Imagine during your routine checkup a doctor randomly suggests you to take a test, which would tell (with 99% accuracy) if you have a rather rare disease (only 0.1% people have it). Is it worth to be curious? What would you do if you test positive? Panic? Shock? The test is 99% accurate!

Now imagine that you have 1000 people. Only one of them (0.1%) will have this disease and there is a rather significant probability that he will test positive. Likely 10 people, out of 999 healthy people, will also test positive (1%). So in the end you have 11 positive results and only 1 sick person in the group. So actually there is only 9% chance that you actually have this disease.

This "paradox" is not a mathematical paradox, nor its a mathematical trick. It is actually Bayesian trap! More on it in the following video by Veritasium.

Did you notice that maps use just but a few colors, while we have more than 200 countries here on Earth. Obviously it would not be rational to use different color per each different country. It is more convenient for our human vision, then there is as few colors as possible as then we can pick contrasting colors. Wouldn't it be interesting to know if there is some minimal number of differing colors necessary to create colored maps? More about this in this Numberphile video.

Human behavior is perplexingly complex. Why their collective behavior is so well described by rather general mathematical models using very few parameters? Why do they not need deeper insight into the human psychology or decision making? One of the simple answers - if the statistical signature is not present in the data, which is usually aggregated at least to some degree, we can do nothing about it. Namely, usually we do not observe individuals making decisions and as such we are will not be able to differentiate between different mechanisms of human decision making. There are two major mechanisms of human decision making - homophily (selecting your peers) and peer pressure (adopting your peers behavior). Mathematically there usually will be no difference between them, both mechanisms can be be described using the same Kirman's model.

In this text we will consider Bass diffusion model with heterogeneous agents (each of them having his own independent parameters). We will show that the heterogeneous model produces similar macroscopic dynamics as homogeneous model. To simplify matter even further we will use unidirectional Kirman's model.

Let us now return to the Voter model. In the original model we had agents occupying two possible states. They chose their state simply by copying the choice made by their neighbors. Yet in most elections around the world more than two parties compete for the electoral vote. Furthermore it is hardly believable that any established supporter of any party would switch to following the opposing party over night. One way to account for these zealous supporters would be to introduce "agents with fixed state." Yet some strongly opinionated individuals do changes their beliefs, thus this would not be an ideal solution. Alternative approach was considered in [1]. In this paper a three state model is proposed, where the third state serves as intermediate stop for the agents switching between the two main states.

Previously discussed Granovetter threshold model is just one of the numerous simple collective action model. This time we continue the same topic by considering another, yet a bit more complex, riot model, which was proposed by Epstein in [1]. This model is rather interesting in a sense that it is not static as original Granovetter model is. It has interesting temporal dynamics builtin. In a recent paper by British mathematicians [2] this model was applied to explain the patterns observed in 2013 London riots.