Higher-Order Spectral Clustering for Geometric Graphs

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.


Introduction
Graph clustering-the task of identifying groups of tightly connected nodes in a graph-is a widely studied unsupervised learning problem, with applications in computer science, statistics, biology, economy or social sciences [7].
In particular, spectral clustering is one of the key graph clustering methods [15].In its most basic form, this algorithm consists in partitioning a graph into two communities using the eigenvector associated with the second smallest eigenvalue of the graph's Laplacian matrix (the socalled Fiedler vector [6]).Spectral clustering is popular, as it is an efficient relaxation of the NP-hard problem of cutting the graph into two balanced clusters so that the weight between the two clusters is minimal [15].
In particular, spectral clustering is consistent in the Stochastic Block Model (SBM) for a large set of parameters [1], [11].The SBM is a natural basic model with community structure.It is also the most studied one [1].In this model each node is assigned to one cluster, and edges between node pairs are drawn independently and with probability depending only on the community assignment of the edge endpoints.
However, in many situations, nodes also have geometric attributes (a position in a metric space).Thus, the interaction between a pair of nodes depends not only on the community labelling, but also on the distance between the two nodes.We can model this by assigning to each node a position, chosen in a metric space.Then, the probability of an edge appearance between two nodes will depend both on the community labelling and on the positions of these nodes.Recent proposals of random geometric graphs with community structure include the Geometric Block Model (GBM) [8] and Euclidean random geometric graphs [2].The nodes' interactions in geometric models are no longer independent: two interacting nodes are likely to have many common neighbors.While this is more realistic ('friends of my friends are my friends'), this also renders the theoretical study more challenging.
Albeit spectral clustering was shown to be consistent in some specific geometric graphs [13], the geometric structure can also heavily handicap cut-based approach.Indeed, one could partition space into regions such that nodes between two different regions interact very sparsely.Thus, the Fiedler vector of a geometric graph might be associated only with a geometric configuration, and bear no information about the latent community labelling.Moreover, the common technique of regularization [18], which aims to penalize small size communities in order to bring back the vector associated with the community structure in the second position, will not work in geometric graphs as the regions of space can contain a balanced number of nodes.Nonetheless, this observation does not automatically renders spectral clustering useless.Indeed, as we shall see, in some situations there is still one eigenvector associated with the community labelling.Thus, it is now necessary to distinguish the eigenvectors corresponding to a geometric cut-hence potentially useless for cluster recovery-from the one corresponding to the community labelling.In other words, to achieve a good performance with spectral clustering in such a setting, one needs to select carefully the correct eigenvector, which may no longer be associated with the second smallest eigenvalue.
Our working model of geometric graphs with clustering structure will be the Soft Geometric Block Model (SGBM).It is a block generalization of soft random geometric graphs and includes as particular cases the SBM and the GBM.Another important example is the Waxman Block Model (WBM) where the edge probabilities decrease exponentially with the distance.The SGBM is similar to the model of [2], but importantly we do not assume the knowledge of nodes' positions.
In this paper, we propose a generalization of standard spectral clustering based on a higherorder eigenvector of the adjacency matrix.This eigenvector is selected using the average intraand inter-community degrees, and is not necessarily the Fiedler vector.The goal of the present work is to show that this algorithm performs well both theoretically and practically on SGBM graphs.
Our specific contributions are as follows.We establish the weak consistency of higher-order spectral clustering on the SGBM in the dense regime, where the average degrees are proportional to the number of nodes.With a simple additional step, we also establish strong consistency.One important ingredient of the proof is the characterization of the spectrum of the clustered geometric graphs, and can be of an independent interest.In particular, it shows that the limiting spectral measure can be expressed in terms of the Fourier transform of the connectivity probability functions.Additionally, our numerical simulations show that our method is effective and efficient even for graphs of modest size.Besides, we also illustrate by a numerical example the unsuitability of the Fiedler vector for community recovery in some situations.
Let us describe the structure of the paper.We introduce in Section 2 the Soft Geometric Block Model and the main notations.The characterization of the limiting spectrum is given in Section 3.This characterization will be used in Section 4 to establish the consistency of higherorder spectral clustering in dense SGBM graphs.Finally, Section 5 shows numerical results and Section 6 concludes the paper with a number of interesting future research directions.For a measurable function F :

