Why is graph theory important

Together, these results demonstrate that our adaptation of graph theory is a viable quantitative methodology to examine group discussions.

Collaboration and small-group discussions form the foundation for many evidence-based instructional practices and are effective means of enhancing student learning in science, technology, engineering, and mathematics STEM.

Learning theories such as constructivism provide broad explanations for the theoretical basis of group discussions National Research Council, ; Chi, ; Chi and Wylie, Empirically, group discussions help students develop cognitive skills such as critical thinking Webb, b ; Gokhale, ; Bligh, , problem solving Heller et al.

Small-group discussions in STEM learning. Student discussions can be influenced by a number of factors, including group composition, sense of belonging, and values and behaviors related to collaborative activities. The dynamics and quality of these discussions can affect student outcomes, such as cognitive learning, development of process skills, affect, and persistence. In the existing literature, quality of small-group discussions is typically analyzed by discourse analysis.

In this study, we adapt graph theory methodologies to examine the dynamics of these discussions. Citations are available in the body of the text. The effectiveness of discussions depends on how the members of a group interact with one another, and many factors can influence group dynamics Figure 1 , left. Some of these factors are related to group composition, including academic preparedness Hillyard et al.

Other factors involve what students value and how they behave. Group discussions are only effective when students find the activities useful Blumenfeld et al. Similarly, prior experience and attitudes while working in groups Forrest and Miller, ; Hillyard et al. Community also plays an important role.

Positive or negative influences on group dynamics are affected by a strong sense of belonging Anderman, ; Freeman et al. In the literature, the substance of group discussions is commonly studied using qualitative methods, specifically discourse analysis Figure 1 , middle. Typical applications of discourse analysis in this area include understanding student comprehension, knowledge construction, and cognition King, ; Fall et al.

However, most of these methodologies capture group discussions only for short durations for in-depth qualitative analyses and have certain limits in tracking how the conversations progress over time in a quantifiable manner. The dynamics of how students interact and talk with one another in groups is at the crux of many different active-learning strategies and is also critical for equity for and inclusion of all students.

To understand how different factors contribute to group dynamics and how different interactions lead to different student outcomes, it is imperative to be able to quantify how students participate and engage in groups Figure 1. By quantifying how students interact and talk with one another in groups, we can identify factors that contribute to how marginalized and minoritized students may or may not be able to engage in groups.

Furthermore, understanding the dynamics of student group discussions will help elucidate the mechanisms by which different types of interactions contribute to different student outcomes.

Currently, there are not sufficient quantitative tools to examine the dynamics of student group discussions. In this paper, we adapt graph theory to track how students communicate with one another in groups by recording the order in which each participant talks and analyzing these talk-turn patterns in a quantitative manner. Our methodology is developed and tested through three iterations of data collection and two major refinements of the mathematical calculations.

Case studies are selected to demonstrate the potential patterns observed and highlight the utility of this methodology in biology education research. There are several learning theories that deal with the fundamental basis of how people learn. We focused on social constructivism because of its relevance to group learning, and we also used cultural-historical activity theory CHAT to understand how students interact to make a collaborative group effective. Social constructivism posits that learning is a social process, emphasizing how student interactions in a group or classroom setting contribute to how they learn, think, and converse within the academic community Hirtle, ; Adams, ; Powell and Kalina, Vygotsky postulated that people learn by social interaction, and Dewey believed that learners are part of a greater community that teaches and enriches all of its members Hirtle, From Vygotsky and Dewey, it can be said that an open environment where students are able to collaborate with one another is essential for knowledge building Powell and Kalina, This social process of learning forms the foundation of active-learning strategies, which have been shown to be effective across STEM disciplines and settings Freeman et al.

Social constructivists strive to provide an open environment for students to share their thoughts freely and to give students democratic control over their learning to foster a sense of deeper inquiry and learning Davydov, ; Hirtle, ; Adams, In this environment, instructors serve as facilitators in the discussions and provide scaffolds for students whenever necessary Davydov, ; Adams, ; Powell and Kalina, To truly understand learning in the social constructivist view, we need to examine how students interact with one another and with their instructors.

Active engagement with spoken or written language is an important medium for learning, according to the social constructivist perspective Hirtle, When students feel welcomed and their communication styles acknowledged, they are more willing to engage and get more out of activities in the classroom Hirtle, ; Powell and Kalina, In addition, a welcoming and inclusive environment allows students to freely contribute different perspectives and experiences, which can help enhance student understanding of the subject matter Davydov, ; Adams, ; Powell and Kalina, However, differences in communication styles can also bring another set of challenges, which may arise based on how students view other ethnicities and how willing they are to work with others Atwater, ; Powell and Kalina, To foster an inclusive classroom, it is imperative to be able to quantify how different students may or may not engage with the group learning environments, so we can understand the potential biases that are present, among other factors that contribute to an effective collaboration in the learning process.

