6.3.1 Overall graph metrics

Navigate to the Overall Metrics worksheet, which summarizes some of the key properties of the entire network (Figure 6.2). These metrics include the following:

• • Graph type. Undirected or directed.
• • Vertices. The number of total vertices (i.e., rows on the Vertices worksheet).
• • Unique edges. The number of unique edges found in the Edges worksheet.
• • Edges with duplicates. The number of repeated vertex pairs on the Edges worksheet. Duplicate vertex pairs may occur, as, for example in a Twitter network when person A mentions person B in multiple tweets. Duplicate vertex pairs are treated as a single edge for most metrics, since NodeXL metrics currently do not support weighted networks. The ABCD network does not include any duplicate edges.
• • Total edges. The number of total edges (i.e., rows on the Edges worksheet), which is the sum of the Unique Edges and Edges With Duplicates.
• • Self-loops. The number of edges that connect a vertex with itself. A self-loop occurs when the edge list includes the same exact name in the vertex 1 and vertex 2 columns on the Edges tab (i.e., a person is connected to themselves). This may happen when, for example, in an email list a person replies to his or her own email. Self-loops are represented visually in the graph pane by a circular edge that comes out of a vertex and returns to that same vertex. Metrics do not include self-loops in their calculations unless otherwise stated in the description of the metric (e.g., see degree, in-degree, and out-degree).
• • Reciprocated vertex pair ratio. This is only applicable for directed networks. It represents the percent of connected vertex pairs that are connected by directed edges pointing in both directions. It is calculated as: (the number of vertex pairs that are connected to each other in both directions)/(the number of vertex pairs that are connected by one or two edges). In a Twitter network, where edges represent Mentions, a high Vertex Pair Ratio indicates that most of the time when a user mentions a second user, the second user also mentions the first user.
• • Reciprocated edge ratio. This is also only applicable for directed networks. It represents the percentage of edges that are reciprocated (i.e., an edge that has a companion edge pointing in the opposite direction between the same two vertices). It is calculated as: (the number of edges that are reciprocated)/(the number of total edges). While this value is correlated with the Reciprocated Vertex Pair Ratio, it is not the same. It will typically be higher because a single reciprocated vertex pair consists of two reciprocated edges, making the numerator twice as high for the reciprocated edge ratio. While the denominator is also higher for the reciprocated edges calculation, it won’t be double that of the reciprocated vertex pair ratio.
• • Connected components. The number of connected components (i.e., clusters of vertices that are connected to each other but separate from other vertices in the graph). In the ABCD network there is only one connected component because you can get from one vertex to all other vertices. If Ava was not connected to Ethan, then there would be two connected components (see Figure 6.3).

  
  

• • Single-vertex connected components. The number of isolated vertices that are not connected to any other vertices in the graph. There are no isolated vertices in the ABCD network. If Dmitri was not connected to Ava, he would not be connected to anyone in the network and would then become a single-vertex connected component.
• • Maximum vertices in a connected component. The number of vertices in the connected component with the most vertices. This is equal to the number of vertices (13) in the ABCD network, because they are all part of the only connected component.
• • Maximum edges in a connected component. The number of edges in the connected component with the most edges. This is equal to the number of edges (18) in the ABCD network, because they are all part of the only connected component.
• • Maximum geodesic distance (diameter). The geodesic distance is the length of the shortest path between two people. If you think of the edges as roads and the vertices as houses, the geodesic distance would be the number of roads someone must take to get from one house to another, assuming that the person is traveling on the shortest path possible. The maximum geodesic distance, or diameter of a network, is the largest geodesic distance in the network, or the distance between the two vertices that are farthest from each other. In the ABCD network, this value is 4. For example, the shortest path between Liu and Kate is 4; similarly the shortest path from Camila to Ji-yoo, Hassan, Matt, and Kate is also 4. All other geodesic distances are smaller. For example, the shortest path between Gabe and Fay is 1.
• • Average geodesic distance. This is the average length of the geodesic distances between all pairs of vertices. It gives a sense of how “close” community members are from one another. For example, in the ABCD network, a value of 2.22, suggesting that many people in the network know others directly or through a friend of a friend. Interestingly, many large social networks retain a relatively small average geodesic distance. For example, the Facebook network showed an average geodesic distance of only 4.57 with over 1.59 billion Facebook users in 2016.¹
• • Graph density. The graph density is a number between 0 and 1 that indicates the percentage of possible edges that are realized. It is a measure of how interconnected the vertices are in the network. The specific formula for calculating graph density is: (number of actual edges in the network)/(number of possible edges in the network). For undirected graphs the numerator is multiplied by 2. The number of possible edges in the network is based on the total number of vertices (n) in the network. Specifically, it is n*(n − 1). For example, the numerator for the ABCD network is 36 (i.e., 2*18) since it is an undirected network with 18 edges. The denominator is 156 (i.e., 13*12) since there are 13 vertices. Thus, the Graph Density is 0.23 (36/156). If more of the 18 employees became connected, this number would increase. Larger social networks tend to have lower graph densities, all things being equal, so be careful comparing this metric across different networks.
• • Modularity. The modularity metric is only calculated when working with subgroups, which are discussed in Chapter 7. Modularity measures how distinct different subgroups of vertices are from the rest of the network [1]. It can be used to measure the “quality” of the separation of vertices into subgroups. Networks with high modularity have dense connections between vertices that are part of the same subgroup (i.e., module), but sparse connections between vertices that are part of different subgroups. More specifically, modularity is the fraction of within-group edges minus the expected fraction of edges if edges were distributed at random. It will be a value between − 1 and 1, where positive values indicate that the number of within-group edges is higher than the expected number of edges based on chance.
• • NodeXL version. Indicates the version of NodeXL in use when metrics were calculated.

