Daniil Musatov’s research while affiliated with Moscow Institute of Physics and Technology and other places

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Publications (16)


Schematic illustration of matrix centrality. Upper panel: a hypergraph consisting of three hyper-edges and six nodes is presented. Nodes may have two roles: red and green. Precisely, node A has role green in the hyper-edge d and role red in the hyper-edge e, node B has role red in both hyper-edges d and f, node C has role green in hyper-edge f and role red in hyper-edge e. Lower panel: the matrix centralities of nodes A, B, and C. Positions of positive elements in the respective matrices are marked. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Examples of modified sunflower hypergraphs with differentiated node roles. (a) A hypergraph consisting of two connected star-like sub-hypergraphs with two different roles: "red" and "green" indicated by the respective colors. Node C has role "red" in the left hyper-edge and role "green" in the right one. (b) A hypergraph consisting of two connected star-like sub-hypergraphs with three different roles: "from", "through" and "to" indicated by red, blue and green colors respectively. Node C has role "to" in the left hyper-edge and role "from" in the right one. Role centralities r_i ir
i​ of the nodes are shown for both hypergraphs. In all cases, the color of the numbers reported in the vector matches the corresponding role. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A modified sunflower hypergraph with two different node roles (“red” and “green”) and different sizes of petals. Node A has role “red” in hyper-edges r₁, r₂ and d and “green” in the rest ones, node B has role “red” in u and d and “green” in the others, node C has role “green” in u and d and “red” in the others. We omit zero rows corresponding to absent hyper-edges sizes (all but 4 and 8) in matrices Cᵢ, i = A, B, C and vectors c_i

i​, i = A, B, C. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
μ measure and Kendall rank correlation coefficient. 

μ measure (panels (a) – (e)) and KRC (panels (f) – (j)) (see ext for definition) for the centrality vectors corresponding to different roles vs. the number of top nodes selected. Five datasets (enron, stack-overflow, math-overflow, scopus-multilayer and movielens) are considered in our study, and are indicated on top of each pair of panels. The right panels contain moreover legends with the color code used for drawing the different curves. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Matrix centrality for annotated hypergraphs
  • Article

August 2024

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109 Reads

Chaos Solitons & Fractals

Ekaterina Vasilyeva

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K. Kovalenko

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The identification of central nodes within networks constitutes a task of fundamental importance in various disciplines, and it is an extensively explored problem within the scientific community. Several scalar metrics have been proposed for classic networks with dyadic connections, and many of them have later been extended to networks with higher-order interactions. We here introduce two novel measures for annotated hypergraphs: that of matrix centrality and that of role centrality. These concepts are formulated for hypergraphs where the roles of nodes within hyper-edges are explicitly delineated. Matrix centrality entails the assignment of a matrix to each node, whose dimensions are determined by the size of the largest hyper-edge in the hypergraph and the number of roles defined by the annotated hypergraph’s labeling function. This formulation facilitates the simultaneous ranking of nodes based on both hyper-edge size and role type. The second concept, role centrality, involves assigning a vector to each node, the dimension of which equals the number of roles specified. This metric enables the identification of pivotal nodes across different roles without distinguishing hyper-edge sizes. Through the application of these novel centrality measures to a range of synthetic and real-world examples, we demonstrate their efficacy in providing enhanced insights into the structural characteristics of the systems under consideration.


Approximating Kolmogorov complexity

September 2023

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33 Reads

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1 Citation

Computability

It is well known that the Kolmogorov complexity function (the minimal length of a program producing a given string, when an optimal programming language is used) is not computable and, moreover, does not have computable lower bounds. In this paper we investigate a more general question: can this function be approximated? By approximation we mean two things: firstly, some (small) difference between the values of the complexity function and its approximation is allowed; secondly, at some (rare) points the values of the approximating function may be arbitrary. For some values of the parameters such approximation is trivial (e.g., the length function is an approximation with error d except for a O ( 2 − d ) fraction of inputs). However, if we require a significantly better approximation, the approximation problem becomes hard, and we prove it in several settings. Firstly, we show that a finite table that provides good approximations for Kolmogorov complexities of n-bit strings, necessarily has high complexity. Secondly, we show that there is no good computable approximation for Kolmogorov complexity of all strings. In particular, Kolmogorov complexity function is neither generically nor coarsely computable, as well as its approximations, and the time-bounded Kolmogorov complexity (for any computable time bound) deviates significantly from the unbounded complexity function. We also prove hardness of Kolmogorov complexity approximation in another setting: the mass problem whose solutions are good approximations for Kolmogorov complexity function is above the halting problem in the Medvedev lattice. Finally, we mention some proof-theoretic counterparts of these results. A preliminary version of this paper was presented at CiE 2019 conference (In Computing with Foresight and Industry – 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings (2019) 230–239 Springer).


