A faster algorithm for detecting network motifs

A faster algorithm for detecting network motifs
A faster algorithm for detecting network motifs

A Faster Algorithm

for Detecting Network Motifs

Sebastian Wernicke?

Institut f¨u r Informatik,Friedrich-Schiller-Universit¨a t Jena

Ernst-Abbe-Platz2,D-07743Jena,Germany

wernicke@minet.uni-jena.de

Abstract.Motifs in a network are small connected subnetworks that

occur in signi?cantly higher frequencies than in random networks.They

have recently gathered much attention as a useful concept to uncover

structural design principles of complex networks.Kashtan et al.[Bioin-

formatics,2004]proposed a sampling algorithm for e?ciently performing

the computationally challenging task of detecting network motifs.How-

ever,among other drawbacks,this algorithm su?ers from sampling bias

and is only e?cient when the motifs are small(3or4nodes).Based

on a detailed analysis of the previous algorithm,we present a new al-

gorithm for network motif detection which overcomes these drawbacks.

Experiments on a testbed of biological networks show our algorithm to

be orders of magnitude faster than previous approaches.This allows for

the detection of larger motifs in bigger networks than was previously

possible,facilitating deeper insight into the?eld.

1Introduction

Motivation.Based on the idea that“evolution preserves modules that de?ne speci?c[...]functions”[20],Milo et al.[14,15]propose to uncover the struc-tural design principles of biological networks1by detecting small subnetworks which occur in signi?cantly higher frequencies than in random networks.These “topological modules”[20]are called network motifs.2

Some excitement has surrounded the network motif approach with the origi-nal paper by Milo et al.[15]being cited well over40times in some major scienti?c journals as of June2005.The analysis of network motifs has led to interesting results(of which we only name a few here),e.g.,in the areas of protein-protein interaction prediction[1]and hierarchical network decomposition[7].The tran-scriptional network of Escherichia Coli displays motifs to which speci?c function-alities such as the generation of temporal expression programs or the response to ?Supported by Deutsche Telekom Stiftung and Studienstiftung des deutschen Volkes. 1We use the terms“network”and“node”for?elds outside mathematics and computer

science.The terms“graph”and“vertex”are used for discussing algorithmic aspects. 2Note that the term“network motif”has been used in other contexts as well and,

e.g.,may also refer to a common subnetwork in a set of given networks[17]or to

any small labeled subnetwork(without considering connectivity or isomorphy)[5].

?uctuating external signals can be attributed[15,18],suggesting that network motifs play key information processing roles in this type of network[9].The same motifs as in the transcriptional interaction network of E.Coli were also identi?ed for the yeast Saccharomyces Cerevisiae,possibly hinting that common network function implies the sharing of common motifs[11].

To put motif research in proper perspective,it should be noted that it has also been met with some criticism.Artzy-Randrup et al.[2]found that certain random network models lead to a display of motifs although there is no explicit selection mechanism for local structures(Milo et al.answer this criticism in[13]). V′a zquez et al.[19]demonstrated that global network features such as the clus-tering coe?cient also in?uence local features such as the abundance of certain subgraphs.

Previous Work.Much work related to network motifs has been spent on inter-preting and applying the general concept,but considerably less on the involved algorithmics.Finding network motifs consists of three subtasks:

1.Find which subgraphs occur in the input graph(and in which number).

2.Determine which of these subgraphs are topologically equivalent(i.e.,iso-

morphic)and group them into subgraph classes accordingly.

3.Determine which subgraph classes are displayed at a much higher frequency

than in random graphs(under a speci?ed random graph model). Performing the?rst subtask by explicitly enumerating all subgraphs of a certain size can be time consuming due to their potentially large number even in small, sparse networks.For this reason,Kashtan et al.[9]propose an algorithm that estimates subgraph occurrences from a randomly sampled set of subgraphs.We discuss this algorithm in full detail in Section2,mentioning only in passing here that it provides only biased sampling.This leads to considerable drawbacks such as an inconsistent sampling quality and the need for a computationally expensive bias correction.Besides[9],we are only aware of the work by Duke et al.[6]on approximating the number of size-k subgraphs in a given graph.Their algorithm,however,has no practical relevance since the input graph has to be astronomically large(as compared to k)in order to ensure a reasonable quality of approximation.

