An individual forms opinions based on how he can himself assimilate the facts around him and also based on what opinions his friends and other people he interacts with hold. This makes things complicated and intriguing enough that this has been an active area of research since decades.

One question is, can we formulate simple enough models that match with the data we get from real life experiments? If we could, then we would get some insight into human behavior and a tool for making useful predictions.

The simplest model that has been studied is this:

An opinion is just a real number. Each person starts with an initial opinion. Next, in each time step, he looks at the opinions held by his friends and updates his own opinion to the average of his old opinion and the opinions of his friends. It doesn’t have to be a simple average. A person may have different trusts for different friends and thus he might want to take a weighted average instead. However the model doesn’t allow individuals to change the weights at any step. The weights chosen in the beginning have to be the weights always.

Using simple linear algebra tricks and borrowing known results from the Markov Chain literature, it can be shown that this kind of system converges to an equilibrium in most natural cases. An equilibrium here means a set of opinions for which the averaging step doesn’t lead to any change, i.e., for all individuals, the new opinion remains the same as the old one. In fact, it can be proved that the equilibrium that’s reached is a consensus, i.e., every individual has the same opinion.

An objection to this model that one might have is the simple representation of an opinion. Can it really be represented by just a single real number?

Anyway, DeGroot, the person who introduced this model also showed that the same thing happens if the opinions are drawn from any vector space and in each time step a person updates his opinion to some convex combination of the opinions of his friends (including himself).

That’s something.

The only issue is that in real life, people don’t reach consensus. So what’s going on?

Of course, the model seems too simple to resemble real life accurately. For one, the weights (or trust) we assign to people changes over time depending on various factors. For example, if a person seems to be changing his mind every minute, we will probably assign a lesser weight to his opinion.

Also, even though this process of repeated averaging has been shown to always converge to a consensus, we don’t really know the amount of time it takes to reach there. From what I know by quickly glancing through Bernard Chazelle’s new work on bird flocking, the time taken by a community whose size is close to the population of a country to reach a consensus is probably way more than the age of the universe.

Anyway. Friedkin and Johnsen modified this model a bit to make it more realistic. In their model, an individual has a fixed internal opinion that doesn’t change with time and during an averaging step, he takes a weighted average of the opinions of his friends (including himself) and the fixed internal opinion. Because the internal opinion can be different for different people, this system will obviously not reach a consensus always.

The system does have an equilibrium though and Friedkin and Johnsen proved that the equilibrium is almost always reached.

However, their model is different from DeGroot’s simpler model in a fundamental way. Let me explain.

Given a set of numbers , the mean is the number that minimizes the function . Thus the averaging step above can be seen as a step where a person is trying to minimize the cost incurred with respect to the cost function . Here represents the neighborhood of , i.e., the set of friends of .

With the above definition of cost, we can measure the quality of a certain opinion vector. For example, we can say that the sum of costs incurred by each person is the social cost of the whole group. And then given an opinion vector, we can decide how good it is by measuring how far it is from the opinion vector that minimizes the social cost. In particular, we can measure the quality of the opinion vector that the group converges to in equilibrium.

The fundamental difference between DeGroot’s model and Friedkin and Johnsen’s model is that in DeGroot’s model, the equilibrium reached also minimizes the total social cost but in Friedkin and Johnsen’s model, it does not necessarily.

David Bindel, Jon Kleinberg and Sigal Oren prove in their FOCS ’11 paper that the situation is not that bad. Even though the total cost at equilibrium may not minimize the total social cost, it can be worse at most by a factor of 9/8. That’s pretty cool.