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2013-09-24T22:23:39Z

69.249.206.47: /* Proof */

{{Orphan|date=July 2013}}

'''DeGroot learning''' refers to a rule-of-thumb type of social learning process. The idea was stated in its general form by the American statistician [[Morris DeGroot|Morris H. DeGroot]];<ref name=DeGroot74/> antecedents were articulated by John R. P. French<ref name=French56/> and Frank Harary.<ref name=Harary59/> The model has been used in [[physics]], [[computer science]] and most widely in the theory of [[Social network analysis|social networks]].<ref name = Jackson2008/>

== Setup and the learning process ==
Take a society of <math> n </math> agents where everybody has an opinion on a subject, represented by a vector of probabilities <math> p(0) = (p_1(0), \dots, p_n(0) ) </math>. Agents obtain no new information based on which they can update their opinions but they communicate with other agents. Links between agents (who knows whom) and the weight they put on each other's opinions is represented by a trust matrix <math> T </math> where <math> T_{ij} </math> is the weight that agent <math> i </math> puts on agent <math> j </math>'s opinion. The trust matrix is thus in a one-to-one relationship with a [[Weighted graph|weighted]], [[directed graph]] where there is an edge between <math> i </math> and <math> j </math> if and only if <math> T_{ij} > 0 </math>. The trust matrix is [[Stochastic matrix|stochastic]], its rows consists of nonnegative real numbers, with each row summing to 1.

Formally, the beliefs are updated in each period as

:<math>
p(t) = T p(t-1)
</math>

so the <math> t </math> th period opinions are related to the initial opinions by

:<math>
p(t) = T^t p(0)
</math>

== Convergence of beliefs and consensus ==

An important question is whether beliefs converge to a limit and to each other in the long run.
As the trust matrix is [[Stochastic matrix|stochastic]], standard results in [[Markov chain]] theory can be used to state conditions under which the limit

:<math>
p(\infty) = \lim_{t \to \infty} p(t) = \lim_{t \to \infty} T^t p(0)
</math>

exists for any initial beliefs <math> p(0) \in [0, 1]^n </math>. The following cases are treated in Golub and Jackson
<ref name = GolJack2010/> (2010).

=== Strongly connected case ===

If the social network graph (represented by the trust matrix) is [[strongly connected]], convergence of beliefs is equivalent to each of the followings
* the graph represented by <math> T </math> is [[Aperiodic graph|aperiodic]]
* there is a unique left [[Eigenvalues and eigenvectors|eigenvector]] <math> s </math> of <math> T </math> corresponding to [[Eigenvalues and eigenvectors|eigenvalue]] 1 whose
entries sum to 1 such that, for every <math> p \in [0, 1]^n </math>, <math> \left( \lim_{t \to \infty} T^t p \right)_i = s \cdot p </math>

for every <math> i \in \{1, \dots, n \} </math> where <math> \cdot </math> denotes the [[dot product]].

=== General case ===

It is not necessary to have a [[strongly connected]] social network to have convergent beliefs, however,
the equality of limiting beliefs does not hold in general.

We say that a group of agents <math> C \subseteq \{1, \dots, n \} </math> is ''closed'' if for any <math> i \in C </math>, <math> T_{ij} > 0 </math> only if <math> j \in C </math> . Beliefs are convergent if and only if every set of nodes (representing individuals) that is strongly connected and closed is also [[Aperiodic graph|aperiodic]].

=== Consensus ===

A group <math> C </math> of individuals is said to reach a ''consensus'' if <math> p_i(\infty)= p_j(\infty) </math> for any <math> i, j \in C </math>. This means that, as a result of the learning process, in the limit they have the same belief on the subject.

With a [[strongly connected]] and [[Aperiodic graph|aperiodic]] network the whole group reaches a consensus.
In general, any strongly connected and closed group <math> C </math> of individuals reaches a consensus for every initial vector of beliefs if and only if it is aperiodic. If, for example, there are two groups satisfying these assumptions, they reach a consensus inside the groups but there is not necessarily a consensus at the society level.

== Social influence ==

Take a [[strongly connected]] and [[Aperiodic graph|aperiodic]] social network. In this case, the common limiting beliefs are determined by the initial beliefs through

:<math>
p(\infty) = s \cdot p(0)
</math>

where <math> s </math> is the unique unit length [[Eigenvalues and eigenvectors|left eigenvector]] of <math> T </math> corresponding to the [[Eigenvalues and eigenvectors|eigenvalue]] 1. The vector <math> s </math> shows the weights that agents put on each other's initial beliefs in the consensus limit. Thus, the higher is <math> s_i </math>, the more ''influence'' individual <math> i </math> has on the consensus belief.

The eigenvector property <math> s = s T </math> implies that
:<math> s_i = \sum_{j=1}^n T_{ji} s_j </math>

This means that the influence of <math> i </math> is a weighted average of those agents' influence <math> s_j </math> who pay attention to <math> i </math>, with weights of their level of trust. Hence influential agents are characterized by being trusted by other individuals with high influence.

