Research in Graph Theory

Method for classification of the computational problems on the basis of the multifractal division of the complexity classes

Artem Potebnia — 2020-11-18

This paper proposes the method of the multifractal division of the computational complexity classes, which is formalized by introducing the special equivalence relations on these classes. Exposing the self-similarity properties of the complexity classes structure, this method allows performing the accurate classification of the problems and demonstrates the capability of adaptation to the new advances in the computational complexity theory.

Representation of the greedy algorithms applicability for solving the combinatorial optimization problems based on the hypergraph mathematical structure

Artem Potebnia — 2020-11-18

This paper deals with representing the structural organization of the combinatorial optimization problems in terms of the hypergraphs, whose hyperedges reflect the solutions of the original problem and its nested subproblems. By using the achievements of the matroid theory, the paper analyzes the parameters of such hypergraphs that determine the suitability of the corresponding problems for being processed by the greedy algorithms. In addition, the study contains the examples of the hypergraph structures constructed for the instance of the minimum spanning tree problem.

Formation of the multifractal hypergraph structure reflecting the self-similarity properties of the computational complexity classes

Artem Potebnia — 2020-11-18

In the context of classifying the graph structures from the viewpoint of their self-similarity, this paper introduces the original concepts of the fractal and multifractal hypergraphs. The key idea underlying their definitions consists in the iterative procedure of generating the hyperedges applied to the specified initiator structure. The paper discusses the main properties of such hypergraphs and presents the simplified examples of their instances. Against this background, the investigation proposes the approach for representing the self-similarity properties of the computational complexity classes in terms of the multifractal hypergraph structure.

Innovative concept of the strict line hypergraph as the basis for specifying the duality relation between the vertex separators and cuts

Artem Potebnia — 2020-11-18

This article presents the original approach for establishing the duality relation between the vertex separators and cuts in hypergraphs. The analysis of the existing mathematical structures providing the edge-focused representation of graphs is followed by the identification of the fundamental drawbacks obstructing the formation of the duality relation on the basis of these structures. With a view to filling this research gap, the article formulates the core concept of the strict line hypergraph and introduces the new notions of the degenerate connected components and vertex separators. The proposed structures underlie the construction of the duality relation between the vertex separators and cuts in the form of theorems accompanied with their rigorous proofs and discussions of the specific situations. Finally, the article considers the application of the established relation for linking the combinatorial optimization problems whose solution spaces are composed of the vertex separators and cuts.

Creation of the mathematical apparatus for establishing the duality relation between the vertex separators and cuts in hypergraphs

Artem Potebnia — 2020-11-18

The article introduces the pioneered concept of the strict line hypergraph and proves that the function of its formation is an involution. This background forms the basis for specifying the duality relation between the vertex separators and cuts in hypergraphs. All significant considerations are presented in the form of theorems accompanied with their rigorous proofs and discussions of the specific situations.

Construction of the comprehensive multi-layer graph model of the search spaces associated with the combinatorial optimization problems

Artem Potebnia — 2020-11-20

This article presents the original model of the search spaces associated with the combinatorial optimization problems. The proposed model is characterized by the enhanced descriptive potential allowing to evaluate the reasonableness of applying the different techniques for finding the optimal solutions. With a view to forming the theoretical background required for formalizing such model, the article introduces the mathematical apparatus of the multi-layer graphs. In addition, the article constructs the concrete example of the model for the instance of the traveling salesman problem.

Algebra for transforming the structure of the search spaces associated with the combinatorial optimization problems

Artem Potebnia — 2020-11-24

This article proposes the classification of the search spaces associated with the combinatorial optimization problems based on the type of their constituent solutions. The spaces belonging to each identified class are accompanied by the corresponding graph models. Against this background, the article introduces the original algebra allowing the representation of the search spaces in the unified homogeneous form. The proposed algebra consists of a set of transformations given in an analytical form and illustrated by the modifications of the graph models constructed for the search spaces.

