NodeXL provides access to a powerful set of quantitative network metrics. Aggregate network metrics characterize the network as a whole and include graph density, diameter, reciprocated vertex pair ratio, number of connected components, etc. Vertex metrics also called centrality metrics identify important and unique individuals. They include degree, in-degree, out-degree, betweenness centrality, eigenvector centrality, closeness centrality, PageRank, and clustering coefficient. These metrics, along with other attribute data (i.e., data describing the people or connections) can be mapped onto visual properties to create meaningful network visualizations. NodeXL also provides text analysis features, time series analyses, and identifies top items such as those who are followed the most on Twitter. Network metrics are illustrated using the ABCD network, the Marvel Cinematic Universe network connecting movies- to-characters, and a Twitter Network surrounding the CSCS conference. Instructions on creating affiliation networks from a bimodal network are also provided.
Centrality metrics; Degree; In-degree; Out-degree; Betweenness centrality; Eigenvector centrality; PageRank; Closeness centrality; Clustering coefficient; Density; Network diameter; Connected component; Vertex pair ratio; Time series
When trying to understand networks, analysts often want to identify important vertices, locate subgroups, or get a sense of how interconnected a network is compared to other networks. Although visualization itself can help do this, it is often helpful to use the rich set of quantitative network metrics, also called network graph metrics, which have been developed by social network analysis researchers (Chapter 3).
Network graph metrics can describe an entire network, subgroups, or specific actors within a single network. Aggregate graph metrics such as network density can be used to systematically compare communities, helping analysts decide which communities are highly connected and which are sparse. Tracking aggregate graph metrics over time can determine the effectiveness of interventions on the network as a whole. For example, you would expect the total number of edges to grow, increasing the density of the graph, after a photo sharing activity designed to introduce people to those they don’t know.
Individual person-level metrics provide insights about a person’s position within the network, helping to identify important or “central” people. For example, network graph metrics help identify people who are bridge spanners or who are popular in a network. Once identified, analysts and managers can better know who to contact or influence or bring to the table when trying to implement new programs or gain broader understanding. Metrics can also be used to identify cliques or persistent social roles that show up in many communities. Understanding the mix of social roles that exist within a particular network can help analysts determine if they have a healthy mix of social types or who may be a good candidate to replace an outgoing leader.
NodeXL calculates several network graph metrics. Once calculated, you can use these metrics to change the visual display of your network graphs in powerful ways as shown in this chapter. You can also filter out vertices or edges based on network metrics as discussed later (Chapter 7).
To better understand the meaning of each graph metric, start by opening the ABCD network visualized in the last chapter (or download it from https://www.smrfoundation.org/nodexl/teaching-with-nodexl/teaching-resources/). This network was designed specifically to illustrate the differences between several key metrics. If you download the version online, the layout shown in the book will be reproduced since the vertex positions are locked in place - the Locked? column values are set to Yes (see Advanced topic: Using hidden layout columns in Chapter 4). It should also be set to an undirected network type. Instructions on importing a NodeXL file made on another device can be found at the end of Chapter 4.
To calculate graph metrics, first click on the Graph Metrics button on the Analysis section of the NodeXL ribbon. This opens the Graph Metrics dialog (Figure 6.1). Select the metrics you want to calculate by checking in the boxes next to them. Details about the metric that is selected (e.g., Vertex clustering coefficient) are shown in the box below (see Figure 6.1). Some metrics allow you to customize various options by clicking on the Options… button on the right-hand side. Check the boxes next to the metrics shown in Figure 6.1 and then click Calculate Metrics. Some of the graph metrics can take a while to calculate when working with large networks, so a status bar is used to show progress. NodeXL will create a new Overall Metrics worksheet and take you there to show summary information for the entire network. It also populates a set of Graph Metrics columns on the Vertices worksheet that shows vertex- specific metrics, such as centrality 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:
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.
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.
The degree of a vertex (sometimes called degree centrality) is a count of the number of unique edges that are connected to it. Fay has a degree of 6 because she is directly connected to 6 other individuals. In comparison, Kate has a degree of only 1 because she is connected to only one other person. If the edges represented strong friendship connections between employees at ABCD, we might say that Fay is the most popular person in the network and Kate is one of the least popular. If you were analyzing a directed graph, the single degree metric would be split into two metrics: (1) In-degree, which measures the number of edges that point toward the vertex of interest (i.e., number of people who have received endorsements from others), and (2) Out-degree, which measures the number of edges that the vertex of interest points toward (i.e., the number of people the person has endorsed). In the ABCD network, NodeXL only calculates degree since the network is specified as containing undirected ties.