Distances in Higher-Order Networks and the Metric Structure of Hypergraphs
  • Article
  • Full-text available

June 2023

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247 Reads

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7 Citations

Entropy

We explore the metric structure of networks with higher-order interactions and introduce a novel definition of distance for hypergraphs that extends the classic methods reported in the literature. The new metric incorporates two critical factors: (1) the inter-node distance within each hyperedge, and (2) the distance between hyperedges in the network. As such, it involves the computation of distances in a weighted line graph of the hypergraph. The approach is illustrated with several ad hoc synthetic hypergraphs, where the structural information unveiled by the novel metric is highlighted. Moreover, the method’s performance and effectiveness are shown through computations on large real-world hypergraphs, which indeed reveal new insights into the structural features of networks beyond pairwise interactions. Namely, using the new distance measure, we generalize the definitions of efficiency, closeness and betweenness centrality for the case of hypergraphs. Comparing the values of these generalized measures with their analogs calculated for the hypergraph clique projections, we show that our measures provide significantly different assessments on the characteristics (and roles) of the nodes from the information-transferability point of view. The difference is brighter for hypergraphs in which hyperedges of large sizes are frequent, and nodes relating to these hyperedges are rarely connected by other hyperedges of smaller sizes.

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FIG. 1. The game theoretical framework. The structure of a social network evolves following simple rules of a game. (a) At each step of the game, the individuals forming part of the network (like the red woman in the picture) have to decide whether to stay with the neighborhood formed by their actual friends or to change to another neighborhood formed by potential new friends. The current and new neighborhoods may overlap (in our picture, the blue man and the yellow woman are members of both sets). The decision is based on a careful evaluation of the cost incurred and of the benefit gained with the change. (b) The decision is merely utilitarian. If the benefit is not overcoming the cost, then individuals maintain their current neighborhood (left-hand picture). If, on the contrary, the payoff exceeds the cost, then individuals relinquish their current neighborhood and move to the new one (right-hand picture). The structure of the network then evolves until converging to its Nash equilibrium (if it exists), i.e., to the configuration where no changes of neighborhood are allowed, as no individual has anything to gain in abandoning acquaintances.
Why Are There Six Degrees of Separation in a Social Network?

May 2023

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577 Reads

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19 Citations

Physical Review X

A wealth of evidence shows that real-world networks are endowed with the small-world property, i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultrasmall-world organization, whereby the graph’s diameter is independent of the network size over several orders of magnitude, is still unknown. We show that the “six degrees of separation” is the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. We show, moreover, that the emergence of such a regularity is compatible with all other features, such as clustering and scale-freeness, that normally characterize the structure of social networks. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.


Why are there six degrees of separation in a social network?

November 2022

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352 Reads

A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-small world organization, whereby the graph's diameter is independent of the network size over several orders of magnitude, is still unknown. Here we show that the 'six degrees of separation' are the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.


Vector centrality in hypergraphs

September 2022

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90 Reads

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33 Citations

Chaos Solitons & Fractals

Identifying the most influential nodes in networked systems is of vital importance to optimize their function and control. Several scalar metrics have been proposed to that effect, but the recent shift in focus towards network structures which go beyond a simple collection of dyadic interactions has rendered them void of performance guarantees. We here introduce a new measure of node's centrality, which is no longer a scalar value, but a vector with dimension one lower than the highest order of interaction in a hypergraph. Such a vectorial measure is linked to the eigenvector centrality for networks containing only dyadic interactions, but it has a significant added value in all other situations where interactions occur at higher-orders. In particular, it is able to unveil different roles which may be played by the same node at different orders of interactions – information that is otherwise impossible to retrieve by single scalar measures. We demonstrate the efficacy of our measure with applications to synthetic networks and to three real world hypergraphs, and compare our results with those obtained by applying other scalar measures of centrality proposed in the literature.


