Translartion. Region: Russians Fedetion –
Source: State University Higher School of Economics – State University Higher School of Economics –
Ivan Samoylenko studies graph theory and in his third year he came up with an idea that formed the basis of a scientific article with a very high citation rate in the media. In an interview with the Young Scientists of the HSE project, he spoke about the Watts-Strogatz small world model, singing in the children’s choir of the Bolshoi Theater, and choosing between science and industry.
How I got into science
I am a graduate of the specialized mathematics school #57 in Moscow. I attended math clubs there since high school, and in the 9th grade I transferred to a specialized math class. I got acquainted with some mathematical disciplines at a fairly serious level there. At that time, my attention was drawn to graphs – perhaps because many life questions are clearly formulated in their language. After school, I entered the mathematics department of the Higher School of Economics and am currently mainly engaged in graph theory.
At HSE, I work in two laboratories. In the International Laboratory of Game Theory and Decision Making at HSE in St. Petersburg, I study applications of graphs to game-theoretic problems. And at the Faculty of Mathematics, we created the Scientific and Educational Laboratory of Complex Networks, Hypergraphs, and Their Applications. There, as you can tell from the name of the laboratory, I study both graphs and their generalized version — hypergraphs. And not only from the point of view of theory, but also from the point of view of the possible application of these structures to solving problems from a wide variety of areas — biology, medicine, data analysis, etc.
What is a graph
For clarity, a graph can be represented as a set of points (vertices) connected by lines (edges). The main feature of graph theory is that almost any system can be represented as a set of objects and some interactions between them. For example, when a journalist interviews me, this is also a graph, and a directed one at that. But in this particular example, it is not very clear why the graph is needed – it does not provide any new information about what is happening. However, if many different journalists interview different scientists, with the help of graph theory, you can compare the structural characteristics of the vertices (people) and make unobvious (at first glance) general conclusions.
About the history of graph theory
The father of graph theory is considered to be the mathematician Leonard Euler, who published a solution to the problem of the Königsberg bridges in 1736. He proved that it is impossible to cross all seven Königsberg bridges without crossing any of them twice and return to the starting point. Later, with the development of technology and the emergence of large data sets, graph theory increasingly occupied the minds of mathematicians and was embodied in various fields of knowledge.
Another famous graph problem is the four-color conjecture, the assertion that any map on a plane can be correctly colored in no more than four colors. Although the problem is formulated in a language understandable even to a schoolchild and is easily illustrated with understandable pictures, it took humanity more than 100 years to solve it. And when in 1976 a solution was found (by the way, not at all simple: one of the steps of this solution is to try out almost 2000 options), an important break in the history of all mathematics occurred: this was the first theorem completely proven with the help of a computer.
In general, major breakthroughs and milestones in the history of graph theory are inextricably linked with the development of information technology. Thus, graph theory gained particular popularity with the emergence of a clear example of a very large irregular (which cannot be fully described by a small set of rules) graph — the Internet. In general, the emergence of the Internet led to the emergence of a major branch of graph theory — the theory of complex networks.
The two major modern works in complex network theory are papers describing the mechanisms by which complex networks emerge in the real world: the Watts-Strogatz small-world model and the Barabasi-Albert preferential attachment model. These papers have a great many citations, which is rare in mathematics. The Watts-Strogatz model is even in the top 100 most cited scientific papers of all time.
When large amounts of data appear, it is interesting to identify structural patterns. And now there is a lot of data, you can build informative graph systems in almost any area. For example, I saw a study on how the graph of interactions of British composers of the 20th century is structured. By calculating the characteristics of this graph, for example, some centralities, you can draw a conclusion about which specific composers were structurally important for the development of British music. And from different points of view: someone as an independent actor or founder of a school, and someone as a link, allowing more successful colleagues to interact with each other.
In general, in the language of graph theory, one can formulate models – probabilistic, game-theoretic – and prove their properties with strict mathematical theorems. So this is both an applied and fundamental area of mathematics.
What I am proud of
I came up with a game-theoretic model that describes why the social networks we see in the real world follow the six-handshake rule. It has been described before why there should be relatively few handshakes, but I was able to show where the magic number 6 comes from. A paper about this, based on my bachelor’s thesis, was published in Physical Review X in 2023.
In the language of graph theory, it is easy to formulate what a social network is. The vertices are people, and the relationships between them (for example, acquaintance or friendship) are edges. In this context, the six-handshake rule can be thought of as follows: if we take two random people registered in a social network, then with a probability close to one, the path from one to the other along the “friend” edges will be no longer than six steps.
The Watts and Strogatz paper that I mentioned proposed a random graph model in which a similar phenomenon could be observed. And I came up with a model that, on the one hand, somehow justified why this model was reasonable, and on the other hand, theoretically proved that if it so happened that we had two people in the system who were more than six handshakes apart, then such a system would not be very stable under sufficiently weak constraints.
It was fortunate that our article came out 25 years after Watts and Strogatz’s article. And Strogatz himself wrote about our article on his social networks. He is quite a media person, so such a mention greatly promoted our article; at some point, journalists from different countries even wrote to me to get comments. As a result, according to my calculations, according to the altmetrics indicator, which is responsible for mentions in the world media and social networks, among articles where the first author has affiliation with the HSE, mine is the most mentioned.
How I Got Published in a Top-Rated Magazine
Getting published in high-ranking journals is a separate art (or rather, a craft). Even if you are a young genius, but do not know how to write articles, present material in a format acceptable for your domain, then you most likely will not publish anything in serious journals.
