Scale-space module detection for random fields observed on a graph non embedded in a metric space Bernard Chalmond ∗†

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Abstract

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October 2014

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In the spirit of Lindeberg’s approach for image analysis on regular lattice, we adapt from a statistical viewpoint, the blob detection procedure for graphs non embedded in a metric space. We treat data observed on such a graph in the goal of detecting salient modules. This task consists in seeking subgraphs whose activity is strong or weak compared to those of their neighbors. This is performed by analyzing nodes activity at multi-scale levels. To do that, data are seen as the occurrence of a univariate random field, for which we propose a multi-scale graphical modeling. In the framework of diffusion processes, the covariance matrix of the random field is decomposed into a weighted sum of graph Laplacians at different scales. Under the assumption of Gaussian law, the maximum likelihood estimation of the weights is performed that provides a set of relevant scales. As a result, we obtain a multi-scale decomposition of the random field on which the module detection is based. This method is experimentally analyzed on simulated data and biological networks.

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Keywords. Blob Detection, Module Detection, Network Activity, Graphical Modeling, Scale-space Random Field, Graph Laplacian, Diffusion Kernel, Multiscale Decomposition, Scale Selection

∗ CMLA,

UMR CNRS 8536, ENS Cachan, France University, France

† Cergy-Pontoise

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Scale-space module detection for random fields

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1 Introduction This paper is related to the following general issue : given an undirected graph G = (V, E) with only one component non embedded in a metric space, and an observation x of a univariate real random field X indexed by the nodes V of this graph, one seeks subgraphs {Mk } in V for which the respective observations {xMk } appear as salient profiles, in comparison to their surrounding. Such a subgraph with its profile (Mk , xMk ) is called module. In summary, we have the following schema : [{xi }i∈V , G = (V, E)] ; {(Mk , xMk )} ,

(1)

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where xi ∈ R. This concept depends on the context and what we seek. We use the term module in a broad sense. However, we focus in the following on a particular module called blob or spot, depending of the context.

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Concretely, the problem is as follows. Consider Fig. 3 that shows an image {xi , i ∈ V } where V denotes the nodes of a sampling grid L included in R2 . This image shows a multitude of dark spots of various sizes that a scale-space algorithm has detected. This detection uses the graph L = (V, E) where E are the nearest neighbor connections. Here, our visual perception clearly distinguishes the spots. While keeping the values {xi }, suppose now that we replace L by a graph G whose nodes are not in a metric space. Displaying x requires to represent the graph in the plane, which implies to choose a particular layout. This is illustrated in Fig. 7 and Fig. 8(c) that display an "image" on such a graph with three different layouts. This image contains five distinct spots that it is difficult to recognize, although these figures display the same image x. Similarly, in the case of Fig. 3, our perception of spots would be greatly disturbed, and the detection algorithm would not work because it requires a metric space. Dealing with this problem is the subject of our article. To our knowledge it has not been still treated.

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Although different, this problem suggests another problem that needs to be presented in order to avoid some confusions. Let a set of points {ξi , i ∈ V } in space Rp with p > 1, allowing to define a similarity matrix W between these points, e.g. from their correlations or distances. This matrix is then used to infer a connectivity structure E, typically by connecting highly correlated or spatially close nodes. The detection of modules is then performed on the graph G = (V, E), for example by using the concept of betweenness [19]. Here, the objective is the determination of {Mk }. In summary, we have the following two-step schema : (i)

{ξi }i∈V ; W ; E ,

(ii)

G = (V, E) ; {Mk } .

(2)

This second schema is further discussed in Section 1.1. In a nutshell, we can say that (2) seeks sub-networks in G whereas (1) seeks active sub-networks with respect to x. The tackled issue is resulting from molecular biology for which a vast literature exists. With respect to our concerns, a few references are [17, 9, 35, 27, 11, 22]. This

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issue arises also in other fields, like social networks where modules refer to communities [1, 26]. One crucial step when studying the structure and dynamics of these networks is to identify modules/communities. However, these studies are mainly devoted to the schema (2) for biological networks and (2-ii) for social networks, whereas we are interested by the schema (1) due to the nature of our data, which come from a univariate random field.

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The connectivity of G is summarized in the graph Laplacian matrix L, which plays a central role in our context : ⎧ i∼j ⎨ −1 if Li,j = di if i=j (3) ⎩ 0 otherwise ,  where i ∼ j denotes the edge (i, j) ∈ E, and di = j 1j∼i is the degree of node i, i.e. the number of edges connected to i. L is a symmetric positive semi-definite matrix, which can be written as L=D−A

where D = diag{di } and A is the binary adjacency matrix ai,j = 1 if i ∼ j. Note that the graph Laplacian appears when one considers the local variation energy, called bending energy :  U (x) = (xi − xj )2 = x Lx, (4) i∼j

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where the sum is over the edges (i, j) ∈ E. In (3) each edge i ∼ j carries the value ai,j = 1. This definition can be extended to the weighted case, where the nonnegative weights are not necessarily all equal to 1. In both cases, we have di = j ai,j . We continue this introduction by positioning our contribution with respect to previous works. Then, the model and the methodology are presented in Section 2. The multi-scale decomposition and the module detection are tested on simulated random fields and on real data in Section 3, where diagnostic tools are introduced. We invite the reader to have a look at Fig. 8 that illustrates and summarizes the method.

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1.1 Graph partitioning

Although module detection is not a partitioning task, some aspects of the related problem of spectral partitioning could lead to confusions. There is a large literature on spectral clustering for graph partitioning [36, 30, 28, 10, 3] among many others. In spectral clustering [30, 36], given a graph as in (2-ii) we compute eigenvectors u1 , ..., um associated to the m smallest eigenvalues μ1 , ..., μm of L, and assign to every node i the vector m {uk (i)}m k=1 in R . Then, graph partitioning is the outcome of a vector space clustering algorithm such as k-means applied to the resulting vectors. In background of this procedure, there is an important property. If the graph is composed with c connected components, then the first c eigenvalues of L are zero, and the corresponding eigenvectors are the indicator vectors of the connected components. Fig.5 illustrates such a graph with 4 components.

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This approach also works when one replaces the adjacent/ nonadjacent coefficients ai,j by a similarity or closeness measure : ai,j = w(i, j). The multiplicity of the eigenvalue 0 is the number of connected components of the underlying graph where nodes i, j are adjacent when w(i, j) > 0. Two examples illustrate the closeness measure for the schema (2). When the graph Laplacian represents a 3D discrete surface (mesh), every node i ∈ V is associated with a 3D coordinate point ξi in R3 , also denoted vi [31]. The weight of an edge i ∼ j is defined by the Gaussian function w(i, j) = exp −(vi − vj 2 /σv2 ). Hence, the geometric structure of the mesh is encoded in the weights. The second example concerns graph based image segmentation [28]. The image is {xi , i ∈ V } where V are the nodes of a 2D regular grid embedded in R2 . Every node i ∈ V is associated with a 3D vector ξi = (vi , xi ) where vi is a 2D coordinate point. The Gaussian weight function is rewritten w(i, j) = exp −(vi − vj 2 /σv2 + |xi − xj |2 /σx2 ). Other weighting functions were proposed in the literature. Two pixels are connected if they are within distance δ : w(i, j) = 0 if vi − vj  < δ. But how to chose the graph connection radius δ ? In [10], from heuristic considerations, the graph weights are segmented into different scales : W = W1 + W2 + ... + Wr ,

(5)

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where Ws corresponds to a specific spatial separation range : ws (i, j) = 0 if δs−1 < vi − vj  < δs . In our case, the graph G is not embedded in an Euclidean space as the mesh in the examples above. Although non uniform weights ai,j can be chosen, these weights are not necessarily associated to a distance. To perform blob extraction, we use the diffusion property based on the graph Laplacian that does not require to have an explicit closeness measure. Since diffusion is a multi-scale process, we take advantage of this property to define a decomposition of the affinity matrix. This decomposition is related to generalized additive models that bring a theoretical base [37, 14].

