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Diﬀerential Community Detection in Paired Biological Networks

Raghvendra Mall, Ehsan Ullah, Khalid Kunjia and Halima Bensmail

Qatar Computing Research Institute, Hamad Bin Khalifa University

Doha, Qatar

Fulvio D’Angelo

Department of Neurology, Department of Pathology,

Institute for Cancer Genetics,

Columbia University Medical Center, New York, U.S.A,

Michele Ceccarelli

BioGeM, Institute of Genetic Research “Gaetano Salvatore” &

Department of Science and Technology, University of Sannio

Ariano Irpino & Benevento, Italy

June 7, 2017

Abstract

Motivation: Biological networks unravel the inher-

ent structure of molecular interactions which can lead

to discovery of driver genes and meaningful pathways

especially in cancer context. Often due to gene muta-

tions, the gene expression undergoes changes and the

corresponding gene regulatory network sustains some

amount of localized re-wiring. The ability to identify

signiﬁcant changes in the interaction patterns caused

by the progression of the disease can lead to the rev-

elation of novel relevant signatures.

Methods: The task of identifying diﬀeren-

tial sub-networks in paired biological networks

(A:control,B:case) can be re-phrased as one of ﬁnd-

ing dense communities in a single noisy diﬀerential

topological (DT) graph constructed by taking abso-

lute diﬀerence between the topological graphs of A

and B. In this paper, we propose a fast two-stage

approach, namely Diﬀerential Community Detection

(DCD), to identify diﬀerential sub-networks as dif-

ferential communities in a de-noised version of the

DT graph. In the ﬁrst stage, we iteratively re-order

the nodes of the DT graph to determine approximate

block diagonals present in the DT adjacency matrix

using neighbourhood information of the nodes and

Jaccard similarity. In the second stage, the ordered

DT adjacency matrix is traversed along the diagonal

to remove all the edges associated with a node, if that

node has no immediate edges within a window. We

then apply community detection methods on this de-

noised DT graph to discover diﬀerential sub-networks

as communities.

Results: Our proposed DCD approach can eﬀec-

tively locate diﬀerential sub-networks in several sim-

ulated paired random-geometric networks and vari-

ous paired scale-free graphs with diﬀerent power-law

exponents. The DCD approach easily outperforms

community detection methods applied on the origi-

nal noisy DT graph and recent statistical techniques

in simulation studies. We applied DCD method on

two real datasets: a) Ovarian cancer dataset to dis-

cover diﬀerential DNA co-methylation sub-networks

in patients and controls; b) Glioma cancer dataset

to discover the diﬀerence between the regulatory

networks of IDH-mutant and IDH-wild-type. We

1

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demonstrate the potential beneﬁts of DCD for ﬁnd-

ing network-inferred bio-markers/pathways associ-

ated with a trait of interest.

Conclusion: The proposed DCD approach over-

comes the limitations of previous statistical tech-

niques and the issues associated with identify-

ing diﬀerential sub-networks by use of commu-

nity detection methods on the noisy DT graph.

This is reﬂected in the superior performance of

the DCD method with respect to various met-

rics like Precision, Accuracy, Kappa and Speciﬁcity.

The code implementing proposed DCD method

is available at https://sites.google.com/site/

raghvendramallmlresearcher/codes.

1 Background

In the modern era complex networks are ubiquitous.

Their omnipresence is reﬂected in a myriad of do-

mains including web graphs [6], road graphs [11], so-

cial networks [24, 42], ﬁnancial networks [4] and bi-

ological networks [22, 27, 43]. Here we focus on bi-

ological networks but the caveats introduced in this

paper apply to networks in other domains.

In network biology, particularly in cancer research,

comparisons are performed on gene regulatory net-

works [57] and DNA co-methylation networks [56]

obtained from the gene expression and DNA methy-

lation proﬁles respectively of healthy and diseased

tissues. The goal is to identify genes whose expres-

sion or methylation levels are signiﬁcantly diﬀerent

between the conditions and can lead to discovery of

novel molecular diagnostic and prognostic signatures.

It was shown in [53, 1, 9] that the gene regulatory net-

works undergo some amount of localized re-wirings as

cancer progresses.

One of the primary problems in cell biology is to in-

fer regulatory networks, that capture the interactions

between molecular entities from high-throughput

data. An important challenge that needs to be ad-

dressed is how the cell changes its behaviour in re-

sponse to changes in copy number or alterations such

as driver somatic mutations or an external stimuli.

The gene expression and methylation levels change

due to the downstream eﬀect of the de-regulation of

the global behaviour of the cell in diﬀerent conditions,

for example diﬀerent cancer subtypes [9]. Hence,

it can be suggested that driver mutations regulate

functional pathways described by diﬀerent local re-

wirings in the intrinsic gene regulatory networks.

The problem of detecting signiﬁcant changes in

paired biological networks is diﬀerent from popular

graph theory problems like graph isomorphism [46]

and sub-graph matching [51] for which various graph

matching and graph similarity algorithms [5, 30] exist

and have been utilized in biological networks[55, 45].

This problem has primarily been addressed either in

a statistical framework [37, 21, 50, 33] or from a com-

munity detection perspective [33, 10, 54, 23, 14, 32]

in literature.

In statistics, a common statistic used to distin-

guish one graph from another is the Mean Absolute

Diﬀerence (MAD), which is deﬁned as: d(A, B) =

1

N(N−1) Pi6=j|aij −bij |. Here aij and bij are edge

weights corresponding to the topological graphs of

networks Aand B. A topological graph captures

ﬁrst order interactions between the nodes in the net-

work and can better apprehend subtle changes be-

tween two networks [49]. The MAD distance is

equivalent to the Hamming distance [18] which has

been widely used for comparing networks [7, 15].

The Quadratic Assignment Procedure (QAP) [37]

deﬁned as: Q(A, B) = 1

N(N−1) Pi=1 Pj=1 aij bij

is another statistic used to identify association be-

tween networks. These statistics are often used in

permutation-based procedures to detect signiﬁcant

diﬀerence between two networks. Ruan et al [50]

showed that these statistics are not always sensi-

tive to subtle topological variations and proposed a

Generalized Hamming Distance (GHD) based statis-

tic to measure the distance between paired biological

graphs which outperforms MAD and QAP.

