dense subgraphs with restrictions & applications to gene annotations graphs

Samir KhullerUniversity of Maryland

Joint Work withBarna Saha, Allie Hoch, Louiqa Raschid, Xiao-Ning Zhang

RECOMB 2010

Sitting in a talk on community detection….. How do we define a community?

Perhaps we want to capture a group of individuals with strong interactions within the group?

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1The density of {1,2,3,4,5,6,7} = 9/7 = 1.28

The density of {1,2,3,4} = 6/4 = 1.5

The densest subgraph is {1,2,3,4}.

How do we compute the densest subgraph?

Surprisingly, this can be solved optimally in polynomial time!

[Goldberg 84, Lawler 76, Queyranne 75, GGT]

Extends to weighted graphs.

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sum of weights of edges in the induced subgraphSubgraph density = number of nodes in the induced subgraph

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sum of weights of edges in the induced subgraphSubgraph density = number of nodes in the induced subgraph

Density of entire graph is 13/8 > 1.5

Are all dense subgraphs meaningful ?◦ How do we allow some control over the kind of dense

subgraphs that are found?◦ Putting size constraints makes the problem intractable

immediately. Densest subgraph of size >=k. NP-hard, 2 approximation [Anderson][Khuller,

Saha]. Greedily Union densest subgraphs…..

Densest subgraph of size <=k (or =k). NP-hard and some approximations known [Feige,

Kortsarz, Peleg] [Charikar et al].

Are Dense Subgraphs Useful?

Goldberg’s algorithm: a new flow network, is created with “directed” edges. ◦ An s-t min cut is computed in order to find the

densest subgraph after guessing the max density. Nodes on the “s” side of the cut are part of a densest subgraph.

◦ GGT speeds everything up to a single flow computation!

Lawler’s algorithm: slightly different flow construction, more intuitive.

Greedy algorithm: recursively delete low degree nodes. Gives a 2 approximation to density! Fast!

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V2We use s-t min cuts

Original Graph:

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Edges from source to nodes: m’= sum of all edges in graph

Edge from node i to sink: m’ + 2g – d(i)

g = guess = 2

CUT = m’|V| +2|V1| (g-D1)where V1 are the source side nodes

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Since the cut is <21, the guess is too low.

g = guess = 2

CUT = m’|V| +2|V1| (g-D1)where V1 are the source side nodes

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Since the min cut is the trivial cut (and unique), the guess is too high.

g = guess = 4

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Since the cut is smaller than 21, the guess is too low.

g = guess = 2 1/3

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Since the cut has value 21 and V1 is not empty, guess is correct!

g = guess = 2 1/2

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What can we do with this weapon?

Consider gene annotation data from TAIR. For large networks we can use the fast

greedy approximation (gave us the densest subgraph every time!).

AT1G15550GA4

GO:0016707 gibberellin 3-beta-dioxygenase activity

GO:0009686 gibberellin biosynthetic process

GO:0009739 response to gibberellin stimulus

GO:0009639 response to red or far red light

GO:0008134 transcription factor binding

GO:0010114 response to red light

PO:0019018 embryo axis

PO:0009046 flower

PO:0009005 root

PO:0009001 fruit

PO:0020001 ovary placenta

PO:0020148 shoot apical meristem

PO:0020030 cotyledon

PO:0009064 receptacle

PO:0003011 root vascular system

PO:0000014 rosette leaf

PO:0004723 sepal vascular system

PO:0009047 stem

PO:0020141 stem node

PO:0009009 embryo

PO:0004714 terminal floral bud

PO:0009025 leaf

PO:0007057 0 germination

PO:0007131 seedling growth

PO:0009067 filament

GO:0009740 gibberellic acid mediated signalling

GO:0005737 cytoplasm

GO-(gene)-PO tri-partite graph






GO:0008135 biological process

GO OntologyGO Ontology

PO:0019018 embryo axisPO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryoPO:0004714 terminal floral bud

PO:0009025 leaf

PO:0009067 filament

Plant structurePO OntologyPO Ontology

Gene Annotation GraphGene Annotation Graph

Construct graphs for each gene using their GO, PO annotations

Combine the graphs of several genes into one single weighted graph

Gene 1

Gene 2

Gene 3

Gene 4

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

Scientists like to find patterns in gene annotation graphs – but these are huge!

Need to allow some control over the kind of patterns that are computed

Would like to find biologically meaningful patterns Gene

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Gene 3

Gene 4

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

Node

Edge

AT1G15550GA4








PO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryo


PO:0009025 leaf



PO:0009067 filament



GO-(gene)-PO tri-partite graph








PO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryo


PO:0009025 leaf



PO:0009067 filament



GO-PO bipartite graph

Gene Annotation GraphGene Annotation Graph

Construct complete bipartite graph for each gene using their GO, PO annotations

Combine the bipartite graphs of several genes into one single weighted graph

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

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How can we extract knowledge? Cliques – these might give us some

biological information – but this is a stringent reqmt.

However clique finding is well known to be really hard (NP-hard, hard to approximate).

Why not look for “dense regions”? Note that the notion of density could be

defined for hyper-edges as well, but for our purposes this does not do as well.

Dense Subgraphs in Gene Dense Subgraphs in Gene Annotation GraphAnnotation Graph A collection of GO-PO terms that appear together in the

underlying genes.

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GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

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(GO3,PO1),(GO3,PO2),(GO3,PO4),(GO4,PO1),(GO4,PO2),(GO4,PO4) appear frequently in the 4 genes

Are all dense subgraphs biologically meaningful ?◦ How do we allow some control over the kind of dense

subgraphs that are computed.◦ In fact we can impose both restrictions at the same time!