1.2 Blob detection On a regular mesh

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In the image analysis domain, when G is simply a regular grid embedded in R2 , the problem of salient area detection has received much attention and in particular for blob detection [24], as illustrated in Fig.3. In this figure, the detected blobs are localized by squares whose size (scale) is adapted to the width of the blobs. A blob is regarded as a spot and a simple model is given by the Gaussian profile [29]. In this case, we have the following result on which the scale-space blob detection is based. Consider the simple image x = {xi , i ∈ V } representing a Gaussian spot characterized by a width parameter λ0 and centered at a point v0 on the grid : xi ∝ (λ) exp −vi − v0 2 /2λ0 , for every i ∈ V . Consider a smooth version  x of the image

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obtained by convolution with Gaussian kernel Gλ :  (λ)  xi = xi Gλ (i, i ) = Gλ (i, .)  x , i

(6)

Gλ (i, i ) ∝ exp −(vi − vi 2 /2λ) . [24] gives the following property that the spot center satisfies :  d (λ)  λ[Δ x ]i0  = 0, dλ λ=λ0

(7)

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where Δ denotes the discretized Laplacian operator on the grid. In the image processing literature, ΔGλ called Laplacian of Gaussian, is used for multi-resolution representations [33]. Laplacians of Gaussian have mathematical properties, which have been widely studied in the scale-space community. (7) tell us that the derivative of λΔGλ is able to select the width λ0 of the Gaussian spot. Essentially, λΔ quantifies in some sense a curvature of the smoothed spot, and this curvature is optimal when λ = λ0 . This property is used to detect blobs in the images : the detected blob centers are the local extrema of the discretized scale-space volume (λ)

{λ[Δ x

]i , i ∈ V, λ ∈ Λ} ,

(8)

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where Λ is a finite set of scales corresponding to an increasingly coarse sub-sampling of the regular grid. For every detected blob, the optimization of (8) returns a scale, which characterizes the width of the blob. From mesh to graph non embedded in a metric space

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In this paper, module refers to the extension of the blob concept to graphs. In a first step, this extension is straightforward since (6) is the solution of the heat equation on Z2 whose extension to graphs is well known [20]. However, extending the blob detection to non geometric graphs requires some modifications with respect to scale and space. While for a mesh, it is natural to choose Λ from a sub-sampling of the grid ([12], Chap. 10), for non geometric graph this choice is much less trivial since the relevant scales are irregularly spread in R+ , and the scale has no explicit dimension. In this goal, the multi-scale representation { x(λ) , λ ∈ Λ} must be (λ) revisited in order to get a sparse representation denoted {x , λ ∈ Λ} yielding a nonredundant decomposition of x in term of reconstruction : λ x(λ) = x, a property that (λ) does not satisfy. This property of non-redundancy is necessary for the identification x of the right scales.

1.3 Module, semantic module and related works In image analysis, the module detection is used first for extracting areas of interest without using any strong prior information. These areas are then interpreted with greater precision or extended using high-level information in order to obtain semantic modules, as for

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object recognition [8]. This remark holds also in systemic biology where semantic modules correspond to biological modules (see [17, 9] among many others). The definition of biological modules does not rely solely on areas and profiles, but also uses complex biological knowledge. In several papers, the detection of biological modules operates in two stages: firstly, detection of module seeds, or more simply modules , and secondly refinement of the detected modules to finally obtain meaningful biological modules [35]. We comment some main approaches for module detection as introduced in the bioinformatic literature. Given a scoring function that allows to compute the importance of every sub-network, finding the maximal-scoring connected subgraph is an NP-problem. In the seminal work [17], the main limitation is that node scores are treated independently since the sub-network score is calculated as a sum of the node scores. To overcome this limitation, [9] proposes an inverse problem approach in which the node scores are modeled by a hidden Markov random field model under a constraint of regularity that is expressed by a bending energy as (4). Two major well-known drawbacks are inherent to this approach [5] : the data-driven determination of the regularity scale (the trade-off parameter), and the energy minimization that requires stochastic optimization, a difficult computation task, already encountered in [17]. But conceptually, the main limitation of the Markovian model is that it is mono-scale, which is not suitable when the size of the modules is varying. Instead of using the bending energy at a single scale, we propose to use it with a multi-scale formulation in order to adapt the scale to the module sizes. Technically, the advantage of this approach is twofold. First, the set of relevant scales can be estimated efficiently from the data. Second, we avoid the huge computation burden of the stochastic optimization. The computation is limited to scan a multi-scale representation of type (6) by searching the differential local extrema as it is done for blob detection on a grid L.

2 Models and Method

2.1 Random Field and Diffusion Process

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This section summarizes a set of fundamental results on graph Laplacian and diffusion kernels. Consider a random field X = (X1 , ..., Xn ) observed on an undirected graph G = (V, E). V denotes the node set and E the edges connecting them. The dependency structure between the random variables {Xi } depends on the topological structure given by E. This dependency structure is here limited to a covariance structure modeled by a diffusion kernel [25], a choice explored in many domains and especially in pattern recognition, biological networks analysis and image processing [2, 32, 39]. We seek to represent X by a random field model on G, denoted Y(λ), whose covariance stucture depends on a scale parameter λ > 0. Essentially, this model is obtained by equalizing the variations due to a change of scale, with the spatial variations as follows :  Yi (λ + dλ) − Yi (λ) = ˙ dλ (Yj (λ) − Yi (λ)) , (9) j∈V : j∼i

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and in vector form : Y(λ + dλ) − Y(λ) = −dλ L Y(λ) , L=D−A,

(10)

where the graph Laplacian L is defined in (3). The equation (10) is the discretized version on G of the classical heat differential equation : 1 d dλ Y(λ) = −L Y(λ) , (11) Y(0) =X. whose solution is = Kλ X .

(12)

Kλ

= e−λL ,

(13)

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Y(λ)

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∞ i Kλ is a matrix exponential whose definition is eM = i=0 Mi! . For every node, one has :  Yi (λ) = Kλ (i, j)Xj = Kλ (i, .)X , (14) j: j∼i

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which is the generalization of (6). The exponential of a symmetric matrix providing a semi-definite positive matrix, the matrix Kλ , which is called diffusion kernel, can be used as a covariance matrix for modeling the covariance between the random variables {Xi }. The more λ is large, the more the off-diagonal effects in Kλ increase. λ is interpreted as a scale parameter and Yi (λ) as a scale-space random field on V × R+ . By nature, the diffusion kernel has a multi-scale property that is well identified, and especially for dimensionality reduction applications [23]. However, the choice of its scale parameter λ remains a difficulty [11]. For small λ, Kλ (i, i) reflects local properties of G around the node i, while for large λ it captures some global structures. For instance, in the geometry processing field, the diagonal term Kλ (i, i) has been used as a shape descriptor [31] by considering that for every λ, the local spatial extrema of this function provide a feature-based scale-space representation of shapes, useful for shape matching.

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2.2 Graphical Modeling

The outstanding issue at the end of the previous modeling step is the choice of λ. In other words what is the scale λ the most representative of the observed profile x. In fact, several scales may explain this profile. Therefore, a natural approach consists of decomposing X into r independent random fields according to a discrete set of relevant scales Λ = {λ1 < ... < λr } : X=

r 

X(j) + X(0) ,

j=1 1 In

the classic case of diffusion in R2 , λ is a time parameter.

(15)

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where X(j) denotes the random field at scale λj and X(0) a residual [16, 34]. Following the idea of Fourier decomposition, for every profile x, the {x(j) }rj=1 can be seen as frequency components of x, from high to low frequencies. The decomposition (15) is related to the additive spline models whose theoretical foundation can be traced back to [37] Chap.10, (see also [14]) and later reintroduced under the name of multiple kernel in the machinelearning community [21]. Note that X(j) does not match Y(λj ) in (12), since the sum j Y(λj ) over a given set of scales does not reconstruct X. In our approach, the covariance matrix Cov(X(j) ) of every component is modeled from the diffusion kernel (13). So we use r kernels {Kλ1 , ..., Kλr } denoted {K1 , ..., Kr }, such that Cov(X(j) ) = Kj = σj2 κj where κj is given by (13) at scale λj . Due to the independence of the components, the covariance matrix Cov(X) is the following multiscale diffusion kernel : r  j=0

r 

σj2 κj

j=0

σj2 e−λj L

(16)

σ02 In

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r 

Kj =

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¯ σ,Λ = K =

+

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Each kernel κj is weighted by a positive parameter σj2 that is all the more great than the scale λj significantly contributes to the random field X. The covariance matrix K0 is that of a white noise. As we said above, the more λj is large, the more the off-diagonal effects in Kj increase. In other words, when λj increases, the components {X(j) } are increasingly smooth. The passage from X(j) to X(j+1) implies that some details in X(j) are attenuated. If we assume that the dependency structure of the random variables {Xi } is uniquely described by its kernel, then it is legitimate to consider that X is distributed according to the Gaussian law ¯ σ,Λ ) . X ∼ N (0, K (17)

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The scales {λj }rj=0 and their associated weights σ ={σ ˙ j2 }rj=0 are unknown parameters that are estimated using the maximum likelihood principle 2 . Although the theoretical mean of X is zero, the empirical mean of each observed subprofile xMk is not necessarily zero, as for instance in Fig. 9. This is due to high scales that create long range correlations, or in other words low frequencies. Understanding the diffusion kernel is not a trivial task, this requires to call the graph spectral theory [25]. Note also that the choice of the diffusion kernel as covariance matrix arises as a necessity because we have only one observation of X. If we could have many observations then the covariance matrix could be estimated. In comparison with the heuristic decomposition (5), which uses a hard multi-scale separation of the weights, the multi-scale representation (16) appears as a soft decomposition based on the overall structure of the graph via L, and moreover allows statistical estimation of each component contribution. 2 For

notational convenience, we introduce λ0 = 0 that is associated to K0 .

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• Weight estimation. For a given Λ, let (σ|Λ) = log(pσ,Λ (x)) be the log-likelihood of σ, where pσ,Λ denotes the probability density of x. Given an observation x and the ¯ σ,Λ ), the log-likelihood is Gaussian N (0, K ¯ σ,Λ | − x K ¯ −1 x + Cte , (σ|Λ) = − log |K σ,Λ

(18)

where Cte denotes a constant term. The maximum likelihood estimate is computed under the constraint of positivity of the parameters σ : σ ˆ (λ) = argmax (σ|Λ) under the constraint σ > 0 .

(19)

σ

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For moderate sizes of n, the non-linear programming algorithms using gradient descent techniques are operational. For larger dimensions, the computation of the determinant ¯ σ,Λ | and the inverse K ¯ −1 becomes more difficult [18]. To reduce the amount of compu|K σ,Λ ¯ σ,Λ | tation, one might also wonder whether it would be possible to remove the term log |K  ¯ −1 in the likelihood, in order to work only with the generalized least-squares x Kσ,Λ x. Theoretically, we know that this estimate is not statistically consistent [13]. Our experiments have confirmed this defect, by showing severe aberrations in the multi-scale decompositions (cf. Section 3.1). • Scale estimation. A procedure for selecting the set Λ is now required. Given a uniform discretization Λ0 of the scale domain in R, the scale selection procedure estimates a subset Λ of scales irregularly distributed in Λ0 , which explains the profile of x according to a given criteria :

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Λ0 ={λ0j = jδ, j = 1, ..., r0 } ; Λ={λ0j1 , ..., λ0jr } ,

where δ is the discretization stepsize. First, the estimation σ ˆ (Λ0 ) is computed according to (19). To determine r, we perform a diagonalization of the covariance matrix 0

 = K

r 

0

σ ˆj2 (Λ0 )e−λj L ,

(20)

j=1

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of which we retain only the r largest eigenvalues ν1 ≥ ... ≥ νr according to the criteria r i=1 νi =1−, (21) r 0 i=1 νi

where  is a positive parameter chosen close to 0, typically  = 0.01 or 0.025. This criterion is related to that used in Principal Component Analysis [15]. It means the dispersion of X can be approximatively represented by r linearly independent components with an information loss determined by . Finally, from the estimated r, we can then achieve the selection of relevant scales Λ. These scales are associated with the r largest σ ˆj2 (Λ0 ), i.e. the scales whose components are the most involved in the dispersion of X. These scales are denoted {λ0j1 , ..., λ0jr } or more simply {λ1 , ..., λr }. As a consequence of (21), the

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estimates σ ˆj2 associated to the scales in Λ0 \Λ are much lower than those in Λ, and even close to 0. This selection achieves a pruning of non-significant scales. • Statistical Multi-scale Decomposition. This task concerns the estimation of the r components X(j) of the multi-scale decomposition of X. Component estimation is  = U Dν the spectral equation of closely linked to the scale selection problem. Denote KU the previous diagonalization, where Dν is the diagonal matrix of the r largest eigenvalues  Because of (21), assume that the eigenvalues λ0 \ λ are approximately equal to Λ of K. zero. This is especially true when  is very small. Beyond r0 , the eigenvalues νr0 +1 ≥ ... ≥ νn are smaller and we can consider they are all close to 0. In this case, one can write  = U B where B are the coordinates of X  on the eigenvectors U . To estimate the scale X components, we take into account the importance of each eigenvalue, and this, using a Bayesian estimation with a prior distribution related on these eigenvalues.