The GHD permutation distribution follows a nor-

mal distribution under the null hypothesis that net-

works Aand Bare independent for scale-free net-

works whose power-law exponent αshould strictly

satisfy: 1 ≤α≤2 or α≥3. They also gen-

erated closed-form expression for p-values and de-

vised a diﬀerential sub-network identiﬁcation tech-

nique, namely dGHD, where they iteratively remove

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least diﬀerent node. This is unlike previous diﬀer-

ential network analysis techniques [15, 14, 17] and

generate p-values by comparing the remaining sub-

networks. Recently, a Closed-Form approach was

proposed in [33] which is faster and more accurate

than the dGHD technique for identifying statistically

signiﬁcant changes between paired networks as diﬀer-

ential sub-networks. However, these statistical tech-

niques are still computationally expensive and suﬀer

from strict restrictions on the exponent of power-law

for scale-free graphs. It was shown in [38] that biolog-

ical networks are scale-free and usually have power-

law exponents that satisﬁes: 0 < α ≤2 which is not

always within the restrictions acceptable for dGHD

and Closed-Form techniques.

The problem of community detection in graphs

has received wide attention from several perspectives

[16, 3, 48, 47, 44, 36, 34, 35, 29] and have also been

applied to biological networks. Methods such as jAc-

tiveModules [10] and the Spinglass algorithm [47]

have been applied to discover biologically meaning-

ful modules such as protein complexes, disease asso-

ciated clusters of genes, etc. as shown in [54, 23].

The problem of identifying diﬀerential sub-networks

in paired biological networks can be re-formulated

as one of ﬁnding heavy sub-networks, or dense mod-

ules, on a single diﬀerential topological (DT) graph

obtained by taking the absolute diﬀerence in the

edge weights between the topological graph of net-

work A and the topological graph of network B i.e.

DT(A, B)ij =|aij −bij |,∀i, j ∈V. This problem

is equivalent to identifying communities in the DT

graph. The notion of communities mean that nodes

within one community are densely connected to each

other and sparsely connected to nodes outside that

community. Large-scale networks consist of several

such communities. Hence, community detection is

equivalent to ﬁnding dense block diagonals in the DT

adjacency matrix. However, the DT graph can suf-

fer from noise caused by interactions between nodes

which are not part of diﬀerential sub-networks (re-

ferred further as non-diﬀerential nodes) and nodes

which are part of diﬀerential sub-networks (referred

further as diﬀerential nodes) which are just one hop

away in either network A or B but not in both. This

leads to spurious connections around the block diag-

onals present in the DT adjacency matrix. Commu-

nity detection techniques like Louvain [3], Infomap

[48] and Spectral [34] method can be applied to the

obtain communities/modules with diﬀerential nodes

with having perfect recall but suﬀer from very low

precision due to false recognition of non-diﬀerential

nodes as part of diﬀerential sub-networks.

The problem of identifying communities in the DT

graph such that the nodes comprising the commu-

nities are part of diﬀerential sub-networks between

paired biological networks (A, B) is unlike the tradi-

tional module based diﬀerential network analysis as

shown in [14, 32]. In traditional module based diﬀer-

ential network analysis, modules are detected at ﬁrst

in weighted gene co-expression networks (WGCNA)

[14] obtained from gene expression data for case and

controls. The modules are then compared using ei-

ther Jaccard co-eﬃcient (MOda) [32] or additional

genetic marker data (WGCNA) [14] is utilized to dif-

ferentiate the modules. The advantage of these meth-

ods is that by focusing on modules rather than on in-

dividual gene expressions, they can greatly alleviate

the multiple-testing problem inherent in micro-array

data analysis. However, our goal is to identify the

diﬀerence between the paired biological networks as

dense modules/communities rather than comparing

the modules in the paired biological networks. For ex-

ample, say minor localized changes within two mod-

ules in the original biological networks together form

a diﬀerential sub-network. The method proposed in

this paper will be able to identify these changes as a

diﬀerential community which might otherwise not be

detected by WGCNA or MOda.

In this paper, we propose a novel two-stage ap-

proach, namely Diﬀerential Community Detection

(DCD), to identify diﬀerential sub-networks in paired

biological networks as communities from the origi-

nal nosiy DT graph. The proposed DCD method

overcomes the restrictions on power-law exponents

for scale-free graphs implied by statistical techniques

and retains the advantage of greatly reducing the

burden of multiple-testing from module based diﬀer-

ential network analysis techniques. We applied our

DCD method on two real datasets, an ovarian cancer

dataset to discover diﬀerential DNA co-methylation

sub-networks in patients and controls, and a glioma

3

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certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted June 8, 2017. ; https://doi.org/10.1101/147538doi: bioRxiv preprint

cancer dataset to discover the diﬀerence between the

regulatory networks of IDH-mutant and IDH-wild-

type.

2 Method

The proposed DCD approach consists of two pri-

mary stages: In the ﬁrst stage of DCD, the proposed

method re-orders the nodes of the DT graph to gen-

erate approximate block diagonals inherently present

in the DT adjacency matrix. It utilizes the neigh-

bourhood information from the DT graph for all the

nodes and a notion of similarity based on the Jac-

card index [31]. In the second stage of DCD, the

ordered yet noisy DT adjacency matrix is traversed

along the diagonal to remove all the edges associated

with a node, if that node has no immediate edges

within a window. This is because the ordered DT

adjacency matrix is already comprised of block diag-

onals and nodes which are not part of block diagonals

are the ones causing spurious connections in the DT

graph. We then pick out such nodes and remove all

the edges associated with these nodes. Finally, we

apply community detection techniques like Louvain

[3], Infomap [48] and Spectral [34] methods on this

de-noised DT graph to discover the diﬀerential sub-

networks as communities. Figure 1 illustrates all the

steps involved in the DCD algorithm and its com-

parison with direct application of community detec-

tion techniques on noisy DT graph to locate diﬀer-

ential sub-networks on a pair of simulated random-

geometric (RG) networks.

