Structure of Business Firm Networks and Scale-Free Models

We study the structure of business firm networks and scale-free models with degree distribution $P(q)\propto (q+c{)}^{-\lambda }$ using the method of $k$-shell decomposition.We find that the Life Sciences industry network consist of three components: a nucleus,'' which is a small well connected subgraph, tendrils,‘’ which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a ``bulk body’’ which consists of the majority of nodes. At the same time we do not observe the above structure in the Information and Communication Technology sector of industry. We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils decrease as $\lambda$ increases and disappear for $\lambda \ge 3$. We compare the $k$-shell structure of random scale-free model networks with two real world business firm networks in the Life Sciences and in the Information and Communication Technology sectors. Our results suggest that the observed behavior of the $k$-shell structure in the two industries is consistent with a recently proposed growth model that assumes the coexistence of both preferential and random agreements in the evolution of industrial networks.