Model definition and notations
x dx the Fourier transform of F .The Fourier series of F is given by For two integrable functions F, G :

Soft Geometric Block Model
A Soft Geometric Block Model (SGBM) is defined by a dimension d, a number of nodes n, and a set of blocks K.The node set is taken as V = [n].The model is parametrized by a node labelling σ : V → K, a node position X : V → T d , and a measurable positive function The probability of appearance of an edge between nodes i and j is defined by F (||X i − X j ||, σ i , σ j ).Consequently, the model parameters specify the distribution of the adjacency matrix A = (A ij ) of a random graph.
Furthermore, for this work we assume that the model has only two equal size blocks, i. e., K = {1, 2}, and n i=1 1 The labels are assigned randomly, that is, the set {i ∈ [n] : σ i = 1} is chosen randomly over all the n 2 -subsets of [n].We assume that the entries of X and σ are independent and ∀i ∈ V , X i is uniformly distributed over T d .Finally, suppose that for any where F in , F out : T d → [0, 1] are two measurable functions.We call these functions connectivity probability functions.
The average intra-and inter-community edge density are denoted by µ in and µ out .Their expression is given by the first Fourier mode of F in and F out : These quantities will play an important role in the following, as they represent the intensities of interactions between nodes in the same community and nodes in different communities.In particular, the average inside community degree is n 2 − 1 µ in , and the average outside community degree is n 2 µ out .
Example 1.An SGBM where F in (x) = p in and F out (x) = p out with p in , p out being constants is an instance of the Stochastic Block Model.
Example 2. An SGBM where Example 3. We call Waxman Block Model (WBM) an SGBM with F in (x) = min(1, q in e −s in ||x|| ), F out (x) = min(1, q out e −sout||x|| ).This is a clustered version of the Waxman model [16], which is a particular case of soft geometric random graphs [12].
Formally, clustering or community recovery problem is the following problem: given the observation of the adjacency matrix A and the knowledge of F in , F out , we want to recover the latent community labelling σ.Given an estimator σ of σ, we define the loss ℓ as the ratio of misclassified nodes, up to a global permutation π of the labels: ℓ (σ, σ) = 1 n min π∈S 2 i 1 (σ i = π • σ i ) .Then, σ is said to be weakly consistent (or equivalently, achieves almost exact recovery) if ∀ǫ > 0 : lim n→∞ P (ℓ (σ, σ) > ǫ) = 0, and strongly consistent (equivalently, achieves exact recovery) if 3 The analysis of limiting spectrum 3.1 Limit of the spectral measure Theorem 1.Consider an SGBM defined by (1)- (2).Assume that F in (0), F out (0) are equal to the Fourier series of F in (•), F out (•) evaluated at 0. Let λ 1 , . . ., λ n be the eigenvalues of A, and the spectral measure of the matrix 1  n A. Then, for all Borel sets B with µ (∂B) = 0 and 0 ∈ B, a.s., lim where µ is the following measure: .
Remark 1.The limiting measure µ is composed of two terms.The first term, corresponds to the spectrum of a random graph with no community structure, and where edges between two nodes at distance x is drawn with probability F in +Fout 2 (x).In other words, it is the null-model of the considered SGBM.Hence, the eigenvectors associated with those eigenvalues bear no community information, but only geometric features.
On the contrary, the second term corresponds to the difference between intra-and inter-community edges.In particular, the ideal eigenvector for clustering is associated with the eigenvalue λ closest to Other eigenvectors might mix some geometric and community features and hence are harder to analyze.
Last, the eigenvalue λ is not necessarily the second largest eigenvalue, as the ordering of eigenvalues here depends on the Fourier coefficients F in (k) and F out (k), and is in general non trivial.
Proof.The outline of the proof of Theorem 1 follows closely [4].First, we show that ∀m ∈ N, lim n→∞ E µ n (P m ) = µ(P m ) where P m (t) = t m .Second, we use Talagrand's concentration inequality to prove that µ n (P m ) is not far from its mean, and conclude with Borel-Cantelli lemma.
(i) By Lemma 1 in the Appendix, in order to establish the desired convergence it is enough to show that lim n→∞ Eµ n (P m ) = µ(P m ) for any m ∈ N. First, By definition, n iff α is an m-tuple without repetition.We have, where We first bound the quantity R m .Since , Moreover, where Let us first show that the value of G(α) depends only on the number of consecutive indices corresponding to the nodes from the same community.More precisely, let us define the set S(α) = {j ∈ [m] : σ i j = σ i j+1 }.Using Lemma 2 in the Appendix and the fact that the convolution is commutative, we have

