1.1 Places attributes

We first explore how the students (Elements) are distributed among places using the table() function in R base:

table(Result1_df$NbElements)

  1   2   3   4   5   7  10  11  12  15  16  18 
179  15  12   4   1   3   1   1   1   2   2   2 

We can utilize ggplot2 package to visualize this distribution:

library(ggplot2)

ggplot(data = Result1_df, aes(x = NbElements)) +
  geom_histogram(binwidth = 1, fill = "blue", color = "black", alpha = 0.7) +
  labs(title = "Students by place",
       x = "Number of Students",
       y = "Frequency") +
  theme_minimal()

Explanation:

• ggplot(data = Result1_df, aes(x = NbElements)): Initializes the ggplot object, specifying the data frame Result1_df and the aesthetic mapping x = NbElements.
• geom_histogram(binwidth = 1, fill = "blue", color = "black", alpha = 0.7): Adds the histogram layer to the plot. binwidth sets the width of the bins, fill and color set the fill and border colors, and alpha controls the transparency.
• labs(title = "Students by place", x = "Number of Elements on X-axis", y = "Frequency"): Sets the title and axis labels.
• theme_minimal(): Applies a minimal theme to the plot, but you can customize the theme based on your preferences.

The table and histogram below reveal that most places (179, or 80%) consist of unique trajectories centered on a single student. These places are perfectly aligned with individual students. This is an intriguing finding in itself. It suggests something significant about the structure of the network, which can be attributed by historical circumstances. Specifically, the prevalence of these idiosyncratic places reflects the widespread adoption of the elective system in American higher education institutions during the late 19th century.

Similarly, we can explore the distribution the universities (Sets) among places.

table(Result1_df$NbSets)

  1   2   3   4 
 79 119  21   4 

ggplot(data = Result1_df, aes(x = NbSets)) +
  geom_histogram(binwidth = 1, fill = "blue", color = "black", alpha = 0.7) +
  labs(title = "Universities by place",
       x = "Number of Universities",
       y = "Frequency") +
  theme_minimal()

Most places contain a maximum of two universities, meaning that the majority of students attended a maximum of two different universities. This suggests that many students were relatively mobile during their studies and transferred to a different institution to complete their training. Fewer students (25 places) attended more than two universities during their studies. More specifically, 21 places involved students who attended 3 and 4 places involved students who attended 4 different universities.

1.2 Most significant places

Beyond crude statistics, we want to gain deeper insights into the students and universities that define each place. The “PlaceDetail” column provides this information. Since examining all 223 places individually would be time-consuming, we can start by focusing on the most populated places, which include a minimum of 2 students and 2 colleges. In this approach, we choose to discard idiosyncratic places that involve only one student or unique curricula. Using the filter() function, 13 such places are found.

n2 <- Result1_df %>% 
  filter(NbElements > 1 & NbSets > 1)

kable(n2, caption = "The 13 most significant places") %>%
  kable_styling(bootstrap_options = "striped", full_width = TRUE, position = "left")

In subsequent steps, it is recommended to carefully examine the list of places and their associated details, beginning with the most significant and progressively broadening the selection to encompass less populous places. It is beyond the scope of this paper to extensively describe the places found in this dataset, but I have contributed an in-depth research paper that demonstrates how places can be used to identify typical educational paths and track students who followed similar trajectories (Armand 2024). Interested readers are encouraged to consult this comprehensive research paper for further information.

1.3 Typology of places

If the data includes qualitative attributes, these attributes can be used to further characterize the places and build a typology. In our example, for instance, we considered the students’ field of study and the time of graduation to establish the relative strength of places, as shown in the table below (see also (Armand 2024) for an in-depth analysis of the different types of places).

Type A places represent the strongest potential for direct interaction, as they involve students who enrolled in the same programs at the same time, likely attending the same classes. In our population, there are four places of this type, all featuring students in sciences or engineering who graduated during or after World War I (P032, P033, P037, P173).
Type B places are characterized by potential interactions outside the classroom setting. While students in these places may not have enrolled in the same courses, their paths could have crossed on campus through various extracurricular activities. An illustrative example of a Type B place is represented by physician Yan Fuqing (顔福慶) (Yen Fu-ching) and businessman Cao Maoxiang (曹蝥祥) (Y.S. Tsao), who attended Yale and Harvard between 1909 and 1914 (P038).
Type C places refer to students who attended the same universities and graduated in the same disciplines but at different periods of time. Although these students did not have the opportunity to physically interact on campus, their shared educational background created a potential for future collaborations and intergenerational connections. For instance, economist Ma Yinchu (馬寅初) graduated from Yale and Columbia in 1910–1914, a decade before banker Lin Zhang (林障) completed his studies at the same institutions in 1920–1922 (P034).
Type D places represent the weakest form of association, bringing together students from different generations who pursued diverse academic disciplines at the same alma mater. For instance, place P161 exemplifies a Type D place where three graduates from Stanford University, spanning a period between 1905 and 1922, pursued distinct fields of study ranging from engineering to the humanities. These multigenerational and multidisciplinary places encompass a significant number of students, often exceeding ten individuals, who share a common educational institution.

While it is beyond the scope of this study to detail the methodology used for classifying places, we provide the typology below as an illustration, used to compute the number of places in each category. We hope this case study inspires other researchers to create their own typology of places, tailored to their research questions and data attributes.

# load data 
placetypo <- read_delim("data/placetypo.csv", 
    delim = "\t", escape_double = FALSE, 
    col_types = cols(...1 = col_skip()), 
    trim_ws = TRUE)

kable(head(placetypo)) %>%
  kable_styling(bootstrap_options = "striped", full_width = TRUE, position = "left")

We count the number of places for each type using the group_by() and count() functions:

placetypo %>% 
  group_by(Type) %>% 
  count(sort = TRUE)

# A tibble: 4 × 2
# Groups:   Type [4]
  Type      n
  <chr> <int>
1 TypeD    25
2 TypeB     8
3 TypeC     7
4 TypeA     4

In the following section, we will demonstrate how to construct and analyze networks of places in order to investigate the structure and dynamics of two-mode networks, taking Sino-American alumni networks as a specific case.

2.1 Creating Networks

As highlighted earlier, the result of place detection includes an edge list of places linked by sets (designated as “Edgelist”). We can take advantage of this list to build a network of places linked by universities (Sets) and its transposed network of universities (Sets) linked by places. This involves performing a dual projection, referred to as “reduction” in Figure 1, but applied to the places and sets rather than the students and universities.

To build a network of places linked by universities, we begin by creating an adjacency matrix⁶ from the edge list:

bimod <- table(Result1$Edgelist$Places, Result1$Edgelist$Set) 
PlacesMatrix <- bimod %*% t(bimod)
diag(PlacesMatrix) <- 0 

In essence, the chunk above creates a two-mode network representation using a contingency table (here bimod) and then computes the cross-product of this bimodal matrix with its transpose to create a square matrix (PlacesMatrix). Finally, it sets the diagonal elements of this matrix to \(0\). The resulting matrix may be used for analyzing relationships or patterns between places and sets in the context of the data stored in Result1$Edgelist. Below, we provide a detailed explanation of the code, line by line:

• table(Result1$Edgelist$Places, Result1$Edgelist$Set): Uses the table() function to create a contingency table (cross-tabulation) of the occurrences of each combination of values in the vectors Result1$Edgelist$Places and Result1$Edgelist$Set. The result, assigned to the object bimod, is a two-dimensional table.
• bimod %*% t(bimod): Performs matrix multiplication. The %*% operator is used for matrix multiplication, and t(bimod) transposes the matrix bimod. The result is a matrix called PlacesMatrix that represents the cross-product of bimod and its transpose.

