**4. The second-order ties**

Second-order ties imply the existence of a hierarchical position from which the link is connected in an exclusive derivation or sharing with other elements directly connected to junction or branch points. This structural condition, displayed in **Figure 3b**, leads to the principle of "Graicunas Span of Control" (GSOC) [31] in the organization of the network, which Graicunas shared with Lyndall Urwick, one of his best estimators and master of scientific management in 30 years [32]. This principle was also listed in the "Ten Principles of Management" recently reconsidered [33]. The GSOC principle defines seven as the number of direct subordinates a manager can adequately manage.

In organizational structures, this problem led to the assumption of a hierarchicalfunctional order and authority managed with intermediate positions whose number was related to the organizational dimensions and whose order depended on the dimensions of the organizational network and on the managerial structure [34].

This leads to the consideration that, while the number of network connections, which a position oversees, can theoretically be infinite, in practice the organizational performance control capabilities are limited.

**Figure 6** shows the hyperbolic growth of second-order ties correspondent to the first ones as calculated (**Table 1**) by Graicuna's Equation [Eq. (1)] in which R is the number of second-order ties and n is the correspondent number of first-order ties.

$$R = n \left[ \frac{2n}{2} + (n - 1) \right] \tag{1}$$

After its codification in organizational principles, GSOC was submitted to several critics and cited by Simon [35] as an example of an inconsistent criterion, at best considered a proverb. Particularly, Simon underlined that the GSOC has a relative appliance, as it is adequate only to restricted situations in which other elements, such as technology, information systems, and the managerial process, allow its appliance [36].

Considering the connection between March and Simon, we note that in the Handbook of Organizations edited by March [37], the principle was mentioned and significantly quoted by Urwick [38, 39] for any case limited to closed systems [40].

The last topic in the evolution of networks brings the Graicunas principle back up to date with the limits of control systems in the face of the growing complexity of organizations and with an extension to processes of its interpretative potential.
