# GSQL Graph Algorithm Library

Graph algorithms are functions for measuring characteristics of graphs, vertices, or relationships. Graph algorithms can provide insights into the role or relevance of individual entities in a graph. For example: How centrally located is this vertex? How much influence does this vertex exert over the others?

Some graph algorithms measure or identify global characteristics: What are the natural community groupings in the graph? What is the density of connections?

• New algorithms, over the past few months:

• Approximate Closeness Centrality

• Greedy Graph Coloring

• Jaccard Similarity, All-Pairs in Batch mode

• Cosine Similarity, All-Pairs in Batch mode

# Using the GSQL Graph Algorithm Library

The GSQL Graph Algorithm Library is a collection of expertly written GSQL queries, each of which implements a standard graph algorithm. Each algorithm is ready to be installed and used, either as a stand-alone query or as a building block of a larger analytics application.

GSQL running on the TigerGraph platform is particularly well-suited for graph algorithms for several reasons:

• Turing-complete with full support for imperative and procedural programming, ideal for algorithmic computation.

• Parallel and Distributed Processing, enabling computations on larger graphs.

• User-Extensible. Because the algorithms are written in standard GSQL and compiled by the user, they are easy to modify and customize.

• Open-Source. Users can study the GSQL implementations to learn by example, and they can develop and submit additions to the library.

## Library Structure

The library contains two main sections: algorithms and tests. Within the algorithms folder are four subfolders:

• schema-free: This contains algorithms that are ready to use (e.g. INSTALL QUERY) as-is.

• templates: This contains algorithms that need to be prepared with the install.sh script to target them for a specific graph schema.

• generated: This contains the algorithms converted from template to graph-specific format by the install.sh script.

• examples: This contains examples of generated algorithms

The tests folder contains small sample graphs that you can use to experiment with the algorithms. In this document, we use the test graphs to show you the expected result for each algorithm. The graphs are small enough that you can manually calculate and sometimes intuitively see what the answers should be.

### Release Branches

Starting with TigerGraph product version 2.6, the GSQL Graph Algorithm Library has release branches:

• Product version branches (2.6, 3.0, etc.) are snapshots created shortly after a product version is released. They contain the best version of the graph algorithm library at the time of that product version's initial release. They will not be updated, except to fix bugs.

• Master branch: the newest released version. This should be at least as new as the newest. It may contain new or improved algorithms.

• Other branches are development branches.

It is possible to run newer algorithms on an older product version, as long as the algorithm does not rely on features available only in newer product versions.

## Schema-Free Algorithms

Most GSQL graph algorithms are schema-free, which means they are ready to use with any graph, regardless of the graph's data model or schema. Schema-free algorithms have run-time input parameters for the vertex type(s), edge type(s), and attributes which the user wishes to use.

To use a schema-free algorithm, the algorithm (GSQL query) must first be installed. If your database is on a distributed cluster, you should use the DISTRIBUTED option when installing the query to install it in Distributed Query Mode.

## Installing Template Algorithms

Remember that GSQL graph algorithms are simply GSQL queries. A few algorithms make use of GSQL features that do not yet accept run-time parameters. Instead, these algorithms are in template format. A script is needed to personalize these algorithms before they are installed.

Make sure that install.sh is owned by the tigergraph user.

1. Within the algorithms folder is a script install.sh. When you run the script, it will first ask you which graph schema you wish to work on. (The TigerGraph platform supports multiple concurrent graphs.)

2. It then asks you to choose from a menu of available algorithms.

3. After knowing your graph schema and your algorithm, the installer will ask you some questions for that particular algorithm:

• the installer will guide you in selecting appropriate vertex types and edge types. Note this does not have to be all the vertex or edge types in your graph. For example, you may have a social graph with three categories of persons and five types of relationships. You might decide to compute PageRank using Member and Guest vertices and Recommended edges.

• Some algorithms use edge weights as input information (such as Shortest Path where each edge has a weight meaning the "length" of that edge. The installer will ask for the name of that edge attribute.

4. Single Node Mode or Distributed Mode? Queries that analyze the entire graph (such as PageRank and Community Detection) will run better in Distributed Mode if you have a cluster of machines.

5. It will then ask you what type of output you would like. It will proceed to create up to three versions of your algorithm, based on the three ways of receiving the algorithm's output:

• Stream the output in JSON format, the default behavior for most GSQL queries.

• Save the output value(s) in CSV format to a file. For some algorithms, this option will add an input parameter to the query, to let the user specify how many total values to output.

• Store the results as vertex or edge attribute values. The attributes must already exist in the graph schema, and the installer will ask you which attributes to use.

6. After creating queries for one algorithm, the installer will loop back to let you choose another algorithm (returning to step 2 above).

7. If you choose to exit, the installer makes a last request: Do you want to install your queries? Installation is when the code is compiled and bound into the query engine. It takes a few minutes, so it is best to create all your personalized queries at once and then install them as a group.

Example:

$bash install.sh *** GSQL Graph Algorithm Installer ***Available graphs: - Graph social(Person:v, Friend:e, Also_Friend:e, Coworker:e)Graph name? social​Please enter the number of the algorithm to install: 1) EXIT 2) Weighted PageRank 3) Personalized PageRank 4) Triangle Counting(minimal memory) 5) Triangle Counting(fast, more memory) 6) Cosine Neighbor Similarity (single vertex) 7) Cosine Neighbor Similarity (all vertices) 8) Jaccard Neighbor Similarity (single vertex) 9) Jaccard Neighbor Similarity (all vertices)#? 2 Weighted pageRank() works on directed edges​Available vertex and edge types: - VERTEX Person(PRIMARY_ID id STRING, name STRING, score FLOAT, tag STRING) WITH STATS="OUTDEGREE_BY_EDGETYPE" - DIRECTED EDGE Friend(FROM Person, TO Person, weight FLOAT, tag STRING) WITH REVERSE_EDGE="Also_Friend" - DIRECTED EDGE Also_Friend(FROM Person, TO Person, weight FLOAT, tag STRING) WITH REVERSE_EDGE="Friend" - UNDIRECTED EDGE Coworker(FROM Person, TO Person, weight FLOAT, tag STRING)​Please enter the vertex type(s) and edge type(s) for running PageRank. Use commas to separate multiple types [ex: type1, type2] Leaving this blank will select all available types Similarity algorithms only take single vertex type​Vertex types: PersonEdge types: FriendThe query pageRank is dropped.The query pageRank_file is dropped.The query pageRank_attr is dropped.​Please choose query mode:1) Single Node Mode2) Distributed Mode#? 1​Please choose a way to show result:1) Show JSON result 3) Save to Attribute/Insert Edge2) Write to File 4) All of the above#? 4​gsql -g social ./templates/pageRank.gsqlThe query pageRank has been added!​gsql -g social ./templates/pageRank_file.gsqlThe query pageRank_file has been added!​If your graph schema has appropriate vertex or edge attributes, you can update the graph with your results.Do you want to update the graph [yn]? yVertex attribute to store FLOAT result (e.g. pageRank): scoregsql -g social ./templates/pageRank_attr.gsqlThe query pageRank_attr has been added!Created the following algorithms: - pageRank(float maxChange, int maxIter, float damping, bool display, int outputLimit) - pageRank_attr(float maxChange, int maxIter, float damping, bool display) - pageRank_file(float maxChange, int maxIter, float damping, bool display, file f) ​​Please enter the number of the algorithm to install:1) EXIT2) Closeness Centrality3) Connected Components4) Label Propagation5) Community detection: Louvain6) PageRank7) Shortest Path, Single-Source, Any Weight8) Triangle Counting(minimal memory)9) Triangle Counting(fast, more memory)#? 1ExitingAlgorithm files have been created. Do want to install them now [yn]? yStart installing queries, about 1 minute ...cpageRank query: curl -X GET 'http://127.0.0.1:9000/query/social/pageRank?maxChange=VALUE&maxIter=VALUE&damping=VALUE&display=VALUE&outputLimit=VALUE'. Add -H "Authorization: Bearer TOKEN" if authentication is enabled.pageRank_file query: curl -X GET 'http://127.0.0.1:9000/query/social/pageRank_file?maxChange=VALUE&maxIter=VALUE&damping=VALUE&display=VALUE&f=VALUE'. Add -H "Authorization: Bearer TOKEN" if authentication is enabled.pageRank_attr query: curl -X GET 'http://127.0.0.1:9000/query/social/pageRank_attr?maxChange=VALUE&maxIter=VALUE&damping=VALUE&display=VALUE'. Add -H "Authorization: Bearer TOKEN" if authentication is enabled.​[======================================================================================================] 100% (3/3)$

After the algorithms are installed, you will see them listed among the rest of your GSQL queries.

