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Modeling Discrete Combinatorial Systems as Alphabetic Bipartite Networks: Theory and Applications

Life and language are discrete combinatorial systems (DCSs) in which the basic building blocks are finite sets of elementary units: nucleotides or codons in a DNA sequence and letters or words in a language. Different combinations of these finite units give rise to potentially infinite numbers of genes or sentences. This type of DCS can be represented as an Alphabetic Bipartite Network ($α$-BiN) where there are two kinds of nodes, one type represents the elementary units while the other type represents their combinations. There is an edge between a node corresponding to an elementary unit $u$ and a node corresponding to a particular combination $v$ if $u$ is present in $v$. Naturally, the partition consisting of the nodes representing elementary units is fixed, while the other partition is allowed to grow unboundedly. Here, we extend recently analytical findings for $α$-BiNs derived in [Peruani et al., Europhys. Lett. 79, 28001 (2007)] and empirically investigate two real world systems: the codon-gene network and the phoneme-language network. The evolution equations for $α$-BiNs under different growth rules are derived, and the corresponding degree distributions computed. It is shown that asymptotically the degree distribution of $α$-BiNs can be described as a family of beta distributions. The one-mode projections of the theoretical as well as the real world $α$-BiNs are also studied. We propose a comparison of the real world degree distributions and our theoretical predictions as a means for inferring the mechanisms underlying the growth of real world systems.