*4.4. Subgroup-Level Analysis*

A network consists of several subgroups. Exploring the subgroup structure based on the NMA method helps gain a deeper understanding of the characteristics and evolution of the owner–contractor collaborative network. Unlike SNA, which focuses on the overall network structure and the role of nodes, NMA focuses on investigating the subgroup structure of the network. According to the number of award-winning projects and the number of organizations involved in the network, an average of three to four organizations (owners and contractors) are involved in each project. Therefore, we focus on the three-node subgroups and four-node subgroups of the owner–contractor collaborative network.

For the undirected unweighted network, there are two structural forms for the threenode subgroup and six structural forms for the four-node subgroup. Their topologies are shown in Figure 9.

**Figure 9.** Topology of three-node and four-node subgroups.

We imported the data for each snapshot point into the Mfinder 1.2 software and performed 100 iterations, producing motif results for different subgraphs. The number of occurrences of a certain type of subgroup in the real network and the random network is shown in Figure 10. The Z-Scores of the three-node subgroups and the four-node subgroups in different snapshots are shown in Table 5.

**Figure 10.** *Cont*.

**Figure 10.** The number of three-node subgroups and four-node subgroups in real networks and random networks: (**a**) subgroup 3-1; (**b**) subgroup 3-2; (**c**) subgroup 4-1; (**d**) subgroup 4-2; (**e**) subgroup 4-3; (**f**) subgroup 4-4; (**g**) subgroup 4-5; (**h**) subgroup 4-6.


**Table 5.** Z-Score of all three-node subgroups and four-node subgroups in different snapshots.

It can be seen in Figure 10 that the number of three-node subgroups and four-node subgroups both increased in the real network and the random network during the study period. This means that the structure of owner–contractor collaborative networks in China's construction industry is becoming increasingly complex, and the collaboration between organizations is becoming more and more diverse. For the three-node subgroup, subgroup 3-1 always appeared more frequently in the random network than in the real network, while subgroup 3-2 is on the contrary. Table 4 shows that the Z-Score of subgroup 3-2 is positive, while that of subgroup 3-1 is negative. Thus, subgroup 3-2 is the network motif, while subgroup 3-1 is the network anti-motif. For the four-node subgroups, subgroup 4-1, subgroup 4-2, and subgroup 4-4 always appeared more frequently in the random network than in the real network, while subgroup 4-3, subgroup 4-5, and subgroup 4-6 appeared much more frequently in the real network than in the random network. It can also be seen in Table 5 that the Z-Scores of subgroup 4-3, subgroup 4-5, and subgroup 4-6 are all greater than 0, while those of subgroup 4-1, subgroup 4-2, and subgroup 4-4 are all less than 0. Therefore, for the four-node subgroup, subgroups 4-3, 4-5, and 4-6 are network motifs, and subgroups 4-1, 4-2, and 4-4 are network anti-motifs.

Motifs are fundamental patterns that recur in networks, and their frequency in real networks is significantly higher than in random networks with the same number of nodes and connections. Anti-motifs are just the opposite. The above results show that subgroups 3-2, 4-3, 4-5, and 4-6 are the motifs in the owner–contractor collaborative relationship network, that is, there are many local collaborative relationships of these forms in the network. Among them, subgroups 3-2 have the largest Z-Score in the three-node subgroup, and subgroups 4-6 have the largest Z-Score in the four-node subgroup. That is, subgroups 3-2 and 4-6 are the two most dominant subgroup structures in the owner–contractor collaborative network. As seen in Figure 9, these four forms are all generated based on the

complete collaboration of the three organizations (containing at least one triangle). This subgroup structure facilitates the organization's efficient collaboration and the network's development. Subgroups 3-1, 4-1, 4-2, and 4-4 are anti-motifs in the owner–contractor collaborative relationship network. As can be seen in Figure 9, these four forms are mainly binary cooperation between organizations (containing no triangle) and do not have the basis for multiple collaborations. They are undesirable because they reduce the connectivity and cohesiveness of the network.