Much work has already been done concerning the second subtask and we rely on the nauty algorithm[12]for performing it in practice.

As to the third subtask,the standard approach for determining subgraph sig-ni?cance so far has been to explicitly generate an ensemble of random graphs un-der a given random graph model.One popular of these random graph models—which we also focus on in this work—is that of random graphs which preserve the degree sequence of the original graph.(Alternative choices,e.g.,additionally preserve the number of bidirectional edges.)While there has been some research concerning the properties of graphs with prescribed degree sequence(such as the average path length[16]),the problem of subgraph distribution within such graphs has only been studied for directed sparse random graphs with expected degree sequences[8].

Contribution and Structure of this Work.We give signi?cant improvements for

the?rst and third subtask of motif detection.Based on a comprehensive anal-

ysis of the drawbacks encountered when using the subgraph sampling approach

proposed by Kashtan et al.[9],Section2presents a new algorithm for subgraph

sampling which does not su?er from these drawbacks(and has some additional

useful features).While this comes at the price of only being able to control the

expected number of samples,our proposed algorithm is much easier to implement

and experiments in Section4reveal it to be orders of magnitude faster than the

algorithm of Kashtan et al.As to the task of determining subgraph signi?cance,

Section3proposes a new approach that does not require the explicit generation

of random graphs with a prescribed degree sequence.This approach leads to a

faster algortithm that is moreover able to focus on determining the signi?cance

of speci?c subgraphs(which is not possible with previous approaches).

The proposed new algorithms have been implemented in C++,the source code

is freely available online at http://www.minet.uni-jena.de/?wernicke/motifs/.

We show in Section4that in a testbed of biological networks,our algorithm

detects network motifs signi?cantly faster than the implementation of Kashtan

et al.This enables the analysis of larger networks and more complex motifs than

previously possible.

2A Faster Algorithm for Subgraph Sampling Introduction.The algorithm for subgraph sampling suggested by Kashtan et

al.[9]is based on the idea that we start by selecting a random edge in the

input graph and then randomly extend this subgraph until we obtain a con-

nected subgraph with the desired number of vertices.Subsection2.1discusses

this approach and its main drawbacks.We present a new approach to subgraph

sampling(which is based on randomized enumeration)in Subsection2.2.Note

that,due to lack of space,we omit the proofs of the theorems and lemmas

presented in this section.

Notation.Basic familiarity with graph-theoretic terminology is assumed.Given

a graph G=(V,E)(which can be directed),we let n:=|V|and assume that

all vertices in V are uniquely labeled by the integers1,...,n.We write“u>v”

to abbreviate“label(u)>label(v).”For a set V′?V of vertices,its neighbor-hood N(V′)is the set of all vertices from V\V′which are adjacent to at least

one vertex in V′.

A connected subgraph that is induced by a vertex set of cardinality k is

called size-k subgraph.For a given integer k,the set of all size-k subgraphs

in G can be partitioned into sets S i k(G)called subgraph classes where two

size-k subgraphs belong to the same subgraph class if and only if they are

isomorphic.The concentration C i k(G)of a subgraph class S i k(G)is de?ned as C i k(G):=|S i k(G)|·( j|S j k(G)|)?1.For a graph G,an integer k,and a set R of size-k subgraphs that were randomly sampled in G by an algorithm A,a map-ping?C i k:(R,G)→[0,1]is called an estimator for C i k(G).We say that?C i k(R,G)

G 1G 2

Fig.1.1and G 2have an equal number of (connected)size-3subgraphs.

The subgraph occurs once in each of them.As outlined in the text,esa

oversamples in both G

1and G 2.The oversampling is worse for G 1.is unbiased (with respect to A )if the expected value of ?C i k (R ,G )equals C i k (G )and biased otherwise.

2.1The Previous Approach:Edge Sampling

For a given graph G =(V,E )and an integer k ≥3,Kashtan et al.[9]suggest to sample a random subgraph by starting with a randomly chosen edge and then adding neighboring vertices until a subgraph of the desired size k is obtained:

Algorithm:Edge Sampling(G,k )(esa)

Input:A graph G =(V,E )and an integer 2≤k ≤|V |.

Output:Vertices of a randomly chosen size-k subgraph in G .

