Download presentation
Presentation is loading. Please wait.
Published byDebra Ross Modified over 9 years ago
1
Liang Ge
2
Introduction Important Concepts in MCL Algorithm MCL Algorithm The Features of MCL Algorithm Summary
3
Simualtion of Random Flow in graph Two Operations: Expansion and Inflation Intrinsic relationship between MCL process result and cluster structure
4
Popular Description: partition into graph so that Intra-partition similarity is the highest Inter-partition similarity is the lowest
5
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
6
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
7
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
8
Markov Chain Random Walk on Graph Some Definitions in MCL
9
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.
11
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.
12
Simple Graph Simple graph is undirected graph in which every nonzero weight equals 1.
13
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 )
14
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
16
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
18
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
19
(M G +I) 2 MGMG
20
MGMG
22
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.
23
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.
28
Expansion Operation: power of matrix, expansion of dense region Inflation Operation: mention aboved, elimination of unfavoured region
32
http://www.micans.org/mcl/ani/mcl- animation.html http://www.micans.org/mcl/ani/mcl- animation.html
33
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.
34
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}
35
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
36
R denotes the inflation operation constants. A denotes the loop weight.
37
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.