Introduction

Important Concepts in MCL Algorithm

MCL Algorithm

The Features of MCL Algorithm

Summary

Graph Clustering Intuition:

◦ High connected nodes could be in one cluster◦ Low connected nodes could be in different

clusters. Model:

◦ A random walk may start at any node ◦ Starting at node r, if a random walk will reach

node t with high probability, then r and t should be clustered together.

Markov Clustering (MCL) Markov process

◦ The probability that a random will take an edge at node u only depends on u and the given edge.

◦ It does not depend on its previous route.◦ This assumption simplifies the computation.

MCL Flow network is used to approximate the

partition There is an initial amount of flow injected

into each node. At each step, a percentage of flow will goes

from a node to its neighbors via the outgoing edges.

MCL Edge Weight

◦ Similarity between two nodes◦ Considered as the bandwidth or connectivity.◦ If an edge has higher weight than the other, then

more flow will be flown over the edge.◦ The amount of flow is proportional to the edge

weight.◦ If there is no edge weight, then we can assign the

same weight to all edges.

Intuition of MCL Two natural clusters

When the flow reaches the border points, it is likely to return back, than cross the border.

A B

MCL When the flow reaches A, it has four

possible outcomes.◦ Three back into the cluster, one leak out.◦ ¾ of flow will return, only ¼ leaks.

Flow will accumulate in the center of a cluster (island).

The border nodes will starve.

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

Definitions nxn Adjacency matrix A.

◦ A(i,j) = weight on edge from i to j◦ If the graph is undirected A(i,j)=A(j,i), i.e. A is symmetric

nxn Transition matrix P.◦ P is row stochastic◦ P(i,j) = probability of stepping on node j from node i = A(i,j)/∑iA(i,j)

nxn Laplacian Matrix L.◦ L(i,j)=∑iA(i,j)-A(i,j)◦ Symmetric positive semi-definite for undirected graphs◦ Singular

Definitions

Adjacency matrix A Transition matrix P

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1

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What is a random walk

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t=0

What is a random walk

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1/2

1/21

t=0 t=1

What is a random walk

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t=0 t=1

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t=2

What is a random walk

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t=0 t=1

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t=2

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t=3

Probability Distributions

xt(i) = probability that the surfer is at node i at time t

xt+1(i) = ∑j(Probability of being at node j)*Pr(j->i) =∑jxt(j)*P(j,i)

xt+1 = xtP = xt-1*P*P= xt-2*P*P*P = …=x0 Pt

What happens when the surfer keeps walking for a long time?

Flow Formulation

• Flow: Transition probability from a node to another node.• Flow matrix: Matrix with the flows among all nodes; ith

column represents flows out of ith node. Each column sums to 1.

1 2 3

1 2 3

0.5 0.5

1 1

1 2 3

1 0 0.5 0

2 1.0 0 1.0

3 0 0.5 0

Flow

Matrix

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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 MG ,is defined by setting the entry (MG)pq equal to w(vp,vq)

Markov Matrix

The Markov matrix associated with a graph G is denoted by TG and is formally defined by letting its qth column be the qth 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 (Mk)pq has a straightforward interpretation as the number of paths of length k between vp and vq

(MG+I)2

MG

MG

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

The MCL algorithm

Expand: M := M*M

Inflate: M := M.^r (r usually 2), renormalize columns

Converged?

Input: A, Adjacency matrixInitialize M to MG, the canonical transition matrix M:= MG:= (A+I) D-1

Yes

Output clusters

No

Prune

Enhances flow to well-connected nodes as well as to new nodes.

Increases inequality in each column. “Rich get richer, poor get poorer.”

Saves memory by removing entries close to zero.

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Multi-level Regularized MCL

Input Graph

Intermediate Graph

Intermediate Graph

Coarsest Graph

. . . . . .

Coarsen

Coarsen

Coarsen

Run Curtailed R-MCL,project flow.

Run Curtailed R-MCL, project flow.

Input Graph

Run R-MCL to convergence, output clusters.

Faster to run on smaller graphs

first

Captures global topology of

graph

Initializes flow matrix of refined

graph

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http://www.micans.org/mcl/ani/mcl-animation.html

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 overlapping 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 before 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