2.1 Ordering the Noisy DT graph

The goal of ﬁrst stage of DCD method is to detect

sets of nodes which have higher similarity with each

other in comparison to other nodes by following an

iterative procedure to order the nodes in the adja-

cency matrix of the original DT graph G(V, E ). The

total number of nodes in the DT graph is represented

as N=|V|. This iterative process is essential as

nodes are not usually ordered in the G(V, E) and the

inherent block diagonals have to be discovered. It

is important to locate approximate block diagonals

as it is a necessary condition for the second stage

of DCD approach. We deﬁne d(vi, V t) as degree of

the node vi∈Vt, where Vtrepresents the set of

nodes to be investigated at iteration t. During the

ﬁrst iteration, we identify the node with highest de-

gree i.e. vt

max = argmaxd(v, V t) using the topology of

G(V, E ) and calculate its Jaccard similarity w.r.t. all

the nodes in DT graph. Mathematically, it is deﬁned

as:

J(vt

max, vi) = |n(vt

max)∩n(vi)|

|n(vt

max)∪n(vi)|(1)

Here vt

max is the node with highest degree during iter-

ation t,vi∈V,n(·) represents the immediate neigh-

bourhood set of a node and |·| represents the cardinal-

ity function. The Jaccard co-eﬃcient of all the nodes

that don’t share a speciﬁed number of neighbours (θ)

with vt

max is set to 0. This threshold θis a tunable

parameter representing the minimum size of a block

diagonal to be considered as a diﬀerential community

in the DT graph. We then sort all the nodes having

non-zero Jaccard similarity with vt

max in decreasing

order and break ties based on degree where higher

degree nodes are placed closer to vt

max. These or-

dered nodes and their corresponding edges results in

the ﬁrst approximate block diagonal ABDtwhich is

preserved in ODT, representing the adjacency matrix

of ordered noisy DT graph. ABDtis an approximate

block diagonal because nodes with spurious connec-

tions are still present and associated with ABDtas

highlighted in Figure 1g.

During further iterations (t > 1), an additional

step is performed to re-order the nodes which are

common between the ABDt−1and ABDt. The or-

der of common nodes whose Jaccard similarity was

higher with the previous vt−1

max are unchanged and

these nodes are removed from ABDt. However,

nodes which are common with ABDt−1but have

higher Jaccard similarity with vt

max are removed from

ABDt−1while their order is retained in ABDt. This

iterative process is greedy by nature, as in any iter-

ation twe compare only ABDt−1with ABDt, and

stop when either all the nodes in the G(V, E) are

part of some approximate block-diagonal or degree

of vt

max is 0, which means we are left with only iso-

lated nodes in the G(V, E). Algorithm 1 summarizes

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this procedure.

Algorithm 1: Ordering Noisy DT graph

Data: Noisy DT graph G(V, E) and threshold θ.

Result: Ordered noisy DT adjacency matrix ODT.

Initialize t= 1, Vt=Vand an all zero adjacency matrix

ODT ∈RN×N.

while Vt6=∅do

Select node with highest degree as vt

max from Vt.

if d(vt

max, V t)=0then

Break out of loop. // Only isolated nodes left in

Vt.

end

Calculate J(vt

max, vi), ∀vi∈Vusing Eq. 1.

Set J(vt

max, vi) = 0, {∀vi∈V||n(vt

max)∩n(vi)|< θ}.

Order viwith non-zero J(vt

max, vi) in decreasing order.

Ordered set of nodes and corresponding edges generate

ABDt.

if t > 1then

Identify common nodes cas

c={vi|vi∈ABDt−1∩vi∈AB Dt}.

Remove nodes and corresponding edges from

ABDt−1and its preserved copy in ODT s.t.

J(vt−1

max, vi)< J (vt

max, vi), ∀vi∈c.

Remove those nodes and corresponding edges from

ABDts.t. J(vt−1

max, vi)> J (vt

max, vi), ∀vi∈c.

Keep remaining set of ordered nodes and their edges

as ABDt.

/* A node can only be part of one approximate

block diagonal. */

end

Add ABDtrelated info to ODT .

Vt=Vt\s, such that s={vi∈ABDt}.

t=t+ 1.

end

if Vt6=∅then

// Still isolated nodes are left.

Maintain isolated nodes vi∈Vtas isolated in ODT.

end

2.2 De-noising the DT graph

Once we have obtained ODT as shown in Figure 1g,

we prune out spurious edges associated with nodes

which are falsely recognized as part of block diag-

onals in the previous step. We traverse the land-

scape of the ODT matrix, for example in Figure 1g

from left to right and bottom to up, along the di-

agonal. Since we have already identiﬁed approxi-

mate block diagonals (ABD’s) in ODT, our premise

is that if we traverse along the diagonal and pick a

node viat random, there should be some immedi-

ate edges within θto the left and to the right (below

and above due to symmetry) in the landscape of ODT

for it to be a diﬀerential node in ABD. This means

that d(vi, Vi−θ) and d(vi, Vi+θ) have to be non-zero

at the same time. Here Vi−θand Vi+θrepresent the

neighbourhood up to θnodes to the left and right

of vi. A non-diﬀerential node can be part of ABD

due to spurious connections with the diﬀerential set

of nodes present in ABD. We then remove all the

edges associated with such nodes from ODT to gen-

erate the de-noised ordered DT graph i.e. DDT. The

proposed process leads to de-noised block diagonals

BD in DDT instead of having ABD as shown in Fig-

ure 1h. Algorithm 2 summarizes the de-noising pro-

cedure.

Algorithm 2: De-noising the DT graph

Data: Ordered DT adjacency matrix ODT and parameter θ.

Result: De-noised ordered DT adjacency matrix DDT .

Initialize an all 0 adjacency matrix DDT ∈RN×N, where

nodes have same order as in ODT.

for i= 1 to Ndo

if (i≤θ&d(vi, Vi+θ)=0) or (i≥N−θ&

d(vi, Vi−θ)=0) or (d(vi, Vi+θ)=0&d(vi, Vi−θ)=0)

then

Set all edge-weights associated to viin ODT to 0.

// These nodes are non-differential nodes.

end

else

Copy all edge-weights associated to viin ODT to

DDT.

/* Node viis part of a differential community.

*/

end

end

We can now run state-of-the-art community detec-

tion algorithms [34, 3, 48] to distinguish the BD’s in

DDT as diﬀerential communities in paired biological

networks. The overall time complexity of proposed

steps is O(tN log N+tEdµ), where tis number of

iterations in Algorithm 1, Erepresents number of

edges and dµrepresents the average degree of a node

in DT graph. Algorithm 3 provides an overview of

the proposed DCD approach.