== Examples ==
These examples appear in Jackson <ref name=Jackson2008 /> (2008).

=== Convergence of beliefs ===
[[File:De Groot learning convergent example.png|200px|right|thumb|A society with convergent beliefs]]
Consider a three-individual society with the following trust matrix:

:<math>
T =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{pmatrix}
</math>

Hence the first person weights the beliefs of the other two with equally, while
the second listens only to the first, the third only to the second individual.
For this social trust structure, the limit exists and equals

:<math>
\lim_{t \to \infty} T^t p(0) = \left(\lim_{t \to \infty} T^t\right) p(0) = \begin{pmatrix}
2/5 & 2/5 & 1/5 \\
2/5 & 2/5 & 1/5 \\
2/5 & 2/5 & 1/5 \\
\end{pmatrix} p(0)
</math>

so the influence vector is <math> s = \left( 2/5, 2/5, 1/5 \right) </math> and the consensus belief is
<math> 2/5 p_1(0) + 2/5 p_2(0) + 1/5 p_3(0) </math>. In words, independently of the initial beliefs,
individuals reach a consensus where the initial belief of the first and the second person has twice as
high influence than the third one's.

=== Non-convergent beliefs ===
[[File:De groot learning nonconvergent beliefs.png|200px|right|thumb|A society with non-convergent beliefs]]
If we change the previous example such that the third person also listens exclusively to the first
one, we have the following trust matrix:

:<math>
T =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
1 & 0 & 0 \\
\end{pmatrix}
</math>

In this case for any <math> k \geq 1 </math> we have

:<math>
T^{2k - 1} =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
1 & 0 & 0 \\
\end{pmatrix}
</math>

and
:<math>
T^{2k} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1/2 & 1/2 \\
0 & 1/2 & 1/2 \\
\end{pmatrix}
</math>

so <math> \lim_{t \to \infty} T^t </math> does not exist and
beliefs do not converge in the limit. Intuitively, 1 is updating based on 2 and 3's beliefs while
2 and 3 update solely based on 1's belief so they interchange their beliefs in each period.

== Asymptotic properties in large societies: wisdom ==

It is possible to examine the outcome of the DeGroot learning process in large societies,
that is, in the <math> n \to \infty </math> limit.

Let the subject on which people have opinions be a "true state" <math> \mu \in [0, 1] </math>. Assume that individuals
have [[Independence (probability theory)|independent]] noisy signals <math> p_i^{(0)}(n) </math> of <math> \mu </math>
(now superscript refers to time, the argument to the size of the society).
Assume that for all <math> n </math> the trust matrix <math> T(n) </math> is such that the
limiting beliefs <math> p_i^{(\infty)}(n) </math> exists independently from the initial beliefs.
Then the sequence of societies <math> \left( T(n) \right)_{n = 1}^{\infty} </math> is called ''wise'' if

:<math>
\max_{i \leq n} | p_i^{(\infty)} - \mu | \xrightarrow{\ p\ } 0
</math>

where <math> \xrightarrow{\ p\ } </math> denotes [[Convergence of random variables|convergence in probability]].
This means that if the society grows without bound, over time they will have a common and accurate
belief on the uncertain subject.

A necessary and sufficient condition for wisdom
can be given with the help of [[#Social influence|influence vectors]]. A sequence of societies is wise if and only
if
:<math>
\lim_{n \to \infty} \max_{i \leq n} s_i(n) = 0
</math>
that is, the society is wise precisely when even the most influential individual's influence vanishes in the
large society limit. For further characterization and examples see Golub and Jackson<ref name=GolJack2010 /> (2010).

== References ==
<references>
* <ref name = DeGroot74>DeGroot, Morris H. 1974. “[http://www.jstor.org/stable/10.2307/2285509 Reaching a Consensus.]” ''Journal of the American Statistical Association'', 69(345): 118–21.</ref>

* <ref name = French56>French, John R. P. 1956. “A Formal Theory of Social Power” ''Psychological Review'', 63: 181–94.</ref>

* <ref name = GolJack2010>Golub, Benjamin & Matthew O. Jackson 2010. "[http://www.aeaweb.org/articles.php?doi=10.1257/mic.2.1.112 Naïve Learning in Social Networks and the Wisdom of Crowds,]" American Economic Journal: Microeconomics, American Economic Association, vol. 2(1), pages 112-49, February.</ref>

* <ref name=Jackson2008>Jackson, Matthew O. 2008. ''Social and Economic Networks.'' Princeton University Press.</ref>

* <ref name = Harary59>Harary, Frank. 1959. “A Criterion for Unanimity in French's Theory of Social Power” in Dorwin Cartwright (ed.), ''Studies in Social Power'', Ann Arbor, MI: Institute for Social Research.</ref>

</references>

[[Category:Social learning theory]]

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