Abdelrahman Said
384 lines
18 KiB
HTML
384 lines
18 KiB
HTML
<html>
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<title>Chapter 26. Graphlets</title>
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<link rel="prev" href="igraph-Community.html" title="Chapter 25. Detecting community structure">
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<link rel="chapter" href="igraph-Installation.html" title="Chapter 2. Installation">
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<link rel="chapter" href="igraph-Memory.html" title="Chapter 6. Memory (de)allocation">
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<link rel="chapter" href="igraph-Data-structures.html" title="Chapter 7. Data structure library: vector, matrix, other data types">
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<link rel="chapter" href="igraph-Random.html" title="Chapter 8. Random numbers">
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<link rel="chapter" href="igraph-Iterators.html" title="Chapter 9. Vertex and edge selectors and sequences, iterators">
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<link rel="chapter" href="igraph-Attributes.html" title="Chapter 10. Graph, vertex and edge attributes">
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<link rel="chapter" href="igraph-Generators.html" title="Chapter 11. Deterministic graph generators">
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<link rel="chapter" href="igraph-Games.html" title='Chapter 12. Stochastic graph generators ("games")'>
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<link rel="chapter" href="igraph-Bipartite.html" title="Chapter 13. Bipartite, i.e. two-mode graphs">
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<link rel="chapter" href="igraph-Spatial.html" title="Chapter 14. Spatial graphs">
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<link rel="chapter" href="igraph-Operators.html" title="Chapter 15. Graph operators">
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<link rel="chapter" href="igraph-Visitors.html" title="Chapter 16. Graph visitors">
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<link rel="chapter" href="igraph-Structural.html" title="Chapter 17. Structural properties of graphs">
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<link rel="chapter" href="igraph-Cycles.html" title="Chapter 18. Graph cycles">
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<link rel="chapter" href="igraph-Cliques.html" title="Chapter 19. Cliques and independent vertex sets">
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<link rel="chapter" href="igraph-Motifs.html" title="Chapter 20. Graph motifs, dyad census and triad census">
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<link rel="chapter" href="igraph-Isomorphism.html" title="Chapter 21. Graph isomorphism">
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<link rel="chapter" href="igraph-Coloring.html" title="Chapter 22. Graph coloring">
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<link rel="chapter" href="igraph-Flows.html" title="Chapter 23. Maximum flows, minimum cuts and related measures">
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<link rel="chapter" href="igraph-Separators.html" title="Chapter 24. Vertex separators">
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<link rel="chapter" href="igraph-Community.html" title="Chapter 25. Detecting community structure">
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<link rel="chapter" href="igraph-Graphlets.html" title="Chapter 26. Graphlets">
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<link rel="chapter" href="igraph-HRG.html" title="Chapter 27. Hierarchical random graphs">
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<link rel="chapter" href="igraph-Embedding.html" title="Chapter 28. Embedding of graphs">
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<link rel="chapter" href="igraph-Layout.html" title="Chapter 29. Generating layouts for graph drawing">
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<link rel="chapter" href="igraph-Processes.html" title="Chapter 30. Processes on graphs">
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<link rel="chapter" href="igraph-Foreign.html" title="Chapter 31. Reading and writing graphs from and to files">
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<link rel="chapter" href="igraph-Linalg.html" title="Chapter 32. Using BLAS, LAPACK and ARPACK for igraph matrices and graphs">
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<link rel="chapter" href="igraph-Nongraph.html" title="Chapter 33. Non-graph related functions">
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<link rel="chapter" href="igraph-Advanced.html" title="Chapter 34. Advanced igraph programming">
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<link rel="chapter" href="igraph-Glossary.html" title="Chapter 35. Glossary">
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<link rel="chapter" href="igraph-Licenses.html" title="Chapter 36. Licenses for igraph and this manual">
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</div></div>
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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="igraph-Graphlets"></a>Chapter 26. Graphlets</h1></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="igraph-Graphlets.html#about-graphlets">1. Introduction</a></span></dt>
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<dt><span class="section"><a href="igraph-Graphlets.html#performing-graphlet-decomposition">2. Performing graphlet decomposition</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="about-graphlets"></a>1. Introduction</h2></div></div></div>
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<p>
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Graphlet decomposition models a weighted undirected graph
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via the union of potentially overlapping dense social groups.
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This is done by a two-step algorithm. In the first step, a candidate
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set of groups (a candidate basis) is created by finding cliques
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in the thresholded input graph. In the second step,
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the graph is projected onto the candidate basis, resulting in a
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weight coefficient for each clique in the candidate basis.