CHAT is another theoretical framework relevant to student learning and especially articulates the connection between what people think and what people do Roth, ; Roth et al. Specifically, second-generation CHAT considers how the relationships among people the subjects , the activity the object , tools, rules, community, and division of labor can all affect the final outcome Figure 2 , and a core idea of CHAT is the interconnectedness of these various components Roth, ; Roth et al.

In the literature, CHAT has been used to observe student—student relationships and student—instructor relationships by examining the division of labor, the learning community, and the unwritten rules guiding these relationships Nussbaumer, Connections defined across the different components in CHAT cannot necessarily be easily seen directly Roth et al. Cultural-historical activity theory CHAT. In this study, we used CHAT to consider how the relationships among students the subjects , the learning activity the object , tools, rules, community, and division of labor in small groups can contribute to the final learning outcome.

CHAT emphasizes the interconnectedness of these various components. These connections are not always easily observable, thus necessitating a methodology that can quantify some of these connections. Specifically, we developed a methodology based on graph theory to quantify the division of labor, the interactions among students and peer facilitators in small groups the community , and potentially hidden rules that guide how different students may or may not engage with the activity the object.

We chose graph theory to model the order in which students talk in a group, which we consider a proxy for the dynamics of the discussion. Graph theory uses a set of mathematical principles and formulas to examine the relationships among objects of interest Zweig, In its simplest form, a graph consists of nodes and edges Godsil and Royle, : Nodes represent the objects of interest, and edges represent the connections between them Figure 3A. In our methodology, we model the participants of the group as nodes.

When one participant talks after another, an edge is connected between them, and we define such as an edge as a talk-turn. There are several interpretations for edges. An edge does not necessarily suggest that two participants talk directly to each other; in fact, when a participant talks, everyone else in the group may be listening, but only one person talks in the next turn. Thus, an edge only indicates that one participant talks after the other.

Relevant graph theory parameters. A In a graph, nodes represent the objects of interest, and edges represent the connections between them. The graph shown here contains five nodes and four edges. B Edges can be weighted, typically to represent frequency of some sort, or unweighted. C Graphs can be directed with edges pointing from one node to another or undirected with edges simply connecting two nodes. Directed graphs have more information, as they show how much reciprocation is present between a pair of nodes.

D Degree indicates the number of edges connected to a node. The degree of each node is labeled. E Three graphs with their associated densities: As more edges are added, density increases.

F Degree centralities of a graph with five nodes: The degree centrality is exactly the degree of each node. G The centralization of two graphs that both have five nodes: The left graph has a higher centralization, because it is more centralized on node 1, while the right graph is less centralized. Edges have additional important features. First, edges can be weighted, usually to present frequency Godsil and Royle, ; Figure 3B. We use edge weight to represent the number of times one participant talks after another participant, capturing the frequency of talk-turns between any two participants.

Second, edges may be directed pointing from one node to another or undirected simply connecting two nodes; Godsil and Royle, ; Figure 3C. In our methodology, the edges in a directed graph track the sequential order in which participants talk.

We used a directed graph rather than an undirected graph, because we can track the reciprocation between a pair of nodes; that is, if one person responds more after another person but not the other way around.

On the other hand, an undirected graph shows only that there was a talk-turn between the two nodes. Graphs can have many mathematical parameters, and we selected relevant parameters to capture information on the dynamics of group discussions Table 1.

Degree and density are related parameters dealing with the number of connections that nodes have with one another Figure 3, D and E ; here, these parameters represent how many participants talk after another participant.

Degree is a parameter of individual nodes and measures the number of edges connected to a node Zafarani et al. Density is a parameter for the entire graph and is the total number of edges in a graph normalized to the maximum number of possible edges Borgatti et al.

Density for a given graph ranges from 0 to 1 in value and is calculated as. TABLE 1. Graph theory parameters used in the development of our methodology. Nodes with higher degrees indicate participants who engage in talk-turns with or between more people. Graphs with higher density values indicate greater overall diversity in participants talking after one another; in other words, participants are talking after different people more often.

Centrality and centralization are another pair of related parameters for individual nodes and the entire graph respectively Figure 3, F and G. Centrality captures the notion that some nodes are more important to the connections of edges in a graph than others Zafarani et al.

Centrality can be estimated using a variety of methods that emphasize different interpretations for what an edge means in a graph. Many types of centrality deal with connections of edges beyond two nodes and are often used to examine the flow of information across many people.

In this study, we model talk-turns between two participants as the smallest unit of analysis; we also do not imply that information is flowing only from one participant to the next, as everyone in the group can be listening to the information.