We introduce the following equivalence relationship in
Let us now calculate the cardinal of each equivalence class with |S(α)| = p.First of all, we choose the set S(α) which can be done in m p ways if m − p is even and cannot be done if m − p is odd.The set S(α) defines the community labels up to the flip of communities since σ i j = σ i j+1 for any j ∈ S(α) and Let N 1 (α) be the number of indices i j with σ i j = 1.Consider first the case σ i 1 = 1 and note that N 1 (α) is totally defined by the set S(α).There is n 2 possible choices for i 1 .Now we have two possibilities.If then the index i 2 can be chosen in n 2 ways.Resuming the above operation, we choose N 1 (α) indices from the first community, and it can be done in n/2(n/2 − 1) . . .(n/2 − N 1 (α)) ways.The indices from the second community can be chosen The same reasoning applies if σ i 1 = 2. Hence, when n goes to infinity, the cardinal of each equivalence class is This can be rewritten as Hence, Therefore, equations (3), ( 4) and (5) give us: Finally, since F in , F out are equal to their Fourier series at 0, and using (ii) For each m ≥ 1, and n fixed, we define Let A, A be two adjacency matrices.We denote the Hamming distance by Let M m be the median of Q m .Talagrand's concentration inequality [14, Proposition 2.1] states that which after integrating over all t gives Thus, using again inequality (8), we have for all s > Cm n 2 , However, by (6), lim n→∞ Eµ n (P m ) = µ(P m ) with probability 1. Hence µ n (P m ) converges in probability to µ(P m ).Let s n = 1 n κ with κ > 0, and Since n∈N ǫ n < ∞ when κ < 2, an application of Borel-Cantelli lemma shows that the convergence holds in fact almost surely.This concludes the proof.

Conditions for the isolation of the ideal eigenvalue
As noticed in Remark 1, the ideal eigenvector for clustering is associated with the eigenvalue of the adjacency matrix A closest to n µ in −µout

2
(remind that µ in = F in (0) and µ out = F out (0)).The following Proposition 1 shows that this ideal eigenvalue is isolated from other eigenvalues under certain conditions.Proposition 1.Consider the adjacency matrix A of an SGBM defined by (1)-( 2), and assume that: with µ in = µ out .Then, the eigenvalue of A the closest to n µ in −µout 2 is of multiplicity one.Moreover, there exists ǫ > 0 such that for large enough n every other eigenvalue is at a distance at least ǫn.

2
. We will show that there exists ǫ > 0 such that for large enough n, we have for all i = i * : Due to condition (9), and the fact that there is some fixed ǫ 1 > 0 such that Similarly, condition (10) ensures the existence of ǫ 2 > 0 such that Denote ǫ 3 = |µ in −µout|

4
. Let ǫ = min (ǫ 1 , ǫ 2 , ǫ 3 ), and consider the interval Therefore, for n large enough the only eigenvalue of A in the interval B is λ i * .
The following Proposition shows that conditions ( 9) and ( 10) of Proposition 1 are almost always verified for a GBM.
Then the set B = B + ∪ B − of 'bad' parameters is of zero Lebesgue measure: Thus, it is enough to show that Leb(B + ) = 0 and Leb(B − ) = 0. We shall establish the first equality, and the second equality can be proved similarly.By Lemma 3 in the Appendix, the condition (9) for given functions F in and F out is as follows: Notice that lim k j →∞ sinc(2πr in k j ) = 0 and lim k j →∞ sinc(2πr out k j ) = 0 while the right-hand side of the above equation is fixed.Therefore, this equation can hold only for k from a finite set K. Let us fix some k ∈ K and denote Let us now fix r in , and consider the condition defining B + k as an equation on r out .Define the functions Then the mentioned equation takes the form Consider the function h k : C → R: Clearly, this function coincides with f k on R.Moreover, it is holomorphic in C, as sinc(z) is holomorphic (it can be represented by the series ∞ n=0 (−1) n (2n+1)!z 2n ), and the product of holomorphic functions is again holomorphic.But then the derivative h ′ k (z) is also holomorphic, therefore, it has at most countable number of zeros in C. Clearly, h ′ k ≡ f ′ k on R, which yields that f ′ k has at most countable number of zeros in R.
Hence, R + is divided into at most countable number of intervals on which the function is correctly defined and, as far as is measurable as well.Consequently, there is a unique solution is the graph of some measurable function in R 2 + .Since such a graph has a zero Lebesgue measure (see e.g., [17, Lemma 5.3]), we have: Carrying out similar argumentation for B − completes the proof.