GSQL > ls...Queries:   - cc_subquery(vertex v, int numVert, int maxHops) (installed v2)  - closeness_cent(bool display, int outputLimit) (installed v2)  - closeness_cent_attr(bool display) (installed v2)  - closeness_cent_file(bool display, file f) (installed v2)  - conn_comp() (installed v2)  - conn_comp_attr() (installed v2)  - conn_comp_file(file f) (installed v2)  - label_prop(int maxIter) (installed v2)  - label_prop_attr(int maxIter) (installed v2)  - label_prop_file(int maxIter, file f) (installed v2)  - louvain() (installed v2)  - louvain_attr() (installed v2)  - louvain_file(file f) (installed v2)  - pageRank(float maxChange, int maxIter, float damping, bool display, int outputLimit) (installed v2)  - pageRank_attr(float maxChange, int maxIter, float damping, bool display) (installed v2)  - pageRank_file(float maxChange, int maxIter, float damping, bool display, file f) (installed v2)  - tri_count() (installed v2)  - tri_count_fast() (installed v2)​

## Running an Algorithm

Running an algorithm is the same as running a GSQL query. For example, if you selected the JSON option for pageRank, you could run it from GSQL as below:

GSQL > RUN QUERY pageRank("Page","Links_to",_,30,_,50,_,_,_)

Installing a query also creates a REST endpoint. The same query could be run thus:

curl -X GET 'http://127.0.0.1:9000/query/alg_graph/pageRank?v_type=Page&e_type=Links_to&max_iter=30&top_k=50'

GSQL lets you run queries from within other queries. This means you can use a library algorithm as a building block for more complex analytics.

# Library Overview

The following algorithms are currently available. The algorithms are grouped into five classes:

• Path

• Centrality

• Community

• Similarity

• Classification

Some algorithms are only appropriate for certain types of graphs. For example, Strong Connected Components (SCC) is designed for graphs with directed edges.

• Coming soon means that TigerGraph plans to release this variant of the algorithm soon.

• n/a means that this variant of the algorithm is typically not used

 Algorithm Class UndirectedEdges DirectedEdges WeightedEdges Single-Source Shortest Path Path Yes Yes Yes All Pairs Shortest Path Path Yes Yes Yes Minimum Spanning Tree Path Yes n/a Yes Minimum Spanning Forest Path Yes n/a Yes Maximal Independent Set Path Yes Coming Soon n/a Cycle Detection Path no Yes n/a Estimated Diameter Path Yes n/a n/a PageRank Centrality n/a Yes n/a Weighted PageRank Centrality n/a Yes Yes Personalized PageRank Centrality n/a Yes Coming soon Closeness Centrality Centrality Yes n/a Coming soon Approximate Closeness Centrality (NEW) Centrality Yes n/a Coming soon Betweenness Centrality Centrality Yes n/a Coming soon Connected Components Community Yes n/a n/a Strongly Connected Components Community n/a Yes n/a K-Core Community Yes n/a n/a Label Propagation Community Yes n/a n/a Louvain Modularity Community Yes n/a n/a Triangle Counting Community Yes n/a n/a Cosine Similarity of Neighborhoods (single-source, all-pairs and batch (NEW)) Similarity Yes Yes Yes Jaccard Similarity of Neighborhoods (single-source, all-pairs and batch (NEW)) Similarity Yes Yes No Greedy Graph Coloring (NEW) Classification Yes Yes Yes K-Nearest Neighbors (with cosine similarity for "nearness") Classification Yes Yes Yes

## Computational Complexity

Computational Complexity is a formal mathematical term, referring to how an algorithm's requirements scale according to the size of the data or other key parameters. Computational complexity is useful for comparing one algorithm to another, but it does not describe speed in absolute terms.

For graphs, there are two key data parameters:

• V (or sometimes n), the number of vertices

• E (or sometimes m), the number of edges

The notation O(V^2) (read "big O V squared") means that when V is large, the computational time is proportional to V^2.

Time complexity describes how the execution time is expected to vary with the data size and other key parameters. Normally, time complexity is based on simplified and idealized computer architecture: memory accesses and arithmetic operations always take one unit of time.

Memory complexity describes how the run-time memory usage scales with the data size and other key parameters.

## Standard Parameters

The GSQL Algorithm library has consistent parameter names and order. Input parameters come first, parameters for the body of the algorithm come next, and output configuration parameters come last.

In GSQL, to accept a default parameter value, use _ E.g., closeness_cent(["Person", "Organization"], ["Likes"], _, _, _, _, _, _, _)

### Parameters for the Graph Schema

Schema-free algorithms need to know the name of the vertex types, edge types, and the edge weight attribute for weighted edge algorithms.

 Parameter Type and Name Description SET v_type The name(s) of the vertex types to include. SET e_type The name(s) of the edge types to include. STRING wt_edge The name of the edge weight attribute to use. STRING wt_type The data type of the edge weight. Must be INT, FLOAT, or DOUBLE.

Set notation for GSQL parameters

Use square brackets to enclose a set-type parameter, even if there is just a single item in the set, e.g. closeness_cent(["Person", "Organization"], ["Likes"], _, _, _, _, _, _, _)

### Parameters for Output Options

There are usually three options for output:

• Send JSON output to standard output.

• Write results to an output file in CSV format.

• Store the output values in a user-specified attribute of a vertex or edge type.

Beginning with v3.0, each of the options is selected independently by setting appropriate parameters. More than one option may be selected:

 Parameter type and name Default Description BOOL print_accum true If true, the output will be in JSON format INT output_limit -1 If output_limit >= 0, limit the number of vertices in the JSON output to this value.If output_limit < 0, then do not limit JSON output. STRING result_attr Emptystring The name of an attribute. If not the empty string, take the algorithm's output values and store them in the given attribute. STRING file_path Emptystring The path to the output file. If not the empty string, write output to this file. BOOL display_edges false If true, and if print_accum is true, include relevant edges in the JSON output, so that the graph can be displayed.

# Path Algorithms

These algorithms help find the shortest path or evaluate the availability and quality of routes.

## Single-Source Shortest Path, Unweighted

This algorithm finds an unweighted shortest path from one source vertex to each possible destination vertex in the graph. That is, it finds n paths.

If your graph has weighted edges, see the next algorithm. With weighted edges, it is necessary to search the whole graph, whether you want the path for just one destination or for all destinations.

### Description and Uses

If a graph has unweighted edges, then finding the shortest path from one vertex to another is the same as finding the path with the fewest hops. Think of Six Degrees of Separation and Friend of a Friend. Unweighted Shortest Path answers the question "How are you two related?" The two entities do not have to be persons. Shortest Path is useful in a host of applications, from estimating influences or knowledge transfer, to criminal investigation.

