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| <div class="title">PageRank<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div> </div> |
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| <div class="contents"> |
| <div class="toc"><b>Contents</b> <ul> |
| <li> |
| <a href="#pagerank">PageRank</a> </li> |
| <li> |
| <a href="#examples">Examples</a> </li> |
| <li> |
| <a href="#literature">Literature</a> </li> |
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| </div><p>Given a graph, the PageRank algorithm outputs a probability distribution representing the likelihood that a person randomly traversing the graph will arrive at any particular vertex. This algorithm was originally used by Google to rank websites where the World Wide Web was modeled as a directed graph with the vertices representing the websites. The PageRank algorithm initially proposed by Larry Page and Sergey Brin is implemented here [1].</p> |
| <p>We also implement personalized PageRank, in which a notion of importance provides personalization to a query. For example, importance scores can be biased according to a specified set of vertices in the graph that are of interest or special in some way [2].</p> |
| <p><a class="anchor" id="pagerank"></a></p><dl class="section user"><dt>PageRank</dt><dd><pre class="syntax"> |
| pagerank( vertex_table, |
| vertex_id, |
| edge_table, |
| edge_args, |
| out_table, |
| damping_factor, |
| max_iter, |
| threshold, |
| grouping_cols, |
| personalization_vertices |
| ) |
| </pre></dd></dl> |
| <p><b>Arguments</b> </p><dl class="arglist"> |
| <dt>vertex_table </dt> |
| <dd><p class="startdd">TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>vertex_id </dt> |
| <dd><p class="startdd">TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids are of type INTEGER with no duplicates. They do not need to be contiguous.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>edge_table </dt> |
| <dd><p class="startdd">TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex and destination vertex.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>edge_args </dt> |
| <dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:</p><ul> |
| <li>src (INTEGER): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.</li> |
| <li>dest (INTEGER): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.</li> |
| </ul> |
| <p class="enddd"></p> |
| </dd> |
| <dt>out_table </dt> |
| <dd><p class="startdd">TEXT. Name of the table to store the result of PageRank. It will contain a row for every vertex from 'vertex_table' with the following columns:</p><ul> |
| <li>vertex_id : The id of a vertex. Will use the input parameter 'vertex_id' for column naming.</li> |
| <li>pagerank : The vertex's PageRank.</li> |
| <li>grouping_cols : Grouping column (if any) values associated with the vertex_id.</li> |
| </ul> |
| <p>A summary table is also created that contains information regarding the number of iterations required for convergence. It is named by adding the suffix '_summary' to the 'out_table' parameter.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>damping_factor (optional) </dt> |
| <dd><p class="startdd">FLOAT8, default 0.85. The probability, at any step, that a user will continue following the links in a random surfer model.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>max_iter (optional) </dt> |
| <dd><p class="startdd">INTEGER, default: 100. The maximum number of iterations allowed.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>threshold (optional) </dt> |
| <dd><p class="startdd">FLOAT8, default: (1/number of vertices * 1000). If the difference between the PageRank of every vertex of two consecutive iterations is smaller than 'threshold', or the iteration number is larger than 'max_iter', the computation stops. If you set the threshold to zero, then you will force the algorithm to run for the full number of iterations specified in 'max_iter'. It is advisable to set threshold to a value lower than 1/(number of vertices in the graph) since the PageRank value of nodes is initialized to that value.</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>grouping_cols (optional) </dt> |
| <dd>TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one distribution per group. When this value is NULL, no grouping is used and a single model is generated for all data. <dl class="section note"><dt>Note</dt><dd>Expressions are not currently supported for 'grouping_cols'.</dd></dl> |
| </dd> |
| <dt>personalization_vertices (optional) </dt> |
| <dd>INTEGER[], default: NULL. A comma separated list of vertices or nodes for personalized PageRank. When this parameter is provided, personalized PageRank will run. In the absence of this parameter, regular PageRank will run. </dd> |
| </dl> |
| <dl class="section note"><dt>Note</dt><dd>On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.</dd></dl> |
| <p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl> |
| <ol type="1"> |
| <li>Create vertex and edge tables to represent the graph: <pre class="syntax"> |
| DROP TABLE IF EXISTS vertex, edge; |
| CREATE TABLE vertex( |
| id INTEGER |
| ); |
| CREATE TABLE edge( |
| src INTEGER, |
| dest INTEGER, |
| user_id INTEGER |
| ); |
| INSERT INTO vertex VALUES |
| (0), |
| (1), |
| (2), |
| (3), |