3 Simulated Experiments &

Results

We performed multiple simulated experiments on

paired random-geometric (RG) and paired scale-free

networks under diﬀerent experimental settings. All

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Algorithm 3: Diﬀerential Community Detection

(DCD) approach for paired biological networks

Data: Paired biological networks (A,B) and threshold θ.

Result: Diﬀerential sub-networks identiﬁed as diﬀerential

communities.

Create topological graphs for networks Aand B.

Generate the noisy DT graph: DT(A, B)ij =|aij −bij |,

∀i, j ∈V.

Use G(V, E ) and θto generate ODT as shown in Algorithm 1.

Use ODT and θto generate DDT using Algorithm 2.

Use either Louvain [3], Infomap [48] or Spectral [34]

community detection technique on DDT to identify

communities Ci∈C,i= 1,...,k.

if |Ci|< θ then

Remove Cifrom C.

end

All remaining communities in Care marked as diﬀerential

communities.

/* A differential community represents the set of nodes

whose corresponding edges form the differential

sub-networks. */

the experiments were repeated 10 times for each ex-

perimental setting.

In an RG network nodes are generated by uni-

formly sampling Npoints on [0,1]2. An edge is drawn

between points if the euclidean distance between the

points is less than a parameter ν. This parameter ν

controls the density of the RG network where smaller

values of νresult in sparse networks while larger val-

ues of νresult in dense networks. We performed two

set of experiments on RG networks. In the ﬁrst case,

we generated RG network A1with N= 1,000 and

ν= 0.15. Network Bis obtained by permuting ﬁrst

100 nodes in network A. Thus, these ﬁrst 100 nodes

form the diﬀerential sub-network for the paired RG

networks A1and B1.

In the second case, we again used N= 1,000 and

ν= 0.15 to generate network A2. We then cre-

ate a small dense RG network with 100 nodes us-

ing ˆν= 0.3. Network B2was generated by replac-

ing ﬁrst 100 nodes in network A2with the small

dense sub-network. These 100 nodes form the dif-

ferential sub-network for the paired networks A2and

B2. Such a mechanism can appear in real-life net-

works, for example, in case of cancer the transcrip-

tion activity of some set of genes might get enhanced

or suppressed generating more or fewer edges in a

sub-network of the gene or DNA methylation net-

work. We performed similar set of experiments using

density parameter ν= 0.3 and permuting ﬁrst 100

nodes, using density parameter ν= 0.3 and adding

more edges to ﬁrst 100 nodes using revised density

parameter ˆν= 0.5 on paired RG networks.

We also conducted experiments on undirected

scale-free graphs, hereby referred as Power-Law (PL)

networks, using N= 1000 and E= 10,000 with vary-

ing power-law exponents α={1,1.5,2}respectively.

We permuted the ﬁrst 100 nodes of each PL network

(A) to form the permuted network (B). The pro-

posed DCD method has one tunable parameter θ. In

Figure 2, we illustrate the eﬀect of θon the area under

the precision-recall curve. From Figures 2a, 2b, 2e,

2f, 2i and 2j, we can observe that for smaller values

of θ({3,5}), the area under precision-recall curves

are relatively lower in comparison to those for higher

values of θ. This is due to the fact that for smaller

values of θ, we are allowing smaller sized communities

to be distinguished as diﬀerential sub-networks. This

can force to break the natural block diagonals inher-

ently present in the DT graph and reduce the number

of true positives (i.e. nodes which are actually part

of diﬀerential sub-networks) leading to lower preci-

sion and recall. At the same time, smaller values of

θallow non-diﬀerential nodes with few spurious con-

nections to diﬀerential nodes to be falsely identiﬁed

as part of diﬀerential sub-networks resulting in lower

precision. For higher values of θ({7,9}), the area

under precision-recall curves shows less variance and

converges to nearly perfect result (≈1) as depicted

in Figures 2c, 2g,2h, 2k and 2l.

Table 1 encapsulates a comprehensive comparison

of the proposed DCD approach, where the commu-

nity detection technique used in DCD is either Lou-

vain [3] or Infomap [48] or Spectral [34], with sta-

tistical techniques like dGHD [50] and Closed-Form

[33] approach and direct application of community

detection methods like Louvain, Infomap and Spec-

tral on the noisy DT graph to detect diﬀerential sub-

networks in the simulated experiments. We used the

threshold θ= 7 in the DCD approach for all compar-

isons as the area under precision-recall curves shows

less variance and converges to nearly perfect value

(≈1) in all the simulated experimental settings for

this threshold as depicted in Figure 2. For nearly

all PL graph experiments, if we directly apply com-

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munity detection methods on the noisy DT graph,

they identify all the nodes in the network as part of

diﬀerential sub-network as depicted from evaluation

metrics in Table 1.

4 Application to co-

methylation networks in

ovarian cancer

We applied our proposed DCD approach, with pa-

rameter θset to 7, on co-methylation networks gen-

erated from ovarian cancer dataset [52]. Thus, the

smallest community in DT graph should comprise

at least 7 nodes. The ovarian cancer dataset con-

sists of methylation proﬁles for 27,578 CpG islands

of 540 women, of which 266 cases were from post-

menopausal women with ovarian cancer and 274 were

healthy controls with similar age as that of cases.

In our analysis, we have compared case and control

DNA co-methylation networks to identify diﬀerential

sub-networks.

The pre-processed dataset was downloaded from

GEO (repository number GSE19711). The original

data was collected using Illumina Inﬁnium 27k Hu-

man DNA methylation Beadchip v1.2. Since there

were no missing or negative values for the intensity

of the methylated (M) and unmethylated (U) alle-

les, beta values corresponding to each CpG probe

were computed as: β=M

M+Uas in [50]. We fol-

lowed the quality control procedure as originally in-

troduced in [52]. Then principal component analysis

(PCA) was applied to the beta values for detection

and removal of outliers. After quality control, 243

case samples and 214 control samples remained for

further analysis. Networks for case and control sam-

ples were created by treating each probe as a node.

Edges between the nodes represent strong correlation

and were inferred following [19]. Adjacency measure

Ωij was computed for each pair of nodes (iand j)

as Ωij =

1+cor(βi,βj)

2

b

, where cor(βi, βj) represents

Pearson’s correlation coeﬃcient between beta values

observed at ith and jth CpG sites. The exponent

bwas set to 12 to emphasize more on higher posi-

tive correlations [57]. An edge exists if Ωij value was

higher than 0.2. The resulting control network has

73,145 edges and case network has 102,799 edges.

