Fruchterman-Reingold

The visualization of graphs is an area where aesthetics and functionality come together to provide comprehensible representations of abstract structures. One popular approach for graph visualization is the use of force-directed algorithms. Among these, the Fruchterman-Reingold algorithm stands out for its simplicity and its capacity to produce aesthetically pleasing layouts.

Introduction

Force-directed algorithms consider graphs as physical systems, where nodes behave like atomic particles and edges act like springs. The Fruchterman-Reingold algorithm uses this concept to model attractive and repulsive forces among nodes, aiming to reach an equilibrium that gives a pleasing display.

Forces in the Fruchterman-Reingold Algorithm

The algorithm essentially models two primary forces:

1. Attractive Force

This force acts between nodes that are connected by an edge. It's analogical to a spring force where nodes are attracted towards each other. The attractive force () between two nodes and at a distance is computed as:Where:

• is the Euclidean distance between nodes and .
• is the normalized vector pointing from node to node .
• is a constant determining the strength of the spring force.
• is the ideal distance between the two nodes.

2. Repulsive Force

This force acts between every pair of nodes, ensuring that nodes repel each other and the graph spreads out. This force can be imagined similar to how like-charged particles repel in a magnetic field. The repulsive force () between two nodes and at a distance is computed as:Where:

• is the Euclidean distance between nodes and .
• is the normalized vector pointing from node to node .
• is a constant determining the strength of the repulsive force.

Pseudocode of the Algorithm

class ForceDirectedLayout {
    graph: Graph
    C_rep: number
    C_spring: number
    l: number
    verticesPositions: Array of Vectors
    maxIterations: number
    thresholdForce: number
    iterations: number
    
    Initialize(graph: Graph) {
        // Store the graph
        // Set constants C_rep, C_spring, and l
        // Initialize verticesPositions with random positions
        // Set maxIterations, thresholdForce, and iterations to predefined values
    }

     calculateRepulsiveForce(u: Vertex, v: Vertex) -> Vector {
        // Get the coordinates of u and v
        // Calculate the direction vector from v to u
        // Calculate the distance between u and v
        // Return the repulsive force using the formula
     }

     calculateAttractiveForce(u: Vertex, v: Vertex) -> Vector {
        // Get the coordinates of u and v
        // Calculate the direction vector from u to v
        // Calculate the distance between u and v
        // Return the attractive force using the formula
     }

    calculateForce(u: Vertex) -> Vector {
        // Initialize a zero vector for repulsiveForces
        // For each vertex v in the graph:
        //     If u is not v:
        //         Add the repulsive force between u and v to repulsiveForces
        
        // Initialize a zero vector for attractiveForces
        // For each vertex v adjacent to u:
        //     If u is not v:
        //         Add the attractive force between u and v to attractiveForces
        
        // Return the sum of repulsiveForces and attractiveForces
    }

    calculateForces() -> Array of Vectors {
        // Initialize an empty array for forces
        // For each vertex in the graph:
        //     Calculate the net force acting on the vertex
        //     Append the net force to the forces array
        
        // Return forces
    }

    processOneStep() {
        // Calculate forces for all vertices
        // For each vertex in the graph:
        //     Calculate the new position by adding the force to the current position
        //     Update the position of the vertex in verticesPositions
        
        // Increment the iteration counter
        // Calculate the maximum force magnitude across all vertices
        
        // Return the maximum force magnitude
    }

    processAllSteps() {
        // Reset the iteration counter
        // Initialize a variable for the maximum force magnitude
        // Repeat until the maximum force magnitude is below the threshold or until the maximum iterations is reached:
        //     Calculate the maximum force magnitude using processOneStep
        
        // Return the final verticesPositions
    }
}

Visualizing the Algorithm: Playground

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