Next, we use the graph_from_adjacency_matrix() function included in the igraph package to transform the matrix into a network of places linked by universities.

library(igraph)
Net1 <- graph_from_adjacency_matrix(adjmatrix = PlacesMatrix, mode="undirected", weighted = TRUE)

We apply the same method for building the transposed network of universities:

bimod2 <- table(Result1$Edgelist$Set, Result1$Edgelist$Places)
PlacesMat2 <- bimod2 %*% t(bimod2)
diag(PlacesMat2) <- 0

Net2 <- graph_from_adjacency_matrix(adjmatrix = PlacesMat2, mode="undirected", weighted = TRUE)

An alternative method of projection is provided below.

# Creating a network from the list of Places links
Net <- graph_from_data_frame(Result1$Edgelist, directed = FALSE)
# Transformation into a 2-mode network
V(Net)$type <- bipartite_mapping(Net)$type
# Projection
projNet <- bipartite_projection(Net, multiplicity = TRUE)
Net1 <- projNet$proj1  # Network of elements/places
Net2 <- projNet$proj2  # Network of sets (universities)

# Convert igraph objects into edge lists (not run in this session)
  # edgelist1 <- as_edgelist(Net1)
  # edgelist2 <- as_edgelist(Net2)
# Export edge lists and node lists as csv files (not run in this session)
  # write.csv(edgelist1, "edgelist1.csv")
  # write.csv(Result1_df, "nodelist1.csv")
  # write.csv(edgelist2, "edgelist2.csv")

2.2 Visualizing Networks

Let’s plot the network of places linked by universities:

plot(Net1, 
     vertex.size = 5, 
     vertex.color = "orange", 
     vertex.label.color = "black", 
     vertex.label.cex = 0.3, 
     main = "Network of places linked by universities")

In the network of places linked by sets (universities), isolated nodes refer to places with only one student who attended only one university, such as P200(1-1), which refers to N.E. Lurton, who studied exclusively at Benton University.

plot(Net2, 
     vertex.size = 5, 
     vertex.color = "light blue", 
     vertex.label.color = "black", 
     vertex.label.cex = 0.3, 
     main = "Network of universities linked by places")

In the network of sets (universities) linked by places, isolated nodes refer to universities attended by students who did not study at any other university.

The plot() function from the igraph package includes various arguments. The first argument is required, as it specifies the network object to be plotted. The other arguments are optional. In the above example, we specified the following arguments:

• vertex.size: Size of vertices (or nodes)
• vertex.color: Color of vertices
• vertex.label.color: Color of vertices labels
• vertex.label.cex: Size of vertices labels
• main: Title for the graph.

We can remove the labels to improve legibility and adjust the node sizes according to the number of students in each place, to add meaningful information:

plot(Net1, 
     vertex.size = Result1_df$NbElements,
     vertex.color = "orange", 
     vertex.label = NA,
     layout = layout_components,
     main = "Network of places linked by universities")

For the graph of universities linked by places, we can adjust the node sizes according to the number of students attending each university:

univ_count <- auc %>% 
  group_by(University) %>% 
  count()

plot(Net2, 
     vertex.size = univ_count$n/2, 
     vertex.color = "light blue", 
     vertex.label = NA, 
     layout = layout_components,
     main = "Network of universities linked by places")

As evident from the plots, the two networks are each made up of a large, densely connected component surrounded by a myriad of isolated nodes and smaller components, which refer to the singular curricula described in the previous section. To substantiate this preliminary visual exploration, it is recommended to turn to network metrics.