Each of these networks follows a scale-free network

model as shown in Figure 3.

Our approach detected diﬀerential sub-networks

comprising of a total of 1,893 nodes. We used Lou-

vain [3] method for detection of communities in the

diﬀerential case and control sub-networks. Nine com-

munities were detected in the case diﬀerential sub-

network out of which seven are also present in the

control diﬀerential sub-network as shown in Figure

4.

We investigated the biological meaning of the

sub-networks by identifying enriched Gene Ontol-

ogy (GO) terms. We used R package GOstats [13]

to identify Biological Processes (BP) and Molecu-

lar Functions (MF). The hypergeometric test de-

tected 711 BP and 100 MP statistically signiﬁcant

terms enriched in the sub-networks at 5% signiﬁ-

cance level. The top three BPs were regulation of

myeloid cell apoptotic process, myeloid cell apoptotic

process, and establishment of protein localization to

organelle. The top three MFs were protein binding,

peroxidase activity and glycosaminoglycan binding.

Furthermore, we identiﬁed 16 signiﬁcantly enriched

KEGG pathways at 5% signiﬁcance level including

transcriptional mis-regulation in cancer, hematopoi-

etic cell lineage, and pathways in cancer using DAVID

[20].

We detected probes with signiﬁcant changes in

mean methylation levels using the t-test. We found

5,098 signiﬁcantly diﬀerentially methylated CpGs at

5% signiﬁcance level after FDR correction for mul-

tiple testing [2]. Table 2 summarizes the number of

probes, diﬀerentially methylated probes (qi), density

ratio between control and case sub-networks (Ri),

and distribution of enriched GO terms and KEGG

pathways in the identiﬁed communities.

5 Application in Glioma Can-

cer

We also applied the DCD approach, with parame-

ter θset to 7, on gene regulatory networks (GRN)

7

.CC-BY-NC-ND 4.0 International licensea

Graph Parameters MethodDT graph AUC ROC Precision Recall Accuracy Speciﬁcity Kappa Time

Mean ±Sd Mean ±Sd Mean ±Sd Mean ±Sd Mean ±Sd Mean ±Sd Mean

RG:Permute ν=0.15 Closed-Form Noisy 0.935 ±0.051 0.849 ±0.037 0.846 ±0.102 0.969 ±0.011 0.983 ±0.004 0.828 ±0.068 0.078

RG:Permute ν=0.15 dGHD Noisy 0.926 ±0.018 0.793 ±0.021 0.878 ±0.036 0.965 ±0.005 0.974 ±0.003 0.813 ±0.026 1.0

RG:Permute ν=0.15 Louvain Noisy 0.5885 ±0.012 0.3425 ±0.007 1.0 ±0.0 0.424 ±0.017 0.114 ±0.017 0.177 ±0.024 0.012

RG:Permute ν=0.15 Infomap Noisy 0.589 ±0.012 0.343 ±0.006 1.0 ±0.0 0.425 ±0.016 0.115 ±0.0168 0.178 ±0.024 0.018

RG:Permute ν=0.15 Spectral Noisy 0.5884 ±0.012 0.3425 ±0.007 1.0 ±0.0 0.424 ±0.017 0.114 ±0.017 0.177 ±0.024 0.015

RG:Permute ν=0.15 DCD (Louvain) De-noised 0.990 ±0.007 1.0 ±0.0 0.980 ±0.0176 0.994 ±0.004 0.986 ±0.013 1.0 ±0.0 0.014

RG:Permute ν=0.15 DCD (Infomap) De-noised 0.990 ±0.008 1.0 ±0.0 0.980 ±0.0176 0.994 ±0.005 0.986 ±0.012 1.0 ±0.0 0.021

RG:Permute ν=0.15 DCD (Spectral) De-noised 0.990 ±0.007 1.0 ±0.0 0.980 ±0.0176 0.994 ±0.004 0.986 ±0.014 1.0 ±0.0 0.018

RG:Dense ν=0.15, ˆν= 0.3Closed-Form Noisy 0.927 ±0.048 0.839 ±0.031 0.862 ±0.098 0.969 ±0.008 0.982 ±0.005 0.825 ±0.054 0.081

RG:Dense ν=0.15, ˆν= 0.3dGHD Noisy 0.922 ±0.022 0.806 ±0.027 0.868 ±0.045 0.966 ±0.006 0.977 ±0.004 0.816 ±0.032 1.0

RG:Dense ν=0.15, ˆν= 0.3Louvain Noisy 0.599 ±0.008 0.349 ±0.004 0.999 ±0.002 0.440 ±0.011 0.130 ±0.011 0.199 ±0.0015 0.013

RG:Dense ν=0.15, ˆν= 0.3Infomap Noisy 0.602 ±0.005 0.350 ±0.003 0.999 ±0.002 0.444 ±0.007 0.134 ±0.008 0.205 ±0.011 0.020

RG:Dense ν=0.15, ˆν= 0.3Spectral Noisy 0.600 ±0.007 0.348 ±0.004 1.0 ±0.0 0.440 ±0.011 0.131 ±0.011 0.200 ±0.015 0.016

RG:Dense ν=0.15, ˆν= 0.3DCD (Louvain) De-noised 0.998 ±0.002 1.0 ±0.0 0.995 ±0.005 0.999 ±0.001 0.997 ±0.003 1.0 ±0.0 0.015

RG:Dense ν=0.15, ˆν= 0.3DCD (Infomap) De-noised 0.998 ±0.003 1.0 ±0.0 0.995 ±0.006 0.999 ±0.003 0.997 ±0.002 1.0 ±0.0 0.0124

RG:Dense ν=0.15, ˆν= 0.3DCD (Spectral) De-noised 0.998 ±0.003 1.0 ±0.0 0.995 ±0.005 0.999 ±0.002 0.997 ±0.002 1.0 ±0.0 0.019

RG:Permute ν= 0.3Closed-Form Noisy 0.877 ±0.067 0.714 ±0.075 0.789 ±0.135 0.947 ±0.016 0.975 ±0.011 0.716 ±0.099 0.083

RG:Permute ν=0.3dGHD Noisy 0.724 ±0.029 0.645 ±0.049 0.577 ±0.059 0.921 ±0.007 0.971 ±0.006 0.504 ±0.051 1.0