Global step:
Let λ be the eigenvalue of A closest to λ * = (µ in −µout) 2 n, and v be the associated eigenvector.

Consistency of higher-order spectral clustering
In this section we show that spectral clustering based on the ideal eigenvector (see Algorithm 1) is weakly consistent for SGBM (Theorem 2).We then show that a simple extra step can in fact achieve strong consistency.
Remark 2. The worst case complexity of the eigenvalue factorization is O n 3 [5].However, when the matrix is sufficiently sparse and the eigenvalues are well separated, the empirical complexity can be close to O(kn), where k is the number of required eigenvalues [5].Moreover, since Algorithm 1 uses only the sign of eigenvector elements, a quite rough accuracy can be sufficient for classification purposes.

Weak consistency of higher-order spectral clustering
Theorem 2. Let us consider the d-dimensional SGBM with connectivity probability functions F in and F out satisfying conditions (9)- (10).Then Algorithm 1 is weakly consistent.
Proof.Let us introduce some notations.Recall that µ in = F in (0) and µ out = F out (0).In the limiting spectrum, the ideal eigenvalue for clustering is We consider the closest eigenvalue of A to λ * : Also, let v be the normalized eigenvector corresponding to λ.In this proof, the Euclidian norm • 2 is used.The plan of the proof is as follows.We consider the vector ) T , where we supposed without loss of generality that the n/2 first nodes are in Cluster 1, and the n/2 last nodes are in Cluster 2. The vector v * gives the perfect recovery by the sign of its coordinates.We shall show that with high probability for some constant C > 0 We say that an event occurs with high probability (w.h.ṗ.) if its probability goes to 1 as n → ∞.
With the bounding (12), we shall then show that at most o(n) of entries of v have a sign that differ from the sign of the respective entry in v * ; hence v retrieves almost exact recovery.
In order to establish inequality (12) we will use the following theorem from [10].
, δ is the separation of ρ from the next closest eigenvalue and v is the eigenvector corresponding to λ, then First we deal with ρ(v * ).Since v * is normalized and real-valued (by the symmetry of A), we have ρ(v * ) = v T * Av * .Denote u = Av * .Then, obviously, It is clear that each entry A ij with i = j is a Bernoulli random variable with the probability of success either µ in or µ out .This can be illustrated by the element-wise expectation of the adjacency matrix: Let us consider the first term in (13) for i ≤ n/2.Since A ij are independent for fixed i, it is easy to see that Then we can use the Chernoff bound to estimate a possible deviation from the mean.For any Let us take t = 2 . Then for large enough n Similarly, and Denote γ n = 2(µ in + µ out ) log n and notice that γ n = Θ( √ log n).By the union bound, we have for large enough n By the same argumentation, Now let us calculate ρ(v * ): We already established that More precisely, by (16), In the same way, by (17), Finally, Now let us denote As we already know, Then A similar bound can be derived for the case i > n/2.Taking into account that ρ(v * ) does not depend on i, using the union bound and equations ( 15) and ( 18), we get that One can readily see that Thus, we finally can bound the Euclidean norm of the vector w: Now we can use Theorem 3.According to this result, where δ = min i |λ i (A) − ρ(v * )| over all λ i = λ.By Proposition 1, δ > ǫn.Then, as far as v * is normalized, a simple geometric consideration guarantees that Let us denote the number of errors by If we now remember that the vector v * consists of ± 1 √ n , it is clear that for any i with sign((v The number of such coordinates is r, therefore, Then, by (19), the following chain of inequalities holds: Hence, with high probability Thus, the vector v provides almost exact recovery.This completes the proof.

