DeGroot learning

The idea was stated in its general form by the American statistician Morris H. DeGroot;[1] antecedents were articulated by John R. P. French[2] and Frank Harary.

[3] The model has been used in physics, computer science and most widely in the theory of social networks.

agents where everybody has an opinion on a subject, represented by a vector of probabilities

Links between agents (who knows whom) and the weight they put on each other's opinions is represented by a trust matrix

The trust matrix is thus in a one-to-one relationship with a weighted, directed graph where there is an edge between

The trust matrix is stochastic, its rows consists of nonnegative real numbers, with each row summing to 1.

th period opinions are related to the initial opinions by An important question is whether beliefs converge to a limit and to each other in the long run.

As the trust matrix is stochastic, standard results in Markov chain theory can be used to state conditions under which the limit exists for any initial beliefs

If the social network graph (represented by the trust matrix) is strongly connected, convergence of beliefs is equivalent to each of the following properties: The equivalence between the last two is a direct consequence from Perron–Frobenius theorem.

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.

Beliefs are convergent if and only if every set of nodes (representing individuals) that is strongly connected and closed is also aperiodic.

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 network the whole group reaches a consensus.

In general, any strongly connected and closed group

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 a strongly connected and aperiodic social network.

is the unique unit length left eigenvector of

shows the weights that agents put on each other's initial beliefs in the consensus limit.

Hence influential agents are characterized by being trusted by other individuals with high influence.

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

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.

If we change the previous example such that the third person also listens exclusively to the first one, we have the following trust matrix: In this case for any

It is possible to examine the outcome of the DeGroot learning process in large societies, that is, in the

Let the subject on which people have opinions be a "true state"

Assume that individuals have independent noisy signals

(now superscript refers to time, the argument to the size of the society).

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 influence vectors.