Our article, published in the journal, consists of two parts. This is the main, “selling” part, which should be read by a completely non-technical person, and the additional part, which provides technical details and detailed evidence. As the author of the concept and idea, I wrote almost all the additional material (with detailed evidence), while a team of several leading scientists worked on the first part. First of all, Stefano Bocaletti, who was introduced to me by my supervisor in the graduate school of MIPT, Andrei Mikhailovich Raigorodsky, made a significant contribution to the release of this publication.
He was the first person who was able to read my drafts and believed in the concept I proposed (it should be noted that in 2021, when I started writing this work, there were no good LLM chats yet, and my English was so bad that even at local competitions of the Faculty of Mathematics my work did not take prizes; then I accidentally found out that one of the reasons was the inability to read it normally).
Then Stefano, for some time, invited his friends, also very strong network scientists, to join our team so that they could help us illuminate and explore our problem: what experiments to conduct, where to place emphasis so that the work could be published in a major interdisciplinary journal. And everything worked out: our article has a fairly good citation rate both in the media and in other scientific publications. So it’s one thing to discover a phenomenon, and quite another to successfully convey your results to the scientific community. Moreover, the criteria for an interesting publication are different for different domains. For example, I know that my fellow economists from the Game Theory Laboratory did not really like the format of my work. I have yet to master writing good economic articles.
On the lack of time, but not ideas
I keep a document with tasks that can be done and where minimal progress has been made. There are more than 20 of them. There is no shortage of ideas, there is a shortage of time, and sometimes there is a shortage of workers.
With semi-applied ideas, it is often unclear in advance whether they are good or not; this can only be determined by conducting an experiment. In theory, it sometimes happens that you come up with something — and it is immediately clear that it is a good idea. Even its refutation can be informative and interesting. In the context of applied methods, everything is different: if something does not work, it is no longer so interesting. But on the other hand, if you know the result in advance, then what kind of science is it? You research, and if something works out — that’s great.
What I dream about
I would like young Russian scientists to have an easier life. So that they could not only survive, doing exclusively or mainly science. The presence of specialized specialists who have the opportunity to fully devote their time to research is critically important from the point of view of the development of science and technology. To explain my understanding of the problem, I would like to give an example from game theory. There is such a concept as a “rational agent”. Let’s say a young man (or woman) as a rational agent chooses where to go to work. In theory, if in science, there will be less money, but the work will be more free. If in industry, vice versa. Such a trade-off with clear alternatives: for each person, you can figuratively imagine a payoff function depending on these two factors, and each chooses one of the two paths depending on which factor is more important for the person.
However, this model is relevant only if the economic difference is not too big. In practice (this is not only our problem, but in Russia it is felt especially acutely) the gap is colossal. In some situations it is more reasonable and simpler to go to work in a corporation, and in your free time to get together with friends and discuss science, and some people do just that.
Another important issue is time constraints. Many scientific projects/grants/programs are very heavy and unwieldy from a bureaucratic point of view. The project setup activities may begin when a student, say, has just entered a master’s program, and the launch — when he or she is already finishing the last pages of his or her diploma.
In such conditions, a young scientist will have to look for part-time work/other jobs, be in a state of constant uncertainty, which leads to constant stress. So many, even among those who are really interested in a scientific career, cannot cope and simply leave science. If we attract young scientists and administrative personnel (in my understanding, a scientist should not be busy writing papers, he should be engaged in science, if he does not have additional paid administrative duties) on more market-based terms, it seems to me that much more interesting and breakthrough work could be done.
If I hadn’t become a mathematician
The simplest answer is that I would go into IT, because that’s how I make money. But, in principle, I could become anyone, mathematics is not about theorems, but rather about a way of thinking. I don’t know who I could become. I could even do music, I even sang in the children’s choir of the Bolshoi Theater. Many opera productions have parts where children sing, and opera houses have children’s choirs.
So that there is no feeling that I am Luciano Pavarotti, I should clarify that it is easier for boys to get into the Bolshoi Choir. The Bolshoi Children’s Choir consisted (at least when I was there) mainly of girls, and any boy there is a great success; there are fewer of them in music in general, and in early adolescence many leave because of voice failure. We had a situation when three people stopped taking part in performances at once. Because when two boys almost six feet tall and a third, also quite a large fellow with the nickname Horse stand next to a soloist shorter than them by a head and a half and have to portray small children, a noticeable dissonance arises.
What I was interested in at school
I was interested in history. I was even closer to the final stage of the All-Russian in history than in mathematics. I also played a lot of “What? Where? When?” and continued to do so as a student, although a little less actively. Now, unfortunately, I have almost no time for this: I have to work in industry, do science, and I also have a social and organizational load in the laboratories where I work.
Who would I like to meet?
With John Conway. I have a close relationship with his attitude to mathematics: he saw it in various everyday things, and although he became famous mainly for the game “Life”, he was in fact an amazingly versatile scientist with a large number of important works in various areas of mathematics. I was very upset when I read the news of his death at the very beginning of the covid pandemic. It would also be interesting to talk to mathematicians from the golden age of the Faculty of Mechanics and Mathematics – for example, Andrey Kolmogorov, the author of the axiomatics of probability theory.
What are my hobbies besides science?
I am a curious person and try to get acquainted with different things, to find out what is happening in the world. Sometimes I watch history channels, sometimes I can watch something about football or a strange documentary. In general, almost any information is interesting to me. But all this is irregular. I work systematically, slept – good, did not sleep – well, what to do.
Advice to young scientists
Think carefully about choosing your future track. I can also wish you patience and strength, mental and physical – you will definitely need it.
Favorite place in Moscow
I really like Moscow as a whole. I’ve been to different cities and I can’t say that even one of them is close to Moscow in terms of comfort (I have a certain sense of being a Muscovite, of course). If I have to name a specific place, I can simply say that I love the Moscow metro – it’s very practical (and at the old stations, it’s also aesthetically pleasing).
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