01

{u,v }←random edge from E 02

V ′←{u,v }03

while |V ′|=k do 04

{u,v }←random edge from V ′×N (V ′)05

V ′←V ′∪{u }∪{v }06return V ′

As already noted in [9],esa has a bias for sampling certain subgraphs more often than others.Figure 1shows a concrete example we have constructed to illustrate this.The total number of connected size-3subgraphs both in G 1and G 2

is 28.Since the subgraph occurs exactly once each in G

1and G 2,we should expect that esa samples with probability 1

28within both graphs.However,Pr[esa samples in G

119·1+29·28=16and Pr[esa samples in G 2]=3

12·28=116.illustrates some crucial problems of esa :The is oversampled and—as a direct consequence—the only other occurring subgraph is undersampled.The oversampling of is worse for G

1than it is for G 2it is possible to show (using an of the above example)that the magnitude of the oversampling cannot be estimated simply from the number of edges neighboring the oversampled subgraph.Given a set R of size-k subgraphs that were randomly sampled using esa ,the demonstrated bias can be overcome by using the following (unbiased)estimator [9]:?C i k (R ,G ):= {G ′∈R |G ′∈S i k (G )}(Pr[G ′is sampled by esa ])?1 G ′∈R

(Pr[G ′is sampled by esa ])?1.(1)The main idea here is that each subgraph is (ex post facto)scored inversely pro-portional to the probability that esa samples it.While it is possible to correctly estimate C i k (G )in this way,several disadvantages remain:

–The bias itself remains.E.g.,subgraphs which appear in low concentration and are at the same time undersampled by esa are hardly ever found.3

–Computing(1)is expensive since the calculation of each single probability can require as much as O(k k)time[9].

–We have no estimate as to what fraction of subgraphs has been sampled.–esa can sample the same subgraph multiple times.

In the next subsection we suggest a new approach to subgraph sampling that overcomes these problems.

2.2The New Approach:Randomized Enumeration

The idea here is to start with an algorithm that e?ciently enumerates all size-k subgraphs.This algorithm is then modi?ed to randomly“skip”some of these subgraphs during its execution,yielding an unbiased subgraph sampling algo-rithm.

Enumerating all size-k subgraphs.Given a graph G=(V,E),the following algo-rithm enumerates all of its size-k subgraphs(with N excl(v,V′):=N({v})\N(V′) being the exclusive neighborhood of v with respect to V′?V):

Algorithm:EnumerateSubgraphs(G,k)(esu)

Input:A graph G=(V,E)and an integer1≤k≤|V|.

Output:All size-k subgraphs in G.

01for each vertex v∈V do

02V Extension←{u∈N({v})|u>v}

03call ExtendSubgraph({v},V Extension,v)

04endfor

ExtendSubgraph(V Subgraph,V Extension,v)

E1if|V Subgraph|=k then output G[V Subgraph]and return

E2while V Extension=?do

E3Remove an arbitrarily chosen vertex w from V Extension

E4V′Extension←V Extension∪{u∈N excl(w,V Subgraph)|u>v}

E5call ExtendSubgraph(V Subgraph∪{w},V′Extension,v)

E6return

The basic idea of esu is that—starting with a vertex v from the input graph—we add only those vertices to the V Extension set that have two properties:Their label must be larger than that of v and they must not be neighbor to a vertex in V Subgraph(other than the newly added vertex w).Some more insight into the structure of esu can be gained by the following visualization.

3Kashtan et al.[9]observe that esa can accurately estimate the concentration of S i k(G)with less than(C i k(G))?1samples for subgraphs which are oversam-pled.In return however,other subgraphs might be missed completely for far more than(C i k(G))?1samples and would consistently be overlooked as motif candidates.

({1},{

123

1421521621723413513813911542632732832932763891234

789G

Fig.2.The above esu -tree corresponds to calling EnumerateSubgraphs(G,3).The tree has 16leafs which correspond to the 16size-3subgraphs in G .

De?nition 1.With a call to EnumerateSubgraphs(G,k ),we associate a tree-graph called esu -tree which represents the recursive function calls.The root at depth 0represents the call of EnumerateSubgraphs(G,k ).Each call of ExtendSubgraph(V Subgraph ,V Extension ,v )is represented by an edge from the vertex representing the caller function to a vertex representing the callee.The callee vertex is labeled (V Subgraph ,V Extension )and located at depth |V Subgraph |.