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</p>
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<p>
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For more information on graphlet decomposition, see
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Hossein Azari Soufiani and Edoardo M Airoldi: "Graphlet decomposition of a weighted network",
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<a class="ulink" href="https://arxiv.org/abs/1203.2821" target="_top">https://arxiv.org/abs/1203.2821</a> and <a class="ulink" href="http://proceedings.mlr.press/v22/azari12/azari12.pdf" target="_top">http://proceedings.mlr.press/v22/azari12/azari12.pdf</a>
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</p>
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<p>
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igraph contains three functions for performing the graphlet
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decomponsition of a graph. The first is <a class="link" href="igraph-Graphlets.html#igraph_graphlets" title="2.1. igraph_graphlets — Calculate graphlets basis and project the graph on it."><code class="function">igraph_graphlets()</code></a>, which
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performs both steps of the method and returns a list of subgraphs
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with their corresponding weights. The other two functions
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correspond to the first and second steps of the algorithm, and they are
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useful if the user wishes to perform them individually:
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<a class="link" href="igraph-Graphlets.html#igraph_graphlets_candidate_basis" title="2.2. igraph_graphlets_candidate_basis — Calculate a candidate graphlets basis"><code class="function">igraph_graphlets_candidate_basis()</code></a> and
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<a class="link" href="igraph-Graphlets.html#igraph_graphlets_project" title="2.3. igraph_graphlets_project — Project a graph on a graphlets basis."><code class="function">igraph_graphlets_project()</code></a>.
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</p>
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<p>
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<em><span class="remark">
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Note: The term "graphlet" is used for several unrelated concepts
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in the literature. If you are looking to count induced subgraphs, see
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<a class="link" href="igraph-Motifs.html#igraph_motifs_randesu" title="4.1. igraph_motifs_randesu — Count the number of motifs in a graph."><code class="function">igraph_motifs_randesu()</code></a> and <a class="link" href="igraph-Isomorphism.html#igraph_subisomorphic_lad" title="4.1. igraph_subisomorphic_lad — Check subgraph isomorphism with the LAD algorithm"><code class="function">igraph_subisomorphic_lad()</code></a>.
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</span></em>
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="performing-graphlet-decomposition"></a>2. Performing graphlet decomposition</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="igraph-Graphlets.html#igraph_graphlets">2.1. <code class="function">igraph_graphlets</code> — Calculate graphlets basis and project the graph on it.</a></span></dt>
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<dt><span class="section"><a href="igraph-Graphlets.html#igraph_graphlets_candidate_basis">2.2. <code class="function">igraph_graphlets_candidate_basis</code> — Calculate a candidate graphlets basis</a></span></dt>
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<dt><span class="section"><a href="igraph-Graphlets.html#igraph_graphlets_project">2.3. <code class="function">igraph_graphlets_project</code> — Project a graph on a graphlets basis.</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_graphlets"></a>2.1. <code class="function">igraph_graphlets</code> — Calculate graphlets basis and project the graph on it.</h3></div></div></div>
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<a class="indexterm" name="id-1.27.3.2.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_graphlets(const igraph_t *graph,
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const igraph_vector_t *weights,
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igraph_vector_int_list_t *cliques,
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igraph_vector_t *Mu, igraph_int_t niter);
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</pre></div>
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<p>
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</p>
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<p>
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This function simply calls <a class="link" href="igraph-Graphlets.html#igraph_graphlets_candidate_basis" title="2.2. igraph_graphlets_candidate_basis — Calculate a candidate graphlets basis"><code class="function">igraph_graphlets_candidate_basis()</code></a>
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and <a class="link" href="igraph-Graphlets.html#igraph_graphlets_project" title="2.3. igraph_graphlets_project — Project a graph on a graphlets basis."><code class="function">igraph_graphlets_project()</code></a>, and then orders the graphlets
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according to decreasing weights.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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The input graph, it must be a simple graph, edge directions are
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ignored.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>weights</code></em>:</span></p></td>
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<td><p>
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Weights of the edges, a vector.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>cliques</code></em>:</span></p></td>
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<td><p>
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An initialized list of integer vectors. The graphlet basis is
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stored here. Each element of the list is an integer vector of
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vertex IDs, encoding a single basis subgraph.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>Mu</code></em>:</span></p></td>
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<td><p>
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An initialized vector, the weights of the graphlets will
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be stored here.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>niter</code></em>:</span></p></td>
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<td><p>
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The number of iterations to perform for the projection step.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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See also: <a class="link" href="igraph-Graphlets.html#igraph_graphlets_candidate_basis" title="2.2. igraph_graphlets_candidate_basis — Calculate a candidate graphlets basis"><code class="function">igraph_graphlets_candidate_basis()</code></a> and