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Proposition 1 Given an observation x and the prior distribution B ∼ N(0, Dν ) the Bayesian estimation provides the scale components : (j) ˆ , ∀j = 1, ..., r

x = Kj U Dν−1 b

= (σ 2 D−1 + Ir )−1 U  x . b 0 ν

(22) (23)

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The proof is given in Appendix 5.1. In this proof, if we replace the spectral equation  0 = U 0 D0 , then Proposition 1 is still relative to Λ by the equation relative to Λ0 : KU ν valid. This is useful when we do not assume that all eigenvalues Λ0 \ Λ are negligible. In this case, the Bayesian estimation is more justified because of the high difference between the values of Λ and Λ0 \ Λ.

2.3 Module Detection

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Given an undirected graph G and an observation x of the random field, we first compute the estimated scales {λj , j = 1, ..., r} and the associated decomposition (22) {x(j) , j = 1, ..., r} as presented in the previous section. Rather than considering directly the components, we consider their spatial variations with respect to the graph Laplacian L :  (j) (Lx(j) )v = (xv(j) − xi ), ∀v ∈ V i: i∼v

= dv xv(j) −



(j)

xi

(24) .

i: i∼v

This specifies the regularity of each component. Lx(j) is all the more great positively (j) (resp. negatively) than the expression xv of the node v is strongly increasing (resp. decreasing) with respect to its neighbors. Therefore we look for nodes that are most differentially expressed with respect to L, and this by examining the expression of the (j) components at different scales. Since the amplitude of variations of L x decreases when

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the scale increases, a specific normalization is required. As in the case of blob detection (j) (8) on a lattice L, an efficient normalization is λj L x . A scale λj for which λj (Lx(j) )v is a local extremum with respect to scale and space, is seen as reflecting a module at position v and scale λj . This implies the following procedure. For any node v ∈ V , we denote by Nvk ⊂ V the relative nodes of v of order k, (k = 1, ..., κ). k = 1 means the nearest neighbors (NN), k = 2 means the nearest neighbors of v to which their NN are added, etc... The module detection consists in searching local optima of the components with respect to the neighborhoods as follows3 : ∀ v ∈ V : (j(v), k(v), v  (v)) = argopt λj (Lx(j) )v ; j,k,v  ∈Nvk

(25)



if v (v) = v, then v is a module center at scale λj(v) . k(v)

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When a module center is detected at v, its area Mv is defined by the subgraph {v, Nv ◦

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In the next section, we denote V the nodes corresponding to the detected module centers, and therefore the set of detected areas is written as : ◦

{Mv , v ∈ V } .

3 Experiments

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Recall that a module is an active subgraph denoted xM where M is a subgraph of G. A regular lattices L do not show particular structure like stars or clusters, unlike the case of irregular graphs G. Let us give an example of graph showing a particular structure. The graph structure is organized around known subgraphs {Rk = (Vk , Ek )} called regulons (or hubs). A regulon is a set of nodes Vk ⊂ V connected to one or several common nodes, called regulators. A regulon can be connected to several regulators and a regulator can be connected to several regulons. Fig.5 shows a graph with four regulons and four regulators. In practice, such regulons can be used a posteriori for interpreting the detected modules or inferring semantic modules. Depending on the profile of x, the area M of a module can be simply a regulon, a subregulon or the union of several regulons. Fig.6 shows a short time-series {x(1), x(2), x(3)} of a random field X observed on the graph of Fig.5. The colors depict the output of the scale-space module detection performed on every x(t). Successively, 3, 3 and 2 modules were detected, while the graph is composed of 4 regulons.

3.1 Evaluation on Simulated Data 3.1.1 Simulation Procedure

For phenomena of high complexity, simulated data are an important preliminary support for modeling when we do not have data with sufficient knowledge of the "ground truth". 3 v  (v)

also depends on k, what is omitted to simplify the writing.

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The simulation of the random field X requires to give the ground truth, consisting of a graph G = (V, E) and the parameters (Λ, σ). In our procedure, G is organized in regulons : G = m k=1 Rk . Here, we assume for simplicity that each regulon Rk is associated to only one regulator rk . The symbol + in indicates that the regulons are mutually connected. This high-level of connection is equivalent to a graph GB between the regulators : GB = ({rk }, Er ). The simulation is done in two steps. First, the simulation of a graph G consisting of m ¯ Λ,σ ) is regulons is done as described in Appendix 5.2. Second, a sample x of X ∼ N (0, K ¯ Λ,σ = drawn. To do that, we simply simulate α ∼ N (0, In ) since the diagonalization K ¯ Λ,σ ). V DV  implies V D1/2 α ∼ N (0, K 3.1.2 Simulated Data

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Fig.8(a) shows the inter-regulon graph GB and Fig.8(b) the graph G. Each regulon has its own color. Fig.8(c) displays an observation x of the random field X on G, and Fig.9 shows its 1-D profile. In this experiment, each regulon is a potential module since the simulation procedure is based on a regulon structure. x was simulated using the multi-scale kernel (16) with 3 scales : λ1 = 8, λ2 = 14, λ3 = 24 and σ1 = σ2 = σ3 = 1. Although the theoretical mean of X is zero, the mean of each observed regulon in Fig.9 is not zero. However, due to the correlation between regulators, two regulons may have similar mean levels. This situation is favorable to the concept of module. This is consistent with Fig. 6 wherein there are 3 detected modules for 4 regulons. Note that in the absence of observation x, the spectral partitioning as recalled in Introduction, detects 4 modules corresponding to the 4 regulons. 3.1.3 Data Analysis

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The estimation and detection tasks are illustrated in the figures 8, 9, 10 and 11. The maximum likelihood estimation (19) was performed using the scale domain Λ0 = {2k, k = 0, 1, ..., 15}. Fig.10 displays the statistical multi-scale decomposition. The continuous 15 black line connecting the data points is the sum k=1 ˆx(2k) of all the components except the noise component ˆ x(0) . Since this line interpolates the data points, this means that the estimated noise component is very low. The selected Λ is computed from this decomposition using (21) with  = 0.1. The three main components associated to {λ1 = 8, λ2 = 10, λ3 = 30} are shown in Fig.9. It is interesting to compare this statistical decomposition with the ordinary scale-space decomposition (12) shown in Fig.11. The statistical decomposition has the ability to focus more clearly on the spectral content of x. λ3 = 30 reflects low frequencies whereas λ2 = 10 contributes to high frequencies. But above all, it remedies the redundancy of ordinary scale-space representation, and therefore favors the identification of the right scales. The detected modules shown in Fig.8(d) correspond to the rule (25). In Fig.9, the locations of the detected extrema are indicated by red circles. There is exactly one detected module per regulon. This procedure is statistically assessed by Monte Carlo simulation. The random field X is simulated 200 times under the same conditions as above. From the obtained samples

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4

5 (a)

6

7

8

0

1

2

3 (b)

4

5

t

Figure 1: (a) Histogram of the empirical probabilities {P (|Λ| = k)}8k=1 of the number ◦ of selected scales , (b) Histogram of the empirical probabilities {P (|V | = k)}5k=1 of the number of detected modules.

