SCC (Small-World Optimized)

Supported Graph Characteristics

Unweighted edges

Homogeneous vertex types

Heterogeneous vertex types

Algorithm link: Strongly Connected Components

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.

The algorithm starts by trimming the graph, which removes all vertices whose indegree or outdegree is 0. 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.

Notes

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., BFS and Coloring-based Parallel Algorithms for Strongly Connected Components and Related Problems.

We use a default value of 100000 for the threshold used to choose initial pivot vertices to increase the chances that the initial pivot is contained within the largest SCC.

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)

Parameters

Name Description Data type

Name	Description	Data type
`STRING v_type`	The vertex type to count as part of a strongly connected component	(empty string)
`STRING e_type`	The edge type to traverse	(empty string)
`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.`	(empty string)
`UINT 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.	`100000`
`BOOL to_show_cc_count`	Whether to print the number of vertices in each strongly connected component.	False

STRING v_type

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

(empty string)

STRING e_type

The edge type to traverse

(empty string)

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.

(empty string)

UINT 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.

100000

BOOL to_show_cc_count

Whether to print the number of vertices in each strongly connected component.

False

Output

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

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.

Example

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

Graph with 7 SCCs

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}]
}