Graphology metrics

Miscellaneous metrics to be used with graphology.

Installation

npm install graphology-metrics

Usage

Graph metrics

• Density
• Diameter
• Extent
• Modularity
• Simple size
• Weighted size

Node metrics

• Eccentricity
• Weighted degree

Edge metrics

• Disparity
• Simmelian strength

Centrality

• Betweenness centrality
• Edge betweenness centrality
• Closeness centrality
• Degree centrality
• Eigenvector centrality
• HITS
• Pagerank

Layout quality metrics

• Connected Closeness
• Edge Uniformity
• Neighborhood Preservation
• Stress

Graph metrics

Density

Computes the density of the given graph. Note that multi variants can exceed 0, as it is also the case when considering self loops.

import {density} from 'graphology-metrics/graph/density';

// Passing a graph instance
const d = density(graph);

// Passing the graph's order & size
const d = density(order, size);

// Or to force the kind of density being computed
import {
  mixedDensity,
  directedDensity,
  undirectedDensity,
  multiMixedDensity,
  multiDirectedDensity,
  multiUndirectedDensity
} from 'graphology-metric/graph/density';

const d = undirectedDensity(mixedGraph);

// If you need to chose the kind of density dynamically
import {abstractDensity} from 'graphology-metric/graph/density';

abstractDensity('directed', true, 10, 24);

Arguments

Either:

• graph Graph: target graph.

Or:

• order number: number of nodes in the graph.
• size number: number of edges in the graph.

Abstract version arguments

Either:

• type string: type of density to compute (directed, undirected or mixed).
• multi boolean: whether to compute density for the multi of simple case.
• graph Graph: target graph.

Or:

• type string: type of density to compute (directed, undirected or mixed).
• multi boolean: whether to compute density for the multi of simple case.
• order number: number of nodes in the graph.
• size number: number of edges in the graph.

Diameter

Computes the diameter, i.e the maximum eccentricity of any node of the given graph.

import diameter from 'graphology-metrics/graph/diameter';

const graph = new Graph();
graph.addNode('1');
graph.addNode('2');
graph.addNode('3');
graph.addUndirectedEdge(1, 2);
graph.addUndirectedEdge(2, 3);

diameter(graph);
>>> 2

Arguments

• graph Graph: target graph.

Extent

Computes the extent - min, max - of a node or edge’s attribute.

import {nodeExtent, edgeExtent} from 'graphology-metrics/graph';
// Alternatively, to load only the relevant code:
import {nodeExtent, edgeExtent} from 'graphology-metrics/graph/extent';

// Retrieving a single node attribute's extent
nodeExtent(graph, 'size');
>>> [1, 34]

// Retrieving multiple node attributes' extents
nodeExtent(graph, ['x', 'y']);
>>> {x: [-4, 3], y: [-34, 56]}

// The same for edges
edgeExtent(graph, 'weight');
>>> [0, 5.7]

Arguments

• graph Graph: target graph.

• attributes _string

  array_: single attribute names or array of attribute names.

Modularity

Computes the modularity, given the graph and a node partition. It works on both directed & undirected networks and will return the relevant modularity.

import modularity from 'graphology-metrics/graph/modularity';

// Simplest way
const Q = modularity(graph);

// Custom node partition
const Q = modularity(graph, {
  getNodeCommunity(node, attr) {
    return attr.customPartition;
  }
});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • getNodeCommunity ?string|function community: name of the node community attribute or getter function.
  • getEdgeWeight ?string|function weight: name of the edges’ weight attribute or getter function.
  • resolution ?number: resolution parameter (γ).

Simple size

Computes the simple size of a given graph, i.e. its number of edges if we consider the graph simple, even if it has multiple edges between pairs of nodes.

import {simpleSize} from 'graphology-metrics';
// Alternatively, to load only the relevant code:
import simpleSize from 'graphology-metrics/graph/simple-size';

const graph = new MultiGraph();
graph.mergeEdge(1, 2);
graph.mergeEdge(1, 2);
graph.mergeEdge(4, 3);
graph.mergeUndirectedEdge(5, 6);

simpleSize(graph);
>>> 3

Weighted size

Computes the weighted size, i.e. the sum of the graph’s edges’ weight, of the given graph.

import weightedSize from 'graphology-metrics/graph/weighted-size';

const graph = new Graph();
graph.mergeEdge(1, 2, {weight: 3});
graph.mergeEdge(1, 2, {weight: 1});

// Simplest way
weightedSize(graph);
>>> 4

// With custom weight attribute
weightedSize(graph, 'myWeightAttribute');
>>> 4

// With custom getter
weightedSize(graph, (_, attr) => attr.importance);

Arguments

• graph Graph: target graph.
• getEdgeWeight ?string|function weight: name of the weight attribute or getter function.