RG:Permute ν=0.3Louvain Noisy 0.909 ±0.006 0.702 ±0.013 1.0 ±0.0 0.872 ±0.008 0.730 ±0.0149 0.818 ±0.011 0.013

RG:Permute ν=0.3Infomap Noisy 0.877 ±0.011 0.698 ±0.010 1.0 ±0.0 0.842 ±0.09 0.725 ±0.022 0.807 ±0.009 0.021

RG:Permute ν=0.3Spectral Noisy 0.911 ±0.007 0.708 ±0.017 1.0 ±0.0 0.876 ±0.009 0.736 ±0.018 0.823 ±0.014 0.017

RG:Permute ν=0.3DCD (Louvain) De-noised 0.996 ±0.001 1.0 ±0.0 0.992 ±0.002 0.998 ±0.001 0.995 ±0.002 1.0 ±0.0 0.016

RG:Permute ν=0.3DCD (Infomap) De-noised 0.996 ±0.002 1.0 ±0.0 0.992 ±0.002 0.998 ±0.002 0.995 ±0.001 1.0 ±0.0 0.025

RG:Permute ν=0.3DCD (Spectral) De-noised 0.996 ±0.001 1.0 ±0.0 0.992 ±0.003 0.998 ±0.000 0.995 ±0.003 1.0 ±0.0 0.02

RG:Dense ν=0.3, ˆν= 0.5Closed-Form Noisy 0.979 ±0.005 0.771 ±0.061 0.930 ±0.082 0.965 ±0.012 0.969 ±0.011 0.821 ±0.062 0.09

RG:Dense ν=0.3, ˆν= 0.5dGHD Noisy 0.848 ±0.071 0.700 ±0.038 0.731 ±0.148 0.941 ±0.010 0.964 ±0.009 0.672 ±0.078 1.0

RG:Dense ν= 0.3, ˆν= 0.5Louvain Noisy 0.758 ±0.056 0.353 ±0.086 1.0 ±0.0 0.613 ±0.090 0.310 ±0.125 0.517 ±0.113 0.014

RG:Dense ν=0.3, ˆν= 0.5Infomap Noisy 0.752 ±0.060 0.349 ±0.092 1.0 ±0.0 0.604 ±0.097 0.302 ±0.134 0.505 ±0.121 0.023

RG:Dense ν=0.3, ˆν= 0.5Spectral Noisy 0.750 ±0.087 0.332 ±0.047 1.0 ±0.0 0.589 ±0.099 0.286 ±0.101 0.500 ±0.175 0.02

RG:Dense ν=0.3, ˆν= 0.5DCD (Louvain) De-noised 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 0.017

RG:Dense ν=0.3, ˆν= 0.5DCD (Infomap) De-noised 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 0.027

RG:Dense ν=0.3, ˆν= 0.5DCD (Spectral) De-noised 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 1.0 ±0.0 0.024

PL:Permute α=1Closed-Form Noisy 0.797 ±0.046 0.307 ±0.007 0.799 ±0.049 0.801 ±0.018 0.349 ±0.051 0.802 ±0.022 0.09

PL:Permute α=1dGHD Noisy 0.797 ±0.023 0.294 ±0.009 0.794 ±0.027 0.787 ±0.008 0.333 ±0.045 0.784 ±0.019 1.0

PL:Permute α=1Louvain Noisy 0.500 ±0.001 0.100 ±0.001 1.0 ±0.0 0.100 ±0.001 0.001 ±0.000 0.001 ±0.000 0.015

PL:Permute α= 1 Infomap Noisy 0.501 ±0.002 0.101 ±0.000 1.0 ±0.0 0.101 ±0.000 0.001 ±0.000 0.001 ±0.000 0.026

PL:Permute α=1Spectral Noisy 0.500 ±0.000 0.100 ±0.001 1.0 ±0.0 0.100 ±0.001 0.001 ±0.000 0.001 ±0.000 0.019

PL:Permute α=1DCD (Louvain) De-noised 0.973 ±0.012 1.0 ±0.0 0.945 ±0.023 0.995 ±0.002 0.969 ±0.014 1.0 ±0.0 0.018

PL:Permute α=1DCD (Infomap) De-noised 0.973 ±0.011 1.0 ±0.0 0.945 ±0.024 0.995 ±0.003 0.969 ±0.015 1.0 ±0.0 0.03

PL:Permute α=1DCD (Infomap) De-noised 0.973 ±0.013 1.0 ±0.0 0.945 ±0.022 0.995 ±0.001 0.969 ±0.013 1.0 ±0.0 0.022

PL:Permute α=1.5Closed-Form Noisy 0.811 ±0.045 0.311 ±0.011 0.797 ±0.051 0.807 ±0.022 0.366 ±0.015 0.810 ±0.004 0.088

PL:Permute α=1.5dGHD Noisy 0.809 ±0.043 0.301 ±0.009 0.791 ±0.042 0.797 ±0.015 0.344 ±0.016 0.796 ±0.007 1.0

PL:Permute α= 1.5Louvain Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.016

PL:Permute α=1.5Infomap Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.026

PL:Permute α=1.5Spectral Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.019

PL:Permute α=1.5DCD (Louvain) De-noised 0.989 ±0.005 1.0 ±0.0 0.979 ±0.010 0.998 ±0.001 0.988 ±0.006 1.0 ±00.018

PL:Permute α=1.5DCD (Infomap) De-noised 0.989 ±0.004 1.0 ±0.0 0.979 ±0.010 0.998 ±0.000 0.988 ±0.005 1.0 ±00.03

PL:Permute α=1.5DCD (Spectral) De-noised 0.989 ±0.006 1.0 ±0.0 0.979 ±0.009 0.998 ±0.001 0.988 ±0.007 1.0 ±00.022

PL:Permute α=2Closed-Form Noisy 0.825 ±0.019 0.345 ±0.015 0.825 ±0.035 0.826 ±0.007 0.402 ±0.024 0.826 ±0.004 0.085

PL:Permute α= 2 dGHD Noisy 0.818 ±0.027 0.327 ±0.018 0.799 ±0.050 0.816 ±0.008 0.375 ±0.031 0.817 ±0.004 1.0