Thus, degree centrality is the most appropriate, because it relies only on the degree of a node or the number of edges connected to a node. Degree centrality for a given node is calculated as. A node with high degree centrality means that the participant talks before and after many different people, which is another proxy for active participation. This parameter provides additional information to the frequency of talk-turns edge weights. While centrality is a parameter for individual nodes, centralization is the equivalent parameter for the entire graph and measures whether the graph is centered around a particular node Borgatti et al.

Similarly, we use degree centralization, because it does not involve edges beyond two nodes. Degree centralization for a given graph ranges from 0 to 1 and is calculated as. We use degree centralization to determine to what extent a discussion is dominated by its most active participant.

Finally, subgraphs are smaller graphs within graphs Godsil and Royle, We use subgraphs to determine highly connected subgroups within the larger group of participants based on edges and their relative weights Supplemental Material. High connectedness means that individuals talk more frequently after one another within the subgroup than after participants outside the subgroup.

Within the subgroup, participants may be willing to speak after one another or are more likely to contribute ideas among one another that could be expanded upon or responded to. A similar research methodology, social network analysis, has emerged in recent years in biology and physics education research Grunspan et al. However, social network analysis and graph theory are not the same, even though their names are often used interchangeably in the literature Zweig, Graph theory is a branch of mathematics that seeks to understand how different parameters and graphical structures are related to one another Zweig, , and social network analysis is a specific application of graph theory more focused on relating the properties of the graph to understand the flow of information and social capital, as well as the formation of beliefs and identities, within a group of people Knaub et al.

In this paper, we use graph theory to track the talk-turns among participants in small-group discussions rather than the flow of information in a social network Table 1. This study was conducted at a large, private, not-for-profit, doctoral university highest research activity , with an undergraduate profile that is 4-year, full-time, primarily residential, more selective, and lower transfer-in, as reported by the Carnegie Classification of Institutions of Higher Education McCormick and Zhao, We observed groups of introductory biology students tackling conceptual problems related to their course work in an optional, peer-led academic program Drane et al.

In this program, consistent groups of five to seven students meet weekly to work with peer facilitators who have previously excelled in the course Swarat et al. Our methodology was developed through three iterations of data collection based on observations of students solving problems in groups Figure 4. In the first iteration, qualitative notes and memos were written during observations to track the discussions.

Partial talk-turn data were included as part of the notes. In the second iteration, the relative physical positions of participants in each group were recorded in hand-drawn diagrams. Each talk-turn between any two participants was drawn as a line between them, and the number of talk-turns was tracked by tally marks. This resulted in undirected data for our graph theory calculations.

Three iterations of data collection. The third and final iteration combined both the earlier iterations and also recorded the order of talk-turns.

In addition to the hand-drawn diagrams for physical positions, talk-turn data were recorded in a question-or-response format in a spreadsheet, and qualitative notes and memos were written during observations. Each participant was assigned a number based on the initial order in which he or she first talked in the group. Whenever a participant talked, his or her number was recorded under either the question or response column, which resulted in directed data for our graph theory calculations.

For the purpose of this study, questions were nonrhetorical Smith et al. While we acknowledge that group discussions have complex discourse patterns not captured in this simple format, we wanted to include discourse moves as part of the methodology, so future studies can examine group discussions by combining quality discourse data and our quantitative methodology.

We examined how many questions and responses were provided by each participant peer facilitator and students in a group. Questions and responses per hour were calculated using the following formulas, and scatter plots were generated to visualize the talk patterns of participants. These plots especially allowed us to compare the behaviors of peer facilitator versus students within a group. For comparisons across groups, a normalized talk ratio was calculated based on a fair-share number of turns for each participant, assuming that all participants in the group talked for an equal number of turns.

Normalized talk ratio for a given participant was then calculated as the number of talk-turns by that participant divided by the fair-share number of turns in the group. From our third iteration of data collection with the question-and-response format, we defined an episode in the discussion as the number of talk-turns from a question to the last response immediately before the next question. We reasoned that a question was likely to indicate a new episode, especially in the initiation—reply—evaluation discourse pattern typically observed in a classroom Macbeth, , while acknowledging that many other scenarios may also occur, for example, a non sequitur response that leads to a new and productive direction or episode of the discussion.

Nonetheless, we wanted to establish and test a robust methodology that can handle episodes, a common feature in discourse analysis, for potential future studies. With this operationalized definition of episodes, we calculated the frequency of episodes in different lengths. For analysis in NodeXL Basic, data in the question-or-response format were converted into an edge list, which included participant pairs who engaged in talk-turns, with corresponding weights for each of the edges. Subgroups were identified using the Girvan—Newman algorithm, a hierarchical method designed for small groups Girvan and Newman, Our MATLAB script takes the talk-turn data in the question-and-response format and generates an edge list and a corresponding weight list for the edges.