Innovative metrics for assessing the catastrophic collapse of the complex networks under the greedy attacks on their most important vertices and edges

Artem Potebnia — 2020-11-24

This article proposes the family of the improved metrics for quantifying the damage of the complex networks under the attacks on their vertices and edges. The primary advantage of the proposed metrics compared to the existing ones lies in their ability to describe more thoroughly the distribution of the vertices by the connected components in the damaged network. Another important advance consists in introducing the metrics of the normalized diameter, radius, and average path length describing the structure of the largest connected component in the damaged network with respect to the reference model of the path graph. Finally, the article illustrates the application of the developed metrics for assessing the robustness of the complex networks by performing the experimental analysis of the citation network extracted from the IEEE Xplore database.

Method of graph mining based on the topological anomaly matrix and its application for discovering the structural peculiarities of complex networks

Artem Potebnia — 2020-11-24

The article introduces the mathematical concept of the topological anomaly matrix providing the foundation for the qualitative assessment of the topological organization underlying the large-scale complex networks. The basic idea of the proposed concept consists in translating the distributions of the individual vertex-level characteristics (such as the degree, closeness, and betweenness centrality) into the integrative properties of the overall graph. The article analyzes the lower bounds imposed on the items of the topological anomaly matrix and obtains the new fundamental results enriching the graph theory. With a view to improving the interpretability of these results, the article introduces and proves the theorem regarding the smoothness of the closeness centrality distribution over the graph’s vertices. By performing the series of experiments, the article illustrates the application of the proposed matrix for evaluating the topology of the real-world power grid network and its post-attack damage.

Comprehensive method for discovering and assessing the rich-club organization in multihypergraphs

Artem Potebnia — 2020-11-27

The topological structure of the graph-based models constructed for a wide range of the real-world complex systems is characterized by the clear presence of the rich-club organization. Conceptually, such organization implies the tendency of the most important (in accordance to the prescribed metric) vertices to be tightly interconnected with each other and form the cohesive communities referred to as the rich-clubs. Recently, the rich-club ordering has attracted a considerable attention of investigators due to its impact on the robustness and performance of the modeled system as well as the regime of its functioning. At the same time, the prior studies in this direction are entirely limited to the case of simple graphs. This, in turn, points to the existence of the fundamental research gap associated with the need to develop the method for detecting the rich-club organization in multihypergraphs (i.e. hypergraphs allowing the presence of the parallel hyperedges). With a view to bridging the identified gap, this work introduces the family of the original metrics providing the formal way for determining whether the submultihypergraph induced by the most important nodes of multihypergraph could be properly regarded as its rich-club. The proposed metrics are designed to exhaustively capture the complex nature of relationships established in the considered submultihypergraph and, accordingly, take into account not only the number of its hyperedges but also their cardinality and role in ensuring the connectivity of vertices. Furthermore, the paper elaborates the scheme of normalizing the introduced metrics with respect to the reference ensemble of random multihypergraphs possessing the same sequences of vertex degrees and hyperedge cardinalities as the multihypergraph under investigation. Such normalization allows discovering the intentionally emerged rich-club ordering in multihypergraphs that does not follow merely from the structural restrictions imposed by the local properties of vertices and hyperedges. Finally, the paper illustrates the descriptive potential of the developed method by constructing the multihypergraph-based representation of the scientific co-authorship hypernetwork extracted from the IEEE Xplore database and performing the rigorous experimental analysis of its rich-club organization.

New method for estimating the tree-likeness of graphs and its application for tracing the robustness of complex networks

Artem Potebnia — 2020-11-27

This article presents the original metric of the minimum cyclic reachability distribution providing the foundation for assessing the topological tree-likeness of the large-scale sparse graphs. The conceptual sense enclosed in the proposed metric lies in interpreting the similarity of the analyzed graph to a tree based on the position of each its vertex in the structure of the graph's paths. With a view to demonstrate the theoretical value of the introduced metric, the article deals with its applying for discovering the dependence of the topological tree-likeness of the random graphs on the parameters of their generation. Finally, the article illustrates the usage of the designed metric for estimating the robustness of the complex networks and pursues the experimental analysis of the graph model constructed for the real-world power transmission network.

Устойчивость сложных систем к структурным атакам на основе формирования вершинных сечений [Resistance of complex systems to structural attacks based on the formation of vertex cuts]

Artem Potebnia — 2020-11-29

Предлагается классификация структурных атак на графовые модели сложных систем и сформирован набор параметров для оценки их эффективности. Выполнено экспериментальное исследование фрагмента наукометрической базы IEEE Xplore.