Sergey Melnik

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Sergey Melnik

Here is a link to my homepage:

Projects in mind

Network's shortest paths

Consider a random network consisting of N nodes where each pair of nodes is connected with a given probability p (see Erdos-Renyi random graph). The question is about the shortest path through the network from one node to another (also called geodesic path, or intervertex distance):

Can we calculate analytically (and exactly) the probability distribution of shortest paths in such a network? That is, how likely it is that a pair of nodes chosen at random will be distance D apart. For distance D=1 the answer is obviously p. What are such probabilities for distances D=2, D=3, D=4, etc.?

Note that the network has a finite number of nodes (which is not necessarily large) and there are only 2 parameters here, N and p. This is a simple yet fundamental question and there is much more to follow in terms of immediate applications and new theories once we answer it.

The Monopoly project

Have you ever wondered about the best strategy to play Monopoly?

We can start by answering simple questions such as "is it really better to be the first to throw the dice?", or "what are the most valuable squares on the field, and when?", or "to what extent do your skills help you overcome randomness to win this game?", or "how long would a game last?", or "how does all this changes with the number of players?". We can then go deeper to model negotiations and resource management strategies - plenty of possibilities here.

I am sure those who play this game already have some intuition. But can we precisely quantify these and other effects? I guess not yet. I think there is a lot here that can be modelled (analytically?) and compared with numerical (Monte Carlo?) simulations.

Finally, we can look into creating a new, even more exciting game to be played by the future generations. And surely you will have better calculated chances next time you play Monopoly with your friends!