When the graph is unweighted, we can use a "greedy" approach to find the shortest path. In computer science, a greedy algorithm makes intermediate choices based on the data being considered at the moment, and then does not revisit those choices later on. In this case, once the algorithm finds any path to a vertex T, it is certain that that is a shortest path.

### Specifications

CREATE QUERY shortest_ss_no_wt (VERTEX source, SET<STRING> v_type,  SET<STRING> e_type, INT output_limit = -1, BOOL print_accum =TRUE,  STRING result_attr ="", STRING file_path ="", BOOL display_edges =FALSE)
 Characteristic Value Result Computes a shortest distance (INT) and shortest path (STRING) from vertex source to each other vertex. Input Parameters VERTEX source: ID of the source vertexSET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT output_limit: If >=0, max number of vertices to output to JSON.BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store distance values (INT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(E), E = number of edges Graph Types Directed or Undirected edges, Unweighted edges

### Example

In the below graph, we do not consider the weight on edges. Using vertex A as the source vertex, the algorithm discovers that the shortest path from A to B is A-B, and the shortest path from A to C is A-D-C, etc.

[  {    "ResultSet": [      {        "v_id": "B",        "v_type": "Node",        "attributes": {          "[email protected]": 1,          "[email protected]": [            "A",            "B"          ]        }      },      {        "v_id": "A",        "v_type": "Node",        "attributes": {          "[email protected]": 0,          "[email protected]": [            "A"          ]        }      },      {        "v_id": "C",        "v_type": "Node",        "attributes": {          "[email protected]": 2,          "[email protected]": [            "A",            "D",            "C"          ]        }      },      {        "v_id": "E",        "v_type": "Node",        "attributes": {          "[email protected]": 2,          "[email protected]": [            "A",            "D",            "E"          ]        }      },      {        "v_id": "D",        "v_type": "Node",        "attributes": {          "[email protected]": 1,          "[email protected]": [            "A",            "D"          ]        }      }    ]  }]

## Single-Source Shortest Path, Weighted

### Description and Uses

Finding shortest paths in a graph with weighted edges is algorithmically harder than in an unweighted graph because even after you find a path to a vertex T, you cannot be certain that it is a shortest path. If edge weights are always positive, then you must keep trying until you have considered every in-edge to T. If edge weights can be negative, then it's even harder. You must consider all possible paths.

A classic application for weighted shortest path is finding the shortest travel route to get from A to B. (Think of route planning "GPS" apps.) In general, any application where you are looking for the cheapest route is a possible fit.

### Specifications

The shortest path algorithm can be optimized if we know all the weights are nonnegative. If there can be negative weights, then sometimes a longer path will have a lower cumulative weight. Therefore, we have two versions of this algorithm

shortest_ss_pos_wt (VERTEX source, SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr, STRING wt_type, INT output_limit = -1, BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "", BOOL display_edges = FALSE)
shortest_ss_any_wt (VERTEX source, SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr, STRING wt_type, INT output_limit = -1, BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "", BOOL display_edges = FALSE)
 Characteristic Value Result Computes a shortest distance (INT) and shortest path (STRING) from vertex source to each other vertex. Input Parameters VERTEX source: Id of the source vertexSET v_type: Names of vertex types to useSET e_type: Names of edge types to useSTRING wt_attr: Name of edge weight attributeSTRING wt_type: Data type of edge weight attribute: "INT", "FLOAT", or "DOUBLE"INT output_limit: If >=0, max number of vertices to output to JSON.BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store distance values (INT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(V*E), V = number of vertices, E = number of edges Graph Types Directed or Undirected edges, Weighted edges

The shortest_path_any_wt query is an implementation of the Bellman-Ford algorithm. If there is more than one path with the same total weight, the algorithm returns one of them.

Currently, shortest_path_pos_wt also uses Bellman-Ford. The well-known Dijsktra's algorithm is designed for serial computation and cannot work with GSQL's parallel processing.

### Example

The graph below has only positive edge weights. Using vertex A as the source vertex, the algorithm discovers that the shortest weighted path from A to B is A-D-B, with distance 8. The shortest weighted path from A to C is A-D-B-C with distance 9.

[  {    "ResultSet": [      {        "v_id": "B",        "v_type": "Node",        "attributes": {          "[email protected]": 8,          "[email protected]": [            "D",            "B"          ]        }      },      {        "v_id": "A",        "v_type": "Node",        "attributes": {          "[email protected]": 0,          "[email protected]": []        }      },      {        "v_id": "C",        "v_type": "Node",        "attributes": {          "[email protected]": 9,          "[email protected]": [            "D",            "B",            "C"          ]        }      },      {        "v_id": "E",        "v_type": "Node",        "attributes": {          "[email protected]": 7,          "[email protected]": [            "D",            "E"          ]        }      },      {        "v_id": "D",        "v_type": "Node",        "attributes": {          "[email protected]": 5,          "[email protected]": [            "D"          ]        }      }    ]  }]

The graph below has both positive and negative edge weights. Using vertex A as the source vertex, the algorithm discovers that the shortest weighted path from A to E is A-D-C-B-E, with a cumulative score of 7 - 3 - 2 - 4 = -2.

## Single-Pair Shortest Path

The Single-Pair Shortest Path task seeks the shortest path between a source vertex S and a target vertex T. If the edges are unweighted, then use the query in our tutorial document GSQL Demo Examples.

If the edges are weighted, then use the Single-Source Shortest Path algorithm. In the worst case, it takes the same computational effort to find the shortest path for one pair as to find the shortest paths for all pairs from the same source S. The reason is that you cannot know whether you have found the shortest (least weight) path until you have explored the full graph. If the weights are always positive, however, then a more efficient algorithm is possible. You can stop searching when you have found paths that use each of the in-edges to T.

## All-Pairs Shortest Path

The All-Pairs Shortest Path algorithm is costly for large graphs because the computation time is O(V^3) and the output size is O(V^2). Be cautious about running this on very large graphs.

The All-Pairs Shortest Path (APSP) task seeks to find the shortest paths between every pair of vertices in the entire graph. In principle, this task can be handled by running the Single-Source Shortest Path (SSSP) algorithm for each input vertex, e.g.,

CREATE QUERY all_pairs_shortest(SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr, STRING wt_type, STRING result_attr = "", STRING file_path = ""){  Start = {v_type};  Result = SELECT s FROM Start:s        POST-ACCUM          shortest_ss_any_wt(s, v_type, e_type, wt_attr, wt_type,          result_attr, file_path+s);}

This example highlights one of the strengths of GSQL: treating queries as stored procedures that can be called from within other queries. We only show the result_attr and file_path options, because subqueries cannot send their JSON output.

For large graphs (with millions of vertices or more), however, this is an enormous task. While the massively parallel processing of the TigerGraph platform can speed up the computation by 10x or 100x, consider what it takes just to store or report the results. If there are 1 million vertices, then there are nearly 1 trillion output values.

There are more efficient methods than calling the single-source shortest path algorithm n times, such as the Floyd-Warshall algorithm, which computes APSP in O(V^3) time.

Our recommendation:

• If you have a smaller graph (perhaps thousands or tens of thousands of vertices), the APSP task may be tractable.

• If you have a large graph, avoid using APSP.

## Minimum Spanning Tree (MST)

### Description and Uses

Given an undirected and connected graph, a minimum spanning tree is a set of edges that can connect all the vertices in the graph with the minimal sum of edge weights. The library implements a parallel version of Prim's algorithm:

1. Start with a set A = { a single vertex seed }

2. For all vertices in A, select a vertex y such that

1. y is not in A, and

2. There is an edge from y to a vertex x in A, and

3. The weight of the edge e(x,y) is the smallest among all eligible pairs (x,y).

3. Add y to A, and add the edge (x,y) to MST.

4. Repeat steps 2 and 3 until A has all vertices in the graph.

If the user specifies a source vertex, this will be used as the seed. Otherwise, the algorithm will select a random seed vertex.

