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In the previous post, we started to examine the network of co-attendance during Rotary meetings. Up to this point, our analysis was locally focused on attendees'' influence or intermediarity, and the frequency of co-attending pairs. We now propose to enlarge the scope of observation by clustering the initial network into larger communities of co-attendees.Can we identify significant clusters of co-attendees? On what basis did they group together? Can we detect structural holes between communities and bridges connecting them? Who (which nodes) serve as bridges? How did these clustering patterns change over time?Methodologically, we relied on Newman’s fast greedy algorithm to detect communities within our networks of co-attendance. We chose this method because it works fairly well for weighted networks (Newman, 2004) and returned the highest modularity score (we obtain the same results as with Louvain algorithm, but slightly better ones than other hierarchical clustering methods). The resulting clustered networks are visualized on Fig. 9a, 9b and 9c. Interactive versions of these networks are available on CytNetShare (Petit, Fitch, Harkson).[gallery size="full" columns="1" ids="4041,4043,4045"][gallery size="full" ids="4042,4044,4046"]The algorithm detected 11 communities in Petit’s network [Table F1], 6 in Fitch’s network [Table F2] and 7 in Harkson’s network [Table F3]. Drawing on this method, Petit’s network appears less homogeneous than Fitch and Harkson. The following tables summarize the results of community detection in each network, including the size of communities (number of members) and their global characterization.Communities in Petit's networkNoColorSizeAttributes2yellow21Core4fuchsia17Non-members10green12Charter9purple6Committee11grey3Non-members1, 3, 5, 6, 7, 82Non-membersTable F1. Communities in Petit’s co-attendance network Petit’s network is clustered into two large communities (more than 15 members each), two medium groups (6 to 10 members) and seven small communities (only 2 or 3 members). The largest community (yellow) can be labeled as the “core” group, since it includes a majority of key members (directors, officers, charter members, committee members) (14 of 21). There are no ordinary members in this group, which is mixed in terms of membership status (it includes both key members and non-members) and nationality (3 Chinese). Community 4 (purple) is exclusively composed of non-members. It includes 1 Chinese and 1 Japanese (for 15 Westerners). Community 10 includes a large majority of (Western) charter members (only 2 non-members and 1 undermined). Community 9 includes a majority of committee members (only 1 ordinary member), only men and Westerners. Isolated dyads or tryads involve mostly non-members (or undetermined) and Westerners (except for one Chinese, but Chinese were underrepresented at the time).The core cluster (2-yellow) is connected to all other clusters (except for isolates), i.e. clusters 4 (by two nodes 54 - President Petit - and node 61 - Director Sammons), 9 (by node 4 only) and 10 (by 2 nodes 4 - Secretary Baker - and 54 -President Petit). Two clusters (4 and 10) are connected to the core cluster 2 only. All nodes in cluster 4 (non-members) are connected to both 61 (Director Sammons) and 54 (President Petit) in cluster 2, except for node 67 (Chinese guest Tcheng) which is connected to 61 (Director Sammons) only. The charter cluster (10) is connected to the core cluster by nodes 2, 17, 65 and 69 (1 charter member is connected to the secretary and 3 non-members are connected to the president). All nodes in the committee cluster (9, purple) are connected to the core cluster through secretary Baker. The seven remaining clusters (six dyads and one triangle) are isolated from the rest of the network. To sum up, there are three major hubs in Petit’s network (node 54 joining cluster 2, 4 and 10, node 4 joining cluster 2, 9, 10, node 61 joining 2 and 4), among which two cutpoints (node 4 connecting cluster 2 and 9 and node 2 (Baker) connecting cluster 10 and 2).Communities in Fitch's networkNoColorSizeAttributes1red37Presidential3green17Japanese4fuchsia16Women6kaki12Chinese5turquoise9Non-members2yellow4Gender-mixedTable F2. Communities in Fitch’s co-attendance networkIn Fitch’s network, the largest cluster (red) includes a majority of non-members (24 of 37). Except for the president, the secretary and one director, most Rotarians in this group are ordinary members. This community includes a majority of men (3 women only) and Westerners (31 of 37), except for 4 Chinese and 2 other nationals. Community 3 involves mostly non-members (except for 2 ordinary Rotarians). It is exclusively male but rather heterogenous in terms of nationality (3 Japanese, 1 Chinese). Community 4 includes almost as many non-members (7) as members (9), most of them being ordinary members (except for the treasurer, one director and a senior member). All non-members are women. This group includes a large majority of Westerners (except for one Chinese and one Japanese). Community 6 is exclusively male, with a majority of ordinary members (7+ 1 honorary member) and 4 non-members. This group includes a substantial number of Chinese (5, among which 4 Rotarians). It is rather homogenous in terms of classification (business sector). Most of them are professionals working as newspaper editors, lawyers or economic experts. Cluster 5 includes Western non-members only (except for one ordinary Rotarian), among which 3 women. Cluster 2 is exclusively composed of non-members, mostly Westerners (except for one Chinese). It is a mixed group in terms of gender (2 men, 2 women). Fitch is the only network in which we find no dyadic clusters.Two clusters (3- Japanese and 4-women) are disconnected. Cluster 3 (Japanese) is connected to cluster 1 (presidential), cluster 6 (Chinese cluster) through node 1 (non-member J. Arnold). All nodes in cluster 3 (Japanese) are connected to cluster 6 (kaki) and 1 (red). One node in this cluster (43 - a woman, Mrs. Hawkings) is connected to two other clusters (1 and 6). Yet paradoxically, she is not connected to the women cluster 4 (purple). Cluster 5 (non-member) is connected to cluster 1 (presidential) only. Cluster 6 (Chinese) is connected to three other clusters 1, 3 and 4. All nodes in cluster 6 are connected to cluster 1 and 3, and two nodes (no) are connected to cluster 4. Two nodes in cluster 6 (26 - ordinary members Du Pac - and 51 - non-member Jenkins) are connected to three other clusters (1/presidential, 3/Japanese and 4/women). The remaining cluster (2/gender-mixed) is utterly disconnected from the rest of the network. To summarize, there are two major hubs in Fitch’s network: node 1 (Arnold) joining clusters 1, 3 and 6 and node 30 (president Fitch) who represents the most central hub in the network. He is connected to every (non-isolated) clusters. He is the only connection with cluster 5 (he serves as a bridge for cluster 5 of non-members).  