DeGroot learning refers to a rule-of-thumb type of social learning process. The idea was stated in its general form by the American statisticianMorris H. DeGroot;^[1] antecedents were articulated by John R. P. French^[2] and Frank Harary.^[3] The model has been used inphysics,computer science and most widely in the theory ofsocial networks.^[4][5]

Take a society of${\displaystyle n}$ agents where everybody has an opinion on a subject, represented by a vector of probabilities${\displaystyle p(0)=(p_1(0),\dots ,p_n(0))}$. 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${\displaystyle T}$ where${\displaystyle T_{ij}}$ is the weight that agent${\displaystyle i}$ puts on agent${\displaystyle j}$'s opinion. The trust matrix is thus in a one-to-one relationship with aweighted,directed graph where there is an edge between${\displaystyle i}$ and${\displaystyle j}$ if and only if${\displaystyle T_{ij}>0}$. The trust matrix isstochastic, its rows consists of nonnegative real numbers, with each row summing to 1.

Formally, the beliefs are updated in each period as

${\displaystyle p(t)=Tp(t-1)}$

so the${\displaystyle t}$ th period opinions are related to the initial opinions by

${\displaystyle p(t)=T^{t}p(0)}$

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

${\displaystyle p(\mathrm{\infty })=\underset{t\to \mathrm{\infty }}{lim}p(t)=\underset{t\to \mathrm{\infty }}{lim}{T}^{t}p(0)}$

exists for any initial beliefs${\displaystyle p(0)\in [0,1{]}^n}$. The following cases are treated in Golub and Jackson^[6] (2010).

If the social network graph (represented by the trust matrix) isstrongly connected, convergence of beliefs is equivalent to each of the following properties:

• the graph represented by${\displaystyle T}$ isaperiodic
• there is a unique lefteigenvector${\displaystyle s}$ of${\displaystyle T}$ corresponding toeigenvalue 1 whose entries sum to 1 such that, for every${\displaystyle p\in [0,1{]}^n}$,${\displaystyle {\left(\underset{t\to \mathrm{\infty }}{lim}{T}^{t}p\right)}_i=s\cdot p}$ for every${\displaystyle i\in \{1,\dots ,n\}}$ where${\displaystyle \cdot }$ denotes thedot product.

The equivalence between the last two is a direct consequence fromPerron–Frobenius theorem.

It is not necessary to have astrongly 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${\displaystyle C\subseteq \{1,\dots ,n\}}$ isclosed if for any${\displaystyle i\in C}$,${\displaystyle T_{ij}>0}$ only if${\displaystyle j\in C}$ . Beliefs are convergent if and only if every set of nodes (representing individuals) that is strongly connected and closed is alsoaperiodic.

A group${\displaystyle C}$ of individuals is said to reach aconsensus if${\displaystyle p_i(\mathrm{\infty })=p_j(\mathrm{\infty })}$ for any${\displaystyle i,j\in C}$. This means that, as a result of the learning process, in the limit they have the same belief on the subject.

With astrongly connected andaperiodic network the whole group reaches a consensus.In general, any strongly connected and closed group${\displaystyle C}$ 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.

Take astrongly connected andaperiodic social network. In this case, the common limiting belief is determined by the initial beliefs through

${\displaystyle p(\mathrm{\infty })=s\cdot p(0)}$

where${\displaystyle s}$ is the unique unit lengthleft eigenvector of${\displaystyle T}$ corresponding to theeigenvalue 1. The vector${\displaystyle s}$ shows the weights that agents put on each other's initial beliefs in the consensus limit. Thus, the higher is${\displaystyle s_i}$, the moreinfluence individual${\displaystyle i}$ has on the consensus belief.

The eigenvector property${\displaystyle s=sT}$ implies that

${\displaystyle s_i=\sum _{j=1}^n{T}_{ji}{s}_j}$

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

These examples appear in Jackson^[4] (2008).

Consider a three-individual society with the following trust matrix:

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

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

If we change the previous example such that the third person also listens exclusively to the firstone, we have the following trust matrix:

In this case for any${\displaystyle k\ge 1}$ we have

and

so${\displaystyle \underset{t\to \mathrm{\infty }}{lim}{T}^t}$ does not exist and beliefs do not converge in the limit. Intuitively, 1 is updating based on 2 and 3's beliefs while2 and 3 update solely based on 1's belief so they interchange their beliefs in each period.

It is possible to examine the outcome of the DeGroot learning process in large societies,that is, in the${\displaystyle n\to \mathrm{\infty }}$ limit.

Let the subject on which people have opinions be a "true state"${\displaystyle \mu \in [0,1]}$. Assume that individualshaveindependent noisy signals${\displaystyle p_i^{(0)}(n)}$ of${\displaystyle \mu }$(now superscript refers to time, the argument to the size of the society).Assume that for all${\displaystyle n}$ the trust matrix${\displaystyle T(n)}$ is such that the limiting beliefs${\displaystyle p_i^{(\mathrm{\infty })}(n)}$ exists independently from the initial beliefs. Then the sequence of societies${\displaystyle {\left(T(n)\right)}_{n=1}^{\mathrm{\infty }}}$ is calledwise if

${\displaystyle \underset{i\le n}{max}|p_i^{(\mathrm{\infty })}-\mu |\stackrel{p}{\to }0}$

where${\displaystyle \stackrel{p}{\to }}$ denotesconvergence 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 ofinfluence vectors. A sequence of societies is wise if and onlyif

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\underset{i\le n}{max}{s}_i(n)=0}$

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^[6] (2010).

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