О максимальной модулярности случайных конфигурационных графов
Keywords: configuration random graph; modularity; limit theorem
Configuration graphs with random independent identically distributed vertex degrees are considered. The degrees are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of semiedges pairwise equiprobably to form edges. Such models can be used to adequately describe the topology of transport, electricity, social networks and the Internet. An important characteristic of the structure of a graph is its modularity. It is a measure for graph clustering in the case vertices are divided into groups (clusters). Graphs with high modularity have dense edges between the vertices within clusters but sparse connections between vertices of different clusters. The notion of modularity and its properties in random configuration graphs are discussed. The maximum modularity of a graph is used to describe the level of graph clustering and to find the best division of vertices. The limit theorem for the maximum modularity as the number of vertices tends to infinity is proved.
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