Strong consistency of higher-order spectral clustering with local improvement
In order to derive a strong consistency result, we shall add an extra step to Algorithm 1.Given σ, the prediction of Algorithm 1, we classify each node to the community where it has the most neighbors, according to the labeling σ.We summarize this procedure in Algorithm 2, and Theorem 4 states the exact recovery result.

Global step:
Let σ be the output of Algorithm 1.
Local improvement: Remark 3. The local improvement step runs in O(nd max ) operations, where d max is the maximum degree of the graph.Albeit being convenient for the theoretical proof, we will see in the numerical section (Figure 3) that the local improvement typically leads to a small gain of accuracy in practice.
Theorem 4. Let us consider the d-dimensional SGBM defined by (1)-( 2), and connectivity probability functions F in and F out satisfying conditions (9)- (10).Then Algorithm 2 provides exact recovery for the given SGBM.
Proof.We need to prove that the almost exact recovery of Algorithm 1 (established in Theorem 2) can be transformed into exact recovery by the local improvement step.This step consists in counting neighbours in the obtained communities.For each node i we count the number of neighbours in both supposed communities determined by the sign of the vector v coordinate: According to Algorithm 2, if Z 1 (i) > Z 2 (i), we assign the label σ i = 1 to node i, otherwise we label it as σ i = 2. Suppose that some node i is still misclassified after this procedure and our prediction does not coincide with the true label: σ i = σ i .Let us assume without loss of generality that σ i = 1 and, therefore, Let us denote by Z 1 (i) and Z 2 (i) degrees of node i in the communities defined by the true labels σ: Since sign( x j ) coincides with the true community partition for all but C log n nodes (see the end of the proof of Theorem 2), which implies that Hence, taking into account that Z 2 (i) > Z 1 (i), which means that the inter-cluster degree of node i is asymptotically not greater than its intracluster degree (since Z j (i) = Θ(n) w.h.p.).Intuitively, this should happen very seldom, and Lemma 4 in the Appendix gives an upper bound on the probability of this event.Thus, by Lemma 4, for large n, Then each node is classified correctly with high probability and Theorem 4 is proved.
5 Numerical results

Higher-order spectral clustering on 1-dimensional GBM
Let us consider a 1-dimensional GBM, defined in Example 2. We first emphasize two important points of the theoretical study: the ideal eigenvector for clustering is not necesarily the Fiedler vector, and some eigenvectors, including the Fiedler vector, could correspond to geometric configurations.
Figure 1 shows the accuracy (i.e., the ratio of correctly labeled nodes, up to a global permutation of the labels if needed, divided by the total number of nodes) of each eigenvector for a realization of 1-dimensional GBM.We see that, although the Fiedler vector is not suitable for clustering, there is nonetheless one eigevector that stands out of the crowd.
Then, in Figure 2 we draw the nodes of a GBM according to their respective position.We then show the clusters predicted by some eigenvectors.We see some geometric configurations (Figures 2a and 2c), while the eigenvector leading to the perfect accuracy corresponds to position 4 (Figure 2b).  Figure 3 shows the evolution of the accuracy of Algorithms 1 and 2 when the number of nodes n increases.As expected, the accuracy increases with n.Moreover, we see no significant effect of using the local improvement of Algorithm 2. Thus, we conduct all the rest of numerical experiments with the basic Algorithm 1.
In Figure 4, we illustrate the statement of Proposition 2: for some specific values of the pair (r in , r out ), the Conditions ( 9) and (10) do not hold, and Algorithm 1 is not guaranteed to work.We observe in Figure 4 that these pairs of bad values exactly correspond to the moments when the index of the ideal eigenvector jumps from one value to another.
Finally, we compare in Figure 5 the accuracy of Algorithm 1 with the motif counting algorithms presented in references [8] and [9].Those algorithms consist in counting the number of  common neighbors, and clustering accordingly.We call Motif Counting 1 (resp.Motif Counting 2) the algorithm of reference [8] (resp. of reference [9]).We observed that with present realizations the motif counting algorithms take much more time than HOSC takes.We thank the authors for providing us the code used in their papers.