The structure of the tree is illustrated in an example in Figure 2.Omitting the proof here,it is also the basis to establish the correctness of the esu algorithm.Theorem 2.Given a graph G and k ≥2,esu enumerates all size-k subgraphs in G (each size-k subgraph is output exactly once).??

The tree structure to represent esu exposes some useful properties.E.g.,using a technique by Knuth [10],we can randomly explore paths in the tree in order to quickly estimate the total number of size-k subgraphs in the input graph.Probably the most important feature of the esu -tree,however,is that we can use it to e?ciently sample subgraphs uniform at random (i.e.,without bias).Uniformly sampling size-k subgraphs.The esu algorithm completely traverses its corresponding esu -tree.Where complete traversal is too time-expensive,we can explore only parts of the esu -tree such that each leaf is reached with equal probability.For this purpose,a probability 0

(To simplify the discussion,we will also use this name when all p d are set to1, in which case rand-esu is equivalent to esu.)rand-esu visits each leaf of the esu-tree with equal probability and hence estimating subgraph concentrations from its output is straightforward(the proofs are omitted).

Lemma3.rand-esu visits each leaf in the esu-tree with probability d p d.??Theorem4.Given a graph G,an integer k,and0

–The parameters p d in?uence the distribution of the sampling,i.e.,if p d is small for small d,some local neighborhoods in the input graph are likely not to be explored at all while others will be explored extensively.

–The running time is in?uenced from an amortized point of view:If the p d values are large for small values of d(and hence small for larger d),much of the esu-tree is explored but only comparably few leafs are reached.

As a general rule from these observations,the parameters p d should be larger for small d and become smaller as d increases—as long as the sacri?ce made with respect to the amortized running time per sample is acceptable.This ensures a lower variance for the number of samples and the exploration of many di?erent regions in the input graph.

Concluding this section,while rand-esu—as compared to esa—requires a choice of sampling parameters and only allows for controlling the expected num-ber of samples,it has a lot to o?er in return.Most importantly it is unbiased, which rules out the respective disadvantages of esa.Also,it is much faster(see Section4)and easier to implement since we do not require any bias-correcting parts.Contrary to esa,our new algorithm never samples more subgraphs than the input graph contains and results become exact as the number of samples reaches the total number of size-k subgraphs in the input graph.

3Direct Calculation of Motif Signi?cance

The Previous Approach.As already mentioned in the introduction,motif detec-tion includes the subtask of determining subgraph signi?cance.In this work,we x′=?x·p d?otherwise)and explore exactly these.It can be shown that this does not change the probability of a leaf being explored.

consider the case where the signi?cance of a subgraph is determined by compar-ing its concentration in the given graph G to its mean concentration C i k (G ) in random graphs with the same degree sequence [15,9].It is suggested in [15,9]to estimate C i k (G ) by generating a large ensemble of random graphs (typically at least 1000)with the same degree sequence as the original graph and then sampling subgraphs in these random graphs.The random graphs are generated from the original graph by randomly switching edges between vertices,which requires a lot of switching operations while at the same time it is never certain when proper randomization has been reached.Also with this method,we are likely to spend lots of excess computational e?orts estimating the concentra-tions of subgraph classes we are not interested in.5In this section we propose an algorithm for determining subgraph signi?cance without the need to explic-itly generate random graphs (assuming the background model of random graphs with the same degree sequence).We also gain the ability to focus our estimation of signi?cance on speci?c subgraphs.