In addition, a frequency chart is created for each of the possible vertex-specific graph metrics. These frequency charts are particularly helpful when analyzing large networks. Some basic statistics about the metric distributions are shown under the charts (minimum degree, maximum degree, average degree, and median degree). These help characterize the entire networks and allow for comparisons over time or across networks.

6.3.2 Vertex-specific metrics

The different vertex-specific metrics, also called centrality metrics, help identify who is “important” or “central” to a network. Of course, people are important in different ways. Some may have the most direct connections, while others may be important bridge spanners who connect otherwise disparate parts of the network. Each centrality metric captures a different aspect of importance as described below.

To see the vertex-specific metrics navigate to the Vertices worksheet. You will see the new Graph Metrics columns, which can be hidden later if desired by unchecking Graph Metrics from the Workbook Columns button on the NodeXL ribbon. Each value relates directly to one of the vertices. For example, row 4 shows the graph metrics that are specific to Ava (Figure 6.3).

Vertex metrics can be mapped onto visual attributes (Figure 6.3), which you can recreate by using the Autofill Columns feature found in the NodeXL Visual Properties menu ribbon. The graph legend shows that degree (1–6) is mapped to size and betweenness centrality is mapped to opacity. Edge weight and opacity are also mapped to Shared_Connections (see Chapter 5). In addition, eigenvector centrality is mapped to the tooltip (see Ethan’s score in Figure 6.3) and the labels are set to Vertex and positioned so they don’t cross edges. A description of each metric and how it relates to the ABCD Network are provided below.

It is often useful to sort the spreadsheet columns based on graph metrics. For example, the rows in Figure 6.3 are sorted based upon data in the Degree column as indicated by the downward pointing arrow inside the Degree drop-down menu.

6.4.1 Visualizing and interpreting metrics in a bimodal network

Start by looking at the available data on the Edges and Vertices worksheet. On the Edges worksheet, Vertex 1 includes the names of movies and Vertex 2 includes the names of key characters that appeared in those movies. The Vertex worksheet includes additional data for each vertex (see Figure 6.4) including the Type and corresponding Type_Code (1 is a character and 0 is a movie), Release_Date, Phase (a phase is a set of movies related to each other thematically and chronologically), IMDB_Score (average rating out of 10), Metascore (average rating out of 100), US_Opening (millions of dollars made in the opening weekend in the United States), Worldwide_Opening (millions of dollars made in the opening weekend worldwide), and URL (link to the IMDB page for the movie). Calculate the same Graph Metrics as you did for the ABCD network (Figure 6.1) since this network is also an undirected network.