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<a class="link" href="igraph-Graphlets.html#igraph_graphlets_project" title="2.3. igraph_graphlets_project — Project a graph on a graphlets basis."><code class="function">igraph_graphlets_project()</code></a>.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_graphlets_candidate_basis"></a>2.2. <code class="function">igraph_graphlets_candidate_basis</code> — Calculate a candidate graphlets basis</h3></div></div></div>
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<a class="indexterm" name="id-1.27.3.3.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_graphlets_candidate_basis(const igraph_t *graph,
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const igraph_vector_t *weights,
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igraph_vector_int_list_t *cliques,
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igraph_vector_t *thresholds);
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</pre></div>
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<p>
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</p>
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<p>
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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The input graph, it must be a simple graph, edge directions are
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ignored.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>weights</code></em>:</span></p></td>
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<td><p>
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Weights of the edges, a vector.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>cliques</code></em>:</span></p></td>
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<td><p>
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An initialized list of integer vectors. The graphlet basis is
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stored here. Each element of the list is an integer vector of
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vertex IDs, encoding a single basis subgraph.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>thresholds</code></em>:</span></p></td>
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<td><p>
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An initialized vector, the (highest possible)
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weight thresholds for finding the basis subgraphs are stored
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here.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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See also: <a class="link" href="igraph-Graphlets.html#igraph_graphlets" title="2.1. igraph_graphlets — Calculate graphlets basis and project the graph on it."><code class="function">igraph_graphlets()</code></a> and <a class="link" href="igraph-Graphlets.html#igraph_graphlets_project" title="2.3. igraph_graphlets_project — Project a graph on a graphlets basis."><code class="function">igraph_graphlets_project()</code></a>.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_graphlets_project"></a>2.3. <code class="function">igraph_graphlets_project</code> — Project a graph on a graphlets basis.</h3></div></div></div>
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<a class="indexterm" name="id-1.27.3.4.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_graphlets_project(const igraph_t *graph,
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const igraph_vector_t *weights,
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const igraph_vector_int_list_t *cliques,
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igraph_vector_t *Mu, igraph_bool_t startMu,
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igraph_int_t niter);
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</pre></div>
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<p>
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</p>
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<p>
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Note that the graph projected does not have to be the same that
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was used to calculate the graphlet basis, but it is assumed that
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it has the same number of vertices, and the vertex IDs of the two
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graphs match.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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The input graph, it must be a simple graph, edge directions are
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ignored.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>weights</code></em>:</span></p></td>
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<td><p>
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Weights of the edges in the input graph, a vector.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>cliques</code></em>:</span></p></td>
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<td><p>
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An initialized list of integer vectors. The graphlet basis is
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stored here. Each element of the list is an integer vector of
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vertex IDs, encoding a single basis subgraph.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>Mu</code></em>:</span></p></td>
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<td><p>
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An initialized vector, the weights of the graphlets will
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be stored here. This vector is also used to initialize the
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the weight vector for the iterative algorithm, if the
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<code class="constant">startMu</code> argument is true.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>startMu</code></em>:</span></p></td>
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<td><p>
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If true, then the supplied Mu vector is
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used as the starting point of the iteration. Otherwise a
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constant 1 vector is used.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>niter</code></em>:</span></p></td>
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<td><p>
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The number of iterations to perform.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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See also: <a class="link" href="igraph-Graphlets.html#igraph_graphlets" title="2.1. igraph_graphlets — Calculate graphlets basis and project the graph on it."><code class="function">igraph_graphlets()</code></a> and
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<a class="link" href="igraph-Graphlets.html#igraph_graphlets_candidate_basis" title="2.2. igraph_graphlets_candidate_basis — Calculate a candidate graphlets basis"><code class="function">igraph_graphlets_candidate_basis()</code></a>.
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</p>
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<table class="navigation-footer" width="100%" summary="Navigation footer" cellpadding="2" cellspacing="0"><tr valign="middle">
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<td align="left"><a accesskey="p" href="igraph-Community.html"><b>← Chapter 25. Detecting community structure</b></a></td>
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<td align="right"><a accesskey="n" href="igraph-HRG.html"><b>Chapter 27. Hierarchical random graphs →</b></a></td>
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