Scale-free models for the structure of business firm networks.

We study firm collaborations in the life sciences and the information and communication technology sectors. We propose an approach to characterize industrial leadership using k -shell decomposition, with top-ranking firms in terms of market value in higher k -shell layers. We find that the life sciences industry network consists of three distinct 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. Industrial leaders, i.e., the largest companies in terms of market value, are in the highest k -shells of both networks. The nucleus of the life sciences sector is very stable: once a firm enters the nucleus, it is likely to stay there for a long time. At the same time we do not observe the above three components in the information and communication technology sector. 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 in scale-free networks decrease as the exponent of the power-law degree distribution lambda increases, and disappear for lambda>or=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. We argue that the observed behavior of the k -shell structure in the two industries is consistent with the coexistence of both preferential and random agreements in the evolution of industrial networks.