If the graph contains multiple components (i.e., some vertices are disconnected from the rest of the graph, then the algorithm will span only the component of the seed vertex.

If you do not have a preferred vertex, and the graph might have more than one component, then you should use the Minimum Spanning Forest (MDF) algorithm instead.

### Specifications

mst (VERTEX opt_source, SET<STRING> v_type, SET<STRING> e_type,  STRIN wt_attr, STRING wt_type, INT max_iter = -1,  BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "")
 Characteristic Value Result Computes a minimum spanning tree. If the JSON or file output selected, the output is the set of edges that form the MST. If the result_attr option is selected, the edges which are part of the MST are tagged True; other edges are tagged False. Input Parameters VERTEX opt_source: ID of a source vertex (optional)SET v_type: Names of vertex types to useSET e_type: Names of edge types to useSTRING wt_attr: Name of edge weight attributeSTRING wt_type: Data type of edge weight attribute: "INT", "FLOAT", or "DOUBLE"INT max_iter: Maximum of edges to include. If less than (V-1), then the result is not a true spanning tree. BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store result values (BOOL) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V - 1 = number of vertices - 1 Time Complexity O(V^2) Graph Types Undirected edges and connected

Example

In the graph social10, we consider only the undirected Coworker edges.

This graph has 3 components. Minimum Spanning Tree finds a tree for one component, so which component it will work on depends on what vertex we give as the starting point. If we select Fiona, George, Howard, or Ivy as the start vertex, then it works on the 4-vertex component on the left. You can start from any vertex in the component and get the same or an equivalent MST result.

The figure below shows the result of

# Use _ for default valuesRUN QUERY mst(("Ivy", "Person"), ["Person"], ["Coworker"] "weight", "INT",_, _, _, _)

Note that the value for the one vertex is ("Ivy", "Person"). In GSQL, this 2-tuple format which explicitly gives the vertex type is used when the query is written to accept a vertex of any type.

File output:

From,To,WeightIvy,Fiona,6Ivy,Howard,4Ivy,George,4

The attribute version requires a boolean attribute on the edge, and it will assign the attribute to "true" if that edge is selected in the MST:

## Minimum Spanning Forest (MSF)

### Description and Uses

Given an undirected graph with one or more connected components, a minimum spanning forest is a set of minimum spanning trees, one for each component. The library implements the algorithm in section 6.2 of Qin et al. 2014: http://www-std1.se.cuhk.edu.hk/~hcheng/paper/SIGMOD2014qin.pdf.

### Specifications

msf (SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr, STRING wt_type,BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "")
 Characteristic Value Result Computes a minimum spanning forest. If the JSON or file output selected, the output is the set of edges that form the MSF. If the result_attr option is selected, the edges which are part of the MSF are tagged True; other edges are tagged False. Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useSTRING wt_attr: Name of edge weight attributeSTRING wt_type: Data type of edge weight attribute: "INT", "FLOAT", or "DOUBLE"BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store result values (BOOL) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V - c,V = number of vertices, c = number of components Time Complexity O((V+E) * logV) Graph Types Undirected edges

Example

Refer to the example for the MST algorithm. This graph has 3 components. MSF will find an MST for each of the three components.

## Maximal Independent Set

### Description and Uses

An independent set of vertices does not contain any pair of vertices that are neighbors, i.e., ones which have an edge between them. A maximal independent set is the largest independent set that contains those vertices; you cannot improve upon it unless you start over with a different independent set. However, the search for the largest possible independent set (the maximum independent set, as opposed to the maximal independent set) is an NP-hard problem: there is no known algorithm that can find that answer in polynomial time. So we settle for the maximal independent set.

This algorithm finds use in applications wanting to find the most efficient configuration which "covers" all the necessary cases. For example, it has been used to optimize delivery or transit routes, where each vertex is one transit segment and each edge connects two segments that can NOT be covered by the same vehicle.

### Specifications

maximal_indep_set(STRING v_type, STRING e_type,INT max_iter = 100, BOOL print_accum = TRUE, STRING file_path = "")
 Characteristic Value Result A set of vertices that form a maximal independent set. Input Parameters STRING v_type: Name of vertex type to useSTRING e_type: Name of edge type to useINT max_iter: maximum number of iterations for the searchBOOL print_accum: If True, output JSON to standard outputSTRING file_path: If not empty, write output to this file. Result Size Size of the MIS: unknown. Worst case: If the graph is a set of N unconnected vertices, then the MIS is all N vertices. Time Complexity O(E), E = number of edges Graph Types Undirected edges

Example

Consider our social10 graph, with three components.

It is clear that for each of the two triangles -- (Alex, Bob, Justin) and (Chase, Damon, Eddie) -- we can select one vertex from each triangle to be part of the MIS. For the 4-vertex component (Fiona, George, Howard, Ivy), it is less clear what will happen. If the algorithm selects either George or Ivy, then no other independent vertices remain in the component. However, the algorithm could select both Fiona and Howard; they are independent of one another.

This demonstrates the uncertainty of the Maximal Independent Set algorithm and how it differs from Maximum Independent Set. A maximum independent set algorithm would always select Fiona and Howard, plus 2 others, for a total of 4 vertices. The maximal independent set algorithm relies on chance. It could return either 3 or 4 vertices.

## Cycle Detection

### Description and Uses

The Cycle Detection problem seeks to find all the cycles (loops) in a graph. We apply the usual restriction that the cycles must be "simple cycles", that is, they are paths that start and end at the same vertex but otherwise never visit any vertex twice.

There are two versions of the task: for directed graphs and undirected graphs. The GSQL algorithm library currently supports only directed cycle detection. The Rocha–Thatte algorithm is an efficient distributed algorithm, which detects all the cycles in a directed graph. The algorithm will self-terminate, but it is also possible to stop at k iterations, which finds all the cycles having lengths up to k edges.

The basic idea of the algorithm is to (potentially) traverse every edge in parallel, again and again, forming all possible paths. At each step, if a path forms a cycle, it records it and stops extending it. More specifically: Initialization: For each vertex, record one path consisting of its own id. Mark the vertex as Active.

Iteration steps: For each Active vertex v:

1. Send its list of paths to each of its out-neighbors.

2. Inspect each path P in the list of the paths received:

• If the first id in P is also id(v), a cycle has been found:

• Remove P from its list.

• If id(v) is the least id of any id in P, then add P to the Cycle List. (The purpose is to count each cycle only once.)

• Else, if id(v) is somewhere else in the path, then remove P from the path list (because this cycle must have been counted already).

• Else, append id(v) to the end of each of the remaining paths in its list.

### Specifications

cycle_detection (SET<STRING> v_type, SET<STRING> e_type, INT depth,  BOOL print_accum = TRUE, STRING file_path = "")
 Characteristic Value Result Computes a list of vertex id lists, each of which is a cycle. The result is available in 2 forms:streamed out in JSON formatwritten to a file in tabular format Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT depth: Maximum cycle length to search for = maximum number of iterationsBOOL print_accum: If True, output JSON to standard outputSTRING file_path: If not empty, write output to this file. Result Size Number of cycles * average cycle lengthBoth of these measures are not known in advance. Time Complexity O(E *k), E = number of edges.k = min(max. cycle length, depth parameter) Graph Types Directed

Example

In the social10 graph, there are 5 cycles, all with the Fiona-George-Howard-Ivy cluster.

[  {    "@@cycles": [      [        "Fiona",        "Ivy"      ],      [        "George",        "Ivy"      ],      [        "Fiona",        "George",        "Ivy"      ],      [        "George",        "Howard",        "Ivy"      ],      [        "Fiona",        "George",        "Howard",        "Ivy"      ]    ]  }]

## Estimated Diameter

### Description and Uses

The diameter of a graph is the worst-case length of a shortest path between any pair of vertices in a graph. It is the farthest distance to travel, to get from one vertex to another, if you always take the shortest path. Finding the diameter requires calculating (the lengths of) all shortest paths, which can be quite slow.