Although popularity is important, it is not everything. Betweenness centrality is a measure that captures a completely different type of importance: the extent to which a certain vertex lies on the shortest paths between other vertices. In other words, it helps identify individuals who play a “bridge spanning” role in a network. Consider Ethan in the ABCD network. He is directly related to only four people (i.e., he has a degree of 4). Despite his relatively low degree, his position as a “bridge” between Ava (and indirectly all those who Ava is connected to) and the rest of the group may be of utmost importance. If, for example, information were passed from one person to another, Ethan and Ava would be vital for assuring that Dmitri, Liu, Camila, and Ben could communicate with the rest of the group. In fact, if either Ethan or Ava were removed from the network, those four individuals would be entirely disconnected from the other employees. Thus, Ava and Ethan have high betweenness centrality. In contrast, Dmitri and others on the edge of the network have a betweenness centrality of 0. Even Gabe, who has a degree of 5 and is in the center of the graph, has a relatively low betweenness centrality (6.5) because so many of his edges connect people who are already connected through others. In NodeXL, betweenness centrality scores are doubled for directed networks, though the “shortest paths” do not consider directionality in the calculation.
Another characteristic you may care about is how close each person is to the other people in the network. If information needed to flow through the network, some people would be able to get a message to all the other people relatively quickly (i.e., in few steps), whereas others may require many steps. Closeness centrality is a measure of the average shortest distance from each vertex to each other vertex. Specifically, it is the inverse of the average shortest distance between the vertex and all other vertices in the network. The formula is 1/(average distance to all other vertices). The inverse is used so that a higher closeness centrality indicates a more desirable centrality score (i.e., a shorter average distance to other vertices). For example, in the ABCD network, Ethan has the highest closeness centrality score, because he sits right in the “middle” of the network—not too far away from those in the top half of the network and not too far away from those in the bottom half of the network. In contrast, Dmitry, Liu, Camila, and Ben have the lowest closeness since they are so far removed from the majority of the other vertices. In NodeXL, closeness centrality assumes an undirected network, though it shows the same results for directed networks.
In many cases, a connection to a popular individual is more important than a connection to a lone individual. The eigenvector centrality network metric takes into consideration not only how many connections a vertex has (i.e., its degree), but also the centrality of the vertices that it is connected to. Intuitively, it considers not just “how many people you know,” but also “who you know.” For example, in the ABCD network, Gabe has the highest eigenvector centrality (0.169) because his degree is relatively high (5), but also because those he connects to have high eigenvector centrality scores (e.g., Fay, Ji-yoo, Ethan, Hassan, and Ishita). In contrast, Ava has the same degree (i.e., number of connections) as Gabe, but those she connects with don’t have high eigenvector centrality scores since they have so few connections. As a result, Ava has a low eigenvector centrality score (0.043). In NodeXL, eigenvector centrality assumes an undirected network, though it shows the same results for directed networks.
The PageRank centrality metric is best known as the core metric behind Google’s search engine [3]. It is related to eigenvector centrality, but is designed for directed networks such as the world wide web. PageRank includes three distinct factors that determine the ultimate values for each vertex: (1) The number of vertices that link to the target, (2) the PageRank centrality of the linking vertices, and (3) the link propensity of the linking vertices. Consider a specific vertex representing a webpage called PageX. According to factor 1, the PageRank of PageX will increase if more vertices (i.e., websites) link toward it (i.e., it has a high in-degree). According to factor 2, the PageRank of PageX will increase if those who link to it have high PageRank themselves. This means that links are not all created equal. On the web, a link from cnn.com will increase a webpage’s PageRank far more than a link from a local blogger with a small following. According to factor 3, the PageRank of PageX will increase if those who link to it don’t link to many other vertices. In other words, links coming from “selective” linkers (those who only link to a small number of vertices) are more valuable than those coming from “frequent” linkers (those who link to a large number of vertices). In NodeXL, PageRank assumes a directed network, though it shows the same results for both directed and undirected networks. This network metric is not useful for the undirected ABCD network, but is useful for other networks such as the wikipedia page-to-page directed network (Chapter 14).
In some cases, a person’s friends may be friends with each other. For example, Hassan's three friends Ji-yoo, Gabe, and Fay are all directly connected to one another, creating a clique. More generally, a clique or complete graph occurs when all vertices in a group are directly connected to each other. In other cases, a person’s friends may not be friends with one another. For example, none of Ava’s friends are connected to each other. The clustering coefficient measures how connected a vertex’s neighbors are to one another. More specifically, it is calculated as: (the number of edges connecting a vertex’s neighbors)/(the total number of possible edges between the vertex’s neighbors). For example, Ishita’s neighbors include Ethan, Gabe, and Ji-yoo. There are to edges connecting those three individuals (Ji-yoo to Gabe; Gabe to Ethan). However, there are three possible edges between them (those mentioned plus Ethan to Ji-yoo). This results in a clustering coefficient of 2/3 or 0.667 as shown in Figure 6.3. The value will always be between 0 and 1, since it is the percent of possible edges that are realized. It is the same formula as the overall network density, but only calculated on a subset of vertices.