Before interpreting the graph metrics, create a more informative network visualization, such as the one shown in Figure 6.5. Because this is a bimodal network, it is important to make movies and characters easily distinguishable. To address this, use the Autofill Columns feature with values shown in Figure 6.5. This will make movies solid squares, while characters remain disks. Additionally, set the color based on Phase (i.e., clusters of related movies), size based on eigenvector centrality, and add vertex labels (see Figure 6.5). After you reposition the vertices for graph readability, the bimodal network can be understood much better.

Even with a clear visualization, examining the graph metrics can help highlight important vertices. You can sort on different network metrics to identify important characters or movies. Although the metrics are calculated in the same manner as they are calculated for unimodal networks, because it is a bimodal network, the interpretation is different.

Degree has the most intuitive interpretation. For example, Ant-Man (see row 3 in Figure 6.4) has a degree of 3, which means he is in 3 movies. In contrast, Avengers: Infinity War has a degree of 24 (see row 7 of Figure 6.4), indicating that there were 24 key characters in that movie. Sorting on Degree is an easy way to examine which movies include the most key characters (Avengers: Infinity War; Captain America: Civil War; Avengers: Age of Ultron; Avengers), and which characters were in the most movies (Iron Man; Black Widow; Captain America; War Machine; Thor; Pepper Potts).

Other metrics draw attention to different ways that movies or characters are important. As expected, Avengers: Infinity War has the highest betweenness, closeness, and eigenvector centrality given that it was designed to include all key characters from prior movies. However, these centrality metrics highlight unique positions in the network. For example, Ant-Man shows up with a very high betweenness centrality, because he is the only connection to the Ant-Man 1 and Ant-Man and the Wasp movies and their corresponding characters. Closeness centrality and eigenvector centrality both rank Iron Man and Black Widow highly due to the fact that they are connected to so many of the key movies with many characters. Interestingly, several movies with six or more key characters (e.g., Black Panther, Guardians of the Galaxy, Thor: The Dark Underworld) have fairly low eigenvector and closeness centrality scores because they are connected to characters who do not show up in many other movies. Because this is an affiliation network, no movies directly connect to other movies, and no characters directly connect to other characters. As a result, the clustering coefficient is equal to 0 for all vertices.

6.4.2 Mapping graph metrics to X and Y coordinates

In most layouts, the exact location of the vertices is not meaningful; only their position relative to one another has meaning. However, you may want to map network graph metrics, or other attribute data, to X and Y coordinates to visualize how two metrics interact with one another. Other metrics can be used to adjust visual properties, making it possible to display additional dimensions. For example, Figure 6.6 maps the movies onto the X and Y coordinates based on the degree and betweenness centrality respectively, using color and size to indicate Phase and IMDB score.

To recreate Figure 6.6, use the Autofill Columns feature. First, set the Vertex Label to Vertex so that the name of each character will be shown. Next set Vertex X to Degree, Vertex Y to Betweenness Centrality, making sure to check the box that says Use a logarithmic mapping in the Options for both metrics. Set the Vertex Size to IMDB_Score (range from 1.5 to 20). Set Vertex Shape to Type_Code similar to the prior graph, so movies show up as a square. To only show movies, and not characters, make the Vertex Visibility based on Type_Code with the Options as shown in Figure 6.6. Finally, navigate to the Edges worksheet and enter Hide into all of the Edge Visibility cells. This will hide all of the edges from the graph, making it far more readable. You can make the Legend and Axes visible using the Graph Elements drop-down menu in the NodeXL Ribbon. Try creating a similar network showing the characters by changing the Vertex Visibility options to display characters instead of movies.

6.5.1 Calculating and interpreting directed network metrics

In contrast to the small networks you have examined so far, many social media networks include hundreds or thousands of Vertices. In such cases, graph metrics become particularly important since initial visualizations obscure much of the data. Furthermore, network metrics can be used to filter out less important people as described in Chapter 7. The symbiotic relationship of network metrics and network visualization is extremely powerful, though it is often not used to its full potential.

