AI & Graphs (Cambridge International A-Level Computer Science)

Graphs to aid AI

What is a graph?

• A graph is a set of vertices/nodes connected by edges/arcs

• In AI, graphs are used to model relationships, networks, and decision-making problems

• They can represent things like:

  • Routes between cities (pathfinding)
  • Connections in a neural network
  • State transitions in decision trees or planning

Types of graphs

Representation of graphs

• Graphs can be represented using:

  • Adjacency Matrix – ideal for dense graphs and fast lookup
  • Adjacency List – more space, efficient for sparse graphs (often used in AI search)

• Both can be used in AI to represent problem spaces such as:

  • Game boards
  • Map-based navigation
  • Resource allocation networks

Undirected, unweighted graph

• An undirected, unweighted graph is a type of graph where:

  • Edges have no direction , connections go both ways
  • All edges are equal, there are no weights or costs
  • It is represented with a symmetric adjacency matrix, where a 1 indicates a connection and a 0 means no connection

• In this example:

  • Each node is a town (e.g. Bolton, Dunkirk, Teesside)
  • A 1 in the matrix shows a connection exists between two towns
  • A 0 means no direct connection

• Example:

  • Bolton and Frogmore have a 1 at their intersection → they are connected
  • Dunkirk and Moulton have a 0 → no direct connection

• The matrix is symmetric, confirming that the graph is undirected

AI relevance

• Undirected, unweighted graphs are useful in AI for:

  • Exploring environments, e.g. mapping rooms in a building with equal-cost connections
  • Clustering in machine learning, where the presence of a connection implies similarity
  • Social network analysis, where mutual connections (friendships, links) are undirected

• These graphs are often used in:

  • Depth-first or breadth-first search
  • Graph traversal algorithms to explore all reachable nodes

Directed, unweighted graph

• A directed graph (or digraph) is one where edges have a direction
• This means connections go from one node to another
• Connections cannot be traversed in reverse unless another directed edge exists

• In this example:

  • Each node represents a town
  • Arrows show the direction of travel
  • A 1 in the matrix at (row X, column Y) means there is a directed edge from X to Y
  • A 0 means there is no direct connection

• For instance:

  • B → D is allowed (matrix value = 1)
  • D → C is not allowed (matrix value = 0)

• Because this matrix is not symmetric, it reflects the one-way nature of connections

AI relevance

• Directed, unweighted graphs are essential in AI for:

Undirected, weighted graph

• An undirected, weighted graph is a graph where:

  • Edges go both ways (no direction)

  • Each edge has a weight, which can represent:

    • Cost
    • Distance
    • Time
    • Risk

• If no connection exists between two nodes, the matrix contains ∞ (infinity)

• Nodes represent towns

• Edges represent roads, and weights represent distances or costs

• In the matrix:

  • matrix[B][D] = 15 → Bolton to Dunkirk is 15 units
  • matrix[B][C] = ∞ → Bolton and Cardiff are not directly connected

• The matrix is symmetric, confirming this is an undirected graph

AI relevance

• Undirected, weighted graphs are widely used in AI when:

• In AI, the weights are crucial for deciding the best route or option when multiple paths are available

Directed, weighted graph

• A directed, weighted graph is a graph where:

  • Edges have a direction (e.g. A → B but not B → A)

  • Each edge has a weight, such as:

    • Distance
    • Cost
    • Time
    • Probability

• If there is no edge, the value is ∞ (infinity)

• Nodes = Towns

• Edges = One-way roads

• Weights = Travel cost or distance

• The adjacency matrix:

  • B → D = 15 → there is a road from Bolton to Dunkirk costing 15 units
  • C → H = 17 → Cardiff to Hartlepool has a weight of 17
  • H → T = 31 but T → H = ∞ → you can get from Hartlepool to Teesside, but not the other way

• This matrix is not symmetric, confirming the graph is directed

AI relevance

• Directed, weighted graphs are critical in AI for: