This section describes the Balanced Triads algorithm in the Neo4j Graph Algorithms library.

Balanced Triads algorithm is used to evaluate structural balance of the graph.

It is based on the Balance Theory proposed by Fritz Heider in 1958. Unlike other algorithms where there are only positive relationships available, balance theory differentiates between positive and negative relationships. Certain structures between individuals and objects are perceived as balanced whereas others are not. In general balanced structures are preferred over imbalanced ones.

This section includes:

Balanced triads is an algorithm that counts the number of balanced and unbalanced triads a node is member of. It uses Signed graph model to differentiate between positive and negative relationships using the sign of the weight.

Determining if the triad is balanced is simple math:

+ + + = Balanced - + - = Balanced - + + = Unbalanced - - - = Unbalanced

- Balanced triads algorithm has been used to analyse structural balance of networks in the animal kingdom. Find this study in Structural balance in the social networks of a wild mammal.
- Balanced triads algorithm has been used to evaluate dynamics in the process of "healing" imbalanced triads and restoring structural balance in a social network. Find more details in Dynamics of social balance on networks.
- Balanced triads algorithm and balance theory allow us to better understand consumer to brand relationship and consumer behaviour as shown in the study of Updating Heider’s Balance Theory in Consumer Behaviour by Woodside and Chabet.

This sample will explain the Balanced Triads algorithm, using a simple graph:

The following will create a sample graph:

```
MERGE (a:Person {name:'Anna'})
MERGE (b:Person {name:'Dolores'})
MERGE (c:Person {name:'Matt'})
MERGE (d:Person {name:'Larry'})
MERGE (e:Person {name:'Stefan'})
MERGE (f:Person {name:'Sophia'})
MERGE (g:Person {name:'Robin'})
MERGE (a)-[:TYPE {weight:1.0}]->(b)
MERGE (a)-[:TYPE {weight:-1.0}]->(c)
MERGE (a)-[:TYPE {weight:1.0}]->(d)
MERGE (a)-[:TYPE {weight:-1.0}]->(e)
MERGE (a)-[:TYPE {weight:1.0}]->(f)
MERGE (a)-[:TYPE {weight:-1.0}]->(g)
MERGE (b)-[:TYPE {weight:-1.0}]->(c)
MERGE (c)-[:TYPE {weight:1.0}]->(d)
MERGE (d)-[:TYPE {weight:-1.0}]->(e)
MERGE (e)-[:TYPE {weight:1.0}]->(f)
MERGE (f)-[:TYPE {weight:-1.0}]->(g)
MERGE (g)-[:TYPE {weight:1.0}]->(b);
```

The following will count the number of balanced and unbalanced triads that a node is a member of, and return a stream with
`nodeId`

, `balanced`

and `unbalanced`

:

```
call algo.balancedTriads.stream('Person','TYPE',{weightProperty:'weight'})
YIELD nodeId, balanced, unbalanced
RETURN algo.getNodeById(nodeId).name as person,balanced,unbalanced
ORDER BY balanced + unbalanced DESC
LIMIT 10
```

The following will count the number of balanced and unbalanced triads that a node is member of, and write it back. It will return the total balanced triads and unbalanced triads count of the given graph:

```
CALL algo.balancedTriads('Person', 'TYPE', {weightProperty:'weight'})
YIELD loadMillis, computeMillis, writeMillis, nodeCount, balancedTriadCount, unbalancedTriadCount;
```

nodeId | balanced | unbalanced |
---|---|---|

Anna |
3 |
3 |

Matt |
1 |
1 |

Larry |
1 |
1 |

Stefan |
1 |
1 |

Sophia |
1 |
1 |

Dolores |
1 |
1 |

Anna is a member of six triads out of which three are balanced and three are unbalanced. All others are each members of one balanced and one unbalanced triad or triangle.

The following will count the number of balanced and unbalanced triads that a node is a member of, and return a stream with
`nodeId`

, `balanced`

and 'unbalanced':

```
CALL algo.balancedTriads.stream(label:String, relationship:String, {concurrency:4})
YIELD nodeId, balanced, unbalanced
```

Name | Type | Default | Optional | Description |
---|---|---|---|---|

label |
string |
null |
yes |
The label to load from the graph. If null, load all nodes |

relationship |
string |
null |
yes |
The relationship-type to load from the graph. If null, load all relationships |

weightProperty |
string |
null |
no |
The property name that contains weight. If weight is positive, algorithm treats the relationship as positive. With a negative weight, relationship is treated as negative. Must be numeric. |

concurrency |
int |
available CPUs |
yes |
The number of concurrent threads |

Name | Type | Description |
---|---|---|

nodeId |
int |
The ID of node |

balanced |
int |
The number of balanced triads a node is member of |

unbalanced |
int |
The number of unbalanced a node is member of |

The following will count the number of balanced and unbalanced triads that a node is member of, and write it back. It will return the total balanced triads and unbalanced triads count of the given graph:

```
CALL algo.balancedTriads(label:String, relationship:String,
{concurrency:4, write:true, weightProperty:'weight', balancedProperty:'balanced', unbalancedProperty:'unbalanced'})
YIELD loadMillis, computeMillis, writeMillis, nodeCount, balancedTriadCount, unbalancedTriadCount
```

Name | Type | Default | Optional | Description |
---|---|---|---|---|

label |
string |
null |
yes |
The label to load from the graph. If null, load all nodes |

relationship |
string |
null |
yes |
The relationship-type to load from the graph. If null, load all relationships |

concurrency |
int |
available CPUs |
yes |
The number of concurrent threads |

weightProperty |
string |
null |
no |
The property name that contains weight. If weight is positive, algorithm treats the relationship as positive. With a negative weight, relationship is treated as negative. Must be numeric. |

balancedProperty |
string |
'balanced' |
yes |
The property name written back to the count of balanced triads a node is member of |

unbalancedProperty |
string |
'unbalanced' |
yes |
The property name written back to the count of unbalanced triads a node is member of |

write |
boolean |
true |
yes |
Specifies if the result should be written back as a node property |

Name | Type | Description |
---|---|---|

loadMillis |
int |
Milliseconds for loading data |

computeMillis |
int |
Milliseconds for running the algorithm |

writeMillis |
int |
Milliseconds for writing result data back |

postProcessingMillis |
int |
Milliseconds for computing percentiles and community count |

nodeCount |
int |
The number of nodes considered |

balancedTriadCount |
int |
The number of balanced triads in the given graph |

unbalancedTriadCount |
int |
The number of unbalanced triads in the given graph |

p1 |
double |
The 1 percentile of number of balanced triads. |

p5 |
double |
The 5 percentile of number of balanced triads. |

p10 |
double |
The 10 percentile of number of balanced triads. |

p25 |
double |
The 25 percentile of number of balanced triads. |

p50 |
double |
The 50 percentile of number of balanced triads. |

p75 |
double |
The 75 percentile of number of balanced triads. |

p90 |
double |
The 90 percentile of number of balanced triads. |

p95 |
double |
The 95 percentile of number of balanced triads. |

p99 |
double |
The 99 percentile of number of balanced triads. |

p100 |
double |
The 100 percentile of number of balanced triads. |

write |
boolean |
Specifies if the result was written back as a node property |

balancedProperty |
string |
The property name the number of balanced triads is written to |

unbalancedProperty |
string |
The property name the number of unbalanced triads is written to |

The Balanced Triads algorithm support the following graph type:

- ✓ undirected, weighted