2.3 Applying One-Mode Network Metrics to Places and Sets

In network analysis, we usually distinguish between global metrics, which serve to characterize the overall structure of the network, and local metrics, which characterize the vertices and their relative position in the network.

summary(Net1)

IGRAPH 1f1bc77 UNW- 223 1606 -- 
+ attr: name (v/c), weight (e/n)

summary(Net2)

IGRAPH b868c8e UNW- 146 197 -- 
+ attr: name (v/c), weight (e/n)

edge_density(Net1)

[1] 0.06488102

edge_density(Net2)

[1] 0.01861124

count_components(Net1)

[1] 39

count_components(Net2)

[1] 39

components(Net1)$csize

 [1] 184   1   2   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
[20]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
[39]   1

components(Net2)$csize

 [1] 105   3   2   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
[20]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
[39]   1

table(E(Net1)$weight)

   1    2 
1594   12 

table(E(Net2)$weight)

  1   2   4 
190   6   1 

E(Net1)[weight > 1]

+ 12/1606 edges from 1f1bc77 (vertex names):
 [1] P003(1-4)--P016(1-3) P003(1-4)--P018(1-3) P003(1-4)--P026(4-2)
 [4] P003(1-4)--P121(1-2) P007(1-3)--P019(1-3) P007(1-3)--P085(1-2)
 [7] P015(1-3)--P031(2-2) P015(1-3)--P052(1-2) P016(1-3)--P018(1-3)
[10] P016(1-3)--P026(4-2) P017(1-3)--P072(1-2) P018(1-3)--P026(4-2)

E(Net2)[weight == 2]

+ 6/197 edges from b868c8e (vertex names):
[1] Columbia           --California           
[2] Columbia           --Pomona               
[3] Columbia           --Chicago              
[4] New York University--Pennsylvania         
[5] Pennsylvania       --Hawaii               
[6] Pennsylvania       --St. John's University

E(Net2)[weight == 4]

+ 1/197 edge from b868c8e (vertex names):
[1] Columbia--New York University

# components(Net1) (not run in this session)
# components(Net2) (not run in this session)

Net1MC <- induced_subgraph(Net1, vids = components(Net1)$membership == 1)
Net2MC <- induced_subgraph(Net2, vids = components(Net2)$membership == 1)

Degree2 <- degree(Net2MC, normalized = TRUE) 
Eig2 <- eigen_centrality(Net2MC)$vector 
Betw2 <- betweenness(Net2MC)
univ_metrics <- cbind(Degree2, Eig2, Betw2) 
univ_metrics_df <- as.data.frame(univ_metrics)

head(univ_metrics_df %>% arrange(desc(Degree2)))

                      Degree2      Eig2     Betw2
Columbia            0.3750000 1.0000000 2191.8655
Pennsylvania        0.1923077 0.5089454  695.0678
Chicago             0.1923077 0.4721450 1338.0047
Harvard             0.1730769 0.3653417 1020.2555
California          0.1346154 0.4282910  476.5158
New York University 0.1153846 0.6514840  343.6654

plot(Net2MC,
     vertex.color = "light blue",
     vertex.shape = "circle",
     vertex.size = degree(Net2MC)/2, 
     vertex.label.color = "black", 
     vertex.label.cex = degree(Net2MC)/100, 
     main = "Network of universities",
     sub = "The size of vertices represents their degree centrality.")

3.1 Finding Communities

To detect communities with the Louvain algorithm, we apply the cluster_louvain() function included in igraph. We continue to focus on the main component to avoid the detection of artificial clusters made up of isolated nodes:

lvc1 <- cluster_louvain(Net1MC)
lvc2 <- cluster_louvain(Net2MC)

Let’s inspect the results:

print(lvc1)

IGRAPH clustering multi level, groups: 7, mod: 0.52
+ groups:
  $`1`
   [1] "P001(1-4)"  "P004(1-4)"  "P007(1-3)"  "P012(1-3)"  "P014(1-3)" 
   [6] "P016(1-3)"  "P017(1-3)"  "P018(1-3)"  "P019(1-3)"  "P026(4-2)" 
  [11] "P030(3-2)"  "P034(2-2)"  "P035(2-2)"  "P036(2-2)"  "P037(2-2)" 
  [16] "P048(1-2)"  "P056(1-2)"  "P061(1-2)"  "P070(1-2)"  "P072(1-2)" 
  [21] "P085(1-2)"  "P088(1-2)"  "P091(1-2)"  "P098(1-2)"  "P101(1-2)" 
  [26] "P114(1-2)"  "P119(1-2)"  "P128(1-2)"  "P132(1-2)"  "P136(1-2)" 
  [31] "P138(1-2)"  "P149(15-1)" "P169(2-1)"  "P180(1-1)"  "P221(1-1)" 
  