Network metrics must be interpreted differently depending upon the nature of the network. So far, we have examined a traditional network where the vertices represent people, and the edges represent direct connections between those people. However, many interesting networks connect people to things they are affiliated with (e.g., clubs, wiki pages they have edited, Facebook groups, classes). To better understand these “affiliation networks” and the meaning of network metrics associated with them, you will explore the Marvel Cinematic Universe affiliation network. Download the raw data in the file named Marvel_Movie_to_Character_Raw.xlsx file found at https://www.smrfoundation.org/nodexl/teaching-with-nodexl/teaching-resources/. The network connects Marvel Universe movies to key characters that were in the movies. Data for the network was culled from the Marvel Cinematic Universe Wiki.2 Appearances in post-credit or deleted scenes are not included. As you will see, this bimodal network can be transformed into two unimodal networks: a character- to-character and a movie-to-movie network (see Advanced topic: Transforming a bimodal affiliation network into two unimodal networks).
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.
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.
The final example from this chapter will illustrate some of the metrics associated with directed networks that include textual data. Specifically, you will be analyzing the network of Twitter users who posted at least one tweet that included “cscw” (an acronym that currently stands for a community of researchers studying Computer-Supported Cooperative Work and Social Media). Tweets were gathered from September 1, 2018 until November 23rd, 2018, which included time both before and after the 2018 CSCW conference,3 which occurred November 3–7 in New York City. Even though tweets were not gathered until September 1, some older tweets are included, since they were retweeted or replied to during the data collection timeframe. In Chapter 11, you will learn how to import your own Twitter networks. For now, you can download the file CSCW_2018_Twitter_Raw.xlsx from https://www.smrfoundation.org/nodexl/teaching-with-nodexl/teaching-resources/.
The CSCW network is a good example of an EventGraph, or a “social media network diagram of conversations related to events, such as conferences” [5]. Such graphs can help make sense of the conversations around an event, helping to identify key individuals, subgroups, and general properties of the network compared to others. You will explore the graph metrics as part of this chapter and then use the same network in Chapter 7 to illustrate the value of filtering and grouping to bring clarity to a large network.
The CSCW network is also a good example of a multiplex network (Chapter 3), since it includes three types of edges: Mentions, Replies to, and Tweet. After importing the network, browse through the Edges worksheet. Notice the Relationship column, which specifies the type of edge. If a user Mentions another Twitter user, then Vertex1 will include the sender and Vertex2 will include the user who was mentioned. If a user Replies to another user's tweet, then Vertex1 will include the sender and Vertex2 will include the person being replied to. These are directed edges that “point” from Vertex1 to Vertex2. Otherwise, if a person posts a Tweet that is not connected to any other tweets, the same username will show up in the Vertex1 and Vertex2 column. Graphically, this creates a self-loop, which is a loop that starts and ends at the user's vertex. Additional columns on the Edges worksheet indicate the text of the tweet (Tweet), Tweet Date (UTC), Imported ID (a unique identifier for each tweet), and Edge Weight. It is important to remember that each row in the Edges worksheet does not necessarily map to a single tweet. For example, if UserA mentions UserB and UserC in a single tweet, then two rows will be added to the Edges worksheet. The Imported ID can be used to count the total number of unique tweets, as discussed later.
The Vertices worksheet includes a row for each Twitter user in the network, a profile image, and details about the Twitter user such as the number of Followers they have on Twitter as a whole. All of the extra data is pulled in from the Twitter API when using the Twitter importers described in Chapter 11. Additional worksheets and metrics that have been calculated are explained below.
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.
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.
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.
Social network analysis provides a set of powerful quantitative graph metrics for understanding networks and the individuals and groups within them. These include aggregate network metrics such as graph density, diameter, reciprocated vertex pair ratio, and number of connected components, which characterize the network as a whole. They also include vertex metrics related to networks such as degree, in-degree, out-degree, betweenness centrality, eigenvector centrality, closeness centrality, PageRank, and clustering coefficient that can be used to identify unique or important people within a network. These metrics can be mapped onto visual properties such as size and opacity to help more easily make sense of the data, as was shown for the ABCD network. Affiliation networks, such as the Marvel Cinematic Universe network connecting movies-to-characters, have unique properties and their metrics must be interpreted carefully. Visualizations can combine calculated metrics (e.g., degree, betweenness centrality) with other attribute data (e.g., movie ratings; opening weekend proceeds) to gain insights into networks. NodeXL also provides text analysis features, time series analyses, and identifies top items when working with rich datasets such as the CSCW Twitter Network.
The network metrics in NodeXL are widely used because they reveal important properties of individuals in a network [6–10]. However, their computation can be slow, so research efforts on improved algorithms (e.g., [11]), parallelization of execution using multiple processors, and the use of specialized graphic co-processors to speed computation are important. Improved centrality metrics for different types of graphs, such as bimodal and weighted graphs are also being actively explored (see [10] for initial attempts at some of these). The combination of natural language processing (i.e., text analysis) and social network analysis is providing promising results (e.g., [12]). Other metrics are regularly being created to help discover important vertices, edges, motifs, cycles, and other structural features, such as triangles, cliques, near-cliques, chains, holes, and more. Some are specific to certain platforms (e.g., Twitter [13]), while others are more generic.
3.129.26.108