This algorithm uses a simple heuristic to estimate the diameter. rather than calculating the distance from each vertex to every other vertex, it selects K vertices randomly, where K is a user-provided parameter. It calculates the distances from each of these K vertices to all other vertices. So, instead of calculating V*(V-1) distances, this algorithm only calculates K*(V-1) distances. The higher the value of K, the greater the likelihood of hitting the actual longest shortest path.

The current version only computes unweighted distances.

This algorithm query employs a subquery called max_BFS_depth. Both queries are needed to run the algorithm.

### Specifications

estimate_diameter (SET<STRING> v_type, SET<STRING> e_type, INT seed_set_length,  BOOL print_accum = TRUE, STRING file_path = "", BOOL display = FALSE)
 Characteristic Value Result Returns the estimated value for the diameter of the graph Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT seed_set_length: The number (K) of random seed vertices to useBOOL print_accum: If True, output JSON to standard outputSTRING file_path: If not empty, write output to this file. Result Size one integer Time Complexity O(k*E), E = number of edges, k = number of seed vertices Graph Types Directed

# Centrality Algorithms

Centrality algorithms determine the importance of each vertex within a network. Typical applications:

PageRank is designed for directed edges. The classic interpretation is to find the most "important" web pages, based on hyperlink referrals, but it can be used for another network where entities make positive referrals of one another.

Closeness Centrality and Betweenness Centrality both deal with the idea of "centrally located."

## PageRank

### Description and Uses

The PageRank algorithm measures the influence of each vertex on every other vertex. PageRank influence is defined recursively: a vertex's influence is based on the influence of the vertices which refer to it. A vertex's influence tends to increase if (1) it has more referring vertices or if (2) its referring vertices have higher influence. The analogy to social influence is clear.

A common way of interpreting PageRank value is through the Random Network Surfer model. A vertex's PageRank score is proportional to the probability that a random network surfer will be at that vertex at any given time. A vertex with a high PageRank score is a vertex that is frequently visited, assuming that vertices are visited according to the following Random Surfer scheme:

• Assume a person travels or surfs across a network's structure, moving from vertex to vertex in a long series of rounds.

• The surfer can start anywhere. This start-anywhere property is part of the magic of PageRank, meaning the score is a truly fundamental property of the graph structure itself.

• Each round, the surfer randomly picks one of the outward connections from the surfer's current location. The surfer repeats this random walk for a long time.

• But wait. The surfer doesn't always follow the network's connection structure. There is a probability (1-damping, to be precise), that the surfer will ignore the structure and will magically teleport to a random vertex.

### Specifications

pageRank (STRING v_type, STRING e_type,  FLOAT max_change=0.001, INT max_iter=25, FLOAT damping=0.85, INT top_k = 100,   BOOL print_accum = TRUE, STRING result_attr =  "", STRING file_path = "",   BOOL display_edges = FALSE)
 Characteristic Value Result Computes a PageRank value (FLOAT type) for each vertex. Input Parameters STRING v_type: Names of vertex type to useSTRING e_type: Names of edge type to useFLOAT max_change: PageRank will stop iterating when the largest difference between any vertex's current score and its previous score ≤ max_change. That is, the scores have become very stable and are changing by less than max_change from one iteration to the next.INT max_iter: Maximum number of iterations.FLOAT damping: Fraction of score that is due to the score of neighbors. The balance (1 - damping) is a minimum baseline score that every vertex receives.INT top_k: Sort the scores highest first and output only this many scoresBOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store PageRank values (FLOAT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(E*k), E = number of edges, k = number of iterations.The number of iterations is data-dependent, but the user can set a maximum. Parallel processing reduces the time needed for computation. Graph Types Directed edges

### Example

 # Use _ for default values RUN QUERY pageRank("Person", "Friend", 0.001, 25, 0.85, 100 _, _, _, _)

We ran pageRank on our test10 graph (using Friend edges) with the following parameter values: damping=0.85, max_change=0.001, and max_iter=25. We see that Ivy (center bottom) has the highest pageRank score (1.12). This makes sense since there are 3 neighboring persons who point to Ivy, more than for any other person. Eddie and Justin have scores of exactly 1 because they do not have any out-edges. This is an artifact of our particular version pageRank. Likewise, Alex has a score of 0.15, which is (1-damping), because Alex has no in-edges.

## Weighted PageRank

### Description and Uses

The only difference between weighted PageRank and standard PageRank is that edges have weights, and the influence that a vertex receives from an in-neighbor is multiplied by the weight of the in-edge.

### Specifications

pageRank_wt (SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr,  FLOAT max_change=0.001, INT max_iter=25, FLOAT damping=0.85, INT top_k=100,   BOOL print_accum = TRUE, STRING result_attr =  "", STRING file_path = "",   BOOL display_edges = FALSE)
 Characteristic Value Result Computes a weighted PageRank value (FLOAT type) for each vertex. Input Parameters ​STRING v_type: Names of vertex type to useSTRING e_type: Names of edge type to useSTRING wt_attr: Name of edge weight attributeFLOAT max_change: PageRank will stop iterating when the largest difference between any vertex's current score and its previous score ≤ max_change. That is, the scores have become very stable and are changing by less than max_change from one iteration to the next.INT max_iter: Maximum number of iterations.FLOAT damping: Fraction of score that is due to the score of neighbors. The balance (1 - damping) is a minimum baseline score that every vertex receives.INT top_k: Sort the scores highest first and output only this many scoresBOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store PageRank values (FLOAT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(E*k), E = number of edges, k = number of iterations.The number of iterations is data-dependent, but the user can set a maximum. Parallel processing reduces the time needed for computation. Graph Types Directed edges

## Personalized PageRank

### Description and Uses

In the original PageRank, the damping factor is the probability of the surfer continues browsing at each step. The surfer may also stop browsing and start again from a random vertex. In personalized PageRank, the surfer can only start browsing from a given set of source vertices both at the beginning and after stopping.

### Specifications

pageRank_pers(SET<VERTEX> source, STRING e_type,FLOAT max_change=0.001, INT max_iter=25, FLOAT damping = 0.85, INT top_k = 100BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "")
 Characteristic Value Result Computes a personalized PageRank value (FLOAT type) for each vertex. Input Parameters SET source: Set of seed verticesSTRING e_type: Names of edge type to useFLOAT max_change: PageRank will stop iterating when the largest difference between any vertex's current score and its previous score ≤ max_change. That is, the scores have become very stable and are changing by less than max_change from one iteration to the next.INT max_iter: Maximum number of iterations.FLOAT damping: Fraction of score that is due to the score of neighbors. The balance (1 - damping) is a minimum baseline score that every vertex receives.INT top_k: Sort the scores highest first and output only this many scoresBOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store PageRank values (FLOAT) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V = number of vertices Time Complexity O(E*k), E = number of edges, k = number of iterations.The number of iterations is data-dependent, but the user can set a maximum. Parallel processing reduces the time needed for computation. Graph Types Directed edges

### Example

We ran Personalized PageRank on the graph social10 using Friend edges with the following parameter values:

# Using "_" to use default valuesRUN QUERY pageRank_pers([("Fiona","Person")], "Friend", _, _, _, _, _, _,_)

In this case, the random walker can only start or restart walking from Fiona. In the figure below, we see that Fiona has the highest PageRank score in the result. Ivy and George have the next highest scores because they are direct out-neighbors of Ivy and there are looping paths that lead back to them again. Half of the vertices have a score of 0 since they can not be reached from Fiona.