  $`2`
  + ... omitted several groups/vertices

print(lvc2)

IGRAPH clustering multi level, groups: 8, mod: 0.49
+ groups:
  $`1`
   [1] "Columbia"                     "Haverford"                   
   [3] "New York University"          "General Theological Seminary"
   [5] "Trinity"                      "University of Edinburgh"     
   [7] "Pomona"                       "Colorado"                    
   [9] "Denver"                       "Stanford"                    
  [11] "National (Manila)"            "Philippine"                  
  [13] "Bucknell"                     "Crozen Theological Seminary" 
  [15] "Wake Forest"                  "North Central College"       
  [17] "Rochester"                    "Butler"                      
  + ... omitted several groups/vertices

The algorithm found 7 communities of places and 8 communities of universities. The modularity scores⁷ (“mod” result) are quite satisfactory (0.52 and 0.49, respectively). As shown in the results below, the size of communities ranges from 10 to 42 vertices in the network of places, and from 6 to 21 vertices in the network of universities.

table(sizes(lvc1))

10 20 25 26 35 42 
 1  1  1  2  1  1 

table(sizes(lvc2))

 6 10 11 13 14 16 21 
 1  1  1  1  2  1  1 

In the next section, we will show how to visualize the communities.

3.2 Visualizing Communities

V(Net1MC)$group <- lvc1$membership # create a group for each community
V(Net1MC)$color <- lvc1$membership # node color reflects group membership 

plot(lvc1, Net1MC, 
     vertex.label = NA,
     vertex.label.color = "black", 
     vertex.label.cex = 0.5, 
     vertex.size = 1.8,
     main = "Communities of places", 
     sub = "Louvain method")

V(Net2MC)$group <- lvc2$membership
V(Net2MC)$color <- lvc2$membership

plot(lvc2, Net2MC, 
     vertex.label = NA,
     vertex.label.color = "black", 
     vertex.label.cex = 0.5, 
     vertex.size = 3,
     main = "Communities of universities", 
     sub = "Louvain method")

3.3 Extracting Communities

The following code serves to retrieve the membership information contained in the results of community detection (lvc1$membership) in a coherent data frame. Additionally, we compute the size of communities and we join this data with the detailed description of the places.

place_clusters <- data.frame(lvc1$membership, lvc1$names) %>% 
  group_by(lvc1.membership) %>% 
  add_tally() %>% # add size of clusters
  rename(PlaceLabel = lvc1.names, cluster_no = lvc1.membership, cluster_size = n) %>%
  select(cluster_no, cluster_size, PlaceLabel)

place_clusters <- inner_join(x = place_clusters, y = Result1_df, by = "PlaceLabel") 


kable(head(place_clusters), caption = "Communities of places (first 6 places)") %>%
  kable_styling(bootstrap_options = "striped", full_width = TRUE, position = "left")

We follow the same method for extracting community membership in the network of universities:

univ_clusters <- data.frame(lvc2$membership, lvc2$names) %>% 
  group_by(lvc2.membership) %>%  
  add_tally() %>%
  rename(University = lvc2.names, cluster_no = lvc2.membership, cluster_size = n) %>%
  select(cluster_no, cluster_size, University)

kable(head(univ_clusters), caption = "Communities of universities (first 6 places)") %>%
  kable_styling(bootstrap_options = "striped", full_width = TRUE, position = "left")

In the following steps, we extract the communities of places as individual sub-networks.

