1. SiS: Significant Subnetworks in Massive Number of Network Topologies
Md Mahmudul Hasan, Yusuf Kavurucu and Tamer Kahveci
University of Florida

2. Goal A collection D = {G1, G2, … ,Gm} Two positive integers, n and k.
Goal: Find the n connected subnetworks each containing k edges that appear in the largest number of these networks.
Goal: Find the n size-k subnetworks that appear in most of these networks.
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Md Mahmudul Hasan et al.

3. Gene Regulatory Network
Errors in the measurement of high-throughput experiments.
Gene Regulatory Network
Inference Problem
Different models can explain the data equally well.
Inference Algorithm
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Md Mahmudul Hasan et al.

4. Definitions exists(G’, G1) = 1 exists(G’, G2) = 0 frequency(G’, D) = 2A B A B D C A B A B D C D C D C G1 G2 G3 G4 A
exists(G’, G1) = 1
exists(G’, G2) = 0
frequency(G’, D) = 2
G’ =
G’ = D C
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5. Given D, n and k; find n most frequent size-k subgraphs in D.
Definitions (cont.) A B A B A B A B D C D C D C D C G1 G2 G3 G4
Most Frequent Size-k Subgraph: The most frequent size-k subgraph is a size-k subgraph G' for which frequency(G', D) is maximum.
Given D, n and k; find n most frequent size-k subgraphs in D.
The most frequent size-2 subgraph in D:
frequecy(G’, D) = 4. A B D
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6. Definitions (cont.) A B A B A B A B D C D C D C D C G1 G2 G3 G4
Normalized
Frequency,
fr(e, D) is equal to the probability that a randomly selected graph Gi  D contains e.
frequency(C → A) = 2
fr(C → A) = 0.5
frequency(A → B) = 4
fr(A → B) = 1
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7. Definitions (cont.) Multiplication of probabilities,
For any collection of edges {e1, e2, ..., ek} from the graphs in D, the probability that a randomly drawn graph in D contains all of these edges is as follows:
Multiplication of probabilities,
resulting in a very small number.
The score of a subgraph G’ is:
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8. Template graph G1 G2 G3 G4 A B D C 1.0 0.25 0.5
ψ(e) = -log(fr(e, D)) A B D C
1.0
0.25
0.5
Graph with fr(e, D) values A B D C
0.0
1.38
0.69
Template graph
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9. Frequent subgraphs consist frequent edges.
Frequent edges have smaller -log(fr(e, D)) values.
The more frequent a subgraph, the smaller the score value.
0.0
G’ = A D C B A B
0.69
0.0
0.69 D C
1.38
score(G’, D) = = 0.69
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10. Given D, n and k; find n most probable size-k subgraphs in D.
Definitions (cont.)
Most Probable Size-k Subgraph: The most probable size-k subgraph is a size-k subgraph G’ for which score(G’, D) is minimum.
Given D, n and k; find n most probable size-k subgraphs in D.
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11. Three phases of SiS Pre-processing Exploration
Lower-bound to score(G’, D) calculation
Upper-bound to score(G’, D) calculation
Exploration
Looks for the most probable subgraphs
Post-processing and Extension
Calculates the frequency of the most probable subgraphs
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12. Pre-processing Pre-processing Lower-bound to score(G’, D) calculation
Upper-bound to score(G’, D) calculation
Exploration
Looks for the most probable subgraphs
Post-processing and Extension
Calculates the frequency of the most probable subgraphs
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Md Mahmudul Hasan et al.

13. Pre-processing: Lower-bound
0.0 A B
lower-bound values(i.e., LB) to score(G’, D) for size-k’ subgraphs G’s (k’ = 1… k)
0.1
0.3 D
Relax the connectedness constraint.
0.2
0.3 E C
How to calculate LB[1, 2, 3]:
0.0
LB[1]
LB[2]
LB[3]
0.1
= 0.1 F
= 0.3
LB[i] ≤ score (optimal size-i subgraph)
Template graph
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14. Pre-processing: Upper-bound
Upper-bound for size-3 subgraph:
If we start with the edge {C→F}: A
0.0 B
0.1
0.3
G’ = {C→F}
score(G’, D) = 0.1
G’ = {C→F, C → D, A → D}
score(G’, D) = 0.5
G’ = {C→F, C → D}
score(G’, D) = 0.4 D
0.2
0.3
UB[1] …
UB[2]
UB[3] E C
0.1
0.5 F
Template graph
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15. Pre-processing: Upper-bound
Upper-bound for size-3 subgraph:
Starting with {A→B}: A
0.0 B
0.1
G’ = {A→B, A → D, E → D}
score(G’, D) = 0.3
G’ = {A→B}
score(G’, D) = 0
G’ = {A→B, A → D}
score(G’, D) = 0.1
0.3 D
0.2
0.3
UB[1] …
UB[2]
UB[3] E C
0.1
0.5
0.3 F
Template graph
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16. Exploration Exploration Looks for the most probable subgraphs
Pre-processing
Lower-bound to score(G’, D) calculation
Upper-bound to score(G’, D) calculation
Exploration
Looks for the most probable subgraphs
Post-processing and Extension
Calculates the frequency of the most probable subgraphs
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Md Mahmudul Hasan et al.