## Closeness Centrality

We all have an intuitive understanding when we say a home, an office, or a store is "centrally located." Closeness Centrality provides a precise measure of how "centrally located" a vertex is. The steps below show the steps for one vertex v:

 Step Mathematical Formula 1. Compute the average distance from vertex v to every other vertex: ​$d_{avg}(v) = \sum_{u \ne v} dist(v,u)/(n-1)$ 2. Invert the average distance, so we have average closeness of v: ​$CC(v) = 1/d_{avg}(v)$

These steps are repeated for every vertex in the graph.

This algorithm query employs a subquery called cc_subquery. Both queries are needed to run the algorithm.

### Specifications

closeness_cent (SET<STRING> v_type, SET<STRING> e_type, INT max_hops=10,  INT top_k=100, BOOL wf = TRUE, BOOL print_accum = True, STRING result_attr = "",  STRING file_path = "", BOOL display_edges = FALSE)

Parameters

 Characteristic Value Result Computes a Closeness Centrality value (FLOAT type) for each vertex. Required Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT max_hops: If >=0, look only this far from each vertexINT top_k: Sort the scores highest first and output only this many scoresBOOL wf: If True, use Wasserman-Faust normalization for multi-component graphsBOOL print_accum: If true, output JSON to standard outputSTRING result_attr: If not empty, store centrality values (FLOAT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(E*k), E = number of edges, k = number of iterations.The number of iterations is data-dependent, but the user can set a maximum. Parallel processing reduces the time needed for computation. Graph Types Directed or Undirected edges, Unweighted edges

### Example

Closeness centrality can be measured for either directed edges (from v to others) or for undirected edges. Directed graphs may seem less intuitive, however, because if the distance from Alex to Bob is 1, it does not mean the distance from Bob to Alex is also 1.

For our example, we wanted to use the topology of the Likes graph, but to have undirected edges. We emulated an undirected graph by using both Friend and Also_Friend (reverse-direction) edges.

# Use _ for default valuesRUN QUERY closeness_cent(["Person"], ["Friend", "Also_Friend"], _, _, _, _, _, _, _)

## Approximate Closeness Centrality

In the Closeness Centrality algorithm, to obtain the closeness centrality score for a vertex, we measure the distance from the source vertex to every single vertex in the graph. In large graphs, running this calculation for every vertex can be highly time-consuming.

The Approximate Closeness Centrality algorithm (based on Cohen et al. 2014) calculates the approximate closeness centrality score for each vertex by combining two estimation approaches - sampling and pivoting. This hybrid estimation approach offers near-linear time processing and linear space overhead within a small relative error. It runs on graphs with unweighted edges (directed or undirected).

This query uses another subquerycloseness_cent_approx_sub, which needs to be installed before closeness_approx can be installed.

### Specifications

closeness_approx (    SET<STRING> v_type,     SET<STRING> e_type,		INT k = 100,  # sample num		INT max_hops = 10,  # max BFS explore steps 		DOUBLE epsilon = 0.1,  # error parameter    BOOL print_accum = true, # output to console		STRING file_path = "",  # output file 		INT debug = 0,  # debug flag -- 0: No LOG;1: LOG without the sample-node bfs loop;2: ALL LOG.		INT sample_index = 0,  # random sample group		INT maxsize = 1000,  # max size of connected components using exact closeness algorithm		BOOL wf = True # Wasserman and Faust formula )

### Parameters:

 Name Description v_type Vertex types to calculate approximate closeness centrality for. e_type Edge types to traverse. k Size of the sample. max_hops Upper limit of how many jumps the algorithm will perform from each vertex. epsilon The maximum relative error, between 0 and 1. Setting a lower value produces a more accurate estimate but increases run time. print_accum Boolean value that indicates whether or not to output to console in JSON format. file_path If provided, the algorithm will output to this file path in CSV format debug There are many conditional logging statements inside the query. If the input is 0, nothing will be logged. If the input is 1, everything else but the breadth-first-search process from the sample-node. If the input is 2, everything will be logged. sample_index The algorithm will partition the graph based on the sample size. This index indicates which partition to use when estimating closeness centrality. maxsize If the number of vertices in the graph is lower than maxsize, the exact closeness centrality is calculated instead and nothing will be approximated. wf Boolean value that indicates whether to use the Wasserman and Faust formula to calculate closeness centrality rather than the classic formula.

### Result

The result is a list of all vertices in the graph with their approximate closeness centrality score. It is available both in JSON and CSV format.

### Example

Below is an example of running the algorithm on the social10 test graph and an excerpt of the response.

RUN QUERY closeness_aprox(["Person"], ["Friend", "Coworker"], 6, 3   \0.1, true, "", 0, 0, 100, false)​[  {    "Start": [      {        "attributes": {          "[email protected]": 0.58333        },        "v_id": "Fiona",        "v_type": "Person"      },      {        "attributes": {          "[email protected]": 0.44444        },        "v_id": "Justin",        "v_type": "Person"      },      {        "attributes": {          "[email protected]": 0.53333        },        "v_id": "Bob",        "v_type": "Person"      }]

## Betweenness Centrality

The Betweenness Centrality of a vertex is defined as the number of shortest paths that pass through this vertex, divided by the total number of shortest paths. That is

$BC(v) =\sum_{s \ne v \ne t}PD_{st}(v)= \sum_{s \ne v \ne t} SP_{st}(v)/SP_{st} ,$

where $PD$is called the pair dependency, $SP_{st}$is the total number of shortest paths from node s to node t and $SP_{st}(v)$is the number of those paths that pass through v.

The TigerGraph implementation is based on A Faster Algorithm for Betweenness Centrality by Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, (2001). For every vertex s in the graph, the pair dependency starting from vertex s to all other vertices t via all other vertices v is computed first,

$PD_{s*}(v) = \sum_{t:s \in V} PD_{st}(v)$.

Then betweenness centrality is computed as

$BC(v) =\sum_{s:s \in V}PD_{s*}(v)/2$.

According to Brandes, the accumulated pair dependency can be calculated as

$PD_{s*}(v) =\sum_{w:v \in P_s(w)} SP_{sv}(v)/SP_{sw} \cdot (1+PD_{s*}(w)) ,$

where$P_s(w)$, the set of predecessors of vertex w on shortest paths from s, is defined as

$P_s(w) = \{u \in V: \{u, w\} \in E, dist(s,w) = dist(s,u)+dist(u,w) \} .$

For each single vertex, the algorithm works in two phases. The first phase calculates the number of shortest paths passing through each vertex. Then starting from the vertex on the most outside layer in a non-incremental order with pair dependency initial value of 0, traverse back to the starting vertex

This algorithm query employs a subquery called bc_subquery. Both queries are needed to run the algorithm.

Specifications

betweenness_cent(SET<STRING> v_type, SET<STRING> e_type, INT max_hops = 10,  INT top_k=100, BOOL print_accum=TRUE, STRING result_attr="", STRING file_path="",  BOOL display_edges = FALSE)

Parameters

 Characteristic Value Result Computes a Betweenness Centrality value (FLOAT type) for each vertex. Required Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT max_hops: If >=0, look only this far from each vertexINT top_k: Sort the scores highest first and output only this many scoresBOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store centrality values (FLOAT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL display_edges: If true, include the graph's edges in the JSON output, so that the full graph can be displayed. Result Size V = number of vertices Time Complexity O(E*V), E = number of edges, V = number of vertices.Considering the high time cost of running this algorithm on a big graph, the users can set a maximum number of iterations. Parallel processing reduces the time needed for computation. Graph Types Undirected edges, Unweighted edges

### Example

In the example below, Claire is in the very center of the social graph, and has the highest betweenness centrality. Six shortest paths pass through Sam (i.e. paths from Victor to all other 6 people except for Sam and Victor), so the score of Sam is 6. David also has a score of 6, since Brian has 6 paths to other people that pass through David.