gp1 <- induced_subgraph(graph = Net1MC, vids = V(Net1MC)$group == 1)  
gp2 <- induced_subgraph(Net1MC, V(Net1MC)$group == 2) 
gp3 <- induced_subgraph(Net1MC, V(Net1MC)$group == 3) 
gp4 <- induced_subgraph(Net1MC, V(Net1MC)$group == 4) 
gp5 <- induced_subgraph(Net1MC, V(Net1MC)$group == 5) 
gp6 <- induced_subgraph(Net1MC, V(Net1MC)$group == 6)
gp7 <- induced_subgraph(Net1MC, V(Net1MC)$group == 7)

Similarly, we extract the communities of universities.

gu1 <- induced_subgraph(Net2MC, V(Net2MC)$group == 1)  
gu2 <- induced_subgraph(Net2MC, V(Net2MC)$group == 2) 
gu3 <- induced_subgraph(Net2MC, V(Net2MC)$group == 3) 
gu4 <- induced_subgraph(Net2MC, V(Net2MC)$group == 4) 
gu5 <- induced_subgraph(Net2MC, V(Net2MC)$group == 5) 
gu6 <- induced_subgraph(Net2MC, V(Net2MC)$group == 6)
gu7 <- induced_subgraph(Net2MC, V(Net2MC)$group == 7)
gu8 <- induced_subgraph(Net2MC, V(Net2MC)$group == 8)

To illustrate the duality of place-based networks, we will plot the corresponding communities of places and universities to visually compare their structure.

3.4 Visual Comparisons

Using this method, three main categories of communities can be identified based on their topological structure.

a. Star-like communities are characterized by a prominent and highly influential university that attracts students from a wide range of smaller institutions. Columbia University serves as a prime example, having developed into a comprehensive institution with diverse academic offerings and exceptional resources for postgraduate research. Its reputation and extensive curriculum make it an attractive destination for students seeking a broad educational experience.

The two plots below compare the Columbia-centered community and the corresponding community of places. The hairball structure of the community of places is transposed into the star-like structure of the community of universities:

plot(gp1, 
     vertex.label = V(Net1MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = 0.5, 
     vertex.size = 5,
     main = "Columbia community (places)")

plot(gu1, 
     vertex.label = V(Net2MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = degree(gu1)*0.15, # label size proportionates to node degree (in cluster)*0.15
     vertex.size = degree(gu1)*1.5, # same for node size*1.5
     main = "Columbia community (universities)")

b. Chain-type communities can be observed in two scenarios. First, in specialized curricula like engineering and technical studies, institutions such as MIT and Purdue University form chains where student mobility is limited due to the specialized nature of their programs. Second, in cases when American students are dispersed across numerous peripheral institutions with little curricular coherence, chain structures also emerge, as illustrated by the Princeton community below:

plot(gp2, 
     vertex.label = V(Net1MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = 0.5, 
     vertex.size = 5,
     main = "Princeton Community (places)")

plot(gu2, 
     vertex.label = V(Net2MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = degree(gu2)*0.15, 
     vertex.size = degree(gu2)*1.5,
     main = "Princeton community (universities)")

c. Hybrid structures consist of two equally central institutions that have established their own networks of feeder colleges. These institutions not only exchange students but also complement or compete in terms of program offerings. For example, Cornell and the University of California attracted students who navigated between the two due to their strong programs in science, engineering, and agriculture:

plot(gp5, 
     vertex.label = V(Net1MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = 0.5, 
     vertex.size = 5,
     main = "Cornell Community (places)")

plot(gu6, 
     vertex.label = V(Net2MC)$id,
     vertex.label.color = "black", 
     vertex.label.cex = degree(gu6)*0.3, 
     vertex.size = degree(gu6)*2,
     main = "Cornell community (universities)")

For an in-depth analysis of university and place-based communities, please refer to my aforementioned paper (Armand 2024), where I demonstrate how the joint analysis of communities of places and universities can be used to examine patterns of student mobility within selected groups of universities.