Node metrics

Weighted degree

Computes the weighted degree of nodes. The weighted degree of a node is the sum of its edges’ weights.

import {
  weightedDegree,
  weightedInDegree,
  weightedOutDegree,
  weightedInboundDegree,
  weightedOutboundDegree,
  weightedUndirectedDegree,
  weightedDirectedDegree
} from 'graphology-metrics/node/weighted-degree';

// To compute weighted degree of a node
weightedDegree(graph, 'A');

// To use a custom weight
weightedDegree(graph, 'A', function (_, attr) {
  return attr.importance;
});

Arguments

• graph Graph: target graph.
• node any: desired node.
• getEdgeWeight ?string|function: name of the edge weight attribute or getter function.

Eccentricity

Computes the eccentricity which is the maximum of the shortest paths between the given node and any other node.

import eccentricity from 'graphology-metrics/node/eccentricity';

graph.addNode('1');
graph.addNode('2');
graph.addNode('3');
graph.addNode('4');
graph.addUndirectedEdge(1, 2);
graph.addUndirectedEdge(2, 3);
graph.addUndirectedEdge(3, 1);
graph.addUndirectedEdge(3, 4);

eccentricity(graph, 3) >> 1;

Arguments

• graph Graph: target graph.
• node any: desired node.

Edge metrics

Disparity

Function computing a score for each edge which is necessary to apply a “disparity filter” as described in the following paper:

Serrano, M. Ángeles, Marián Boguná, and Alessandro Vespignani. “Extracting the multiscale backbone of complex weighted networks.” Proceedings of the national academy of sciences 106.16 (2009): 6483-6488.

Note that this metric requires a weighted graph or will return a useless result.

Beware, the results must be interpreted thusly: a lower score means a more relevant edge, as is intuited in the paper’s formulae. This means you can prune edges that have a score greater than a given threshold, as a statistical test. Some other implementations might differ in that they offer the opposite intuition (i.e. greater score = more relevant edge).

import disparity from 'graphology-metrics/edge/disparity';

// To compute strength for every edge:
const disparities = disparity(graph);

// To directly map the result onto edge attributes (`disparity`):
disparity.assign(graph);

// Using custom weights
disparity.assign(graph, {getEdgeWeight: (_, attr) => attr.importance});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • edgeDisparityAttribute ?string disparity: Name of the disparity attribute to assign.
  • getEdgeWeight ?string|function weight: Name of the edge weight attribute or getter function.

Simmelian strength

Function returning the simmelian strength, i.e. the number of triangles an edge is part of, of all the edges in the given graph.

import simmelianStrength from 'graphology-metrics/edge/simmelian-strength';

// To compute strength for every edge:
const strengths = simmelianStrength(graph);

// To directly map the result onto edge attributes (`simmelianStrength`):
simmelianStrength.assign(graph);

Centrality

Betweenness centrality

Computes the betweenness centrality for every node.

import betweennessCentrality from 'graphology-metrics/centrality/betweenness';

// To compute centrality for every node:
const centralities = betweennessCentrality(graph);

// To directly map the result onto nodes' attributes (`betweennessCentrality`):
betweennessCentrality.assign(graph);

// To directly map the result onto a custom attribute:
betweennessCentrality.assign(graph, {nodeCentralityAttribute: 'myCentrality'});

// To ignore weights
const centralities = betweennessCentrality(graph, {getEdgeWeight: null});

// To use a getter function for weights
const centralities = betweennessCentrality(graph, {
  getEdgeWeight: (_, attr) => attr.importance
});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • nodeCentralityAttribute ?string betweennessCentrality: Name of the centrality attribute to assign.
  • getEdgeWeight ?string|function weight: Name of the edge weight attribute or getter function.
  • normalized ?boolean true: should the result be normalized?

Edge betweenness centrality

Computes the betweenness centrality for every edge.

import edgeBetweennessCentrality from 'graphology-metrics/centrality/edge-betweenness';

// To compute centrality for every edge:
const centralities = edgeBetweennessCentrality(graph);

// To directly map the result onto edges' attributes (`edgeBetweennessCentrality`):
edgeBetweennessCentrality.assign(graph);

// To directly map the result onto a custom attribute:
edgeBetweennessCentrality.assign(graph, {
  edgeCentralityAttribute: 'myCentrality'
});

// To ignore weights
const centralities = edgeBetweennessCentrality(graph, {getEdgeWeight: null});

// To use a getter function for weights
const centralities = edgeBetweennessCentrality(graph, {
  getEdgeWeight: (_, attr) => attr.importance
});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • edgeCentralityAttribute ?string betweennessCentrality: Name of the centrality attribute to assign.
  • getEdgeWeight ?string|function weight: Name of the edge weight attribute or getter function.
  • normalized ?boolean true: should the result be normalized?

Closeness centrality

Computes the closeness centrality of a graph’s nodes.

import closenessCentrality from 'graphology-metrics/centrality/closeness';

// To compute the eigenvector centrality and return the score per node:
const scores = closenessCentrality(graph);

// To directly map the result to nodes' attributes:
closenessCentrality.assign(graph);

// Note that you can also pass options to customize the algorithm:
const p = closenessCentrality(graph, {wassermanFaust: true});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • nodeCentralityAttribute ?string closenessCentrality: name of the node attribute that will be assigned the closeness centrality.
  • wassermanFaust ?boolean false: whether to use Wasserman & Faust’s normalization scheme.

