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In order to complement the bimodal approach we conducted in our last post, we now propose a dual-projection analysis. For this purpose, we first projected the initial two-mode networks into two one-mode networks of persons-persons joined by events (co-attendance) and events-events joined by persons (multiple attendance) using the R package “tnet” (project_tm function, Opsahl 2008). We alternatively used and compared two different methods of projection. The “sum” method gives the actual pattern of co-attendance (actual number of co-attended events), while Newman’s method takes the relative importance of events into account. In the following essays, we successively examine networks of co-attendance (10) and of multiple attendance (11).The first set of one-mode networks represents the relations between persons joined by events. In person-to-person networks, nodes represent persons and edges link persons who participate in the same events. Edges represent co-attendance but not necessarily direct interaction. A major limitation of one-mode projection is to create direct ties between persons without discriminating between mere co-presence and active interaction. Two persons may attend the same event without interacting directly. Yet they will be treated in the same way as two attendees engaged in active intercourse. Edge weight measures how frequently two persons co-participate in meetings. In the “sum” projection, 1 means they met only once, 2 that they attend twice the same meeting, 3 three times, and so on. In Newman’s projection, it is more complex since the method aims to rectify the negative effects of size variation – by discounting the largest events (that are the most likely to be attended) and key officers who are more likely to attend than less active members, occasional guests and visitors.Global descriptionThe following table compares the global measures in the three networks.PetitFitchHarksonNodes719539Edges319517124Density0.1280.1160.167Connected components824Centralization0.4120.8380.295Clustering coefficient0.7550.9280.847Max node degree378817Min node degree121Max edge weight (sum)345Min edge weight (sum)111Max edge weight (Newman)1.0561.0831.0Min edge weight (Newman)0.0560.050.111Petit’s network includes 71 nodes and 319 edges. Fitch's network includes 95 nodes and 517 edges. Harkson's network includes 39 nodes and 124 edges. Tie weight ranges from 1 to 3 in Petit’s network, from 1 to 4 in Fitch’s network, and from 1 to 8 in Harkson’s (using the “sum” method). As in two-mode networks, the varying number of nodes, edges and other global measures first reflects variations in the quantity of information available for each period. Fitch is the best-documented, Harkson is the poorest, and Petit stands in-between. Globally, the three networks show three distinct profiles. Fitch is the most centralized but also the most heterogeneous (density of 0.128). Harkson presents the opposite situation. It shows the most homogeneous structure (density of 0.167) with a medium number of connected components (4). Petit’s network situates in-between in terms of centralization (0.412) and density (0.116). Fitch has the minimal number of connected components (2) and the maximal clustering coefficient (0.948). Petit shows the maximal number of connected components (8) and the minimal clustering coefficient (0.755). Harkson presents an intermediate situation with 4 connected components and a clustering coefficient of 0.847.Nodes centralityWho are the most influential persons? As in our two-mode analysis, we use degree centrality to measure the importance of persons in co-attendance networks. The following charts compare the degree distribution in the three networks [Fig. 5a, 5b, 5c].Degree distribution (personal influence)Petit’s network displays a widespread degree distribution, ranging from 1 to 37. There is a strong correlation between maximal degree and status, yet with a significant exception for nodes with a degree of 18. 16 of 17 of them are non-members, all are men and Westerners except for one Chinese and one Japanese. Aside for them, lowest degrees are associated with non-members. The president and one director show the highest degrees of 37 and 26, respectively. Those with a degree between 5 and 18 are all Rotarians, men and Westerners. Degree 4 includes Western non-members only. Lowest degrees (2) are usually associated with Western non-members, except for two Chinese and one Western Rotarians.[caption id="attachment_4064" align="aligncenter" width="680"] 5a. Degree distribution in Petit’s co-attendance network[/caption]In Fitch’s network, degree distribution is extremely skewed, ranging from 2 to 88. This converges with the globally high centralization. As in Petit’s network, there is a strong correlation between degree and status. Except for one non-member, persons with a degree over 25 are all Rotarians, male and Westerners. The president shows the highest degree by far (88.). Below 25, the correlation is less obvious. Persons with a degree between 20 and 25 include almost as many non-members (8 of 18) as Rotarians (10 of 18). They are mostly Westerners, except for two Chinese and one Japanese. This group also includes a significant number of women (6 of 18). The next group of persons with a degree between 6 and 13 includes 16 members and 25 non-members, 33 Westerners and 8 Chinese, and only one woman. Lowest degrees (5 and below) are more systematically associated with non-members (only 4 Rotarians of a total of persons). This group also includes a significant number of non-Westerners (3 Chinese, 3 Japanese and 2 other nationals) and women (6 of 31).[caption id="attachment_4065" align="aligncenter" width="680"] 5b. Degree distribution in Fitch’s co-attendance network[/caption]Harkson’s network presents the most balanced profile regarding degree distribution, ranging from 1 to 17. This confirms the global heterogeneity of the network and the declining centrality of the president we observed earlier in the corresponding two-mode network. Harkson’s term displays the strongest correlation between status and degree. All persons with high degree (over 5) are Rotarians. Two in the top three are key officers (president and vice-president/interim president). The most striking difference with previous networks is that the president himself does not hold the highest degree. As explained earlier, Harkson was absent for some time and replaced by the vice-president (Wolfe). Lowest degrees (5 and below) are associated to non-members. Since the number of Chinese members has increased since 1930, we can no longer establish a clear correlation between degree and nationality. The group with highest degrees includes almost as many Chinese (8 of 19) as Westerners (11 of 19). There are no women reported in this network.[caption id="attachment_4066" align="aligncenter" width="680"] 5c. Degree distribution in Harkson’s co-attendance network[/caption]Degree and betweenness (influence and intermediary)Degree centrality, however, is not sufficient to assess attendees'’ positions within the club. We propose to cross-examine degree with betweenness centrality in order to refine our classification of personal profiles. The following charts compare degree and betweenness distributions in the three networks [Fig. 6a, 6b, 6c].In Petit’s network [Fig. 6a], we observe a direct correlation between degree and betweenness. Those who have the highest number of neighbors also serve as brokers, except for the medium group of non-members with a degree of 18 who show a very low betweenness centrality. On the opposite, two Rotarians (one director and one charter member) who are not particularly influential (degree 12 and 16 respectively) show an exceptionally high betweenness centrality (0.67 and O.4, respectively). In Fitch’s network [Fig. 6b], there is no obvious correlation between degree and betweenness. Except for the president himself, who shows the highest betweenness centrality of 0.8, acting as the main hub in the network, no other person serves as broker in this network. This confirms the exclusive centralization around President Fitch during his term. Under Harkson’s presidency [Fig. 6c], we again observe a strong (yet not perfect) correlation between degree and betweenness. Four persons act as intermediaries in this network, showing both a high degree centrality and high betweenness (including the vice-president Wolfe and the president Harkson). There are two minor exceptions to this general rule, however. The president ranked second in terms of intermediarity, but came only third in degree centrality. One ordinary Chinese member shows a moderate degree of 8 but a fairly high betweenness of 0.29 (which contrasts to the null betweenness of other persons in the same class).[gallery size="full" columns="1" ids="4067,4068,4069"]Positional profilesThe three tables below summarize these observations [Table E1, E2, E3]. Combining degree and betweenness centrality, we design a three-class typology of persons: leaders, brokers and outsiders. Leaders show both high degree and high betweenness. Brokers combine low or medium degree and high betweenness. Outsiders have both a low degree and a low betweenness. The attached tables list the major representatives in each category for each network, with their most significant characteristics. We added a residual category (TBF) for persons with high degree but not particularly high betweenness, and more anecdotic cases that require closer analysis. For each case, we specified whether they acted as hubs, bridges or cutpoints between the clustered communities we will detect in the next essay (10-b).  The tables also compare their respective degree in one-mode and two-mode networks.