◦

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{x( ), = 1, ..., 200}, the probability of the number of selected scales {P (|Λ| = k)}8k=1 were estimated for  = 0.01, as well as the probability of the number of detected module centers {P (|V | = k)}5k=1 , as shown in Fig. 1. The number of detected modules is ran◦

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dom, with a main mode at V = 5. In fact, as noted above, the mean levels of two regulons may be substantially close and therefore be recognized as belonging to the same module if they are connected. ¯ σ,Λ | in the likelihood. Above, we have mentioned the prominent role of the term log |K This is confirmed experimentally. Without this term, the estimation-detection procedure was repeated on the same data as previously. The results are shown in Fig.12. The multiscale decomposition is then quite inaccurate. All weights are very high and the scale components are close to zero.

3.2 Bacillus Subtilis Data

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Fig.4 illustrates the multi-scale decomposition of a field x that represents gene expressions of Bacillus Subtilis. The underlying graph G = (V, E) comes from the regulatory network of the bacterium. V denotes genes, E connections between genes and x gene expressions on V (Fig.4-a) 4 . The entire graph contains 1607 genes, 2345 edges and 132 regulons. Fig.5 displays four connected regulons extracted from this network. In steadystate, gene expressions are assumed to be governed by the model (17). In Fig.4 we see the structuring effects of the method in terms of gene grouping as this had already been shown for other regulatory networks [11]. In many applications, one is interested in studying the change of modules across different conditions [22]. In our example, the expression of the regulons depends on the nutritional environment of the bacteria over time, some of them are over-expressed and 4 The biological network has been simplified by removing the protein level network and therefore in G the regulatory protein-encoding genes and their proteins are confused. Furthermore, the edge directions in E have been deleted. Consequently, we cannot speak strictly of regulation in the sense of regulatory networks. .

Scale-space module detection for random fields

14

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

0

0

t

Figure 2: A toy example with three active regulons : two over-expressed and one under− 3 3 expressed. (a) μ

in blue. Confidence band : {μ+ Rk }k=1 in green, {μRk }k=1 in red. (b) A − + particular configuration x ( ) = {μ+ R1 , μR2 , μR3 }.

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other ones are under-expressed. With the module detection, we search to identify regions of the graph that are particularly expressed through time. Fig.6 illustrates this detection on a short time series of the random field observed at 3 time points. The detection has been done at every time t, independently of the others. Every detected module is composed of one or several regulons. For instance, at time t = 1, there are 3 detected modules, which are respectively depicted in green, yellow and pink. The green module is composed of two regulons, what can favor the semantic interpretation of the module from properties of the regulons. The validation is primarily based on biological aspects. In the considered experience, one examines nutrient change effects : an experimental population of cells grows first in Glucose and then Malate is injected at time t0 such that 1 ≤ t0 ≤ 11. The detected modules should reflect this change. A further biological analysis is beyond the scope of our article, [4, 7]. However, a diagnostic tool is now proposed to help with this analysis. The idea is to generate configurations of X using a confidence band around the obtained decomposition 

(j) , μ

= x j∈Λ

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in order to quantify the stability of the detected modules. We start with the bootstrap confidence interval described in [38] that we recall. As a result of the decomposition, the

(0) = x − μ estimated residuals are : x

, whose empirical variance is σ ˆ02 = 

x(0) 2 /n. Derived from (15), a generative model based on μ

is written as :

(0)

X=μ

+ X(0) ,

where Xi are independent Gaussian random variables LG(0, σ ˆ02 ). Generate X(0) allows 2 to simulate X, which is now distributed according to N (

μ, σ ˆ0 In ). Pretending that μ

is the "true" μ, generate N bootstrap samples {x( )}N , and compute their respective smooth =1 profiles {

μ( )}N =1 . Using these samples, for every node i ∈ V , and for a given confidence

Scale-space module detection for random fields

15

+ γ close to one, a confidence interval of μi denoted [μ− i , μi ] is estimated, which provide a confidence band [μ− , μ+ ] as detailed in [38]. Our validation uses this confidence band to generate configurations. Denote {μ− Rk } + and {μRk } the two confidence profiles viewed from the regulons. For generating a new + configuration x , we draw randomly for each regulon Rk between μ− Rk and μRk , as illus trated in Fig. 2. Repeating this process N  times, we obtain new samples {x ( )}N =1 , on which module detection is performed. Finally, among the N  detected fields {{Mv ( ), v ∈ ◦

V ( )}, = 1 . . . N  }, we compute the proportion of fields that fit with the field {Mv , v ∈ ◦

V } obtained on the original x, cf. (26). This proportion is associated to the confidence γ, providing a quantitative diagnostic tool.

4 Conclusion

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t

The experiments show that module detection puts into light the activated modules and therefore provides a mean to study dynamic random fields. However, module detection on time series has been performed without taking into account time dependence. At every time t, the observation of the random field has been treated independently of the others. Nevertheless, it is well known that Markovian dependence can improve the sensitivity of the detection of isolated low signal. In the related paper [6], we present a Markovian spatio-temporal modeling that generalizes the present model. Doing so, in Fig.1 the ◦ probability Pˆ (|V | = 5)} should be higher.

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This paper proposes and implements a multi-scale graphical modeling for univariate random vectors observed on an undirected graph. The result is a multi-scale decomposition of the random field which provides an analysis tool to deal with specific treatments because it allows to select relevant scales. This tool is especially used for module detection. With hindsight, this detector seems relatively simple. However, emphasis has been put on a coherent modeling without heuristics and with very few tunable parameters.

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5 Appendix

5.1 Proof of Proposition 1

0  = X − X(0) = r X(j) , and recall the spectral equation KU  = U Dν where Let X j=1 0   = r Kj . First, since the columns of U are independent, we can write X  = UB K j=1 where B is a r-random vector. Then, the spectral equation allows to rewrite 0

 = U B = KU  D−1 B X ν

=

r 

0

Kj U Dν−1 B

j=1

where

(j)

X

=

Kj U Dν−1 B

=

r 

X(j) ,

j=1

,

(27)

Scale-space module detection for random fields

16

 = U B implies the covariance matrix which provides the components (22). Second, X   = Dν . Cov(B) = U KU

(28)

For a given observation x, the Bayesian estimation of the occurrence of B consists in maximizing the log-likelihood log p(b | x) = log p(x | b) + log p(b) + Cte. Given the Gaussian laws B ∼ N (0, Dν ) and X(0) ∼ N (0, σ02 In ), this amounts to compute

= argmax − 1 x − U b2 − b D−1 b , b ν σ02 b

(29)

5.2 Graph simulation

t

in (23). Note that (29) is similar to the criterion of the which provides the expression b Ridge regression [15].

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The simulation of a graph G consisting of m regulons : G = m k=1 Rk , is done in three steps. 1. For each set of nodes Vk making up a regulon, a regulon graph Rk = (Vk , Ek ) is simulated. 2. At a larger scale, the m regulons are considered as m nodes of a graph, and thus an inter-regulon graph GB is simulated.

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3. The global graph G is obtained on the basis of these m + 1 graphs, as follows. For each regulon Rk , a regulator rk is drawn uniformly at random in this regulon. This regulator regulates the regulon(s) Rk such that k ∼ k  in GB , 5 . The weight of the connections between rk and nodes v in Rk are given by the probabilities of the Binomial law B(|Rk |, p) where 0 < p < 1. When a weight is below a threshold τ , for example 0.05, the weight is set to zero, then the probability distribution is renormalized. By ruling p and τ , one can modulate the number of edges between the regulator and the regulated unit. In this case, rk regulates a subset of nodes in Rk .

