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<a href="#NAME">NAME</a>
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<a href="#SYNOPSIS">SYNOPSIS</a>
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<a href="#DESCRIPTION">DESCRIPTION</a>
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<a href="#PARAMETERS">PARAMETERS</a>
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<a href="#OUTPUT">OUTPUT</a>
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<a href="#EXAMPLES">EXAMPLES</a>
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<a href="#SEE-ALSO">SEE ALSO</a>
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<a href="#REFERENCES">REFERENCES</a>
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<ol class='man-decor man-head man head'>
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<li class='tl'>johnson_cycles(1)</li>
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<li class='tc'>www.complex-networks.net</li>
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<li class='tr'>johnson_cycles(1)</li>
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</ol>
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<h2 id="NAME">NAME</h2>
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<p class="man-name">
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<code>johnson_cycles</code> - <span class="man-whatis">Enumerate the simple cycles of a graph</span>
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</p>
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<h2 id="SYNOPSIS">SYNOPSIS</h2>
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<p><code>johnson_cycles</code> <var>graph_in</var> [<var>max_length</var> [SHOW]]</p>
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<h2 id="DESCRIPTION">DESCRIPTION</h2>
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<p><code>johnson_cycles</code> enumerates all the simple cycles of the graph given
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on input, and prints the total number of cycles of each length. If
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<var>max_length</var> is provided, <code>johnson_cycles</code> ignores any cycle whose
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length is larger than <var>max_length</var>. If <code>SHOW</code> is given as third
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argument, all the found cycles are printed on STDERR as soon as they
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are found.</p>
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<h2 id="PARAMETERS">PARAMETERS</h2>
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<ul>
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<li><p><var>graph_in</var>:
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input graph (edge list) if equal to <code>-</code> (dash), read the edge list
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from STDIN.</p></li>
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<li><p><var>max_length</var>: The maximum length of cycles to count/print. If not
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provided, <code>johnson_cycles</code> tries to find all the cycles of the
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graph (see the <strong>WARNING</strong> in the section <a href="#OUTPUT" title="OUTPUT" data-bare-link="true">OUTPUT</a> below).</p></li>
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<li><p>SHOW:
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If the third (optional) parameter is equal to <code>SHOW</code>, the program
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will dump on the standard error all the cycles found.</p></li>
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</ul>
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<h2 id="OUTPUT">OUTPUT</h2>
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<p><code>johnson_cycles</code> prints on the standard output the number of cycles of
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each length, in the format:</p>
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<pre><code> 2 N_2
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3 N_3
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4 N_4
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5 N_5
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...
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</code></pre>
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<p>where 2, 3, 4, 5... is the cycle lengths and N_2, N_3, N_4, N_5... is
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(twice) the number of cycles of that length. If <code>SHOW</code> is given, each
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cycle is also printed on STDERR as soon as it is found, in the format:</p>
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<pre><code>node_l node_(l-1) node_(l-2) ... node_0
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</code></pre>
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<p>where <code>node_l</code>, <code>node_(l-1)</code>, etc. are the labels of the nodes
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belonging to the cycle which starts at node <code>node_0</code>.</p>
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<p><strong>WARNING</strong>: If the second parameter <var>max_length</var> is not provided,
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<code>johnson_cycles</code> will try to enumerate <strong>all</strong> the cycles of the
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graph. In general, this might take a time <strong>exponential</strong> in the
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number of nodes and edges of the graph. As a consequence, specifying
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a maximum length is <strong>highly recommended</strong> if you are not interested
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in finding the number of cycles of any length.</p>
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<h2 id="EXAMPLES">EXAMPLES</h2>
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<p>We can count the cycle of any length in the graph of Florentine
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families using the command:</p>
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<pre><code> $ johnson_cycles florentine.net
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2 20
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3 6
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4 4
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5 6
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6 10
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7 20
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8 22
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9 8
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10 2
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11 0
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12 0
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13 0
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14 0
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15 0
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16 0
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$
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</code></pre>
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<p>The output means that the graph has 20 cycles of length 2 (edges),
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6/2=3 cycles of length 3, 4/2=2 cycles of length 4, and so on. We
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could otherwise focus on the cycles of length up to 5 and have each
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cycle printed on output:</p>
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<pre><code> $ johnson_cycles florentine.net 5 SHOW
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8 0
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5 1
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8 15 6 1
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8 12 15 6 1
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6 1
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6 15 12 8 1
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6 15 8 1
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8 1
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8 12 14 4 2
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....
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15 12
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2 20
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3 6
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4 4
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5 6
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$
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</code></pre>
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<p>Apart from the degenerate cycles like "8 0", "5 1", etc.,
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corresponding to the cycles obtained by traversing the same undirected
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edge in the two possible directions, we see in that list some of the
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cycles of length 4 (such as "8 15 6 1") and of length 5 (such as "8 12
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15 6 1").</p>
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<p>The enumeration of all the cycles is normally impractical on larger
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graphs, so it is highly recommended to limit the search to short
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sizes. For instance, the command:</p>
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<pre><code> $ johnson_cycles er_1000_5000.net 6
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2 5000
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3 340
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4 2406
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5 19416
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6 160554
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$
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</code></pre>
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<p>will require less than one second on a modern desktop computer, but
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the command:</p>
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<pre><code> $ johnson_cycles er_1000_5000.net 7
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2 5000
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3 340
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4 2406
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5 19416
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6 160554
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7 1360104
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$
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</code></pre>
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<p>will probably take about 15 seconds, while:</p>
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<pre><code> $ johnson_cycles er_1000_5000.net 8
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2 5000
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3 340
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4 2406
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5 19416
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6 160554
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7 1360104
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8 11743500
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$
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</code></pre>
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<p>will run for more than 2 minutes, and larger cycle lengths will
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require exponentially more time.</p>
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<h2 id="SEE-ALSO">SEE ALSO</h2>
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<p><span class="man-ref">f3m<span class="s">(1)</span></span>, <span class="man-ref">shortest<span class="s">(1)</span></span></p>
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<h2 id="REFERENCES">REFERENCES</h2>
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<ul>
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<li><p>D. B. Johnson. "Finding All the Elementary Circuits of a Directed
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Graph". SIAM J. Comput. 4 (1975), 77-84.</p></li>
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<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
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Methods and Applications", Chapter 8, Cambridge University Press
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(2017)</p></li>
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<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
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Methods and Applications", Appendix 15, Cambridge University Press
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(2017)</p></li>
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</ul>
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<h2 id="AUTHORS">AUTHORS</h2>
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<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code><v.nicosia@qmul.ac.uk></code>.</p>
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<ol class='man-decor man-foot man foot'>
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<li class='tl'>www.complex-networks.net</li>
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<li class='tc'>September 2017</li>
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<li class='tr'>johnson_cycles(1)</li>
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</ol>
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