He is also a major connection for cluster 3 (but not the only one – 43 Hawkings - as well). In sum, he serves as a major bridge connecting peripheral nodes with central nodes.Communities in Harkson's networkNoColorSizeAttributes1red9Pre 19304fuchsia8Fellowship5turquoise8Post 19276kaki5Non-members7blue5Post 19302, 32UndefinedUnder Harkson’s term, the algorithm detected 7 communities. Size variations are less important than in previous networks. The largest community (1) comprises a short majority of members (5) (including the the president and one senior member) and 4 non-members. It is an exclusively male group. It includes a majority of Westerners (except for 2 Chinese). All members in this group joined the club before 1930. Cluster 4 includes Rotarians only, with a parity of Western and Chinese members. Most of them served in the Fellowship committee. Cluster 5 also includes Rotarians only, with a majority of Westerners (5) and 3 Chinese. All joined the club after 1927. Most of them are committee chairmen and serve in the same class of committee (club - program, attendance - or community service - charities, schools). Cluster 6 is exclusively composed of Western male non-members (except for one woman). Cluster 7 includes a majority of Rotarians (except for one non-member), with a short majority of Westerners (3) and 2 Chinese). Interestingly, all joined after 1930. The two remaining groups are isolated dyads without clear attribute pattern.We can identify five hubs (4, 11, 30, 32, 44 (join 5 and 1) and three cutpoints in this network – node 18 (president Harkson) (joining 1 and 5), node 5 (ordinary member Berge) (joining clusters 7 and 5) and node 57 (vice-president Wolfe) (joining clusters 5 and 7). Cluster 5 is the best connected (and most central) cluster in the network. It is connected to three other clusters (1, 4 and 7). The three other clusters are interconnected. Cluster 4 is fully connected to 5 and 7 (all nodes in cluster 4 are connected to cluster 5 through node 57 and to cluster 7 through node 5 – ordinary member Berge), 5 connected to 4 and 7 by node 57 and cluster 7 is connected to 4 and 5 by node 5 (who acts as a cutpoint). Cluster 1 is connected to cluster 5 only (through node 18). All nodes in cluster 4 are connected to cluster 5 and 7. Cluster 6 (non-members) is disconnected from the rest of the network, in part because of the declining importance of the president.In Petits’ and Fitch’s clustered networks, hubs and cutpoints roughly overlap with the leaders and brokers we identified earlier [Tables E1, E2, E3]. While inter-cluster bridges multiply in Harkson’s network, they do not always show a high betweenness centrality.Centralization, decentralizationIn the long-term, these observations confirm the highest centralization under Fitch’s presidential term. Petit’s term differs from subsequent networks in that the cluster of core members is in itself a core cluster (it is connected to all other clusters and serving as a hub for the entire network). By contrast, in Fitch and Harkson’s networks, key members are not part of core clusters. Quite the opposite, they now belong to peripheral clusters, while other clusters serve as bridges. In other words, Petit’s network presents a rather simple core-periphery model. Core members form the core of the networks, whereas other clusters occupy peripheral positions. The latter are connected to core nodes only, but they are not connected together. In sum, the core cluster ensures the consistency of the network. Fitch’s network presents a more complex model. Core members are no longer part of the core cluster. Peripheral clusters are not only connected to core members, but may also be connected together (see for instance clusters green, kaki and purple). Only one cluster (turquoise) is a truly peripheral cluster. In Harkson’s network, the global structure is more fragmented. Core members (pre-1930 ones) do not belong to the core cluster. Key members are no longer central but belong instead to the most peripheral cluster (which is connected to the core cluster only). By contrast, other clusters are not only connected to the core cluster but are also interconnected. The central position is now occupied by more recent members who joined the club after 1927, i.e. after the establishment of the nationalist regime) (cluster 5). Unsurprisingly, this group includes more Chinese members than ever.Although clique detection may produce interesting results, we will not pursue this direction. Bimodal projections tend to artificially produce many more cliques than we would find in “natural” networks (Opsahl, 2011). From a long-term perspective, Fitch’s network contains the largest cliques and Harkson’s network has the largest proportion of minimal cliques, which refer to smaller events or events on which we have minimal information. What is interesting to notice here is that cliques in persons-persons networks reproduce events in the original bimodal networks. Therefore, clique detection in co-attendance networks offers a way to retrieve the information lost during the projection process.Another way to balance the loss of information is to complement the analysis of persons-persons joined by events by the analysis of event-event joined by persons. This is the other side of the dual projection method, which we propose to conduct in the next post.ReferencesNewman, M. E. J. “Analysis of Weighted Networks.” Physical Review E 70, no. 5 (November 24, 2004): 056131.Opsahl, Tore. “Clustering in Two-mode Networks” Tore Opsahl (blog), June 21, 2011. https://toreopsahl.com/tnet/two-mode-networks/clustering/ [Last accessed September 06, 2019]