Acknowledgments

The referees are gratefully thanked. Their comments have improved the manuscript. The author is grateful to Alain Trouvé and Yong Yu for the experience we shared on the multiscale decomposition of images, which has been an inspiration. The author thanks warmly Benno Schwikowski for valuable discussions about the adaptation of Bacillus subtilis to nutritional environments, and Xiaoyi Chen for her help in carrying out experiments on Bacillus subtilis data. 5A

node v ∈ V is regulated by another nodes v if Xv is significantly correlated to Xv .

Scale-space module detection for random fields

17

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yflT bmrU

yraA

yoaG yqfA sppA xpaC

yitD

ydcA

mreBH rsgI ybfQ

ydbS

yycD

yuaI

yjgC

yoxC ydhK

yckC

yvlB ythQ yvlD

yvaK

ywjC

yfkD

ywzA

nhaX

ygxB

400

pgdS

yhcO

yvlA

yfhF yjgB

ycbP yfkI

sodA

ohrB

aag

200

spoIIE

(b)

yobJ yjoB

yvlC ythP

ydaE

yugU

dps

yjgD

folD

yrkO

yppD

gabR

yuaF yaaNydjO ywrE yoaF fosB

yjzE

yfkJ yxkO gtaB

ytaB

yitT yocK ybyB ywmE

ywtG

purE

ylmD

yqzC divIVA

med

ylmH

gabT

gabR

gabD

ycdF csbD yerD ydaF

ycdG

ydbD

xpt

1600

flhO tlpB mcpBlytD

tlpC

mcpA

argB

carA

gabT

nusB

1400

yusD

yjcP hemAT

yvyC

yjfB flhP mcpC fliD

purQ

1200

kinC

carB argC argG

purC

−1

yrkN

yrkQ

600 800 1000 component at scale5

yfmT

fliS

argF

argD

ydaS

yrkP

pdxS

yqcF

400

yfmS yoaH motA

yydA

gspA

ykaA

metS

comZ

ahrC

ywlC

ywiE

ylmE

yvyE

ylmG accAftsE yvhJ pdxT

yfmJ yqzD accD

yusE

dnaN

ftsX tkt

yneF

codV clpY ykfB

yhdN

yrrL yppE yuxH

yerB

ywcH

clpQ

dnaA frlR

aldY

sepF

bkdR

rocB lytF

nfrA ycgA

ylmA

200

0

rocR

rocC

fliZ

ilvA yufO

ypmP frlB frlN yufN frlD yuiA

yfhP

yfhE

yngB

ykcC

ykfC

metQ metN

frlM frlO

csrA

ykcB

bkdAAbuk

ykfD

yurJ

ykfA

guaB

flgC cheB fliJ fliY fliL fliP cheA flgB fliM cheY flhA ylxF ycgN fliI flgE flhF cheW fliE fliR fliF cheD fliQ