Restrictions in dense subgraph computation

Distance Restricted

Subset Restricted

GO terms and similarly PO terms that appear must be biologically related

Certain GO, PO terms must appear in the returned subgraph

Are all dense subgraphs biologically meaningful ?◦ How do we allow some control over the kind of dense

subgraphs that are found?


Distance Restricted

Subset Restricted

GO terms that appear in the densest subgraph must be close in the GO ontology graph and similarly for the PO terms

Distance threshold = 1 This means that some sets of nodes are not allowed to

coexist in the final solution: {GO1 ,GO2}, {GO1,GO4}, {PO1 ,PO4}, {PO1,PO2},{PO2,PO3,}.

The final solution is {GO2, GO3, GO4, PO2, PO4}, which has a density of .8.

GO1

GO2

GO3

GO4

PO1

PO2

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PO3

PO4

PO2

PO3

PO4

GO2

GO1

GO3

GO4

For arbitrary ontology graph structure◦ NP Hard even to approximate it reasonably

Reduction from Independent set problem◦ Factor 2 relaxation of distance threshold is enough to get a

solution with density as high as the optimum Trees, Interval Graphs, Each edge participates in

small number of cycles◦ Polynomial time algorithm to compute the optimum

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GO-OntologyPO-Ontology

Distance Threshold=2

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Guess two nodes in each ontology that appears in the optimum solution and have maximum distance

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GO-Ontology PO-Ontology

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Compute all the nodes which are within distance threshold from both the guessed nodes

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In the gene-annotation bipartite graph consider only the chosen nodes and compute the densest subgraph

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In the gene-annotation bipartite graph consider only the chosen nodes and compute the densest subgraph

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Proof of optimality:Any node not chosen can not be in the optimum solutionAll the nodes chosen are within distance threshold

Guess a small subset of nodes from the optimum Choose candidate nodes by considering distance from the

guessed nodes Compute the densest subgraph by restricting the gene

annotation graph to only the chosen nodes

Are all dense subgraphs biologically meaningful ?◦ How do we some control over the kind of dense

subgraphs that are found ?


Distance Restricted

Subset Restricted

Given a subset of GO, PO terms compute the densest subgraph containing them.

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•This set {5,6} must be in the solution.

•Density of {1,2,3,4} = (3+2+2+2)/4 = 2.25– Doesn’t contain {5,6}

•Density of {5,6,7,8} = 6/4 = 1.5 (Satisfies subset requirement)

•Density of {1,2,3,4,5,6,7,8} = (2+3+2+2+1*7)/8 = 2.0 (Best answer)

Polynomial time algorithm to compute the optimum solution

For this problem we modified Lawler’s method of finding densest subgraphs. Let’s assume that we have a graph in which we want to force {5,6} to be in the final solution.

The guess “g” is iteratively updated, as in Goldberg’s algorithm until the min cut is calculated and there is more than one possible solution, one contains just {s’ and s} and the other specifies the densest subgraph.

A graph may contain multiple subgraphs of equal (or close to equal) density

Computing just one subgraph may not be sufficient Compute all subgraphs close to maximum density Extension of Picard and Queyranne’s result

◦ Polynomial time algorithm to find almost all dense subgraphs given the number of such subgraphs is polynomial in the number of vertices.

◦ Their method encodes all possible s-t min cuts.◦ After a max-flow is found, we lower edges with residual capacity close to

zero, to zero and now used [PQ] method to list all s-t min cuts. Can be extended to consider both distance and subset

restriction

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•Density of {1,2,3,4} = 9/8 = 2.25

•Density of {5,6,7,8,9} = 11/5 = 2.

•Density of {1,2,3,4,5,6,7,8,9} = 21/9 = 2.333

•The entire graph is the densest subgraph, but {1,2,3,4} and {5,6,7,8,9} are “almost” dense subgraphs

10 Photomorphogenesis genes

CIB5 CRY2 HFR1 COP1 PHOT1 PHOT2 HY5 SHB1 CRY1 CIB1

66 GO CV terms. 41 PO CV terms; 2230 GO-PO edges.

Generate distance restricted dense subgraph. GO distance = 2. PO distance = 3. Dense subgraph with 3 GO terms & 13 PO terms

Photomorphogenesis ExperimentPhotomorphogenesis Experiment

HFR1

COP1

PHOT1

PHOT2

HY5

13 PO CV terms 3 GO CV termsSet of 10 genes

CRY2

CIB5

SHB1

CIB1

CRY1

(partial) dense subgraph; 3 GO terms; 13 PO terms; 10 genes

0 annotation edges

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Potential Discovery

Genes CRY2 and PHOT1 are both observed in the dense subgraph with the following two GO and PO combinations: 5773: vacuole: cellular_component 13: cauline leaf; plant_structure 37: shoot apex; plant_structure (5773, 13) (5773, 37) This pattern has not been reported in the literature. Two independent studies [Kang et al. Planta 08, Ohgishi PNAS 04] have suggested that there may be some functional interactions between the members of PHOT1 and CRY2 in vacuole

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Density of entire graph is 13/8 > 1.5

Back to communities…?

Can we use distance based cutoffs to define a sub graph of interest?

Identifying dense subgraphs with distance and subset restriction may help in identifying interesting patterns

Potential Applications in other domains:◦ Distance restricted dense subgraph for community detection◦ Subset restricted dense subgraph in PPI network for deriving protein

complexes Ranking almost dense subgraphs Change the notion of density [K,Mukherjee,Saha]?

dense subgraphs with restrictions & applications to gene annotations graphs

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