1. Computational Molecular Biology
Community Structures

2. What is Community Structure
Definition:
A community is a group of nodes in which:
There are more edges (interactions) between nodes within the group than to nodes outside of it
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3. Why Community Structure (CS)?
Many systems can be expressed by a network, in which nodes represent the objects and edges represent the relations between them:
Social networks: collaboration, online social networks
Technological networks: IP address networks, WWW, software dependency
Biological networks: protein interaction networks, metabolic networks, gene regulatory networks
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4. Why CS? Yeast Protein interaction networks My T. Thai

5. Why CS?
IP address network
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6. Why Community Structure?
Nodes in a community have some common properties
Communities represent some properties of a networks
Examples:
In social networks, represent social groupings based on interest or background
In citation networks, represent related papers on one topic
In metabolic networks, represent cycles and other functional groupings
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7. How to detect a community?
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8. Early Work Using hierarchical clustering Overview of this method:
For each pair (u,v), calculate weight wuv which represents how closely connected u and v are
Initialize G = (V, emptyset)
At each iteration, add an edge with the strongest weight
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9. Early Work
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10. Early Work How to define the weight wuv
Many different methods have been proposed:
Number disjoint paths between u and v
Number of possible paths between u and v
Disadvantages:
Tendency to separate the boundary vertices from the communities (to which they should belong)
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11. An Overview of Recent Work
Disjoint CS
Overlapping CS
Centralized Approach
Define the quantity of modularity and use the greedy algorithms, IP, SDP
Spectral clustering
Random Walk, Clique Percolation
Localized Approach
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12. Edge Betweeness
Focus on the edges which are least central, i.e.,, the edges which are most “between” communities
Instead of adding edge to G = (V, emptyset), progressively removing edges from an original graph G = (V,E)
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13. Edge Betweeness Definition:
For each edge (u,v), the edge betweeness of (u,v) is defined as the number of shortest paths between any pair of nodes in a network that run through (u,v)
betweeness(u,v) = | { Pxy | x, y in V, Pxy is a shortest path between x and y, and (u,v) in Pxy}|
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14. Why Edge Betweeness
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15. Algorithm Initialize G = (V,E) representing a network
while E is not empty
Calculate the betweeness of all edges in G
Remove the edge e with the highest betweeness, G = (V, E – e)
Indeed, we just need to recalculate the betweeness of all edges affected by the removal
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16. Time Complexity Let |V| = n and |E| = m
Calculate the betweeness of all edges: O(mn)
Since we need to recalculate each time we remove an edge: O(m2n)
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17. An Example
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18. Disadvantages/Improvements
Can we improve the time complexity?
The communities are in the hierarchical form, can we find the disjoint communities?
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19. Define the quantity (measurement) of modularity Q and find an approximation algorithm to maximize Q
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20. How to define Q Let A be an adjacency matrix of the network
Fraction of edges that fall within communities
where
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21. How to define Q
What is the problem? If we try to maximize the above equation, then we may put all nodes into one single community
How to fix it?
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22. How to define Q? Let kv be the degree of node v
The term kv kw /2m represents the probability of an edge existing between vertices v and w if connections are made random but respecting vertex degrees
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23. Greedy Algorithm
Initially, we have n communities (each node is a community)
At each step, join two communities whose the hierarchical tree has the largest increase in Q
Stop when we left with a single community (run n -1 steps)
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24. Disadvantages It is still a hierarchical approach
Cannot escape a suboptimal maximum
How to avoid the suboptimal maximum?
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25. Local Communities
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26. Overview
Find communities based on local information (not information of entire network)
Two ways:
Detect the communities
Define local modularity, then greedily optimize this function
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27. What We Have Learnt Can we use this Q?
How can we “twist” it to make it work?
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28. Consider this Figure
Suppose we have perfect knowledge of some subgraphs C
Then we should know some neighbors on C, lie in U
Visit some neighbors of nodes in U may extend the knowledge of C
Now, can we re-use Q (defined before) on C?
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29. Some Definitions
Again, define an adjacent matrix A (wrt C) as follows:
Consider this quantity:
where (# of edges in the partial adj matrix)
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30. Relationship between B, C, U
Consider nodes in B
If C is a sharp community, then
Nodes in B have more connections to nodes in C
Nodes in B have less (a few) connections to nodes in U
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31. Definitions Define a boundary-adjacent matrix B as follows:
Define local community R:
where δ(i,j) = 1 iff vi in B and vj in C or vice versa. Otherwise, δ(i,j) = 0.
T: #edges with one ore more endpoints in B
I: #edges in B such that none of their endpoints in U
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32. Properties of R 0 < R < 1
Directly proportional to the sharpness of the boundary given by B
When R is undefined?
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33. A Greedy Algorithm
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34. Overview of Second Method
Start at a vertex, check the degree of each vertex with respect to each one-hop neighbors, two-hop neighbors, …, l-hops neighbors
Why?
If the community is highly connected, the l-hops neighbors tend to revisit the nodes
At the boundary, the number of newly added edges decreases
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35. Some Definitions
kie(j): Emerging degree of a vertex i which is l-hops away vertex j is defined as the number of edges (u,i) where u is not within l-hops away from j
Kjl: Total emerging degree of all nodes that exactly l-hops away from j
where Sjl is the set of all vertices exactly l-hops away from j
Initially, Kj0= degree of node j = kj
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36. Some Definitions The change in total emerging degree My T. Thai