ywfH

yuiB

pbpF

cheC

fliG

yisY

yraE

yqfX sspD yrrD spoVAC ydfR gerKC adhB sspJ splA ykoU

sigD

flhB

ycgM

yraD nfo

spoVAD sspP

bkdB bkdAB

ylxH fliH

yycA

yngA

ptb

lpdV

appB

yoaR

sspK

ybxH gerBA

appD

sspA yraF

gerKB

1600

rin

splB

yhcV

ccpB

yclH

yclI

yitF sspO yndE yndD

1400

1

lytA lytC

pdaA

yabJ

cotH cotZ

cotG yurS

cotU

lytB

rocA

spoVAF

purB

cotX

yybK

spo0A

yscB

gerBB ypzA

ylaJ

purA pbuG

cotB

cgeE cgeC cotV ftsY cgeByxeE

yybN yybM

ybdO

codY

cwlH

cotW

yjcN

skfE spoIIGA

cotY

sspG

sigL

yqxJ

yxbC yxbD

sinI

yxnB

−2

yorB

yocD pnbA

skfC

skfF

asnH dppA

bacF sigA

gerBC

1200

0

ydiPyneB

yobB

slrR skfB

yxbB

dppE dppCkapB yxaM

dppD

bacB

yxkC

yhaM yozL ruvB aprX

sipV

ydgH

sdpA yoqM yfmI

kinB bacC bacA bacE bacD

ymaE

scoC flgM

parE yqjX tagC yhaZ ligA

lytR

yjdG

racA

phrA dppB

yvyF yvyG yuiC

xkdA

gerE

ykuL

xynD

yrpD

sdpB

ansZ

flgK

sigG

spoVAB

exoA sspM

600 800 1000 component at scale4

dnaE

yndF

csgA sspI

yvdQ sleB

400

2

ynzC

levR

acsA

ywbF ydeJ

yomJ

ydgG

ydiO

yneA

levG

yydI

ydjG

yusN

yteA

uvrC pcrA dinB

yhjE

ackA hutH yphF hutI hutG

lysA

spoVAA sspF yozQ sspB yqfU

200

yhjD

sda

ykvR yorC

levF

hutM

hutU

yitG

sspC

uvrB

uvrX uvrA yerH

parC

lexA

acoBacoL acoA

levE sacC levD

yolC ptkA

hutP

spoVT pbpG

yhcQ ypeB

1600

hbs

yydG yqxI yolB

rok

yflB

ydjH

rapC nprE rapE

licR

bdbA

bdbB

yxaL epsK

yydH

yhjC

oppD

yhcN

gerAB gerAA

−2

deoR

licA

albD

albE albA albB sboX sboA

yolJ

sspH

epsI epsC epsB

oppA oppB oppC gpr

ycbG

licB

nrdE

albF albG

sunT

sunA

epsJ yvmC aprE spoIIAA spoIIAB

ywbD

1400

nupC

yorD

acoC

rapG

dacF

oppF

1200

licH

licC

tasA

epsO

ycbJ

arsB yqhG yqcK

600 800 1000 component at scale3

0

yopL malP

glpT

glpP

glpK

yfmG

sinR

epsH arsR

yabT yqhH

400

xylR

pdp

ymaB

glpD

glpF

dltD

dltC

ilvH

sipW

ydjI

yvmB

yclK

leuD leuC

epsA

leuB

ybaJ

yokL

yjbA

yhfW

xynB

treA

gmuA

spoIIQ spoIIR

yphA

seaA

ilvC

leuA epsG

yheJ yokK

yyaC nasB

xylB msmX

cimH

dltAdltB

phy

recA

abrB

bdhA

epsF

epsN yqzG

phrE

spoVG epsE

ilvB

epsM

rsfA

ksgA

ypfB

phrC

yocH

yrzI

yphE ylbC ytfI

yodF

yveA

200

2

licT

bglH

xynPyxiE

treR

kdgA

malA

gmuGgmuB gntK gmuD

epsD phrK

ctaO ywcE

yvcA

glnM

gltB

iolB

treP sucD glpQ

gmuR

gmuC yvnA rbsR amyE gmuE

srfAAcomS srfAB

xylA

kdgK

nrdI

gntR

rbsC

yuaB

wapA

bpr levB

gmuF

rbsD

csn

sacB

sacX

−0.1

iolT

bglS

odhA

cydC nrdF

gntZ

skfA

spo0E

yweA

yvrO

tnrA

lipC sipT glnP ywdK

yokI xynC

yokJ

yvrP

glnR kipA

rbsK

ctaC

ctaF ctaD ctaG

srfAC srfAD

yqaP

ppsB

degU

ykzB

sacY

ctaE

cydB

rbsB gntP

cwlS

yttP

abh

rapK

comA

acuC

iolD

dctP

odhB

bglP

cccA qcrA

cwlD

ylaE

ppsE

ywrD

ssbA

comEC

gltC

ccpA

iolR

iolG

iolI kdgR

yngC

resD

ppsC

degQ

comK glnA

iolJ acuA yvcI ywdA

iolC iolF

yxkF

ywlF alsT

ycsG yqzE glnH

araL

iolH

kduD cotD

kduI

sdpI

yobO qcrC

iolS

citM araB

sacA acuB

nagA

iolE cydA

phrG

ycdA

ydeH yerI

pel

yttA

glnQ ywdJ

noc dprA sbcD yisB

sacP araA araQ pta

araE

rapF

yncM

yydF

yxeD antE

yolA

sacT

ypiF

araD lcfB

etfA

cotC

yhaR kdgT

nasD

ftsZ

ftsA

phrF

ywoF

nin comEB comGF

comN

yflN

abfA araN

fadN

cydD

sdpR

vpr

lytE phrI

spoVS ywhH dacC

1600

araR

sucC

araP fadB lcfA

fadE

yjdB stoA

citB

dhbC

ycnK

guaC

bofC yhcM csbX

yycB

citZ

ctpB

nasF

dhbE yqxD besA dhbB dhbA

tuaF tuaG

ycxA yuzA

pucK pucL pucM

fadA

icd mdh

ccpC fabD

katX

pucJ

sbcC

citT

etfB

resA resE resC

yflA

ykoL plsX fabG

pucR addB

cstA

araM

resB pucI

yxxG

1400

0

galT

ylbP

maf comGD

1200

fadM uxaC

yqhB hmp

rpsR rpsF comFC comGE yhjB ybdK

600 800 1000 component at scale2

acdA

sigX

comGB addAglcR rsmG comFB comGC comC

abnA rpoE

yvgN

ssbB nucA

400

0.1

fadH

minD

pucG

comGA

200

fadF

fadG

minC

feuA

arfM

ahpF

hemB

spoIIIC spoIVCB

perR

yknY

ybbA

comGG comEA

−5

narG mrgA

katA

hemC

fnr

exuR

yoaW

yxzE

yknZ

feuB pucA pucFpucH pucE

zosA

ahpC

scoA

ypuD

rsiW ybfO

pspA

ymzB yvgO yxaB

radC

yqhQ

hemA

yjmC exuT yxjC mmgA mmgC

yjmD uxuA

phoP

ytxG yvyD ytxH

yxjJ

fapR ycnJ

mmgB

ybfM

mcsA

feuC

fabI

hemL

uxaA scoB

rsbV rsbW rsbX

ydjM tagD yjeA

fur

fabHA fabHB

pucD pucC pucB

uxaB

pbpX

tagB ykvT cwlO iseA

acpA

ywjB

narI

hemD

guaD fhuD

clpC

yfmC yclP

yjbC

yfhA

fhuC

yfhC

fhuG

1600

sigM

phoD

fhuB yfmD

yclO yxeB

ykuN

yfiY ykuP yhfQ ycgT yclQ yclN ykuO

yrhH

tuaC

tuaB

tatCD

yfmF yfiZ

yfmE

csoR

1400

0

spoIIID

divIC

ywbO

csbB pstS

t

yotD

copA

1200

bltD

ylxX

tatAD

pstA pstC

pstBB

600 800 1000 component at scale1

5

ymfH ymfF ydfK ymfD

ylxW

murBsbp metA

pstBA copZ

400

yndA

mta

spoIID spoIIIAG

divIB

200

yqfC

coxA spoIIIAD

spoIIIAH bofA spoVE spoIIIAE

yesK

ycgR ycgQ

murF

bcrC

tuaE tuaD

yokU

mreD

ytpB

ytpA

fabF

spsK spoIVCA murGspoVD cotJA spoIIIAA spoIIIAC spoIIIAF yesJ

sigE ypbG

ydaH

phoB ydhF

yfkM

ykvI

yhaX spoIIIAB yhaL coaX

yhdK

ywtF

ywaC

yqjL yceC

disA

ctsR

yodT

ysnD

kamA cotO yhdL

ypuA

ftsH

sigW yceE

ispDyceH

sigI ykuT

radA

ysxC

plsC

spsJcwlJ ytvI

yjcA

bmrR hprT

secDF bmr

yceD yceG yceF yacL

sigB

rnr

ywmF trxA

spoVK yyaD yobW

yodP

yjfA

ddl

tilS

ysdB

gsiB

spoVB

yngI

yfhL

rsbRD

yflH

ysnF

ylxP

yoaA

clpP

clpXlonA

purF

pbuO

glgA

ysxE prkA

yodR

yfkT

yfhD

cypC

yfkS

cotE

dacB

yngE

mbl glgD

yabQ

yxbG yxnA

csbA

ypuB ydeC ctc yhdF yocB

ydaT

ypqA ispG

glgP

spmB

ykgA

yxiS mgsR clpE

gerM

spoVID

yabR

spoIIM safA spoIVFA

ispF

nadE

ydaS

yhxD

ygxB

purE

nusB

purR

spoVR

yqxA

yngF glgB

yjaV asnO

5 0 −5

yngJ

yteV nucB

yodS

yocL spoVMspoIVA

yeaA

ytkL

csbC

yxzF ywiE

yhdN

yraA

aag folD

xpt

yabJ

purN

purH

yybI yhbH sqhC ylbJ spoIVFBykvUglgC yhxC

yuzC

yqeZ

yqfB

yoxB

nhaX purB

purC purL

purS pbuX glyA purM

yitD

ytxC

ydbT

yteJ

yozO pbpE

racX

opuE yaaI

yfkH

aldY

katE

ykgB

ywlB

ydaD

ohrB

purQ

purA pbuG

purDpurK

yoaG yqfA sppA xpaC yuaI

yvrE

ykzI

yjgC ywsB

ydcA

mreBH rsgI ybfQ

ydbS

yycD

ydaG

yflT

yckC

yvlB ythQ yvlD

yvaK

ywjC

yfkD

ywzA

yfkI

sodA

yoxC ydhK

yfhE

ydaP

yfhK

era

yhcO

yvlA

yfhF yjgB

ycbP

yobJ yjoB

yvlC ythP

ydaE

yugU

dps

yjgD

gspA

yjzE

yfkJ yxkO gtaB

ytaB

yitT yocK ybyB ywmE

yuaF yaaNydjO ywrE yoaF fosB

gabD

ycdF csbD yerD ydaF

ycdG

ydbD

yjcP hemAT mcpA

motB

cheV degR

pgdS

flhO tlpB mcpBlytD

tlpC fliT

argF carB argC

argG

argJ

carA

argB argH

(d)

Figure 4: Multi-scale decomposition of gene expression network of Bacillus subtilis (from [7]). (a) Original data. b) Multi-scale decomposition profiles (1-D display). (c-d) Scale components for λ = 2 and λ = 16, respectively. The decomposition has a structuring effect in terms of gene grouping. Despite the use of false colors, it is difficult to distinguish modules, unlike the case of an ordinary image as in Fig. 3.

