# Mark's differentiation model

Let us try to discuss a bit different model by social scientist this time. In [1] a social scientist built a simple model to illustrate that differences between the social groups could be emergent property inherent to how we communicate.

## The model

Once again while implementing this model I take a liberty to interpret it the way I like it. Namely in [1] it is clearly stated that the model should be have sequential agent activation and discrete turns. I do not think such approach is realistic, hence I use random activation in continuous time (implemented using Gillespie algorithm). Due to this change I have also to rearrange the order of the steps executed when activating agents.

Agents themselves differ from each other in information they know. Each of them might recall different set of facts, which may or may not be shared with other agents.

At rate \( \lambda \) we randomly select an agent. Based on the fact it knows we select another agent. The probability for the second agent to be selected is proportional to the number of "facts" these agents share. Note that the first and the second agent can be the same agent.

For example, let us say that agent A knows 3 facts, while agent B knows 5 facts. They know 2 same facts. Agent A does not share any facts with other agents. If agent A is selected as the first agent, then the probability that B will be selected as the second agent is \( 0.4 \), while A will interact with himself with probability \( 0.6 \). Namely the following formula holds (here \( F_{ij} \) is the number of facts that agents \( i \) and \( j \) share):

\begin{equation} P_{ij}=\frac{F_{ij}}{\sum_k F_{ik}} . \end{equation}

In our example case we have \( F_{AB} = 2 \) and \( F_{AA} = 3 \). Hence:

\begin{equation} P_{AB} = \frac{F_{AB}}{F_{AB} + F_{AA}} = \frac{2}{2+3} = 0.4 , \end{equation}

\begin{equation} P_{AA} = \frac{F_{AA}}{F_{AB} + F_{AA}} = \frac{3}{2+3} = 0.6 . \end{equation}

Next the selected agents forget the facts they have not discussed lately. Here, we have decided to use "forgetting" time as our measurement unit of time. Hence it is hard-builtin to be equal 1. Namely, agents forget facts they haven't discussed during the last 1 time tick.

Then the selected agents interact either by discussing (expressing) a fact any one of them knows or by creating a new fact (which will be known only by themselves unless they tell it anybody else). The probabilities to discuss any of the know facts or the new fact are assumed to be equal. Returning to our example both A and B know 6 unique facts (2 of them are known by both) and the possible new fact counts as 7th fact, hence the probabilities are equal to \( 1/7 \).

Afterwards the selection probabilities are updated and the algorithm runs from the start.

## HTML 5 app

This model has a rather strong differentiation built into it. Or at least it works quite strongly when considered in continuous time. It is quite probable for any agent to forget all of his facts and become unable to interact with any other agent. Hence little by little all agents will separate from the main group and become independent.

Here in this app you have just a single parameter \( \lambda \), which sets how often the agents interact in comparison to how fast they forget facts. The plot on the left shows the "interaction matrix": if the cell is dark, then it means that those agents can interact. Note that the agents are always able to interact with themselves, hence the diagonal is always dark. The right figure on the other hand shows the number of groups (blue line) and the size of the largest group (yellow line).

Feel free to explore the models dynamics using an app below. Though my intuition tells me that even if \( \lambda \) is higher, the separation will just become a bit slower, but will not stop until all of the agents are independent. The differentiation mechanics here are very strong.

**Acknowledgment.** This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.

## References

- N. Mark. Beyond individual differences: Social differentiation from first principles. American Socialogical Review 63: 309-330 (1998). doi: 10.2307/2657552.