# Use _ for default valuesRUN QUERY (["Person"], ["Friend"], _, _, _, _, _, _)
[  {    "@@BC": {      "Alice": 0,      "Frank": 0,      "Claire": 17,      "Sam": 6,      "Brian": 0,      "David": 6,      "Richard": 0,      "Victor": 0    }  }]

In the following example, both Charles and David have 9 shortest paths passing through them. Ellen is in a similar position as Charles, but her centrality is weakened due to the path between Frank and Jack.

[  {    "@@BC": {      "Alice": 0,      "Frank": 0,      "Charles": 9,      "Ellen": 8,      "Brian": 0,      "David": 9,      "Jack": 0    }  }]

# Community Algorithms

These algorithms evaluate how a group is clustered or partitioned, as well as its tendency to strengthen or break apart.

## Connected Components

### Description and Uses

A component is the maximal set of vertices, plus their connecting edges, which are interconnected. That is, you can reach each vertex from each other vertex. In the example figure below, there are three components.

This particular algorithm deals with undirected edges. If the same definition (each vertex can reach each other vertex) is applied to directed edges, then the components are called Strongly Connected Components. If you have directed edges but ignore the direction (permitting traversal in either direction), then the algorithm finds Weakly Connected Components.

### Specifications

conn_comp (SET<STRING> v_type, SET<STRING> e_type, INT output_limit = 100,  BOOL print_accum = TRUE, STRING result_attr = "", STRING file_path = "")
 Characteristic Value Result Assigns a component id (INT) to each vertex, such that members of the same component have the same id value. Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT output_limit: If >=0, max number of vertices to output to JSON.BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store community ID values (INT) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V = number of vertices Time Complexity O(E*d), E = number of edges, d = max(diameter of components) Graph Types Undirected edges

### Example

It is easy to see in this small graph that the algorithm correctly groups the vertices:

• Alex, Bob, and Justin all have Community ID = 2097152

• Chase, Damon, and Eddie all have Community ID = 5242880

• Fiona, George, Howard, and Ivy all have Community ID = 0

Our algorithm uses the TigerGraph engine's internal vertex ID numbers; they cannot be predicted.

RUN QUERY conn_comp(["Person"], ["Coworker"], _, _, _, _)

## Strongly Connected Components

### Description and Uses

A strongly connected component (SCC) is a subgraph such that there is a path from any vertex to every other vertex. A graph can contain more than one separate SCC. An SCC algorithm finds the maximal SCCs within a graph. Our implementation is based on the Divide-and-Conquer Strong Components (DCSC) algorithm[1]. In each iteration, pick a pivot vertex v randomly, and find its descendant and predecessor sets, where descendant set D_v is the vertex reachable from v, and predecessor set P_v is the vertices which can reach v (stated another way, reachable from v through reverse edges). The intersection of these two sets is a strongly connected component SCC_v. The graph can be partitioned into 4 sets: SCC_v, descendants D_v excluding SCC_v, predecessors P_v excluding SCC, and the remainders R_v. It is proved that any SCC is a subset of one of the 4 sets [1]. Thus, we can divide the graph into different subsets and detect the SCCs independently and iteratively.

The problem of this algorithm is unbalanced load and slow convergence when there are a lot of small SCCs, which is often the case in real-world use cases [3]. We added two trimming stages to improve the performance: size-1 SCC trimming[2] and weakly connected components[3].

The implementation of this algorithm requires reverse edges for all directed edges considered in the graph.

[1] Fleischer, Lisa K., Bruce Hendrickson, and Ali Pınar. "On identifying strongly connected components in parallel." International Parallel and Distributed Processing Symposium. Springer, Berlin, Heidelberg, 2000.

[2] Mclendon Iii, William, et al. "Finding strongly connected components in distributed graphs." Journal of Parallel and Distributed Computing 65.8 (2005): 901-910.

[3] Hong, Sungpack, Nicole C. Rodia, and Kunle Olukotun. "On fast parallel detection of strongly connected components (SCC) in small-world graphs." Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis. ACM, 2013.

### Specifications

scc (SET<STRING> v_type, SET<STRING> e_type, SET<STRING> rev_e_type,  INT top_k_dist, INT output_limit, INT max_iter = 500, INT iter_wcc = 5,  BOOL print_accum = TRUE, STRING attr= "", STRING file_path="")
 Characteristic Value Result Assigns a component id (INT) to each vertex, such that members of the same component have the same id value. Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useSET rev_e_type: Names of reverse direction edge types to useINT top_k_dist: top k result in SCC distributionINT output_limit: If >=0, max number of vertices to output to JSON.INT max_iter: number of maximum iteration of the algorithmINT iter_wcc: find weakly connected components for the active vertices in this iteration, since the largest SCCs are already found after several iterations; usually a small number(3 to 10)BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store community ID values (INT) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V = number of vertices Time Complexity O(iter*d), d = max(diameter of components) Graph Types Directed edges with reverse direction edges as well

### Example

We ran scc on the social26 graph. A portion of the JSON result is shown below.

[  {    "i": 1  },  {    "trim_set.size()": 8  },  {    "trim_set.size()": 5  },  {    "trim_set.size()": 2  },  {    "trim_set.size()": 2  },  {    "trim_set.size()": 0  },  {    "@@cluster_dist_heap": [      {        "csize": 9,        "num": 1      },      {        "csize": 1,        "num": 17      }    ]  },

The first element "i"=1 means the whole graph is processed in just one iteration. The 5 "trim_set.size()" elements mean there were 5 rounds of size-1 SCC trimming. The final "@@.cluster_dist_heap" object" reports on the size distribution of SCCs.There is one SCC with 9 vertices, and 17 SCCs with only 1 vertex in the graph.

## Label Propagation

### Description and Uses

Label Propagation is a heuristic method for determining communities. The idea is simple: If the plurality of your neighbors all bear the label X, then you should label yourself as also a member of X. The algorithm begins with each vertex having its own unique label. Then we iteratively update labels based on the neighbor influence described above. It is important that the order for updating the vertices be random. The algorithm is favored for its efficiency and simplicity, but it is not guaranteed to produce the same results every time.

In a variant version, some vertices could initially be known to belong to the same community. If they are well-connected to one another, they are likely to preserve their common membership and influence their neighbors,

### Specifications

label_prop (SET<STRING> v_type, SET<STRING> e_type, INT max_iter, INT output_limit,BOOL print_accum = TRUE, STRING file_path = "", STRING attr = "")
 Characteristic Value Result Assigns a component id (INT) to each vertex, such that members of the same component have the same id value. Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useINT max_iter: Number of maximum iteration of the algorithmINT output_limit: If >=0, max number of vertices to output to JSON.BOOL print_accum: If True, output JSON to standard outputSTRING attr: If not empty, store community id values (INT) to this attributeSTRING file_path: If not empty, write output to this file. Result Size V = number of vertices Time Complexity O(E*k), E = number of edges, k = number of iterations. Graph Types Undirected edges

### Example

This is the same graph that was used in the Connected Component example. The results are different, though. The quartet of Fiona, George, Howard, and Ivy have been split into 2 groups:

• (George & Ivy) each connect to (Fiona & Howard) and to one another.

• (Fiona & Howard) each connect to (George & Ivy) but not to one another.

Label Propagation tries to find natural clusters and separations within connected components. That is, it looks at the quality and pattern of connections. The Component Component algorithm simply asks the Yes or No question: Are these two vertices connected?

We set max_iter to 10, but the algorithm reaches a steady state after 3 iterations:

# Use _ for default/empty valuesRUN QUERY(["Person"], ["Coworker"], 10, -1, _, _, _)

## Louvain Method with Parallelism and Refinement

### Description and Uses

The Louvain Method for community detection [1] partitions the vertices in a graph by approximately maximizing the graph's modularity score. The modularity score for a partitioned graph assesses the difference in density of links within a partition vs. the density of links crossing from one partition to another. The assumption is that if a partitioning is good (that is, dividing up the graph into communities or clusters), then the within-density should be high and the inter-density should be low.