Direct Calculation of Subgraph Signi?https://www.360docs.net/doc/ea5032727.html,o et al.observe that the total number of size-k subgraphs within an ensemble of large graphs with the same degree sequence does not vary much (see supplementary online material to [14]for details).This allows us to estimate C i k (G ) by

C i k (G ) ≈ ?C i k (G ) := G ′∈DegSeq (G )|S i k (G ′)| G ′∈DegSeq (G ) i |S i k

(G ′)|(2)where DegSeq (G )is the set of all graphs G ′that have the same degree sequence as G .Since all graphs G ′can be viewed as graphs over the same set of vertices (because they di?er only in their edge sets),Equation (2)can also be written as

?C i k (G ) = {v 1,...,v k }?V |{G ′∈DegSeq (G )|G ′[{v 1,...,v k }]∈S i k }| {v 1,...,v k }?V |{G ′∈DegSeq (G )|G ′[{v 1,...,v k }]connected }|.(3)

Both the nominator and denominator of this equation can be estimated in a Monte Carlo approach—i.e.,by randomly sampling size-k subsets of the vertices in the input graph—as long as we are able to perform the following calculation:Given G and {v 1,...,v k },?nd |{G ′∈DegSeq (G )|G ′[{v 1,...,v k }]∈S i k }|.As it turns out,this number is indeed possible to calculate using two theorems (one for undirected graphs and one for the directed case)due to Bender and Can?eld [3,4].Without going into technical details here,these theorems allow us to calculate for a given degree sequence how many graphs there are which realize exactly this degree sequence under the constraint that a certain subgraph is ?xed.Given a subgraph class S i k and k vertices {v 1,...,v k },we can thus consider all (at most k !)ways in which {v 1,...,v k }can induce a subgraph from S i k and hence estimate the nominator in Equation (3).

5This is especially important for sparse networks where a randomly sampled sub-graph is likely to be a tree.Trees,however,are often considered to be uninteresting motifs [5].

Table1.Number of size-k subgraphs and the number of respective subgraph classes that occur in our test instances for3≤k≤6.(All instances are directed graphs).

nodes edges subgraphs subgraph classes

size-3size-4size-5size-6size-3size-4size-5size-6 coli42351952068389314335022253258441783390

yeast688107913150183174250814932883898733173888 elegans3062345473221394259432560691309307357131977071286375

ythan13559794871697332908118458890398576299339 An analogous approach(considering all ways in which the given vertices can be connected)can be used to estimate the denominator in Equation(3).Omit-ting the details here,it is possible to show that this does not require the explicit consideration of every connected size-k subgraph but only of k k?2subgraphs for undirected graphs and2·(2k)k?2in the directed case.At?rst glance it might seem as if this is prohibitively expensive to calculate,but for two reasons it actu-ally promises a gain in e?ciency:Firstly,the denominator in Equation(3)is the same for all subgraph classes and hence has to be calculated only once.Secondly, the number of occurring subgraph classes is often far less than the total number of subgraphs(see Table1).Experiments which are discussed in the next section con?rm this expected performance gain.

4Experimental Studies

Method and Results.We have implemented our algorithms from Sections2and3 in C++.The source code is freely obtainable online at http://www.minet.uni-jena.de/?wernicke/motifs/.As a comparison,we used the m?nder1.1tool6by Kashtan et al.which implements the esa algorithm.All tests were performed on an AMD Athlon643400+with2.4GHz,512KB cache,and1GB main mem-ory running under the Debian GNU/Linux3.1operating system.Sources were compiled with the GNU gcc/g++3.4.3compiler using the option“-O3.”

The network instances for testing the algorithms were up-to-date versions of the motif detection testbed used by Kashtan et al.[9].The testbed consists of the instances coli(transcriptional network of Escherichia Coli[18]),yeast (transcriptional network of Saccharomyces Cerevisiae[15]),elegans(neuronal network of Caenorhabditis Elegans[9]),and ythan(food web of the Ythan es-tuary[21]).Some properties of these networks are summarized in Table1.The algorithms were compared both for their speed and quality;results and some details as to the experimental setting are shown in Figure3and Table2. Discussion.Most notable in Figure3a,rand-esu is much faster than the esa sampling in m?nder.This amounts to several orders of magnitude for larger subgraphs(k≥5).For small sampling quantities,the“coarse”variant of rand-esu proved to be faster than the“?ne”variant(not explicitly shown in Fig-ure3a).However,Figure3b shows that the resulting sampling quality from 6Source at http://www.weizmann.ac.il/mcb/UriAlon/groupNetworkMotifSW.html

