# SCC (Small-World Optimized)

In addition to the regular strongly connected component algorithm, we also provide a version that is optimized for small-world graphs. A small world graph in this context means the graph has a hub community, where a vast majority of the vertices of the graph are weakly connected.

This version improves upon the performance of the original algorithm when dealing with small-world graphs by combining several different methods used to find connected components in a multi-step process proposed by Slota et al. in BFS and Coloring-based Parallel Algorithms for Strongly Connected Components and Related Problems.

The algorithm starts by trimming the graph, which removes all vertices whose indegree or outdegree is 0. Then in the second phase, the algorithm selects an initial pivot vertex v with a high product of indegree and outdegree. From the initial pivot vertex , the algorithm uses one iteration of the Forward-Backward method to identify all vertices reachable by v (descendants) and all vertices that can reach v (predecessors). The intersection of the descendants and the predecessors form a strongly connected component (SCC). The vertices that are not included in this SCC are passed off to the next step.

After identifying the first SCC, the algorithm uses the coloring method and Tarjan’s serial algorithm to identify the SCCs in the remaining vertices.

## Specifications

```CREATE QUERY tg_scc_small_world(STRING v_type, STRING e_type, STRING re_type,
UINT threshold = 100000, BOOL to_show_cc_count=FALSE)```

### Time complexity

The algorithm has a time complexity of O(V + E), where V is the number of vertices and E is the number of edges in the graph.

## Parameters

Name Description Data type

`v_type`

The vertex type to count as part of a strongly connected component

`STRING`

`e_type`

The edge type to traverse

`STRING`

`re_type`

The reverse edge type to traverse. If the graph is undirected, fill in the name of the undirected edge here as well as for `e_type.`

`STRING`

`threshold`

The threshold used to choose initial pivot vertices. Only vertices whose product of indegree and outdegree exceed this threshold will be considered candidates for the pivot vertex. This is an attempt to increase the chances that the initial pivot is contained within the largest SCC. The default value for this parameter is 100000. It is suggested that you keep this default value when running the algorithm.

`UINT`

`to_show_cc_count`

If set to `TRUE`, the algorithm will return the number of vertices in each strongly connected component.

`BOOL`

## Result

When `to_show_cc_count` is set to true, the algorithm will return the number of strongly connected components in the graph.

## Example

Suppose we have the following graph. We can see there are 7 strongly connected components. With two of them containing more than 1 vertices. The five vertices on the left are each a strongly connected component individually.

Running the algorithm on the graph will return a result of 7:

• GSQL Command

• Result

`RUN QUERY tg_scc_small_world("Person", "Friend", "Friend", _, TRUE)`
```{
"error": false,
"message": "",
"version": {
"schema": 0,
"edition": "enterprise",
"api": "v2"
},
"results": [{"@@CC_count.size()": 7}]
}```