Scale-space module detection for random fields

21

182

189

177

187

180

181

184

186

209

192

194

213

174

185

188

175

208

215 198

206

190

178

183

196 205

214

193

202

203

179

204 212

197

201 200

149

211

148

191

199(rok)

151

166

161

163

139

136

131

169

173

133

171

144

153 165

160

157

138

142

170

155 128

36(codY)

135

158

168

162

156

146

143

164

37(comK)

147

130

145

150 137

132 172 154

195

210

207

176(degU)

134

140 141

129 159

152

108

167

110 111

44

94

24

18

115

76

47

34 77

105 14

32 25

2

26

112

7 59

72

69

71

40

126

84

12

51

75

33

4

46

88

60

66

109

124

79

64

52

9

rin

85

19

58

89

38

87

49

98

13

74

92

68

56

116

17

31

122

1

78 95

106 48

54

91

118

93

22

42 103

28

67

83

100 125

70

39 81 8

29

90

27

113

t

99

114

123 61

16

102

96

53

121

43

3

41

119

73

57

30

11

97

20

50

80

63

21

45

104

10

23

62

5

101

65

107

120

82

117

86

15

6

55

127

35

ep

Figure 5: Graph partioning of G : this graph shows four regulons and four regulators.

182

189

182

189

177

187

180

183

149

149

163

150

165

208

132

166

161

206

190

205

137

198

150

165

205

137

132

202

172

204

199

146

144

153

200

157

138

153

156

157

138

195

210

197

200

162

135

156

164

164

134

214

142

211

170

155

207

167

152

115

61

106

40

25

2

102

96

25

48

31

71

72

74

22

59

70

12

56

1

78

60

19

85

43

3

46

4

88

49

95

51

75

33

87

66

109

73

57

11

97

63

21

45

23

62

5

107

120

117

86

15

19

73

57

66

109

30

63

21

45

23

51 75

33

46

4

88 60

19

85

66

109

73

57

11

97 63

21

45

23

62 5

107

120 117 15

104

10

101

65

86

9

20

50

80

82

124 79

64

52

119 41

6 55

127 35

12 87

30

62 5

117 15

58 89

1 49

3

126

84

38 56

95

92

122 42 103

28

98

74

36

100

13

17 71

22

43

107

120 86

69 116

31

72

70

9

106 48 54

91

118

68

83

112

104

10

101

65 82

26

20

50

80

81 8

29

67 99

114

124 79

64

52

11

97

39

123 61

90

27

113

96

125

4

76

47 53

121 40 2

78

119 41

6

55

127

35

60 85

43 3

46

115

93 32

58

88

49

95

111 24

34 77

25

59

51 75

33

87

191

110

44

94 18

105

102

126

84

108

16 37

14

89

12

56

207

167

152

159

7

38

1

78

104

10

101

65

82

9

92

36

211

170

141

98

13

74

122 42 103

28

214

142

17 71

20

50

80

30

79

64

52

119

41

70

124

116

195

210

197

140

128 129

54

31

100 125

134

168 155

69

22

58

156

164

48

91

118

68

83

72 59

89

38

26

162

135

106

112

7

126

84

122

42

103

28

92

36

100

125

99

114

98

13

17

112

7

116

157

138 143

8

67

54

91

118

68

83

201

193 200

81

29

90

27

113

96

69

pr 26

2

102

67

99

114

61

40

8

29

90

27

113

153 158

39

123

53

121

32

203 204

146

144

76

47

93

14

202

199

130

145

110

115

34

77

105

81

32

208 206

190

205 212

133

111

24

18

39

123

53

121

44

94

76

47

93

108

16

37

111

24

34

77

191

167

152

110

44

94

18

105

207

165 137

147 154

211

170

141

159

108

16

37

14

214

142

129

150

132 172

140

128

141

159

134

168

155

191

140

128

129

194

215 198

196

169

195

210

197

143

209

192

160 136

131

201

193

158

130

145

162

135

143

204

199

163

139

173

203

146

144

201

193

158

130

212

133

154

213

171 166

161

206

190

196

169

147

212

133

168

149 148

151

208

215

160

136

131

203

147

172

145

163

139

173

202

194

213

171

198

196

169

209

192

194

215

160

136

131

154

179

148

151

209

192

213

171

166

161

175

188

178

179

148

151

181 174

176

178

179

139

177

184

185

183

188

176

178

173

180

175

188

176

186

174

185

175

183

187

181

184

186

174

185

182

189

177

187

181

184

186

180

6 55

127 35

Figure 6: Module detection at three time points. This treatment was done without taking into account time dependence. Module detection yields respectively 3, 3 and 2 modules depending on x(t), t = 1, 2, 3.

Scale-space module detection for random fields

22

12

t

11 15 14

10 13

58

53 50 9

60 56

52

7

48

54 55

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Figure 7: Same configuration {x, G} as in Fig. 8(c) but with two different displays : (a) "Edge weighted spring embedded" layout, (b) Hierarchical layout.

Scale-space module detection for random fields

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(c)

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Figure 8: Experiment on simulated data. (a) The regulator graph GB (between regulons). (b) The entire regulon graph G. (c) The observed random field x (displayed with "Force directed" layout). (d) Module detection outcome: there is one detected module per regulon. Given the knowledge of the regulons, we can associate a regulon to each detected module.

Scale-space module detection for random fields

24

lambda: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 1.5 w(30)= 1.45

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Figure 9: Module detection based on the multi-scale decomposition in Fig.10 restricted to Λ. The pink crosses show the 1-D profile of x. The color curves display the three main components for scales {8, 10, 30}. The red circles are the locations of the detected module centers. The color segments at the bottom of the figure locate regulons, their colors are identical to those in Fig.8(a-b-d).

Scale-space module detection for random fields

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lambda: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 1.5

w(30)= 1.45 w(28)= 0.39 w(26)= 0.01 w(24)= 1.31 w(22)= 0.01 w(20)= 0.01 w(18)= 0.27 w(16)= 0.01 w(14)= 0.37 w(12)= 0.72 w(10)= 4.39 w(8)= 1.38 w(6)= 0.63 w(4)= 0.01 w(2)= 0.01

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Figure 10: Statistical multi-scale decomposition for Λ0 = {2, 4, ..., 28, 30}. Scale selection with  = 0.1 selects Λ = {8, 10, 24, 30}. The black curve is the sum of all Λ0 -components.

Scale-space module detection for random fields

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Figure 11: Ordinary scale-space decomposition.

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Scale-space module detection for random fields

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1.5 w(30)= 507 w(28)= 515 w(26)= 523 w(24)= 534 w(22)= 545 w(20)= 559 w(18)= 576 w(16)= 596 w(14)= 621 w(12)= 653 w(10)= 695 w(8)= 750 w(6)= 825 w(4)= 930 w(2)= 1080

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¯ σ,Λ |. All scales are nearly Figure 12: Adverse effect of an estimation made without log |K equal contributions and without informative value.