Because it can take a long time to calculate graph metrics for a network of this size, the file that you downloaded has already run the relevant metrics. Figure 6.7 shows the Graph Metrics settings that were chosen. Options dialog metrics are displayed to indicate specific settings that were added. For example, the Overall Metrics options dialog was used to add Relationship as an edge type (Figure 6.7). This in turn, creates new totals for each edge type (Mentions, Replies to, and Tweet) on the Overall Metrics worksheet (Figure 6.8) with counts of the number of edges for each. Notice the Overall Metrics also includes many metrics that we haven't yet seen. For example, it shows the Number of Edge Types as 3 (i.e., Mentions, Replies to, Tweets). It also shows the Reciprocated Vertex Pair Ratio and the Reciprocated Edge Ratio since it is a directed network. There are 278 self-loops (the same number as there are Tweet edges as expected). There are also many different connected components (106), most of which are single-vertex connected components (65).

Because it is a directed network, all directed network metrics are chosen. Additionally, some undirected network metrics are chosen, such as eigenvector, betweenness, and closeness centrality. It is common for analysts to calculate such metrics, even for directed networks, but the interpretation of them is not exact. For example, betweenness centrality will still identify “bridge spanners,” but they may play that role because many disparate users mention them, or because they mention many disparate users. If you are looking for influencers, then identifying people with high In-Degree or PageRank, both of which are directed metrics, is more useful than identifying people with high Out-Degree or non-directed metrics such as Betweenness Centrality that may be driven by a person’s outbound links.

Two metrics not yet examined include Edge Reciprocation and Vertex reciprocated vertex pair ratio. Notice on the Edges worksheet there is a column called Reciprocated?. If the value is Yes, then there is another edge that exists with the two users’ positions flipped. For example, there is a row where acm_cscw Mentions fcalefato. There is also a row where fcalefato Mentions acm_cscw. As a result, in both of those rows, the Reciprocated? column shows a Yes. A related metric on the Vertices worksheet is shown in the Reciprocated column. This shows the percent of vertex pairs that are reciprocated. It helps identify individuals who are primarily involved in conversations, since the people they reply to or mention also reply to or mention them. For example, some users such as farbandish (0.366), niloufar_s (0.361), and morganklauss (0.409) all had high values because they were actively participating in conversations. Not surprisingly, they also had both high In-Degree and Out-Degree. In contrast, individuals such as katestarbird (0.005) and snaglee2401 (0.027) have low reciprocated edges, in this case because they were mentioned by many users (i.e., had high In-Degree), but mentioned or replied to relatively few (i.e., had low Out-Degree). In general, the more popular someone is, the more difficult it is to have high reciprocation scores. A good example of this is the acm_cscw account, which worked hard to mention and reply to 77 different users (i.e., Out-Degree is 77). However, because they were mentioned or replied to 380 different times, their reciprocation percent is relatively low (0.129). This illustrates the importance of looking holistically at different metrics to fully understand a network.

6.5.2 Examining top items output

When Top Items metrics were calculated, a new worksheet called Top Items was created (Figure 6.9). This includes the metrics that were indicated in the Top Item Metrics Options dialog (Figure 6.7). The first list indicates the users in the network with the highest number of overall Twitter Followers. These can be thought of as global influencers. The second and third list show the top 10 individuals based on In-Degree and PageRank, which help identify the local influencers, or people that the CSCW network is frequently mentioning and retweeting. This includes the official cscw account (acm_cscw), researchers (e.g., katestarbird, informor), academic departments (vt_cs), and topics of discussion (warcraft, the account for the game World of Warcraft which was presented on). Additional lists can be created in the options dialog (Figure 6.7) for items such as the most common hashtags, URLs, words, people replied to or mentioned, or most active tweeters.

6.5.3 Examining time series output

When Time Series metrics were calculated, the results were put into a new Time Series worksheet that includes a pivot table and associated graph (Figure 6.10). The graph shows the total number of unique tweets after they have been bucketed into days (since Days was chosen in the Time Series option box shown in Figure 6.7). Remember the total number of unique tweets is not the same as the total number of edges, since edges are often duplicates (e.g., if user1 mentions user2 and user3 in the same tweet, then two edges are created). However, since the Unique edges by this column was set to Imported ID (Figure 6.7), the graph and corresponding data represent unique tweet counts as desired.

Since Add a slicer for was set to the Relationship column (Figure 6.7), a filtering box is displayed. It allows you to filter based on Mentions, Replies to, and Tweets (Figure 6.10). If you click on one of the types of Relationship, it will filter the graph to only show tweets of that type. You can also choose multiple types. You can add different slicers (Figure 6.7) to examine other factors, such as location or time zone.