These two lists serve as inputs for our R script, which uses the igraph package to calculate graph theory parameters that we define in the Methodological Framework section Kolaczyk and Csardi, All scripts and the source code at the time of publication are available online in the Supplemental Material.

We use a case study approach to highlight the potential utility of our methodology. Case studies are especially useful for two purposes: 1 to examine the range and variations that exist within a setting and 2 to probe particular instances that are problematic or unusual Case and Light, As such, the strength and value of case studies are not about generalizability; rather, case studies can provide insights as exemplars Flyvbjerg, Here, we selected three case studies that demonstrate outcomes in group dynamics that could be observed using our methodology.

Two cases were selected to contrast the extremes of talk-turn behaviors observed in discussions, and a third case was selected to highlight the existence of hidden subgroups. We used the question-and-response data to examine at the talk-turn behaviors of individual participants in groups, comparing peer facilitators with students and students with one another.

From the four groups observed in this iteration of data collection, we identified two extreme patterns Figure 5. First, using the question and response per hour data, we found that the peer facilitators in groups A and B were nearly indistinguishable from students in their respective groups Figure 5 , first row.

In these groups, the peer facilitators and students engaged in similar number question turns and response turns. For example, in group A, the peer facilitator had On the other hand, in groups C and D, the peer facilitators had distinct behaviors compared with students. These peer facilitators engaged in many more talk-turns compared with students in their groups and also had more question turns per hour compared with the peer facilitators in groups A and B.

For example, in group D, the peer facilitator had Characteristics of individual and group talk-turn behaviors. In the first row, questions and responses per hour are shown in scatter plots. In the second row, normalized talk-turn ratios are plotted in descending order for each individual in the group, with Person 1 being the peer facilitator.

Individuals are numbered based on the order in which they first talked. In the third row, histograms represent the distribution of episodes in one recorded session for each group. For the purpose of this study, episodes are defined as the number of talk-turns in between two questions.

To compare across groups more easily, we used the normalized talk ratio defined in the Methods section Figure 5 , second row. Consistent with the question and response per hour data, the peer facilitator in group A had a normalized talk ratio of 1. In contrast, the peer facilitator in group D had a normalized talk ratio of 3. Within each individual group, the peer facilitators had the highest normalized talk ratios. Data Structures. Operating System. Computer Network. Compiler Design. Computer Organization.

Discrete Mathematics. Ethical Hacking. Computer Graphics. Software Engineering. Web Technology. Cyber Security. C Programming. Control System. Data Mining. Data Warehouse. Javatpoint Services JavaTpoint offers too many high quality services. Some of them are given below: 1. Computer Science In computer science graph theory is used for the study of algorithms like: Dijkstra's Algorithm Prims's Algorithm Kruskal's Algorithm Graphs are used to define the flow of computation.

Graphs are used to represent networks of communication. Graphs are used to represent data organization. Graph transformation systems work on rule-based in-memory manipulation of graphs. Graph databases ensure transaction-safe, persistent storing and querying of graph structured data. Graph theory is used to find shortest path in road or a network. In Google Maps , various locations are represented as vertices or nodes and the roads are represented as edges and graph theory is used to find the shortest path between two nodes.

Electrical Engineering In Electrical Engineering , graph theory is used in designing of circuit connections. Linguistics In linguistics , graphs are mostly used for parsing of a language tree and grammar of a language tree. I loved connect-the-dots puzzles when I was a child, but I never would have guessed that there is anything useful about them. But graphs are actually very useful, and not just in various fields of mathematics.

Graphs are useful out here, in the real world! The usefulness of graphs lies in their ability to model many different situations. In our problem, we drew graphs where the vertices represented people and the edges represented friendships.

We talked about imagining actually walking on a graph. So, why not go further: the vertices of a graph could represent towns, and the edges the roads connecting them!

So graphs are a really handy way of representing road maps. And if you imagine a graph where vertices are cities and edges are roads, you might immediately be interested in the shortest distance between two cities, if you have to travel from one city to the other.

A roadmap is an example of a network. A network is just a bunch of interconnected objects. In a roadmap, our objects are cities, and they are connected by roads. If you have a network of things connected to other things, this network can be nicely visualized by using a graph. The things in your network can be represented as vertices, and the connections represented by edges.

And if a graph can model a network, then graphs can model all sorts of important things! For example, the internet is a network! If you think of websites as vertices, and edges telling you which websites are linked to which others, then the World Wide Web is really just a gigantic graph! How about Facebook? Friendships on Facebook can be easily modeled by graphs: each person is a vertex, and an edge connects two people if they are friends.

Researchers study social media for all kinds of reasons, and graph theory is behind much of that research. Since networks are everywhere, graph theory is everywhere, too.