Graph theory gives students a clear way to study relationships by turning people into nodes and connections into edges. In a school social network project, a class can model friendships, follows, study partners, or club memberships using a labeled graph. This matters because the same ideas are used to understand online platforms, disease spread, recommendation systems, and teamwork patterns.
A small network can reveal structure that is hard to see from a list of names alone.
Students can collect a small, ethical sample of data, draw the graph, and calculate measurements such as degree, density, centrality, and cliques. These measurements help identify highly connected students, tightly connected groups, and how information might travel through the network. For an undirected friendship graph, an edge means both people share a connection, while a directed follow graph uses arrows to show direction.
The goal is not to judge people, but to practice mathematical modeling and learn how network structure affects real-world systems.
Understanding Graph Theory and Social Networks Project
Start by deciding exactly what one connection means. A friendship survey can ask whether two people work together often, while a follow graph can record who chooses to follow whom. Do not mix several meanings in one graph.
A person who shares a class with someone is not automatically a friend or follower. Write a clear rule before collecting any information. Use a small group, such as volunteers in one club or a fictional dataset, so the diagram stays readable.
Remove names and use labels such as A, B, and C. Participation should be optional, and nobody should be pressured to reveal private relationships.
The way data is collected changes the result. If each person names up to three study partners, the network may look sparse partly because of that limit. If a survey asks people to name everyone they know, memory and popularity can affect answers.
Two students may describe the same relationship differently. Record these choices in the project report as limits of the model.
A graph is a simplified picture, not a complete description of a social group. It cannot show trust, conflict, time spent together, or the strength of a relationship unless the project deliberately collects that kind of data.
Degree gives a first measure of local connection, but it does not tell the whole story. A student with many links may connect only to one close group. Another student with fewer links may sit between separate groups.
This is where centrality becomes useful. Degree centrality focuses on the number of direct connections. Closeness centrality considers how few steps are needed to reach others.
Betweenness centrality tracks how often a node lies on shortest routes between other nodes. A node with high betweenness can act as a bridge. If that bridge is removed, messages may need a longer route or may not reach another group at all.
Density describes the overall amount of connection, yet it must be interpreted carefully. A network with ten people can have many more possible links than a network with five people. Comparing edge counts alone is therefore unfair.
Density compares the links that exist with the links that could exist. A high density can make information spread quickly, though it can also mean the same information circulates repeatedly within one group. Cliques show fully connected groups, but a clique is not proof of a best friendship group.
It only shows that every required link appears in the chosen data. Check calculations by counting edges carefully, avoid counting an undirected edge twice, and state whether arrows or lines were used. A good final diagram includes a key, readable labels, and a short explanation of what the measurements can and cannot support.
Key Facts
- A graph is written as G = (V, E), where V is the set of vertices or nodes and E is the set of edges.
- In an undirected graph, degree is the number of edges touching a node.
- In a directed graph, in-degree counts incoming arrows and out-degree counts outgoing arrows.
- Density for an undirected simple graph is D = 2E / (N(N - 1)), where N is the number of nodes and E is the number of edges.
- Density for a directed graph with no self-loops is D = E / (N(N - 1)).
- A clique is a group of nodes in which every pair of nodes is connected by an edge.
Vocabulary
- Node
- A node is a point in a graph that represents an object, such as a student profile in a social network.
- Edge
- An edge is a connection between two nodes, such as a friendship link or follow relationship.
- Degree
- Degree is the number of connections a node has in an undirected graph.
- Centrality
- Centrality is a measure of how important or influential a node is based on its position in the network.
- Clique
- A clique is a subgroup in which every member is directly connected to every other member.
Common Mistakes to Avoid
- Counting each undirected edge twice is wrong because a friendship edge between A and B is one connection, not two.
- Using the undirected density formula for a directed follow graph is wrong because directed graphs have more possible edges due to arrow direction.
- Calling any close-looking cluster a clique is wrong because every pair of nodes in the group must be directly connected.
- Treating high degree as the only kind of importance is wrong because a node can be important by connecting groups even if it has fewer total links.
Practice Questions
- 1 A friendship graph has 8 students and 10 undirected friendship edges. What is the density of the graph?
- 2 In a directed follow network, student A follows 5 people and is followed by 3 people. What are A's out-degree and in-degree?
- 3 A student with only 2 connections is the only bridge between two friend groups. Explain why this student might have high importance even without high degree.