*3.2. The Two-Mode or A*ffi*liation Network of Cities and Corporations*

To assess the firms' articulation in the Smart City development through corporate networks, we constructed a two-mode network or affiliation network. In social network analysis, the term "affiliation" is used to refer to membership or participation data, such as actors participating in events, which consist of a set of binary relationships between members of two sets of items [35]. Examples of affiliation data include corporate board memberships [43,44], attendances of events [45] or patterns of authorships and institutional pathways of scientists [46]. Two-mode networks allowed to capture the interaction among cities and firms and to test an urban network formation hypothesis to deepen the understanding of how individual cities and firms interact locally and form observed corporate networks [41].

We built an affiliation matrix where the rows and the columns correspond to the corporations (a total of 66) and the cities (the 65 cities members of the RECI). The matrix allowed us to determine which corporations participate in which cities' smart projects, and which cities are related to which corporations through Smart City projects. The matrix computes z*ij* as 1 when a corporation *i* is present in city *j*, and otherwise as 0. Moreover, we used Gephi to visualize the bipartite graph resulting from the affiliation network. Links connect firms with cities in which they are involved through Smart City initiatives, and there is no direct linkage within the same set of nodes (i.e., between cities or between firms), which enables a clear visualization of linkages between the two sets. In the visualization of the bipartite graph, the nodes' size represents the betweenness centrality in the network [35]. Two-mode betweenness centrality refers to the number of geodesic paths that pass through a given node, weighted inversely by the total number of equivalent paths between the same two nodes, including those that do not pass through the given node [47]. Accordingly, links between firms always include cities, and links between cities always include firms, which means that cities are on geodesic paths between firms and firms on geodesic paths between cities. A city's betweenness centrality increases when the pairs of firms are linked only through that city, and if a firm is involved in that city, all geodesic paths from that firm must include that particular city. In the same vein, a firm gains betweenness centrality when two cities are only connected between them because of the firm's participation in Smart city projects in those cities. Thus, betweenness centrality is an indicator of a city's (or firm's) strategic power by acting as bridge between firm pairs (or city pairs) that would otherwise be unconnected.
