Dynamic hypergraphs of renewal processes in mobile networks.

The characteristics of random geometric hypergraphs are studied as mathematical models of scalable wireless computer networks. An efficient algorithm for finding cliques in geometric graphs, constructing hypergraphs from geometric configurations has been developed. The types of hyper-edges in hypergraphs generated by a scalable configuration have been identified. The influence of random failures of nodes of computer networks and their restorations on the dynamics of hypergraphs of networks is considered. The analysis of the dynamics of the number of active nodes depending on the type of probability distributions of uptime and recovery time is carried out. The dependences of the mathematical expectation of the number of hyper-edges of certain types in the geometric hypergraph of a wireless computer network on the network operation time, on the radii of zones of reliable reception / transmission of a signal, on the ratio of the parameters of local recovery processes are obtained. The presentation of the results is accompanied by charts.

Connectivity and routes in large geometric network hypergraphs

Random geometric hypergraphs are considered as mathematical models of large wireless computer networks. The dependences of the mathematical expectation of the number of hyper-edges in random geometric hypergraphs on the radii of reliable reception / transmission of radio signals by network nodes, as well as on the number of vertices in the hyper- graph are studied. The concepts of the shortest route in a geometric hypergraph are discussed. Calculations of the probabil- ity of connectivity of large random geometric hypergraphs, the mathematical expectation of the diameter of hypergraphs and its change with a change in the radii of the nodes are carried out. The presentation of the results is accompanied by graphs.

Properties of geometric hypergraphs of wireless computer networks

Mathematical models of wireless computer networks are considered, reflecting two types of interaction between nodes of the same network — broadcast and routing. A natural form of representation of such a network is a hypergraph, in which the direct links between nodes that do not require routing are specified by the hyper-edges. The routes are chains of hyper-edges. The concepts of geometric and random hypergraphs are introduced, the dependences of the probabilistic characteristics of random geometric hypergraphs on the number of vertices and radii of reliable reception / transmission of radio signals by network nodes are studied — the mathematical expectation and standard deviation of the number of hyperedges, degrees of hyperedges. The properties of dynamic geometric hypergraphs are discussed, their influence on the connectivity of a computer network, and the requirements for algorithms for managing connectivity in terms of hypergraphs are formulated.

Extremal problems for convex geometric hypergraphs and ordered hypergraphs

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

Tight paths in convex geometric hypergraphs

One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $n$-vertex graph with more than $\frac{k-1}{2}n$ edges contains any $k$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $r$-uniform hypergraph, i.e., a hypergraph where each edge contains $r$ vertices. A tight tree is an $r$-uniform hypergraph such that there is an ordering $v_1,\ldots,v_n$ of its its vertices with the following property: the vertices $v_1,\ldots,v_r$ form an edge and for every $i>r$, there is a single edge $e$ containing the vertex $v_i$ and $r-1$ of the vertices $v_1,\ldots,v_{i-1}$, and $e\setminus\{v_i\}$ is a subset of one of the edges consisting only of vertices from $v_1,\ldots,v_{i-1}$. The conjecture of Kalai asserts that every $n$-vertex $r$-uniform hypergraph with more than $\frac{k-1}{r}\binom{n}{r-1}$ edges contains every $k$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $n$ for every $r$ and $k$. The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $r$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $r$ and $k$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices.

Exploring Cooperative Game Mechanisms of Scientific Coauthorship Networks

Scientific coauthorship, generated by collaborations and competitions among researchers, reflects effective organizations of human resources. Researchers, their expected benefits through collaborations, and their cooperative costs constitute the elements of a game. Hence, we propose a cooperative game model to explore the evolution mechanisms of scientific coauthorship networks. The model generates geometric hypergraphs, where the costs are modelled by space distances, and the benefits are expressed by node reputations, that is, geometric zones that depend on node position in space and time. Modelled cooperative strategies conditioned on positive benefit-minus-cost reflect the spatial reciprocity principle in collaborations and generate high clustering and degree assortativity, two typical features of coauthorship networks. Modelled reputations generate the generalized Poisson parts, and fat tails appeared in specific distributions of empirical data, for example, paper team size distribution. The combined effect of modelled costs and reputations reproduces the transitions that emerged in degree distribution, in the correlation between degree and local clustering coefficient, and so on. The model provides an example of how individual strategies induce network complexity, as well as an application of game theory to social affiliation networks.