Discrete Versions of the KKM Lemma and Their PPAD-Completeness

June 2022

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14 Reads

Lecture Notes in Computer Science

PPAD is the class of computational search problem that are equivalent to the EndOfALine problem: given a succinct representation of a directed graph consisting of chains and cycles and a source in this graph, find a sink or another source. It turns out that this class contains many problems of searching for a fixed point in various frameworks. The complete problems in PPAD include Sperner’s lemma, discrete analogues of Brouwer and Kakutani theorems, Nash equilibrium, market equilibria, cake-cutting and many other models in mathematical economics. In this paper we analyze the Knaster–Kuratowski–Mazurkievicz (KKM) lemma: if an n-dimensional simplex is covered by n+1 closed sets and every face of the simplex is covered by the union of the respective sets, then the intersection of all sets is non-empty. We elaborate a discrete analogue of a covering by closed sets, base on it several discrete analogues of the KKM lemma and prove that the corresponding search problems are PPAD-complete.


Predicting transitions in cooperation levels from network connectivity

September 2021

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87 Reads

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8 Citations

Networks determine our social circles and the way we cooperate with others. We know that topological features like hubs and degree assortativity affect cooperation, and we know that cooperation is favored if the benefit of the altruistic act divided by the cost exceeds the average number of neighbors. However, a simple rule that would predict cooperation transitions on an arbitrary network has not yet been presented. Here we show that the unique sequence of degrees in a network can be used to predict at which game parameters major shifts in the level of cooperation can be expected, including phase transitions from absorbing to mixed strategy phases. We use the evolutionary prisoner’s dilemma game on random and scale-free networks to demonstrate the prediction, as well as its limitations and possible pitfalls. We observe good agreements between the predictions and the results obtained with concurrent and Monte Carlo methods for the update of the strategies, thus providing a simple and fast way to estimate the outcome of evolutionary social dilemmas on arbitrary networks without the need of actually playing the game.


Predicting transitions in cooperation levels from network connectivity

July 2021

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138 Reads

Networks determine our social circles and the way we cooperate with others. We know that topological features like hubs and degree assortativity affect cooperation, and we know that cooperation is favoured if the benefit of the altruistic act divided by the cost exceeds the average number of neighbours. However, a simple rule that would predict cooperation transitions on an arbitrary network has not yet been presented. Here we show that the unique sequence of degrees in a network can be used to predict at which game parameters major shifts in the level of cooperation can be expected, including phase transitions from absorbing to mixed strategy phases. We use the evolutionary prisoner's dilemma game on random and scale-free networks to demonstrate the prediction, as well as its limitations and possible pitfalls. We observe good agreements between the predictions and the results obtained with concurrent and Monte Carlo methods for the update of the strategies, thus providing a simple and fast way to estimate the outcome of evolutionary social dilemmas on arbitrary networks without the need of actually playing the game.


D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios

May 2021

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87 Reads

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37 Citations

Chaos Solitons & Fractals

From biology to social science, the functioning of a wide range of systems is the result of elementary interactions which involve more than two constituents, so that their description has unavoidably to go beyond simple pairwise-relationships. Simplicial complexes are therefore the mathematical objects providing a faithful representation of such systems. We here present a complete theory of synchronization of D-dimensional oscillators obeying an extended Kuramoto model, and interacting by means of 1- and 2- simplices. Not only our theory fully describes and unveils the intimate reasons and mechanisms for what was observed so far with pairwise interactions, but it also offers predictions for a series of rich and novel behaviors in simplicial structures, which include: (a) a discontinuous de-synchronization transition at positive values of the coupling strength for all dimensions, (b) an extra discontinuous transition at zero coupling for all odd dimensions, and (c) the occurrence of partially synchronized states at D=2 (and all odd D) even for negative values of the coupling strength, a feature which is inherently prohibited with pairwise-interactions. Furthermore, our theory untangles several aspects of the emergent behavior: the system can never fully synchronize from disorder, and is characterized by an extreme multi-stability, in that the asymptotic stationary synchronized states depend always on the initial conditions. All our theoretical predictions are fully corroborated by extensive numerical simulations. Our results elucidate the dramatic and novel effects that higher-order interactions may induce in the collective dynamics of ensembles of coupled D-dimensional oscillators, and can therefore be of value and interest for the understanding of many phenomena observed in nature, like for instance the swarming and/or flocking processes unfolding in three or more dimensions.


