the evolution, failing to adjust correctly forrepetition and dependencies in information that they hear multiple times.



Updating

1 Each agent i communicates with other individuals, assigning precisionπij to agent j . Its belief in period t + 1 is assumed to be a weghtedaverage of the beliefs held in period t by all agents.

2 Define the vector p(t) = (p(t)1 , p

(t)2 , p

(t)3 , ...., p

(t)n ) as the vector

including the belief of every individual at time t.

3 Hence, p(t+1) = Tp(t) where T is a n x n stochastic interactionmatrix such that

Tij =πij∑n

k=1 πik

4 Since the updating rule is constant across periods,

p(t) = Tp(t−1) = Ttp(0)



Example 1

p(t+1)1 = 0.6p

(t)1 + 0.2p

(t)2 + 0.2p

(t)3



Convergence

Definition 1

A matrix T is convergent if limt→∞Ttp exists for all vectors p ∈ [0, 1]n

An obvious question which may arise is: under which conditions doesan interaction matrix T converge?

In order to answer such question, we need some further definitions.



Convergence

Definition 2

A walk in T is a sequence of nodes i1, i2, i3, ..., iK , not necessarily distinct,such that Tik,ik+1

> 0 ∀ k ∈ {1, 2, ...,K − 1} .The lenght of the walk is defined to be K − 1.

Definition 3

A cycle in T is a walk i1, i2, i3, ..., iK such that i1 = iK .The lenght of a cycle with K (not necessarly distinct) entries is K − 1.A cycle is simple if the only node appearing twice in the sequence isi1 = iK .



Convergence

Definition 4

The matrix T is strongly connected if there is a path in T going fromevery node to any other node.

Definition 5

The matrix T is aperiodic if the gratest common divisor of the lenght ofits simple cycles is 1.



Example 2: failure of aperiodicity

T =

[0 11 0

]In this case,

Tt =

{I if t is even

T if t is odd



Result 1: Convergence of Beliefs in the De Groot model

Proposition 1

If T is a strongly connected matrix, the following propositions areequivalent(a) T is convergent;(b) T is aperiodic;(c) ∃! left eigenvector s of T corresponding to eigenvalue 1 whose entriessum to 1, s.t. ∀ p ∈ [0, 1]n,

limt→∞

p(t)i =

(limt→∞

Ttp(0))i

= sp(0) ∀i



The influence vector

The ith element of the vector s = (s1, ..., sn) ∈ [0, 1]n can beinterpreted as the influence of agent i .

Since s is a left eigenvector of T, the influence of each agent i can becomputed as a weighted average of the influences of all agents withwhom he is connected, where the weights are given by the importancethat agent j assigns to agent i

si =n∑

j=1

sjTji



Extending the model

Now, we examine a sequence of networks, in which the number n ofagents grows.

A society can be described by a sequence of interaction matrices

(T (n))∞n=1 and of vectors (p(t)(n))∞n=1 indexed by n, where p(t)i (n) is

the belief of agent i in network Γ(n) at time t.

Definition 6: Wisdom

The sequence (T (n))∞n=1 is wise if

plimn→∞maxi≤n

∣∣∣p(∞)i (n)− µ

∣∣∣ = 0



Result 2: Convergence to the truth

Lemma 1

[Application of the Law of large numbers] if (s (n))∞n=1 is any sequence ofinfluence vectors, then

plimn→∞s(n)p(0)(n) = µ

if and only if s1(n)→ 0, where s1(n) = max1≤i≤n si (n)

Proposition 2

If (T (n))∞n=1 is a sequence of convergent stochastic matrices, then it iswise if and only if the associated influence vectors are such that s1 (n)→ 0.

Naıve agents can be misled by highly influential individuals.

The limiting belief of all agents will converge to the truth if nobody’sidiosyncratic error gets a positive weight as the society growsarbitrarily large.



Case of undirected networks with equal weights

We assume that the network can be described by a (symmetric)matrix G which ijth element is equal to 1 if and only if there is an(indirect) link between agents i and j , and equal to 0 otherwise.

We assume also that each individual values equally the opinion ofevery agent with who he is in contact.

Therefore, the stochastic Matrix T (G) is defined by Tij = Gij/di (G) ,where di (G)=

∑nj=1 Gij , is the degree of agent i , i.e. the number of

individuals who have an (indirect) link with agent i .



Case of undirected networks with equal weights

Define a sequence (G (n))∞n=1 of symmetric adjacency matrices.

For each n, the degree of agent i is: di (G (n)) =∑n

i=1 Gij (n) .