[caption id="attachment_4077" align="aligncenter" width="2060"] Table E1. Positional profiles in Petit’s network[/caption][caption id="attachment_4079" align="aligncenter" width="2508"] Table E2. Positional profiles in Fitch’s network[/caption][caption id="attachment_4078" align="alignnone" width="2526"] Table E3. Positional profiles in Harkson’s network[/caption]&#160;Dyads and edge weight (co-attendance frequency)Who are the most frequent co-attendees, i.e. the persons who are the most likely to attend club events together? Relying on edge weights, can we identify the most frequent dyads, i.e. pairs of co-participants? The first analysis of edge weight is based on the “sum” projection. Unsurprisingly, there is a reversely proportional relation between tie weight (number of events co-attended) and number of dyads (co-attending pairs) [Fig. 7a, 7b, 7c]. Those who met regularly are the least frequent. Conversely, pairs who met only once are the most frequent.[gallery size="full" ids="4084,4083,4082"]In Petit’s network [Fig. 7a], tie weight ranges from 1 to 3. A large majority of persons met only once (303 of 319 pairs). Fifteen pairs met twice and only one couple co-attended three events together. Unsurprisingly, co-attendance frequency is directly related to persons’ status and their central position in the club. The most frequent pair is President Petit-Director Sammons. Couples who co-attended two events involve key Rotarians only (officers, directors and charter members), except for one Chinese non-member invited twice as a special guest. All are male and Westerners.In Fitch’s network [Fig. 7b], tie weight ranges from 1 to 4. As in Petit’s network, we observe a similar correlation between tie weight and pair frequency. 496 (of 517) pairs met only once, 16 met twice, 4 met three times and 3 dyads co-attended four events. The three most frequent pairs form a four-node clique including two key officers (president Fitch and secretary Harris) and two non-members but prominent personalities in Shanghai, who were frequently invited to attend Rotary meetings. Their co-attendance is directly related to their active participation in club activities. The second group of most frequent co-attendees (three co-attended events) consists in two dyads with a mutual co-attendee (President Fitch). Fitch met three times with a senior member (Wentworth) and an ordinary member (Peek), separately. As in Petit’s network, co-attendance frequency is strongly linked to attendees' status, gender and nationality. The group of twice-co-attending pairs includes half members and half non-members, but only three Chinese and two women. Interestingly, the strongest ties usually unite mixed pairs in terms of gender and nationality. Only weak ties (weight of 1) link two Chinese or two women.Harkson’s network [Fig. 7c] present a similar correlation between edge weight and dyad frequency. Edge weight ranges from 1 to 5. A large majority (80 of 124) met only once, 39 met twice and 5 co-attended five events. In other words, regular couples tend to meet more often in Harkson’s network than in Fitch and Petit’s networks. This suggests that the Rotary club became a smaller world in wartime. Fitch’s network presents the most complex pattern of co-attendance. At a global scale, it is a larger world than Petit’s, but it contains smaller worlds within it (the 6 pairs than met four times).If we now turn to Newman’s projection, we observe a more complex relation between tie weight and the number of co-attendees. The weakest ties are still the most frequent and the strongest ties are the least frequent ones. But in-between, there is no linear relation between tie weight and the number of dyads. For instance, in Petit’s network, eight pairs have a tie weight of 1, nine have a weight of 0.5 but only one couple shows a tie weight of 0.365. There is no need to pursue the comparison further. We can just conclude that in the “sum” projection, pairs with the strongest ties often differ from those in Newman’s projection. In Petit’s network, for example, the most frequent couple (with a weight of 1.06) is not Petit-Sammons but Petit-Tcheng (a Chinese guest) because they co-attended two highly selective events (events that gathered only a limited number of persons, or events on which we have very few information). In Fitch’s network, the ranking is maintained only for the strongest pair (Fitch-Harris) but not for the next ones. The president is involved in all pairs with a tie weight over 0.5. In Harkson’s network, the most frequent pairs involve four persons who sharply differed from those observed in the sum projection. During this period, the president is involved in pairs with moderate tie weight only (0.6).[gallery size="full" ids="4085,4086,4087"]Up to this point, we have focused solely on co-attending dyads. In the next essay, we propose to enlarge the scope of observation in order to identify larger communities based on members’ co-attendance.ReferencesNewman, M. E. J. “Analysis of Weighted Networks.” Physical Review E 70, no. 5 (November 24, 2004): 056131.Opsahl, Tore. “Projection.” Tore Opsahl (blog), June 21, 2011. https://toreopsahl.com/tnet/two-mode-networks/projection/ [Last accessed August 29, 2019]