PL:Permute α=2Louvain Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.016

PL:Permute α= 2 Infomap Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.026

PL:Permute α=2Spectral Noisy 0.5 ±0.0 0.1 ±0.0 1.0 ±0.0 0.1 ±0.0 0.0 ±0.0 0.0 ±0.0 0.019

PL:Permute α=2DCD (Louvain) De-noised 0.971 ±0.017 1.0 ±0.0 0.941 ±0.033 0.994 ±0.003 0.966 ±0.020 1.0 ±0.0 0.018

PL:Permute α=2DCD (Infomap) De-noised 0.971 ±0.018 1.0 ±0.0 0.941 ±0.032 0.994 ±0.002 0.966 ±0.021 1.0 ±0.0 0.03

PL:Permute α=2DCD (Spectral) De-noised 0.971 ±0.016 1.0 ±0.0 0.941 ±0.033 0.994 ±0.004 0.966 ±0.019 1.0 ±0.0 0.022

Table 1: Comparison of proposed DCD approach with Closed-Form [33] and dGHD [50] statistical techniques

and direct application of community detection methods like Louvain [3], Infomap [48] and Spectral [34]

on nosiy DT graph to identify diﬀerential sub-networks in paired simulated networks for various settings.

Here RG:Permute represents RG networks where ﬁrst 100 nodes are permuted and form diﬀerential sub-

network. Similarly, PL:Permute is used for experiments on PL graphs where ﬁrst 100 nodes are permuted

and constitute the diﬀerential sub-network. RG:Dense depicts RG networks, where ﬁrst 100 nodes have

higher density in network B in comparison to network A and make-up the diﬀerential sub-network. Time

is represented as fraction w.r.t. the computational time of most expensive method (dGHD). Best results

are highlighted in bold. The proposed DCD approach can robustly identify diﬀerential sub-networks in all

simulated experimental settings. It performs the best for evaluation metrics: AUC ROC (area under ROC

curve), Precision, Accuracy, Kappa and Speciﬁcity.

generated from the TCGA pan-glioma dataset [33].

The TCGA pan-glioma dataset includes 1,250 sam-

ples (463 IDH-mutant and 653 IDH-wild-type), 583 of

which were proﬁled with Agilent microarray and 667

with RNA-Seq Illumina HiSeq (REF) downloaded

from the TCGA portal. The batch eﬀects between

the two platforms were corrected using the COM-

BAT algorithm [25]. The ﬁnal gene expression data

includes 12,985 genes and 1,250 samples. From

this data, we inferred the GRN for the two diﬀer-

ent glioma sub-types using the ARACNe [39] algo-

rithm as in [33]. In our analysis, we compared the

GRNs of IDH-mutant and IDH-wild-type to identify

sub-networks of transcription factors (TFs) having a

8

.CC-BY-NC-ND 4.0 International licensea

diﬀerent regulatory program in these two major con-

ditions.

The ARACNe networks were intersected with an

active binding network based on the presence of bind-

ing sites in the promoter of a target gene. The

active binding network is reconstructed for 2,532

unique motifs corresponding to 1,203 unique TFs

[26, 40, 28]. A binding relationship is considered ac-

tive if the TF motif signal is signiﬁcantly (FDR <

0.05) over-represented in the target promoter region

(∓5kbp TSS, hg19) and, in the same position (at

least 1bp overlapping), chromatin state is classiﬁed as

open by Hidden Markov Model proposed in [12]. The

active binding network consists of 6,652,518 overlap-

ping active sites resulting in 1,959,125 unique TF

associations between 1,203 TFs and 51,705 targets.

The ﬁnal pruned networks are then obtained by

considering the common sub-network of active bind-

ing and functional ARACNE networks. They consists

of 13,683 unique connections for IDH-mutant and

14,158 for IDH-wild-type between TF-TF and TF-

target. The number of TFs was reduced to 457 when

intersected with the 12,895 genes of our combined ex-

pression matrix. We then apply the proposed DCD

approach on the noisy DT graph G(V, E) obtained by

taking the absolute diﬀerence between the topologi-

cal graphs of IDH-mutant and IDH-wild-type. The

DCD technique discovered a total of 262 TFs as part

of 7 diﬀerential communities using the Louvain [3]

method in G(V, E).

We further investigated these communities by con-

sidering the regulons of all the TFs associated with

each such community Ciin the corresponding IDH-

mutant and IDH-wild-type GRN. The regulon of a

TF is deﬁned as its neighbourhood in the GRN. We

C1 C2 C3 C4 C5 C6 C7 C8 C9 Total

Probes 825 364 198 155 21 17 11 8 294 1893

qi 5 363 22 140 18 0 1 2 245 5098

Ri 0.82 0.16 0.09 0.11 1.77 0 3.67 0 3.23 0.72

BP 628 542 452 378 195 118 136 124 495 711

MF 86 53 44 37 9 9 16 8 53 100

KEGG 6 4 1 3 0 0 0 0 4 16

Table 2: DNA co-methylation networks: a summary

of diﬀerent communities detected by DCD approach.

probed the regulons of all TFs present in a commu-

nity to detect enriched GO terms using DAVID [25].

We found 15 and 17 statistically signiﬁcant biolog-

ical processes (BP) at a 5% signiﬁcance level using

the regulons of TFs in C1for IDH-mutant and IDH-

wild-type GRNs respectively. We also located 50, 14,

9, 21, 51 and 40 signiﬁcant BPs for C2,C3,C4,C5,C6

and C7respectively in IDH-mutant GRN. Similarly,

we unearthed 71, 11, 4, 20, 48 and 20 signiﬁcant BPs

for C2,C3,C4,C5,C6and C7respectively in IDH-wild-

type GRN.

We utilized the output from DAVID for each Ciin

the IDH-mutant and IDH-wild-type GRN as input

to Enrichment Map tool [41] in Cytoscape. This tool

provides a visualization for functional enrichment as-

sociated with BPs in Ciand allows comparison be-

tween enrichment results for two diﬀerent conditions

(IDH-mutant and IDH-wild-type). Figure 5a illus-

trates the diﬀerence between the enrichment results

of C1in IDH-mutant and IDH-wild-type case. Sim-

ilarly, Figure 5b compares the enrichment results of

C3in IDH-mutant and IDH-wild-type.

