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 $\lambda \ge 3$. 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. 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.

Figure 1

The illustration of the $k$-shell decomposition method. (a) Original network. Nodes are marked by corresponding $k$-shell indices. Note that the $k$-shell index does not coincide with the node degree. (b) The two-crust (left) and the 3-core (right) of the original network.

Figure 2

(Color online) Properties of the LS and the ICT sectors. (a) The growth of the largest connected component of the LS and the ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semilog plot of the ICT network size as a function of time). (b) Cumulative degree distribution, $N\left(q>q_0\right)$, of the LS and the ICT networks. The inset displays cumulative degree distribution $N\left(q>q_0\right)$ of the LS and the ICT network without offset $c$.

Figure 3

(Color online) (a) The $k$-shell structure of the LS network and (b) of the ICT network. Plots of the size of the $k$-crust, $N_0$; the size of the largest connected component of the $k$-crust, $N_1$; and the size of the second largest connected component of the $k$-crust, $N_2$, as a function of the $k$-crust index $k$. The sizes are normalized with the total number of nodes in the network, $N$. $N_2$ is multiplied by 100. The transition of the $k$-crust at $k$-shell $k=18$ of the LS network reveals a jellyfish topology [30]. Since the change in $F_1\equiv N_1/N$ at the $k=k_{\text{max}}$ is smaller in the ICT network than in the previous $k$-shell $k=k_{\text{max}}-1$, the ICT does not have a jellyfish structure. The insets of (a) and (b) show the rate $R$ of the largest connected component change as a function of $k$-shell index $k$. (c) The leadership persistence $P_L\left(t,t_R\right)$ as a function of years in the LS industry. We define $P_L\left(t,t_R\right)$ as a number of firms that were in the nucleus both at time $t_R$ and $t$. Note that most of the LS firms remain in the nucleus after they first enter it.

Figure 4

(Color online) The $k$-shell structure of random SF models with (a) $N=8000$ and (b) $N={10}^6$ nodes. Note that $\lambda =3.4$ curve overlaps with that of $\lambda =3.0$. The rate of the largest connected component change, $R$, as a function of $k$-shell index $k$ for (c) $N=8000$ and (d) $N={10}^6$. Note that in order to avoid the overlap of curves in (c) and (d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as $\lambda$ increases. The nucleus and the tendrils disappear in (a) and (c) for $N=8000$ for $\lambda >2.5$ and in (b) and (d) for $N={10}^6$ for $\lambda >2.7$.

Figure 5

(Color online) Sizes of $k$-shells, $S_k$, for SF random models with $c=0$ (a,b) $\lambda =2.1$ and (c,d) $\lambda =2.5$ as a function of (a,c) $N$ and the $k$-shell index (b,d) $k$. Note that sizes of $k$-shells increases proportionally to $N$. (e,f) Sizes of $k$-shells, $S_k$, for SF networks with $c=5.0$ and (e) $\lambda =2.1$, (f) $\lambda =2.5$. SF models with $c=5.0$ have significantly larger number of $k$-shells, $k_{\text{max}}$, compared to SF models with the same $\lambda$ and $c=0$. $S_k$ in SF models with $c=5.0$ decreases significantly slower as a function of $k$ compared to SF models with the same $\lambda$ and $c=0$ in the small $k$ region. In the large $k$ region both types of SF models with $c=0.0$ and $c=5.0$ seem to have similar behavior $S_k\sim k^{-\delta }$, where $\delta =2/\left(3-\lambda \right)$.

Figure 6

(Color online) (a) The total number of $k$-shells, $k_{\text{max}}$, in SF models as a function of $N$. Note that $k_{\text{max}}\propto N^{\theta }$. (b) The size of the last $k$-shell $S_{\text{max}}$ in the SF model as a function of $N$. Each curve crosses over into a power law regime for $N\ge N_c\left(\lambda \right)$, where $N_c\left(\lambda \right)$ increases with $\lambda$. $N_c\approx 2\ast {10}^4$ for $\lambda \approx 2.4$(c) The calculated exponent $\theta$ as a function of $\lambda$ (symbols). Our calculated values of $\delta$ agree with the mean field theory result $1/\theta \approx \delta =2/\left(3-\lambda \right)$ (solid line).

Figure 7

(Color online) (a) The total number of $k$-shells, $k_{\text{max}}$, of the LS and the ICT networks as a function of $N$ calculated for different years. The number of $k$-shells $k_{\text{max}}$ of the LS network increases approximately as a power-law function of the network size $k_{\text{max}}\sim N^{\theta }$, where $\theta \simeq 0.6$. (b) Size of the nucleus of the LS network, $S_n$, as a function of $N$. $S_n$ exhibits fluctuations as the LS network grows. Unlike in the analyzed SF models, $S_{\text{max}}$ becomes stable for $N>5000$. (c) $S_k$ as a function of the $k$-shell index $k$ for the LS network (squares). $k$-shell sizes $S_k$ as a function of $k$ decrease significantly slower than $S_k\sim k^{-4}$, which is expected for a random SF model with the same $\lambda$ and $c=0$ (triangles). However, the $c=4.0$ offset in the SF degree distribution, $P\left(q\right)$, mimics the $k$-shell structure of the LS network (pluses). $k$-shell sizes, $S_k$, of the randomized LS network (circles) almost exactly match $S_k$ of the original LS network. (d) $S_k$ as a function of $k$-shell index $k$ for the ICT network (squares). SF model with $\lambda =3.4$ and $c=0$ does not possess a $k$-shell structure. However, the introduction of $c=6.0$ in the SF degree distribution yields similar $k$-shell structure (circles), but we attribute it only to finite size effect.