Modeling group interactions of complex systems. Review

Many important properties of complex systems can be described using networks. The analysis of such complex networks is based on graph theory Diestel (2005) and statistical mechanics Albert, Barabasi (2002), which provide universal tools for studying the structural and dynamic properties of the system and its components. The study of complex networks made it possible to reveal such important properties inherent in many real systems as the power law of the distribution of degrees of nodes, belonging to a small world, the presence of a giant component, evolution models. Many methods have been developed to identify important network actors and the presence of communities of actors. Reviews of research methods for complex networks can be found in publications Albert, Barabasi (2002), Newman (2010). In many cases the use of complex networks does not provide complete information about the investigated systems. The real-life systems have many heterogeneous parts with complicated subsystem dependencies that can’t be derived from a simple graph. The concept of “higher-order networks” refers to three main lines of research designed to overcome the limitations associated with traditional models that capture only pairwise connections Battiston etal. (2020), Lambiotte etal. (2019). The first line is based on the assumption that multiple link types can be described in terms of multilayer networks, reviews of the approach can be found in Boccaletti et al. (2014), Bianconi (2018). The second has introduced non-Markovian model of nodes interactions, taking into account the correlation and temporal characteristics to identify the implicit influence of components on each other Holme (2015). The third line generalizes pairwise interactions to arbitrary node sets, these are “networks beyond pairwise interactions” Battiston etal. (2020). This direction is the scope of our short review. To analyze complex data, a number of techniques have been developed that use various mathematical structures to represent “multiple” relationships. Hire we examine three methods of explicit encoding higher order interactions in terms of a social system that includes a certain set of individuals. Any attribute or a number of attributes that can be associated with the individuals of the system can provide a basis for forming a collection of subsets of individuals, where the subset corresponds to the individuals that have the given property. Thus, the structural framework imposed on the set of individuals by such set of subsets is considered. R. H. Atkin in Atkin (1972, 1974, 1976) uses an algebraic approach in which the basis for the study is the concept of a simplex, a geometric figure that is an n-dimensional generalization of a triangle. Analytical methodology is based on the algebraic topology, which uses algebraic methods for studying topological spaces (see, for example, Hatcher (2002)). An abstract p-simplex is a set with p+1 vertices, denoted as (v0, v1,..., vp). The simp lex a = (v0, v1,..., vq ) is a ^-dimensional face (q-face) of the simplex а' = (v0, v-1, ..., vp) if every vertex a is also a vertex ст'. The simplicial complex is the set of simplices closed under the inclusion of all faces of all simplices. Q-analysis is a mathematical framework for describing and analyzing simplicial complexes. In Seidman (1981), a collection of subsets is considered as the backcloth of a set of individuals. To the subset the term “event” (or group), which contributes to the formation of the subset, is applied. The analysis of structures imposed on a set by a collection of subsets is based on the concept of a hypergraph. If X = {x1, x2,..., xn} is a finite set, e = {Ej]i = {1,2,..., m}} - a family of subsets of X, then H = (X, e) is a hypergraph Berge (1976) if Ei = л. i e [mj|; um=iEi = X. S. B. Seidman noted that while the hypergraph focuses on the incidence relationships between vertices and edges (individuals and events), structures representing relationships between individuals (or between events) can be derived from the hypergraph. As an example of using the hypergraph representation, the problem of searching for “anomalies” in the original basic structure is considered. In Wilson (1982), a research method is based on treating data as a bipartite graph. A graph is bipartite if the vertices may be partitioned into exactly two mutually exclusive sets, so that every edge has its ends in different sets: vertices of the same set cannot be adjacent Diestel (2005). In this case the two sets are the set of actors and the set of events. Such an approach, as shown in Wilson (1982), makes it possible, for example, to reveal the division of a set into groups that do not have common individuals and at the same time cover the entire set of individuals or to find sets of groups that allow activity only within the selected set of groups, forming a collective.