*3.3. Co-A*ffi*liation Networks: Inter-Corporation and Inter-City Linkages*

In order to explore the connections between firms and between cities separately, we transformed the affiliation matrix of corporations and cities (66 × 65), in which there were no direct linkages within the same set of nodes, into two one-mode squared matrices of overlaps, through the cross-product method for corporations and cities separately [42]. Applied to our binary matrix, each product is 1 only if two corporations were "present" in a city, and the sum across cities yields the number of cities in which the two corporations participated. Then, the first one-mode matrix obtained for corporations (66 × 66) accounts for the number of cities in which each pair of corporations is present; and the cities (65 × 65) matrix indicates how many corporations were present in each pair of cities. The co-participation (or co-affiliation) of corporations in Smart City projects of the same city can be considered as an indicator of the extent to which firms get involved in these cities. Then, in this case, co-affiliation would be present if the same firm was involved in Smart City projects in two cities. In other words, the more cities in which two firms co-participate, the greater the possibility that these two firms are somehow "inter-linked" [36]. Because most network methods are designed for the analysis of binary data, we followed the procedures by Neal [36] and Field et al. [48] to dichotomize the resulting valued matrices of overlaps. Inter-firm and inter-city relationships were recoded as present if

they were two or more. That is, if two corporations have two or more cities in common, the matrix score will be 1, and 0 if two corporations have less than 2 cities in common. Following the same logic, the inter-city matrix scores were recoded as 1 (present) if two cities had two or more firms in common and 0 otherwise. Setting the threshold of connections at 2 is rather a conservative decision, according to the literature. However, we must be aware of its limitations, particularly when making comparative claims [29]. In order to prevent this overreaching, we tried to emphasize the characterization of the network as a whole while trying to be cautious about its capacity to equalize the structural advantages of cities.

We proceeded by representing the inter-firm and inter-city valued networks, to visualize the most important corporations and cities involved in Smart City projects within the RECI. Subsequently, we computed three normalized measures of centrality degree, closeness and betweenness for each city and firm, the conventional indicators to assess the relative structural position and importance of a node within a network, which in turn creates hierarchies [17,36,49]. Following the usual practice in this field, we calculated normalized measures to allow for the comparability between the different measures of centrality. Degree centrality accounts for the number of direct links within a network. In this case, degree centrality refers to the direct links (both incoming and outgoing) between cities or between firms, which indicates the direct involvement in the network. Closeness centrality captures the extent to which a city is directly connected to other cities in the network or separated from them by only short indirect linkages, reflecting the notion that both direct and indirect linkages contribute to a city's centrality and to opportunities for capital accumulation or innovation diffusion [36]. The closer two firms are in the network, the less dependent one is on intermediary channels. Thus, firms with high closeness centrality can provide their producer clients with a more independent and rapid access to information from a wider ranges of sources [36]. Betweenness centrality focuses on the extent to which a city or a firm serves as an intermediary that facilitates the flow of resources for other cities (or other firms) in the network. A strong position in terms of betweenness would reflect a brokering or gatekeeping position, which would provide an actor with a unique ability to monitor and control resource flows, acting as an interlocking agent. Finally, we also calculated the Gini coefficient as a measure of inequality within each of these centrality measures of hierarchies, with values closer to 0 indicating more equality. Under the logic of network analysis suggested by Neal [36], this inequality index complements the rank provided by centrality measures by reflecting structural inequalities among cities and firms.
