Liang Ge

 Introduction  Important Concepts in MCL Algorithm  MCL Algorithm  The Features of MCL Algorithm  Summary

 Simualtion of Random Flow in graph  Two Operations: Expansion and Inflation  Intrinsic relationship between MCL process result and cluster structure

 Popular Description: partition into graph so that  Intra-partition similarity is the highest  Inter-partition similarity is the lowest

 Observation 1:  The number of Higher-Length paths in G is large for pairs of vertices lying in the same dense cluster  Small for pairs of vertices belonging to different clusters

 Oberservation 2:  A Random Walk in G that visits a dense cluster will likely not leave the cluster until many of its vertices have been visited

 Measure or Sample any of these—high-length paths, random walks and deduce the cluster structure from the behavior of the samples quantities.  Cluster structure will show itself as a peaked distribution of the quantities  A lack of cluster structure will result in a flat distribution

 Markov Chain  Random Walk on Graph  Some Definitions in MCL

 A Random Process with Markov Property  Markov Property: given the present state, future states are independent of the past states  At each step the process may change its state from the current state to another state, or remain in the same state, according to a certain probability distribution.

 A walker takes off on some arbitrary vertex  He successively visits new vertices by selecting arbitrarily one of outgoing edges  There is not much difference between random walk and finite Markov chain.

 Simple Graph  Simple graph is undirected graph in which every nonzero weight equals 1.

 Associated Matrix  The associated matrix of G, denoted M G,is defined by setting the entry (M G ) pq equal to w(v p,v q )

 Markov Matrix  The Markov matrix associated with a graph G is denoted by T G and is formally defined by letting its q th column be the q th column of M normalized

 The associate matrix and markov matrix is actually for matrix M+I  I denotes diagonal matrix with nonzero element equals 1  Adding a loop to every vertex of the graph because for a walker it is possible that he will stay in the same place in his next step

 Find Higher-Length Path  Start Point: In associated matrix that the quantity (M k ) pq has a straightforward interpretation as the number of paths of length k between v p and v q

(M G +I) 2 MGMG

MGMG

 Flow is easier with dense regions than across sparse boundaries,  However, in the long run, this effect disappears.  Power of matrix can be used to find higher- length path but the effect will diminish as the flow goes on.

 Idea: How can we change the distribution of transition probabilities such that prefered neighbours are further favoured and less popular neighbours are demoted.  MCL Solution: raise all the entries in a given column to a certain power greater than 1 (e.g. squaring) and rescaling the column to have the sum 1 again.

 Expansion Operation: power of matrix, expansion of dense region  Inflation Operation: mention aboved, elimination of unfavoured region

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 Find attractor: the node a is an attractor if Maa is nonzero  Find attractor system: If a is an attractor then the set of its neighbours is called an attractor system.  If there is a node who has arc connected to any node of an attractor system, the node will belong to the same cluster as that attractor system.

Attractor Set={1,2,3,4,5,6,7,8,9,10} The Attractor System is {1,2,3},{4,5,6,7},{8,9},{10} The overlaping clusters are {1,2,3,11,12,15},{4,5,6,7,13},{8,9,12,13,14,15},{10,12,13}

 how many steps are requred begore the algorithm converges to a idempoent matrix?  The number is typically somewhere between 10 and 100  The effect of inflation on cluster granularity

R denotes the inflation operation constants. A denotes the loop weight.

 MCL stimulates random walk on graph to find cluster  Expansion promotes dense region;while Inflation demotes the less favoured region  There is intrinsic relationship between MCL result and cluster structure