Self-organization versus hierarchy in open-source social networks

Complex networks emerge under different conditions including design (i.e., top-down decisions) through simple rules of growth and evolution. Such rules are typically local when dealing with biological systems and most social webs. An important deviation from such a scenario is provided by groups, collectives of agents engaged in technology development, such as open-source communities. Here we analyze their network structure, showing that it defines a complex weighted network with scaling laws at different levels, as measured by looking at e-mail exchanges. We also present a simple model of network growth involving nonlocal rules based on betweenness centrality. Our weighted network analysis suggests that a well-defined interplay between the overall goals of the community and the underlying hierarchical organization play a key role in shaping its dynamics.

Figure 1

Social networks of e-mail exchanges in open source communities. Line thickness represents the number of e-mails flowing from the sender to the receiver. Dark depicts active members and frequent communication. (a) Social network $G_{Amavis}$ for the Amavis open-source community. (b) Social network $G_{TCL}$ for the TCL (i.e., Tool Command Language) open-source community with $N=215$ members and $⟨k⟩\approx 3$. In both networks, a few hubs (center dark nodes) route the bulk of information generated by many periphery nodes.

Figure 2

(a) Average betweenness centrality scales with degree $⟨b\left(k\right)⟩\sim k^{\eta }$ with $\eta \approx 1.59$ for the Python OS community. This exponent is close to the theoretical prediction ${\eta }_{BA}\approx (\gamma -1)∕(\delta -1)=1.70$ (see text). (b) Cumulative distribution of undirected degree $P_{>}\left(k\right)\sim k^{-\gamma +1}$ with $\gamma \approx 1.97$. (c) Cumulative distribution of betweenness centrality $P_{>}\left(b\right)\sim b^{-\delta +1}$ with $\delta \approx 1.57$ for $b>{10}^2$.

Figure 3

Correlations and rich-club phenomenon in the Python OS community. (a) Average degree of nearest neighbors vs degree $⟨k_{nn}⟩\sim k^{\theta }$ where $\theta \approx 0.75$ (open circles). Weighted average nearest-neighbors degree is almost constant (closed circles). (b) Rich-club coefficient $\Phi \left(k\right)$ scales with degree, $\Phi \left(k\right)\sim k^2$, for $k<k_c\approx 10$ while it is constant, $\Phi \left(k\right)\approx 1$, for $k>k_c$. The crossover $k_c$ is consistent with the point of decreasing behavior of $⟨k_{nn}\left(k\right)⟩$. (c) Visualization of the rich-club in the Python OS community, where dark nodes depict hubs with $k>k_c$. Peripherical nodes with $k<k_c$ (white balls) are mainly connected to hubs.

Figure 4

Plot of the weighted rich-club coefficient $\Phi (S,k)$ against node degree $k$ for the Python OS network. There is a significant deviation for $k>{10}^2$ that signals the rich-club ordering for this particular community. The subgraphs show the $k$-scaffolds or the predicted rich clubs for different degrees $k>100$. Line thickness indicates the weight attached to the link. We can appreciate how three nodes have a much more stronger internal interaction (i.e., exchange a larger number of e-mails) than with the rest of nodes.

Figure 5

Social network simulation. (a) Linear correlation between node strength $s_i$ and betweenness centrality (or node load) $b_i$ in the Python community. The correlation coefficient is 0.99. This trend has been observed in all communities studied here. (b) Estimation of $\alpha$ in the TCL community (see text). (c) Cumulative degree distribution in the simulated network (open circles) and in the real community (closed squares). All parameters estimated from real data: $N=215$, $m_0=15$, $⟨m⟩=3$, and $\alpha =0.75$. (d) Scaling of average neighbors degree vs degree in the simulated network (open circles) and in the real social network (closed squares). There is very good overlap between the model and data for large $k$. (e) Rendering of the simulated OS network $\Omega$ to be compared with the OS network $G_{TCL}$ in Fig. 1b.