(a)

subgraph size [vertices]1101

102103104105106107

s a m p l i n g s p e e d [s u b g r a p h s p e r s e c o n d ](b)

sampled subgraphs [% of total]0

20406080100s a m p l i n g q u a l i t y [% c o r r e c t e s t i m a t e s ]Fig.3.(a)Sampling speed for di?erent subgraph sizes on a semi-log scale (time mea-surement does not include the grouping of sampled subgraphs into classes).For esu ,the curve shows the mean speed for three di?erent settings of (p 1,...,p k )that lead to sampling of an expected 10%of all subgraphs:(1,...,1,.316,.316),(1,...,1,.5,.2),and (1,...,1,.1).(The speed of the deterministic esu algorithm—not shown here—is slightly faster than that of rand-esu .)(b)Sampling quality for size-5subgraphs (size-4for ythan )versus the percentage p of sampled subgraphs (semi-log scale).We de?ne the sampling quality as the percentage of subgraph classes S k i for which C k i is estimated with at most 20%relative error (considering only those subgraph classes for a given p that we would expect to sample at least 10times on average).rand-esu was run with two di?erent settings of the p d values we refer to as “coarse”(1,...,1,√p,√p )and “?ne”(1,...,1,p ).For our ythan instance,the m?nder tool reproducibly failed to report results for more than 100samples,hence this curve is not https://www.360docs.net/doc/ea5032727.html,ing “coarse”settings for the p d values is relatively low when compared to that of esa .The qualities are roughly equal for the “?ne”variant with esa having a slight advantage for sampling sizes above 1%and close to 100%.(Note that for 100%,rand-esu is equivalent to esu and the results are exact.)Two things are to be noted in this respect,though:Firstly,rand-esu is much faster and can,e.g.,fully enumerate all size-5subgraphs in roughly the same time that esa needs to sample 1%of them.Secondly,the sampling quality of the “?ne”variant appears to be more consistent for di?erent networks,e.g.,in some percentage ranges esa has a very good sampling quality for elegans and a comparably fair one for coli whereas the “?ne”rand-esu remains much more consistent here.Also note that—contrary to esa —statistical estimates about the achieved sampling quality can be made with rand-esu because of its unbiasedness and the ability to estimate the total number of subgraphs (especially with the “?ne”variant where individual samples are fully independent of each other).

As to the estimation of subgraph signi?cance,Table 2shows that for most subgraphs with C i k (G ) >10?5, ?C i k (G ) is a good approximation in our exper-imental setting.Further research should investigate the few exceptions,which might hint that for some subgraphs a larger number of samples is needed.Given that direct calculation with our tool was much faster than the explicit gener-ation of random networks,further investigation in this respect appears to be worthwhile.Also,note that with our new approach,the frequency of some sub-

Table2.For directed size-3subgraphs,the table shows the approximate subgraph

concentrations in random graphs based on the methods discussed in Section3.For

estimating C i k(G) ,10000random graphs were generated.The ?C i k(G) values are based https://www.360docs.net/doc/ea5032727.html,pared to the C i k(G) values,their calculation was a

few hundred times faster on our machine.For most subgraphs with C i k(G) >10?5,

?C i k(G) appears to be a good approximation for C i k(G) (since often here,the ratio C i k(G) / ?C i k(G) is close to one).

graphs could be estimated for which the explicit generation of subgraphs did not give any results due to an extremely low average concentration in the explicitly generated random graphs.

5Conclusion

Based on a detailed analysis of previous approaches we have presented new al-gorithmic techniques which allow for a faster detection of network motifs and o?er useful additional features such as unbiased subgraph sampling and a specif-ically targeted detection of subgraph signi?cance.This enables motif detection for larger motifs and larger networks than was previously possible and hopefully facilitates future research in the?eld.

Further research could improve the presented sampling technique,e.g.,by examining how the labeling of the vertices in the input graph a?ects the sampling quality or seeing if rand-esu can be tweaked to selectively sample“interesting”parts of the input graph.For subgraph signi?cance,we have shown that a direct calculation scheme may serve as a fast and accurate alternative to the explicit generation of random networks.It would be interesting to further explore this path by extending the scheme to classes of random background models other than those that solely preserve the degree sequence.

Acknowledgments.The author is grateful to Jens Gramm(T¨u bingen),Falk H¨u?ner(Jena),and Rolf Niedermeier(Jena)for helpful discussions and com-ments and to an anonymous referee of WABI2005for some insightful remarks on this work.

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an exciting job 翻译

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