Corollary 1

The sequence (G (n))∞n=1 is wise if and only if

max1≤i≤n

si (n) = max1≤i≤n

di (G (n))∑ni=1 di (G (n))

→ 0 as n→∞



Wisdom in terms of social structure

We move now to a more general setting, in which links are notnecessarily reciprocal and individuals may assign different weights todifferent agents.

In order to identify some structural conditions that can ensurewisdom, we introduce some concepts.

Definition 7

A family is a sequence of groups Bn such that Bn ⊂ {1, 2, ...n} ∀ n



Prominent family

TB,C =∑

i∈B. j∈CTij



Prominent family

Definition 8

A group of agents B is prominent in t steps relative to T if(Tt)i ,B > 0 ∀i /∈ B.Call πB (T; t) := mini /∈B (Tt)i ,B the t−step prominence of B relative to T.

Definition 9

The family Bn is uniformly prominent relative to (T (n))∞n=1 if∃ α > 0 s.t. ∀ n there is a t so that Bn is prominent in t steps relative toT (n) with πBn (T (n) ; t) ≥ α.

Definition 10

The family Bn is finite if ∃ q s.t. supn |Bn| < q.



Prominent group: intuition

The group inside the dashed circle is prominent in 2 steps.



Prominent family: intuition



Result 3: Prominent family

Proposition 3

If there is a finite, uniformly prominent family Bn wrt T (n), then thesequence (T (n))∞n=1 is not wise.

Prominence is thus an hindrance to wisdom: families which areheavily influential towards the rest of the society impede convergenceto the truth.



Example 3



Example 3

si (n) =

{1−ε

1−ε+δ if i = 1δ

(1−ε+δ)(n−1) if i > 1



Example 4

limn→∞ s1 (n) can be made arbitrarly close to 1 by choosing a smallδ ∈ (0, 1/2).

si (n) =

(δ

1− δ

)i−1 1−(

δ1−δ

)1−

(δ

1−δ

)n



Result 4: Structural sufficient conditions for wisdom

Theorem 1

If (T (n))∞n=1 is a sequence of convergent stochastic matrices satisfyingbalance and minimal out-dispersion, then it is wise.

Balance Property

If there exists a sequence j (n)→∞ s.t. if |Bn| ≤ j (n), then

supn

TBcn ,Bn (n)

TBnBcn , (n)

<∞

Minimal Out- Dispersion Property

There exists a q ∈ N and r > 0 s.t. if Bn is finite, |Bn| ≥ q and|Cn| /n→ 1, then TBnCn (n) > r for large enough n.



The speed of convergence

Issue: how long does it take Tt to approach its limit (if it exists)? Ingeneral, there is no relationship between speed of convergence andwisdom

Consider the case in which agents weight each other equally: suchsociety will converge immediately to wisdom.

If instead all agents assign a weight equal to 1 to the same agent,then we would have immediate convergence but without wisdom.

Lastly, consider a society in which all individuals place 1− ε weight onthemselves and distribute the rest equally: such society is wise butconverge will happen arbitrarily slowly for small enough ε.



Related literature

De Groot Model (and related literature in physics, computer science)

Sociology: centrality and prestige analysis, relationship between socialstructure and social learning (Katz, 1953, Bonacich, 1987)

Herding Models (ex. Abhijit and Banerjee, 1992): agents converge tosame belief.

Observational Learning Models

Rosenberg et al., (2009): convergence and efficieny learning does notdepend on the network architecture.Bala and Goyal (1998): similar questions, but different setting(observational learning) and not precise calculations of the influence ofeach agent in the network.Acemoglu et al. (2008)

Network-based explanation of political opinion (DeMarzo et al.,2003): different questions, but similar learning results.



Conclusions

1 Small groups of opinion makers who attract a large share of attentionwithout weighting considerably the rest of society are a great obstacleto correct common knowledge.

2 Such peculiarity may be interesting in formulating marketing orelectoral strategies.

3 We can identify networks which are approximately line with the twostructural sufficient conditions identified in the model, in whichtherefore agents will be facilitated in reaching a reasonable consensus.



Discussion

1 Strong behavioral assumptions: real agents are likely to employ moresophisticated belief updating methods.

2 If the society never aknowledges the true value of the parameter,there can be no valuation or revision of an agent’s reliability.

3 The model assumes also that individuals have no cost in gettinginformation.

4 Speed of convergence: if the process of beliefs updating converges tooslowly, will it reach a steady state in ”useful” time? What if µ is notfixed but changes over time?

Although the model does not answer to these issues, it conveys somereasonable results building on a simple framework.


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