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Mikhail Tuzhilin

Publications and source records attributed to Mikhail Tuzhilin.

7 recordsLinked to original sources
arXiv2026

Capability centrality: the next step from scale-free property

In this article we present a new centrality measure called ksi-centrality. We show that ksi-centrality distinguishes real networks from random ones, similar to degree centrality: the ksi-centrality distribution is right-skewed for real networks and centered for random Erdos-Renyi networks, and has linear pattern with a heavy tail on a log plot. Furthermore, the ksi-centrality distribution is centered for models simulating real networks: Barabasi-Albert, Watts-Strogatz, and Boccaletti-Hwang-Latora. Thus, this centrality distribution is an additional and independent property with respect to scale-freeness. We also introduce a normalized version of ksi-centrality and show that it is related to algebraic connectivity and the Chegeer's value of a network. Moreover, the average value of this normalized centrality is in bijective correspondence with the relative number of edges that a new node connects to others in the Barabasi-Albert preferential attachment model, thus answering the question of how to choose the parameter $m$ to model a given real-world network.

cs.SI↗
arXiv2026

A series of real networks invariants

In this article we propose a generalization of two known invariants of real networks: degree and ksi-centrality. More precisely, we found a series of centralities based on Laplacian matrix, that have exponential distributions (power-law for the case $j = 0$) for real networks and different distributions for artificial ones.

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arXiv2025

New exponential law for real networks

In this article we have shown that the distributions of ksi satisfy an exponential law for real networks while the distributions of ksi for random networks are bell-shaped and closer to the normal distribution. The ksi distributions for Barabasi-Albert and Watts-Strogatz networks are similar to the ksi distributions for random networks (bell-shaped) for most parameters, but when these parameters become small enough, the Barabasi-Albert and Watts-Strogatz networks become more realistic with respect to the ksi distributions.

cs.SI↗
arXiv2025

New centrality measure: ksi-centrality

We introduce new centrality measures, called ksi-centrality and normalized ksi-centrality measure the importance of a node up to the importance of its neighbors. First, we show that normalized ksi-centrality can be rewritten in terms of the Laplacian matrix such that its expression is similar to the local clustering coefficient. After that we introduce average normalized ksi-coefficient and show that for a random Erdos-Renyi graph it is almost the same as average clustering coefficient. It also shows behavior similar to the clustering coefficient for the Windmill and Wheel graphs. Finally, we show that the distributions of ksi centrality and normalized ksi centrality distinguish networks based on real data from artificial networks, including the Watts-Strogatz, Barabasi-Albert and Boccaletti-Hwang-Latora small-world networks. Furthermore, we show the relationship between normalized ksi centrality and the average normalized ksi coefficient and the algebraic connectivity of the graph and the Chegeer number.

cs.SI↗
arXiv2024

Introduction to graph theory and basic algorithms

This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications in almost every scientific area: in many branches of mathematics, computer science, physics, chemistry, biology and also in psychology, arts, philosophy and many others. Nowadays, graph theory becomes especially more important because of the rapid development of molecular biology, neural networks and AI fields. One of the aims of writing this book was to give students thorough knowledge about graphs to understand modern scientific fields more deeply. Here we tried to give classical and modern theorems and algorithms in more understandable and simple way. We spent many time to rewrite them and close the gaps in several simplest well-known proofs to provide more precise and accurate material for students.

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