Interestingly, the diﬀerential community C1is en-

riched with functions related to epigenetic changes

such as Chromatin Modiﬁcation and Histone Acety-

lation. Ceccarelli et al showed in [8] that the main

diﬀerence between IDH-mutant and IDH-wild-type

gliomas is the characteristic hyper-methylation phe-

notype (G-CIMP) which has a favourable prognosis

both in high grade and low grade gliomas. Con-

versely, the C3reveals enrichments which are spe-

ciﬁc of IDH-wild-type gliomas such as proliferation

and activation of inﬂammatory response. There-

fore, the DCD approach is not only able to identify

known but also potential novel enrichments which

need to be investigated further, in the two patho-

logical conditions. Additional supplementary infor-

mation is provided at https://sites.google.com/

site/raghvendramallmlresearcher/codes.

6 Conclusion

We propose a fast two-stage DCD approach to iden-

tify diﬀerential sub-networks in paired biological

graphs. The proposed method performs node or-

9

.CC-BY-NC-ND 4.0 International licensea

dering using neighbourhood information of nodes

and Jaccard similarity to detect approximate block-

diagonals. It de-noises the ordered noisy diﬀeren-

tial topological graph by traversing its landscape

along the diagonal. Finally, diﬀerential sub-networks

are identiﬁed using community detection algorithms.

We showcased the eﬀectiveness of proposed approach

w.r.t. various statistical techniques and direct appli-

cation of community detection methods for a myr-

iad experimental settings using evaluation metrics

like Precision, Accuracy, Kappa and Speciﬁcity. The

DCD approach identiﬁed several meaningful biologi-

cal processes and molecular functions on ovarian can-

cer dataset. Similarly, using DCD, we singled out

some functional pathways that are diﬀerent between

the IDH-mutant and IDH-wild-type subtypes in case

of glioma cancer.

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(a) Network A(b) Network B(c) Topological graph of A

(d) Topological graph of B(e) Noisy DT graph (f) Result on noisy DT graph

(g) Ordered noisy DT graph (h) Ordered de-noised DT graph (i) Result on ordered de-noised DT

Figure 1: Illustration of DCD method and its beneﬁt over directly using community detection methods

on noisy DT graph. Figure 1a represents a random-geometric network Awith 1,000 nodes and Figure 1b

represents another random-geometric network Bwhere the nodes 1 to 100 and nodes 500 to 600 have diﬀerent

interaction pattern from network A. Figures 1c and 1d correspond to the topological graphs of network A

and B. Figure 1e shows the noisy diﬀerential topological (DT) graph obtained from topological graphs of

Aand B. Figure 1f evaluates the result of 3 state-of-the-art community detection techniques on the noisy

DT graph to detect diﬀerential sub-networks w.r.t. precision and recall metrics. Figure 1g illustrates the

ordered noisy DT graph obtained from ﬁrst stage of DCD approach. Figure 1h demonstrates the de-noised

DT graph generated after the second stage of DCD method. Figure 1i showcases the eﬃciency of 3 diﬀerent

community detection methods to identify the diﬀerential sub-networks from the de-noised DT graph.

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Recall

Precision

0.2 0.4 0.6 0.8 1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Louvain

Infomap

Spectral

(a) θ= 3, ν= 0.15

Recall

Precision

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Louvain

Infomap

Spectral

(b) θ= 5, ν= 0.15

Recall

Precision

0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Louvain

Infomap

Spectral

(c) θ= 7, ν= 0.15

Recall

Precision

0.4 0.6 0.8 1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Louvain

Infomap

Spectral

(d) θ= 9, ν= 0.15

Recall

Precision

0.5 0.6 0.7 0.8 0.9 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(e) θ= 3, ν= 0.15 & ˆν=

0.3

Recall

Precision

0.5 0.6 0.7 0.8 0.9 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(f) θ= 5, ν= 0.15 & ˆν=

0.3

Recall

Precision

0.5 0.6 0.7 0.8 0.9 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(g) θ= 7, ν= 0.15 & ˆν=

0.3

Recall

Precision

0.5 0.6 0.7 0.8 0.9 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(h) θ= 9, ν= 0.15 & ˆν=

0.3

Recall

Precision

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(i) θ= 3, α= 1.5

Recall

Precision

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(j) θ= 5, α= 1.5

Recall

Precision

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(k) θ= 7, α= 1.5

Recall

Precision

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Louvain

Infomap

Spectral

(l) θ= 9, α= 1.5

Figure 2: Area under the precision-recall curves for diﬀerent values of threshold θfor various experimental

settings. We demonstrate the area under precision-recall curves using the proposed steps of DCD approach

with either Louvain or Infomap or Spectral community detection method. Figures 2a,2b,2c and 2d show the

role of parameter θon precision-recall values for paired RG networks (ν= 0.15) where ﬁrst 100 nodes are

permuted. Figures 2e,2f, 2g and 2h illustrate how the area under precision-recall curves vary with threshold

θfor paired RG networks (ν= 0.15) where the sub-network corresponding to ﬁrst 100 nodes have higher

density (ˆν= 0.5). Similarly, Figures 2i, 2j, 2k and 2l describes the role of variable θon precision-recall

values for paired PL networks (α= 1.5) where the ﬁrst 100 nodes are permuted.

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Figure 3: Degree distribution of nodes for control and case co-methylation networks. Since α < 1 for both the

networks, state-of-the-art statistical techniques cannot be applied on these paired networks for diﬀerential

sub-network analysis.

(a) Diﬀerential sub-networks in controls (b) Diﬀerential sub-networks in case

Figure 4: DNA co-methylation diﬀerential sub-networks. Cluster C7 is a special case. Even though it

comprises of less than 7 nodes in the case sub-network, it consists of 9 nodes in control sub-network and has

very diﬀerent topography in the two sub-networks. As a result, it appears as a diﬀerential community of size

greater than 7 in the de-noised DT graph. Clusters C6 and C8 are not present in the control sub-network.

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(a) Enrichment results for C1(b) Enrichment results for C3

Figure 5: Comparison of enrichment results of IDH-mutant and IDH-wildtype for diﬀerential communities

C1and C3. Here the nodes correspond to the BPs and red circle size is proportional to number of genes

in IDH-mutant associated with that BP. Similarly, the grey circle size in a node (BP) corresponds to the

number of genes in IDH-wild-type related to that BP. Edge size corresponds to the number of genes that

overlap between the two connected BPs. Green edges correspond to IDH-mutant while purple edges represent

interaction between BPs in IDH-wild-type.

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