Unicyclic Components in Random Graphs

Random graphs underly processes such as polymerization [1], percolation [2], and the formation of social networks [3, 4]. Random graphs have been extensively studied, especially in theoretical computer science [5, 6]. Special families of random graphs have been also examined, e.g., planar random graphs appear in combinatorics [7, 8] and in physics [9]. The basic framework for generic random graphs naturally emerged in two different contexts [10]. Flory [1, 11] and Stockmayer [12] modeled a polymerization process in which monomers polymerize via binary chemical reactions until a giant polymer network, namely a gel, emerges. Erdős and Rényi studied an equivalent process in which connected components emerge from ensembles of nodes that are linked sequentially and randomly in pairs [13]. Different methodologies have been employed to characterize random graphs. Kinetic theory, specifically, the rate equation approach, was used to obtain the size distribution of components [14]. Using probability theory, a number of additional characteristics including in particular the complexity of random graphs have been addressed [5, 6]. In this study, we show that the rate equation approach is useful for studying the complexity of random graphs. Our main result asserts that Uk, the average number of unicyclic components of size k in a random evolving graph, is given by