# Conductance metric

## Glossary

Directed

Directed trait. The algorithm is well-defined on a directed graph.

Directed

Directed trait. The algorithm ignores the direction of the graph.

Directed

Directed trait. The algorithm does not run on a directed graph.

Undirected

Undirected trait. The algorithm is well-defined on an undirected graph.

Undirected

Undirected trait. The algorithm ignores the undirectedness of the graph.

Heterogeneous nodes

Heterogeneous nodes fully supported. The algorithm has the ability to distinguish between nodes of different types.

Heterogeneous nodes

Heterogeneous nodes allowed. The algorithm treats all selected nodes similarly regardless of their label.

Heterogeneous relationships

Heterogeneous relationships fully supported. The algorithm has the ability to distinguish between relationships of different types.

Heterogeneous relationships

Heterogeneous relationships allowed. The algorithm treats all selected relationships similarly regardless of their type.

Weighted relationships

Weighted trait. The algorithm supports a relationship property to be used as weight, specified via the relationshipWeightProperty configuration parameter.

Weighted relationships

Weighted trait. The algorithm treats each relationship as equally important, discarding the value of any relationship weight.

## Introduction

Conductance is a metric that allows you to evaluate the quality of a community detection. Relationships of nodes in a community `C` connect to nodes either within `C` or outside `C`. The conductance is the ratio between relationships that point outside `C` and the total number of relationships of `C`. The lower the conductance, the more "well-knit" a community is.

It was shown by Yang and Leskovec in the paper "Defining and Evaluating Network Communities based on Ground-truth" that conductance is a very good metric for evaluating actual communities of real world graphs.

The algorithm runs in time linear to the number of relationships in the graph.

## Syntax

This section covers the syntax used to execute the Conductance algorithm in each of its execution modes. We are describing the named graph variant of the syntax. To learn more about general syntax variants, see Syntax overview.

Example 1. Conductance syntax per mode
Run Conductance in stream mode on a named graph.
``````CALL gds.conductance.stream(
graphName: String,
configuration: Map
) YIELD
community: Integer,
conductance: Float``````
Table 1. Parameters
Name Type Default Optional Description

graphName

String

`n/a`

no

The name of a graph stored in the catalog.

configuration

Map

`{}`

yes

Configuration for algorithm-specifics and/or graph filtering.

Table 2. Configuration
Name Type Default Optional Description

nodeLabels

List of String

`['*']`

yes

Filter the named graph using the given node labels. Nodes with any of the given labels will be included.

relationshipTypes

List of String

`['*']`

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

concurrency

Integer

`4`

yes

The number of concurrent threads used for running the algorithm.

jobId

String

`Generated internally`

yes

An ID that can be provided to more easily track the algorithm’s progress.

logProgress

Boolean

`true`

yes

If disabled the progress percentage will not be logged.

relationshipWeightProperty

String

`null`

yes

Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.

communityProperty

String

`n/a`

no

The node property that holds the community ID as an integer for each node. Note that only non-negative community IDs are considered valid and will have their conductance computed.

Table 3. Results
Name Type Description

community

Integer

Community ID.

conductance

Float

Conductance of the community.

 Only non-negative community IDs are valid for identifying communities. Nodes with a negative community ID will only take part in the computation to the extent that they are connected to nodes in valid communities, and thus contribute to those valid communities' outward relationship counts.

## Examples

 All the examples below should be run in an empty database. The examples use native projections as the norm, although Cypher projections can be used as well.

In this section we will show examples of running the Conductance algorithm on a concrete graph. The intention is to illustrate what the results look like and to provide a guide in how to make use of the algorithm in a real setting. We will do this on a small social network graph of a handful nodes connected in a particular pattern. The example graph looks like this:

The following Cypher statement will create the example graph in the Neo4j database:
``````CREATE
(nAlice:User {name: 'Alice', seed: 42}),
(nBridget:User {name: 'Bridget', seed: 42}),
(nCharles:User {name: 'Charles', seed: 42}),
(nDoug:User {name: 'Doug'}),
(nMark:User {name: 'Mark'}),
(nMichael:User {name: 'Michael'}),

This graph has two clusters of Users, that are closely connected. Between those clusters there is one single edge. The relationships that connect the nodes in each component have a property `weight` which determines the strength of the relationship.

We can now project the graph and store it in the graph catalog. We load the `LINK` relationships with orientation set to `UNDIRECTED` as this works best with the Louvain algorithm which we will use to create the communities that we evaluate using Conductance.

The following statement will project the graph and store it in the graph catalog.
``````CALL gds.graph.project(
'myGraph',
'User',
{
orientation: 'UNDIRECTED'
}
},
{
nodeProperties: 'seed',
relationshipProperties: 'weight'
}
)``````

We now run the Louvain algorithm to create a division of the nodes into communities that we can then evalutate.

The following will run the Louvain algorithm and store the results in `myGraph`:
``````CALL gds.louvain.mutate('myGraph', { mutateProperty: 'community', relationshipWeightProperty: 'weight' })
YIELD communityCount``````
Table 4. Results
communityCount

3

Now our in-memory graph `myGraph` is populated with node properties under the key `community` that we can set as input for our evaluation using Conductance. The nodes are now assigned to communities in the following way:

Table 5. Community assignments
name community

"Alice"

3

"Bridget"

2

"Charles"

2

"Doug"

3

"Mark"

5

"Michael"

5

Please see the stream node properties procedure for how to obtain such an assignment table.

### Stream

Since we now have a community detection, we can evaluate how good it is under the conductance metric. Note that we in this case we use the feature of relationships being weighted by a relationship property.

The Conductance stream procedure returns the conductance for each community. This allows us to inspect the results directly or post-process them in Cypher without any side effects.

For more details on the stream mode in general, see Stream.

The following will run the Conductance algorithm in `stream` mode:
``````CALL gds.conductance.stream('myGraph', { communityProperty: 'community', relationshipWeightProperty: 'weight' })
YIELD community, conductance``````
Table 6. Results
community conductance

2

0.5

3

0.23076923076923078

5

0.2

We can see that the community of the weighted graph with the lowest conductance is community 5. This means that 5 is the community that is most "well-knit" in the sense that most of its relationship weights are internal to the community.