Citations (8)


... The approximation methods of the function describing the size distribution of dust particles was used for the dust collected directly from the center of activity at the construction site, as shown in Figure 7, for example, during the foundation. Based on the hypothesis of Kolmogorov A.N. [42], the process of diameter distribution of building dust particles asymptotically tends to a logarithmically normal distribution law, which looks like the following: ...

Reference:

Dust Pollution in Construction Sites in Point-Pattern Housing Development
Approximating Kolmogorov complexity
  • Citing Article
  • September 2023

Computability

... Despite these advances, characterising shortest paths and connectivity in systems with higher-order interactions remains an open problem. Recently, efforts have been devoted to characterise the concepts of distance [46] and walks [47] and networks in networks with non-dyadic ties, as well as proposing efficient algorithms [48] to extract shortest paths in hypergraphs, limiting the analysis to static systems. ...

Distances in Higher-Order Networks and the Metric Structure of Hypergraphs

Entropy

... Individuals always look for excellent partners in societal networks, meeting friends of friends within their societal circles. A large amount of data shows that in real-world network structures, the link distance between any two individuals does not exceed six degrees [47]. Therefore, we speculate that an individual in a connected network structure can continuously meet friends of friends, actively sending friend requests to those who possess the same or even more societal resources than themselves [48]. ...

Why Are There Six Degrees of Separation in a Social Network?

Physical Review X

... Motivated by these advances, several approaches have been introduced to study the structure of higher-order networks. Some of these mainly extended the traditional network approaches to include group (higher-order) interactions, such as community detection methods based on generalized modularity, 33,34 spectral clustering 35 , bayesian statistics approaches, 36 centrality metrics, 35,37,38 clustering coefficient, 39,40 and k-core decomposition methods. 41 On the other hand, new approaches were appositely proposed to characterize group interactions. ...

Vector centrality in hypergraphs
  • Citing Article
  • September 2022

Chaos Solitons & Fractals

... Instead, we consider all levels of rationality, including fully rational agents. Notably, non-universal dynamic properties can emerge in this context, leading to abrupt cooperationlevel transitions [14,27]. ...

Predicting transitions in cooperation levels from network connectivity

... Other examples can be found in neuroscience [4][5][6][7][8][9], ecology [10,11], biology [12] and social sciences [13][14][15]. Higher order interactions have particularly important consequences in the propagation of epidemics [16][17][18] and synchronization of coupled oscillators [19][20][21][22][23][24][25][26][27][28][29][30][31]. ...

D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios
  • Citing Article
  • May 2021

Chaos Solitons & Fractals

... An example of a network that captures higher-order interactions between different entity types is the bipartite network, where nodes are partitioned by two separate groups, and edges only connect nodes from different groups. Bipartite networks are particularly suitable for modeling systems where two types of entities interact, such as authors and papers in a collaboration network [18], recommendation systems [19] where nodes represent users and the recommending items, and specifically in ...

Multilayer representation of collaboration networks with higher-order interactions

... The structures used to run simulations exhibit scale-free (SF) and Erdős-Rényi-like (ER) degree distributions at both orders m = 1 and m = 2, with their characteristics summarized in Table I. The ER simplicial complex was generated following the methodology in [6], while the SF one was constructed using the model in [46]. In both cases, the GBCM predictions show excellent agreement with the simulated results, demonstrating the capability of the model to capture the behavior of systems with higher-order interactions. ...

Growing scale-free simplices