| (4), |
| (5), |
| (6); |
| INSERT INTO edge VALUES |
| (0, 1, 1), |
| (0, 2, 1), |
| (0, 4, 1), |
| (1, 2, 1), |
| (1, 3, 1), |
| (2, 3, 1), |
| (2, 5, 1), |
| (2, 6, 1), |
| (3, 0, 1), |
| (4, 0, 1), |
| (5, 6, 1), |
| (6, 3, 1), |
| (0, 1, 2), |
| (0, 2, 2), |
| (0, 4, 2), |
| (1, 2, 2), |
| (1, 3, 2), |
| (2, 3, 2), |
| (3, 0, 2), |
| (4, 0, 2), |
| (5, 6, 2), |
| (6, 3, 2); |
| </pre></li> |
| <li>Running PageRank with default values for optional parameters: <pre class="syntax"> |
| DROP TABLE IF EXISTS pagerank_out, pagerank_out_summary; |
| SELECT madlib.pagerank( |
| 'vertex', -- Vertex table |
| 'id', -- Vertix id column |
| 'edge', -- Edge table |
| 'src=src, dest=dest', -- Comma delimted string of edge arguments |
| 'pagerank_out'); -- Output table of PageRank |
| SELECT * FROM pagerank_out ORDER BY pagerank DESC; |
| </pre> <pre class="result"> |
| id | pagerank |
| ----+------------------- |
| 0 | 0.28753749341184 |
| 3 | 0.21016988901855 |
| 2 | 0.14662683454062 |
| 4 | 0.10289614384217 |
| 1 | 0.10289614384217 |
| 6 | 0.09728637768887 |
| 5 | 0.05258711765692 |
| (7 rows) |
| </pre> <pre class="syntax"> |
| SELECT * FROM pagerank_out_summary; |
| </pre> <pre class="result"> |
| __iterations__ |
| ----------------+ |
| 16 |
| (1 row) |
| </pre></li> |
| <li>Running PageRank with a damping factor of 0.5 results in different final values: <pre class="syntax"> |
| DROP TABLE IF EXISTS pagerank_out, pagerank_out_summary; |
| SELECT madlib.pagerank( |
| 'vertex', -- Vertex table |
| 'id', -- Vertix id column |
| 'edge', -- Edge table |
| 'src=src, dest=dest', -- Comma delimted string of edge arguments |
| 'pagerank_out', -- Output table of PageRank |
| 0.5); -- Damping factor |
| SELECT * FROM pagerank_out ORDER BY pagerank DESC; |
| </pre> <pre class="result"> |
| id | pagerank |
| ----+-------------------- |
| 0 | 0.225477161441199 |
| 3 | 0.199090328586664 |
| 2 | 0.136261327206477 |
| 6 | 0.132691559968224 |
| 4 | 0.109009291409508 |
| 1 | 0.109009291409508 |
| 5 | 0.0884610399788161 |
| (7 rows) |
| </pre></li> |
| <li>Now compute the PageRank of vertices associated with each user using the grouping feature: <pre class="syntax"> |
| DROP TABLE IF EXISTS pagerank_out, pagerank_out_summary; |
| SELECT madlib.pagerank( |
| 'vertex', -- Vertex table |
| 'id', -- Vertix id column |
| 'edge', -- Edge table |
| 'src=src, dest=dest', -- Comma delimted string of edge arguments |
| 'pagerank_out', -- Output table of PageRank |
| NULL, -- Default damping factor (0.85) |
| NULL, -- Default max iters (100) |
| 0.00000001, -- Threshold |
| 'user_id'); -- Grouping column name |
| SELECT * FROM pagerank_out ORDER BY user_id, pagerank DESC; |
| </pre> <pre class="result"> |
| user_id | id | pagerank |
| ---------+----+-------------------- |
| 1 | 0 | 0.27825488388552 |
| 1 | 3 | 0.20188114667075 |
| 1 | 2 | 0.14288112346059 |
| 1 | 6 | 0.11453637832147 |
| 1 | 1 | 0.10026745615438 |
| 1 | 4 | 0.10026745615438 |
| 1 | 5 | 0.06191155535288 |
| 2 | 0 | 0.31854625004173 |
| 2 | 3 | 0.23786686773343 |
| 2 | 2 | 0.15914876489397 |
| 2 | 1 | 0.11168334437971 |
| 2 | 4 | 0.11168334437971 |
| 2 | 6 | 0.03964285714285 |
| 2 | 5 | 0.02142857142857 |
| (14 rows) |
| </pre> <pre class="syntax"> |
| SELECT * FROM pagerank_out_summary ORDER BY user_id; |
| </pre> <pre class="result"> |
| user_id | __iterations__ |
| ---------+---------------- |
| 1 | 27 |
| 2 | 31 |
| (2 rows) |
| </pre></li> |
| <li>Personalized PageRank. Here we specify {2,4} as the personalization vertices. This parameter could be specified as ARRAY[2,4] as well. <pre class="syntax"> |
| DROP TABLE IF EXISTS pagerank_out, pagerank_out_summary; |
| SELECT madlib.pagerank( |
| 'vertex', -- Vertex table |
| 'id', -- Vertix id column |
| 'edge', -- Edge table |
| 'src=src, dest=dest', -- Comma delimted string of edge arguments |
| 'pagerank_out', -- Output table of PageRank |
| NULL, -- Default damping factor (0.85) |
| NULL, -- Default max iters (100) |
| NULL, -- Default Threshold |
| NULL, -- No Grouping |
| '{2,4}'); -- Personalization vertices |
| SELECT * FROM pagerank_out ORDER BY pagerank DESC; |
| </pre> <pre class="result"> |
| id | pagerank |
| ----+-------------------- |
| 0 | 0.565232961966315 |
| 2 | 0.378139420991773 |
| 3 | 0.355003292266017 |
| 4 | 0.310111215897626 |
| 1 | 0.160111215897626 |
| 6 | 0.148615315574136 |
| 5 | 0.0803403307142321 |
| (7 rows) |
| </pre> <pre class="syntax"> |
| SELECT * FROM pagerank_out_summary; |
| </pre> <pre class="result"> |
| __iterations__ |
| ----------------+ |
| 37 |
| (1 row) |
| </pre></li> |
| </ol> |
| <p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl> |
| <p>[1] Brin, S. and Page, L. (1998), "The anatomy of a large-scale hypertextual Web search engine", Computer Networks and ISDN Systems. 30: 107–117, <a href="http://infolab.stanford.edu/pub/papers/google.pdf">http://infolab.stanford.edu/pub/papers/google.pdf</a></p> |
| <p>[2] Jeh, Glen and Widom, Jennifer. "Scaling Personalized Web Search", Proceedings of the 12th international conference on World Wide Web, Pages 271-279 Budapest, Hungary, May 20-24, 2003, <a href="http://ilpubs.stanford.edu:8090/530/1/2002-12.pdf">http://ilpubs.stanford.edu:8090/530/1/2002-12.pdf</a> </p> |
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