The most efficient and empirically effective method for calculating modularity was published by a team of researchers at the University of Louvain. The Louvain method uses agglomeration and hierarchical optimization:

1. Optimize modularity for small local communities.

2. Treat each optimized local group as one unit, and repeat the modularity operation for groups of these condensed units.

The original Louvain Method contains two phases. The first phase incrementally calculates the modularity change of moving a vertex into every other community and moves the vertex to the community with the highest modularity change. The second phase coarsens the graph by aggregating the vertices which are assigned in the same community into one vertex. The first phase and second phase make up a pass. The Louvain Method performs the passes iteratively. In other words, the algorithm assigns an initial community label to every vertex, then performs the first phase, during which the community labels are changed until there is no modularity gain. Then it aggregates the vertices with the same labels into one vertex and calculates the aggregated edge weights between new vertices. For the coarsened graph, the algorithm conducts the first phase again to move the vertices into new communities. The algorithm continues until the modularity is not increasing, or runs to the preset iteration limits.

However, phase one is sequential, and thus slow for large graphs. An improved Parallel Louvain Method Louvain Method (PLM) calculates the best community to move to for each vertex in parallel [2]. In Parallel Louvain Method(PLM), the positive modularity gain is not guaranteed, and it may also swap two vertices to each other’s community. After finishing the passes, there is an additional refinement phase, which is running the first phase again on each vertex to do some small adjustments for the resulting communities. [3].

[1] Blondel, Vincent D., et al. "Fast unfolding of communities in large networks." Journal of statistical mechanics: theory and experiment 2008.10 (2008): P10008.

[2] Staudt, Christian L., and Henning Meyerhenke. "Engineering parallel algorithms for community detection in massive networks." IEEE Transactions on Parallel and Distributed Systems 27.1 (2016): 171-184.

[3] Lu, Hao, Mahantesh Halappanavar, and Ananth Kalyanaraman. "Parallel heuristics for scalable community detection." Parallel Computing 47 (2015): 19-37.

### Specifications

louvain_parallel (SET<STRING> v_type, SET<STRING> e_type, STRING wt_attr,  INT iter1=10, INT iter2=10, INT iter3=10, INT split=10, BOOL print_accum = TRUE,  STRING result_attr = "", STRING file_path = "", BOOL comm_by_size = TRUE)
 Characteristic Value Result Assigns a component id (INT) to each vertex, such that members of the same component have the same id value. The JSON output lists every vertex with its community ID value. It also lists community id values, sorted by community size. Input Parameters SET v_type: Names of vertex types to useSET e_type: Names of edge types to useSTRING wt_attr: Name of edge weight attribute (must be FLOAT)INT iter1: Max number of iterations for the first phase. Default value is 10INT iter2: Max number of iterations for the second phase. Default value is 10INT iter3: Max number of iterations for the refinement phase. Default value is 10INT split: Number of splits in phase 1. Increase the number to save memory, at the expense of having a longer running time. Default value is 10.BOOL print_accum: If True, output JSON to standard outputSTRING result_attr: If not empty, store community id values (INT) to this attributeSTRING file_path: If not empty, write output to this file.BOOL comm_by_size: If True, and if print_accum is True, output the membership of each community, with communities arranged by size. Result Size V = number of vertices Time Complexity O(V^2*L), V = number of vertices, L = (iter1 * iter2 + iter3) = total number of iterations Graph Types Undirected, weighted edgesAn edge weight attribute is required.

### Example

If we use louvain_parallel for social10 graph, it will give the same result as the connected components algorithm. The social26 graph is a densely connected graph. The connected components algorithm groups all the vertices into the same community and label propagation does not consider the edge weight. On the contrary, louvain_parallel detects 7 communities in total, and the cluster distribution is shown below (csize is cluster size):

{    "@@clusterDist": [      {        "csize": 2,        "number": 1      },      {        "csize": 3,        "number": 2      },      {        "csize": 4,        "number": 2      },      {        "csize": 5,        "number": 2      }    ]

## Triangle Counting

### Description and Uses

Why triangles? Think of it in terms of a social network:

• If A knows B, and A also knows C, then we complete the triangle if B knows C. If this situation is common, it indicates a community with a lot of interaction.

• The triangle is in fact the smallest multi-edge "complete subgraph," where every vertex connects to every other vertex.

Triangle count (or density) is a measure of community and connectedness. In particular, it addresses the question of transitive relationships: If A--> B and B-->C, then what is the likelihood of A--> C?

Note that it is computing a single number: How many triangles are in this graph? It is not finding communities within a graph.

It is not common to count triangles in directed graphs, though it is certainly possible. If you choose to do so, you need to be very specific about the direction of interest: In a directed graph, If A--> B and B--> C, then

• if A-->C, we have a "shortcut".

• if C-->A, then we have a feedback loop.

### Specifications

The tri_count algorithm is in template format. It is not yet in schema-free format.

We present two different algorithms for counting triangles. The first, tri_count(), is the classic edge-iterator algorithm. For each edge and its two endpoint vertices S and T, count the overlap between S's neighbors and T's neighbors.

tri_count()tri_count_file(FILE filepath)tri_count_attr()

One side effect of the simple edge-iterator algorithm is that it ends up considering each of the three sides of a triangle. The count needs to be divided by 3, meaning we did 3 times more work than a smaller algorithm would have.

tri_count_fast() is a smarter algorithm which does two passes over the edges. In the first pass we mark which of the two endpoint vertices has fewer neighbors. In the second pass, we count the overlap only between marked vertices. The result is that we eliminate 1/3 of the neighborhood matching, the slowest 1/3, but at the cost of some additional memory.

tri_count_fast()tri_count_fast_file(FILE filepath)tri_count_fast_attr()
 Characteristic Value Result Returns the number of triangles in the graph. Input Parameters None Result Size 1 integer Time Complexity O(V * E), V = number of vertices, E = number of edges Graph Types Undirected edges

### Example

In the social10 graph with Coworker edges, there are clearly 4 triangles.

{  "error": false,  "message": "",  "version": {    "edition": "developer",    "schema": 0,    "api": "v2"  },  "results": [    {"num_triangles": 4}  ]}

# Similarity Algorithms

There are many ways to measure the similarity between two vertices in a graph, but all of them compare either (1) the features of the vertices themselves, (2) the relationships of each of the two vertices, or (3) both. We use a graph called movie to demonstrate our similarity algorithms.

## Cosine Similarity of Neighborhoods, Single Source

### Description and Uses

To compare two vertices by cosine similarity, the selected properties of each vertex are first represented as a vector. For example, a property vector for a Person vertex could have the elements age, height, and weight. Then the cosine function is applied to the two vectors.

The cosine similarity of two vectors A and B is defined as follows:

$cos(A,B)=\frac{A\cdot B}{||A||\cdot ||B||} = \frac{\sum_iA_i B_i}{\sqrt{\sum_iA^2_i}\sqrt{\sum_iB^2_i}}$

If A and B are identical, then cos(A, B) = 1. As expected for a cosine function, the value can also be negative or zero. In fact, cosine similarity is closely related to the Pearson correlation coefficient.

For this library function, the feature vector is the set of edge weights between the two vertices and their neighbors.

In the movie graph shown in the figure below, there are Person vertices and Movie vertices. Every person may give a rating to some of the movies. The rating score is stored on the Likes edge using the weight attribute. For example, in the graph below, Alex gives a rating of 10 to the movie "Free Solo".

### Specifications

cosine_nbor_ss (VERTEX source, SET<STRING> e_type