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MCL_Presentation2

MCL_Presentation2 - Clustering on Graphs The Markov Cluster...

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Clustering on Graphs: The Markov Cluster Algorithm (MCL) CS 595D Presentation By Kathy Macropol
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MCL Algorithm s Based on the PhD thesis by Stijn van Dongen Van Dongen, S. (2000) Graph Clustering by Flow Simulation . PhD Thesis, University of Utrecht, The Netherlands. s MCL is a graph clustering algorithm. s MCL is freely available for download at http://www.micans.org/mcl/
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Outline s Background – Clustering – Random Walks – Markov Chains s MCL – Basis – Inflation Operator – Algorithm – Convergence s MCL Analysis – Comparison to Other Graph Clustering Algorithms • RNSC, SPC, MCODE • RRW s Conclusions
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Graph Clustering s Clustering – finding natural groupings of items. s Vector Clustering Graph Clustering Each point has a vector, i.e. • x coordinate • y coordinate • color 1 3 4 4 4 3 4 3 4 2 3 Each vertex is connected to others by (weighted or unweighted) edges.
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Random Walks s Considering a graph, there will be many links within a cluster, and fewer links between clusters. s This means if you were to start at a node, and then randomly travel to a connected node, you’re more likely to stay within a cluster than travel between. s This is what MCL (and several other clustering algorithms) is based on. – Other ways to consider graph clustering may include, for example, looking for cliques. This tends to be sensitive to changes in node degree, however.
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Random Walks s By doing random walks upon the graph, it may be possible to discover where the flow tends to gather, and therefore, where clusters are. s Random Walks on a graph are calculated using “Markov Chains”.
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Markov Chains s To see how this works, an example: s In one time step, a random walker at node 1 has a 33% chance s From node 2, 25% chance for 1, 3, 4, 5 and 0% for 6 and 7. s Creating a transition matrix gives: 1 2 3 4 5 6 7 0 .25 .33 .33 0 0 0 .33 0 .33 .33 .33 0 0 .33 .25 0 .33 0 0 0 .33 .25 .33 0 0 0 0 0 .25 0 0 0 .5 .5 0 0 0 0 .33 0 .5 0 0 0 0 .33 .5 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Also can be looked at as a probability matrix! (notice each column sums to one)
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Markov Chains s A simpler example: s Next time step: .6 .2 .4 .8 .6 .2 .4 .8 .6 .2 .4 .8 = t 0 t 1 t 2 1 1 1 + 1 2 1 .6 * .6 + .4 * .2 = .44 .44 .28 .56 .72 .33 .33 .66 .66 eventually .35 .32 .65 .68 .34 .33 .66 .66
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Markov Chain s Markov Chain: A sequence of variables X 1 , X 2 , X 3 , etc (in our case, the probability matrices) where, given the present state, the past and future states are independent. s Probabilities for the next time step only depend on current probabilities (given the current probability). s A random walk is an example of a Markov Chain, using the transition probability matrices.
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s To turn a weighted graph into a probability (transition) matrix, column normalize. 1
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MCL_Presentation2 - Clustering on Graphs The Markov Cluster...

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