What Is a Network? Defining Nodes, Links, and Systems

Look around you. Connections are everywhere. The friends you play with at recess, the roads that connect your town to others, the websites you visit, even the neurons firing in your brain—all of these form intricate patterns of relationships. Scientists call these patterns networks. At its core, a network is a system of nodes (the points) and links (the connections between them). This simple idea—points and lines—unlocks a powerful language for understanding complexity in our world. In this chapter, we will define these building blocks, explore how we represent them, and see why networks are a universal tool across science and daily life.

What Exactly Is a Node?

A node (also called a *vertex*) is any distinct entity that can be connected to others. Nodes are the “what” in a network. They can be people, cities, computers, words, neurons, or even abstract concepts like ideas. The key is that each node is identifiable and can have relationships with other nodes.

Consider your classroom. Each student is a node. But nodes aren’t limited to people. In a transportation network, nodes are airports or train stations. In the internet, nodes are routers or servers. In a food web, nodes are species—like rabbits, foxes, and plants. Even in a family tree, each person is a node.

Nodes often have attributes—properties that describe them. A person node might have attributes like age, grade, or favorite color. A city node might have population or geographic coordinates. These attributes help us analyze networks beyond just connections. For example, we might ask: Do older students have more friends? (We’ll explore such questions later.)

Importantly, a node doesn’t have to be a single object. Sometimes we group things into a single node. In a school network, an entire club could be one node if we’re studying how clubs interact. This is called aggregation, and it depends on the question we’re asking. The flexibility of what counts as a node is part of what makes network science so widely applicable.

Links: The Bonds That Bind

If nodes are the “who” or “what,” then links (or *edges*) are the “how.” A link represents a relationship, interaction, or flow between two nodes. Links give a network its structure. Without links, we’d just have isolated points.

Links can be undirected or directed, a distinction that fundamentally changes a network’s behavior.

Undirected Links: Mutual Relationships

An undirected link is symmetric. If node A is connected to node B, then node B is connected to node A. It’s a two-way street. Think of a handshake or a friendship where both people consider each other friends. In an undirected network, the link has no arrow; it’s just a line.

Example: In a classroom where students pair up for a project, if Alice works with Bob, then Bob works with Alice. The collaboration link is undirected. Another example: a road between two cities. If you can drive from City X to City Y, you can usually drive back, so the road link is undirected (ignoring one-way streets for now).

Mathematically, an undirected link is represented as an unordered pair: {A, B}. The order doesn’t matter.

Directed Links: One-Way Streets

A directed link (or *arc*) is asymmetric. It goes from one node to another and may not reciprocate. We draw it with an arrow. If node A points to node B, it doesn’t necessarily mean B points back to A.

Examples abound:

• On social media, if Alice *follows* Bob, that’s a directed link from Alice to Bob. Bob might not follow Alice back.
• In a food web, energy flows from plants to herbivores to carnivores. The “eaten by” link is directed: a rabbit is eaten by a fox, but a fox is not eaten by a rabbit.
• In a citation network, when a research paper cites another paper, that’s a directed link from the newer paper to the older one.
• In a workflow, a task might depend on another task; the dependency is directed.

Directed links allow us to model hierarchies, flows, and influence. They make networks richer but also more complex to analyze.

Weighted Links: Strength of Connection

Links can also be weighted, meaning they carry a numerical value representing strength, capacity, frequency, or cost. A friendship might be strong or weak; a road might be a highway or a dirt path; a communication channel might have high or low bandwidth.

In a weighted network, each link has a number $w_{ij} \geq 0$ associated with it. If there’s no link, $w_{ij} = 0$. For undirected weighted networks, $w_{ij} = w_{ji}$; for directed, they can differ.

Example: In a fitness app, you might track how often you exercise with different friends. The link weight could be the number of joint workouts per month. A higher weight means a stronger connection in terms of shared activity.

Link Attributes and Multigraphs

Links themselves can have attributes beyond weight, such as type (e.g., “friendship,” “family,” “colleague”) or timestamp (when the link formed). Sometimes multiple links can exist between the same pair of nodes—for instance, two people might be both colleagues and siblings. This is called a multigraph. For simplicity, many network models collapse multiple links into a single weighted link, but it’s important to recognize the possibility.

Representing Networks: From Pictures to Data

To study networks, we need ways to record them. We can draw them, but computers require structured data. Two common representations are adjacency matrices and adjacency lists.

Visual Representation: The Network Diagram

The most intuitive way is a graph drawing: nodes as dots (often labeled) and links as lines (straight or curved). For directed links, we add arrows. For weighted links, we might make lines thicker or label them with numbers.

```mermaid graph LR A[Alice] -- Friends --> B[Bob] B -- Friends --> C[Charlie] A -- Follows --> D[Dave] D -- Follows --> A C -- Siblings --> B ```

*Figure 1.1: A mixed network with undirected (Friends, Siblings) and directed (Follows) links. Nodes represent people; link labels show relationship type.*

This visual is great for small networks, but as networks grow, drawings become messy—a problem called the “hairball” effect. That’s why we turn to matrices and lists.

Adjacency Matrix: The Grid of Connections

An adjacency matrix is a square table (matrix) where rows and columns represent nodes. The entry at row $i$, column $j$ indicates the link from node $i$ to node $j$.

For an undirected network, the matrix is symmetric: if $i$ is linked to $j$, then $j$ is linked to $i$, so $A_{ij} = A_{ji}$. For a directed network, it may not be symmetric.

Let’s build one for a tiny social network of four people: Alice, Bob, Charlie, and Dave. Suppose we have:

• Alice and Bob are friends (undirected).
• Alice follows Dave on social media (directed).
• Dave follows Alice back (directed).
• Bob and Charlie are siblings (undirected).
• Charlie has no other links.

We assign node indices: 1=Alice, 2=Bob, 3=Charlie, 4=Dave.

The adjacency matrix $A$ (directed version) looks like this:

| From\To | 1 (Alice) | 2 (Bob) | 3 (Charlie) | 4 (Dave) | |---------|-----------|---------|-------------|----------| | 1 (Alice) | 0 | 1 | 0 | 1 | | 2 (Bob) | 1 | 0 | 1 | 0 | | 3 (Charlie)| 0 | 1 | 0 | 0 | | 4 (Dave) | 1 | 0 | 0 | 0 |

Explanation:

• $A_{12}=1$: Alice (1) links to Bob (2) — they are friends, but since friendship is undirected, we also have $A_{21}=1$.
• $A_{14}=1$: Alice follows Dave.
• $A_{41}=1$: Dave follows Alice.
• $A_{23}=1$ and $A_{32}=1$: Bob and Charlie are siblings.
• All other entries are 0 (no link).

For undirected relationships, we could combine them into a single undirected matrix, but it’s common to represent undirected links as two directed ones for consistency.

**Advantages of adjacency matr