17. Example for a size-6 subgraph
LB[1] …
LB[2]
0.5
LB[6]
UB[1] …
UB[2]
UB[6]
2.05
UB[1] …
UB[2]
UB[6]
2.1
Size-6 subgraph, G’
Size-3 subgraph, G’
score(G’’) = 2.05
|E’| = 6
score(G’) = 1.5
|E’| = 3
0.2
score(G’’) = 1.7
|E’| = 4
Tighter upper-bound values, on-the-fly
Template graph
score(G’’) + LB[2] > UB[6]
Example for size-6 subgraph
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18. Exploration Post-processing and Extension
Pre-processing
Lower-bound to score(G’, D) calculation
Upper-bound to score(G’, D) calculation
Exploration
Looks for the most probable subgraphs
Post-processing and Extension
Calculates the frequency of the most probable subgraphs
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Md Mahmudul Hasan et al.

19. Post-processing G’ G’ score(G’, D) frequency(G’, D) 1 2 … n UB[1]
UB[k]
LB[1] …
LB[2]
LB[k]
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20. Extension Maximal frequent subgraph: 11/24/2018
Md Mahmudul Hasan et al.

21. Datasets KEGG database (February, 2012 -- freeze). Dataset
Number of Organisms
Entire Dataset
Template Graph
Nodes
Edges
Eukaryote-G
145
45, 315
78, 499
1, 413
1, 541
Prokaryote-G
1, 486
393, 681
616, 546
1, 676
Eukaryote-AAG
2, 048
2, 951 43 54
Prokaryote-AAG
1, 442
20, 692
27, 334 79
Eukaryote-P
2, 942
5, 757 63
103
Prokaryote-P
31, 130
60, 802
114
KEGG database (February, freeze).
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22. Frequency of top 50 size-k (k = 6 … 20) subgraphs in Global dataset.
Results
Frequency of top 50 size-k (k = 6 … 20) subgraphs in Global dataset.
Frequency of top 50 size-k (k = 6 … 15) subgraphs in eukaryotes for AAG and Pyrimidine network.
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23. Results (cont.)
Correlation of frequent subgraphs in eukaryotes and prokaryotes in Global dataset.
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24. Running time (minutes)
Results (cont.)
Dataset
Support
Running time (minutes)
MULE
SiS
Eukaryote-G
83% 20
0.07
80%
447
4.3
Prokaryote-G
Didn’t run.
0.09 5
Maximal frequent size-25 subgraph in eukaryotes (global map).
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25. Conclusion
A method (SiS) that efficiently discovers significant subnetworks in a collection of networks.
SiS scales to very large datasets (i.e., datasets with over a thousand networks, having a total of hundreds of thousands of nodes and edges) easily.
SiS shows significant improvement over state-of-the-art maximal frequent subnetwork detection algorithm (MULE).
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26. Acknowledgements Yusuf Kavurucu NSF CCF-0829867 NSF IIS-0845439
Tamer Kahveci
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27. Thank You 
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28. Additional - 1
Calculate the upper-bound values(i.e., UB) to score(G’, D) for size-k’ subgraphs G’s (k’ = 1… k)
Greedy Approach:
Start at any edge e  ET and initialize G’ to {e}.
Add an edge e’ adjacent to G’ that has the smallest ψ(e’) values.
Repeat until G’ has k’ edges.
UB[k’] = smallest score(G’, D) among these |ET| values.
score (optimal size-i subgraph) ≤ UB[i]
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29. (let the graph obtained after adding {ej } into E' be G”)
Additional - 2
Initialize G’= (V', E') to a size-1 subgraph (i.e., E' = {ei }
Grow G' by inserting a new edge ej into E' if:
(i) ej is incident to v such that v  V'.
(ii) The index of the new edge (i.e., j) is greater than the index of the first edge in E‘ (i.e., i).
(iii) score(G”, D) + LB[k - |E'| - 1] ≤ UB[k]
(let the graph obtained after adding {ej } into E' be G”)
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30. Additional - 3 A B Z-1 Z-2 C Metabolic Network
Corresponding Enzyme Network
Metabolic networks are downloaded from KEGG database.

31. Additional - 4
A frequent size-20 subgraph in prokaryotes which is infrequent in eukaryotes.
A size-20 subgraph which is frequent both in prokaryotes and eukaryotes.
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