37. Algorithm Randomly choose a starting vertex j
Initially, l = 0, add j to C (C is a community), and K0j = kj
l = l++; add all l-hop neighbors of j to C
Compute ΔKjl. If ΔKjl < α, then return C. Otherwise, repeat step 3
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38. Any Problem?
How to define α? Do we need to define α? If not, what should we change?
What if the starting vertex is the “bridge one”? What can we do?
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39. Impact of α
α = 0, never stop until explore the entire connected subgraph
α is large, stop sooner (l is small), resulting in many small communities
α is too large, return n singleton communities (α > kmax where kmax is the largest degree)
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40. A Small Example Actual CS of the Karate Club Obtained by the Alg
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41. Dynamic Communities
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42. Dynamic Networks Event decomposition A dynamic network
A collection of network snapshots at many time points.
Changes are frequently introduced
Insertions / Removals of nodes
Insertions / Removals of edges
t = 0
t = 1
t = 2
t = 3
Event decomposition
Insertions / Removals of nodes = {Insert/Remove a node}
Insertions / Removals of edges = {Insert/Remove an edge}

43. Recall…
Modularity function
However, max Q is NP-hard

44. Adaptive Solutions An adaptive method: A basic community structure
To maximize the gained modularity with low computational complexity
Locally compute a new structure based on local information after each change of the network
A basic community structure
Only for the first snapshot
Adaptively update the network communities based on this basic structure
Method: Blondel et al (2008)

45. QCA: An Adaptive Method
Input network
Blondel’s method :
Network changes
Need to handle
Node insertion
Edge insertion
Node removal
Edge removal
Basic communities
Updated communities

46. Membership determination
A node actively determines its Membership u
FinS(u) S
FoutC(u) C

47. Introducing a new node C1 C2 C3 Possibilities No new edges
New edges linking with one community
New edges linking multiple communities u C2 C1 C3

48. Handling node insertionu
Join u to the community C
with the highest FoutC(u) C1 C2
FoutC1(u)
FoutC2(u) C3
FoutC3(u)

49. Introducing a new edge Possibilities
A new edge is inside a single community
A new edge is joining two communities u v u v

50. Handling edge insertion
Keep the current community structure intact a b
Find qu,C,D and qv,D,C
Join a to C or D according to qu,C,D and qv,C,D
If a (or b) changes its membership
Check all a’s neighbors a b

51. Time complexity Inserting a new node Inserting a new edge
Visit all neighbors of u at most once
O(du)
Inserting a new edge
Computing qu,C(u),C(v) and in constant time
O(1)

52. Removing an edge Resulting community is either: Remains unchanged
Breaks up into smaller communities
If it contains substructures that are less attractive to the others u v

53. Handling edge removal Strategy Find maximal ‘quasi’-cliques
Let the other singletons determine their best communities

54. Removing a node All edges connected to u will be removed
Resulting community either
Remains unchanged
Breaks up into smaller spices and merged to others

55. Handling node removal Strategy 3-Clique percolation
Let the left over nodes determine their best communities

56. Experimental Results Test our algorithms on real-world data traces
Enron , ArXiv citation and Facebook networks
In comparison with the Blondel’s method at each snapshot
Metrics
Modularity
Number of communities
Normalized Mutual Information (NMI)
Running time

57. ArXiv e-print Citation network
Modularity
# Communities
NMI
Running Time

58. Facebook
Modularity
# Communities
NMI
Running Time