Degree centrality

Computes the degree centrality for every node.

import {
  degreeCentrality,
  inDegreeCentrality,
  outDegreeCentrality
} from 'graphology-metrics/centrality/degree';

// To compute degree centrality for every node:
const centralities = degreeCentrality(graph);

// To directly map the result onto nodes' attributes (`degreeCentrality`):
degreeCentrality.assign(graph);

// To directly map the result onto a custom attribute:
degreeCentrality.assign(graph, {nodeCentralityAttribute: 'myCentrality'});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • nodeCentralityAttribute ?string degreeCentrality: name of the centrality attribute to assign.

Eigenvector centrality

Computes the eigenvector centrality of a graph’s nodes.

import eigenvectorCentrality from 'graphology-metrics/centrality/eigenvector';

// To compute the eigenvector centrality and return the score per node:
const scores = eigenvectorCentrality(graph);

// To directly map the result to nodes' attributes:
eigenvectorCentrality.assign(graph);

// Note that you can also pass options to customize the algorithm:
const p = eigenvectorCentrality(graph, {tolerance: 1e-3});

// To ignore your graph's weights
eigenvectorCentrality.assign(graph, {getEdgeWeight: null});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • nodeCentralityAttribute ?string eigenvectorCentrality: name of the node attribute that will be assigned the eigenvector centrality.
  • getEdgeWeight ?string|function weight: name of the edges’ weight attribute or getter function.
  • maxIterations ?number 100: maximum number of iterations to perform.
  • tolerance ?number 1.e-6: convergence error tolerance.

HITS

Computes the hub/authority metrics for each node using the HITS algorithm.

import hits from 'graphology-metrics/centrality/hits';

// To compute and return the result as 'hubs' & 'authorities':
const {hubs, authorities} = hits(graph);

// To directly map the result to nodes' attributes:
hits.assign(graph);

// Note that you can also pass options to customize the algorithm:
const {hubs, authorities} = hits(graph, {normalize: false});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • getEdgeWeight ?string|function weight: name of the edges’ weight attribute or getter function.
  • nodeHubAttribute ?string hub: name of the node attribute holding hub information.
  • nodeAuthorityAttribute ?string authority: name of the node attribute holding authority information.
  • maxIterations ?number 100: maximum number of iterations to perform.
  • normalize ?boolean true: should the result be normalized by the sum of values.
  • tolerance ?number 1.e-8: convergence error tolerance.

Pagerank

Computes the pagerank metrics for each node.

import pagerank from 'graphology-metrics/centrality/pagerank';

// To compute pagerank and return the score per node:
const scores = pagerank(graph);

// To directly map the result to nodes' attributes:
pagerank.assign(graph);

// Note that you can also pass options to customize the algorithm:
const p = pagerank(graph, {alpha: 0.9});

// To ignore your graph's weights
pagerank.assign(graph, {getEdgeWeight: null});

Arguments

• graph Graph: target graph.
• options ?object: options:
  • nodePagerankAttribute ?string pagerank: name of the node attribute that will be assigned the pagerank score.
  • getEdgeWeight ?string|function weight: name of the edges’ weight attribute or getter function.
  • alpha ?number 0.85: damping parameter of the algorithm.
  • maxIterations ?number 100: maximum number of iterations to perform.
  • tolerance ?number 1.e-6: convergence error tolerance.

Layout quality metrics

Connected Closeness

TODO…

Edge Uniformity

Computes the edge uniformity layout quality metric from the given graph having x and y positions attached to its nodes. Edge uniformity is the normalized standard deviation of edge length of the graph. Lower values should be synonym of better layout according to this particular metric.

Runs in O(E).

import edgeUniformity from 'graphology-metrics/layout-quality/edge-uniformity';

edgeUniformity(graph);
>>> ~1.132

Neighborhood preservation

Computes the “neighborhood preservation” layout quality metric from the given graph having x and y positions attached to its nodes. Neighborhood preservation is the average proportion of node neighborhood being the same both in the graph’s topology and its 2d layout space. The metric is therefore comprised between 0 and 1, 1 being the best, meaning that every node keeps its neighborhood perfectly intact within the layout space.

Runs in approximately O(N * log(N)).

import neighborhoodPreservation from 'graphology-metrics/layout-quality/neighborhood-preservation';

neighborhoodPreservation(graph);
// >>> 0.456

Stress

Computes the “stress” layout quality metric from the given graph having x and y positions attached to its nodes. Stress is the sum of normalized delta between node topology distances and their layout space distances. Lower values should be synonym of better layout according to this particular metric.

Note that this metric does not work very well when the graph has multiple connected components.

Note also that this metric traverses any given graph as an undirected one.

Runs in O(N^2).

import stress from 'graphology-metrics/layout-quality/stress';

stress(graph);
// >>> ~24510.2914