Waxman Block Model
In this Section, we consider the Waxman Block Model introduced in Example 3. Recall that F in (x) = min(1, q in e −s in x ) and F out (x) = min(1, q out e −soutx ), where q in , q out , s in , s out are four positive real numbers.We have the following particular situations: • if s out = 0, then F out (x) = q out and the inter-cluster interactions are independent of the nodes' positions.If s in = 0 as well, we recover the SBM; • if q in = e r in s in and q out = e routsout , then in the limit s in , s out ≫ 1 we recover the 1dimensional GBM.
We show in Figure 6 the accuracy of Algorithm 1 on a WBM.In particular, we see that we do not need µ in > µ out , and we can recover disassociative communities.

Conclusions and future research
In the present paper we devised an effective algorithm for clustering geometric graphs.This algorithm is close in concept to the classical spectral clustering method but it makes use of the eigenvector associated with a higher-order eigenvalue.It provides weak consistency for a wide class of graph models which we call the Soft Geometric Block Model.A small adjustment of the algorithm leads to strong consistency.Moreover, our method was shown to be effective in numerical simulations even for graphs of modest size.Moreover, the problem stated in the current paper might be investigated further in several directions.One of them is a possible study on the SGBM with more than two clusters.The situation here is quite different from the SBM where the spectral clustering algorithm with few eigenvectors associated with the smallest non-zero eigenvalues provides good performance.In the SGBM even the choice of such eigenvectors is not trivial, since the 'optimal' eigenvalue for distinguishing two clusters is often not the smallest one.
Another natural direction of research is the investigation of the sparse regime, since all our theoretical results concern the case of degrees linear in n.In sparser regimes, there are effective algorithms for some particular cases of the SGBM (e. g., for the GBM), but there is no established threshold when exact recovery is theoretically possible.Unfortunately, the method of the current paper without revision is not appropriate for this situation, and the technique will very likely be much more complicated.
Finally, considering weighted geometric graphs could be an interesting task for applications where clustering of weighted graphs is pertinent.For instance, the functions F in and F out can be considered as weights on the edges in a graph.We believe that most of the results of the paper may be easily transferred to this case.

A Auxiliary results
A.1 Hamburger moment problem for the limiting measure Lemma 1. Assume that F : R d → R is such that F (0) is equal to the Fourier series of F (x) evaluated at 0 and 0 ≤ F (x) ≤ 1.Consider the measure µ defined by the function F : Then µ is defined uniquely by the sequence of its moments {M n } ∞ n=1 .
Proof.It is enough to show that Carleman's condition holds true for µ (see [3]): As one can easily see, From the bounds 0 ≤ F (x) ≤ 1 it follows that F (k) ≤ 1.Then it is clear that F n (k) ≤ F (k) for any n ∈ N. We can write Here we used the assumption that F (0) equals its Fourrier series evaluated at 0. Then We notice that Hence,

2. 1 , 1 2 d.
Notations Let T d = R d /Z d be the flat unit torus in dimension d represented by − 1 2 The norm ℓ ∞ in R d naturally induces a norm on T d such that for any vector x = (x 1 , . . ., x d ) ∈ T d we have x = max 1≤i≤d |x i |.

Proposition 2 .
Consider the d-dimensional GBM model, where F in , F out are 1-periodic, and defined on the flat torus T d by F in (x) = 1( x ≤ r in ) and F out (x) = 1( x ≤ r out ), with r in > r out > 0. Denote by B + and B − the sets of parameters r in and r out defined by negation of conditions (9) and (10):

Figure 1 :
Figure 1: Accuracy obtained on a 1-dimensional GBM (n = 2000, r in = 0.08, r out = 0.02) using the different eigenvectors of the adjacency matrix.The eigenvector of index k corresponds to the eigenvector associated with the k-th largest eigenvalue of A.

(a) k = 2 (b) k = 4 (c) k = 8 Figure 2 :
Figure2: Example of clustering done using the eigenvector associated to the k-th largest eigenvalue of the adjacency matrix of a 1-dimensional GBM (n = 150, r in = 0.2, r out = 0.05).For clarity edges are not shown, and nodes are positioned on a circle according to their true positions.The Fiedler vector (k = 2) is associated with a geometric cut, while the 4-th eigenvector corresponds to the true community labelling and leads to the perfect accuracy.The vector k = 8 is associated with yet another geometric cut.

Figure 3 :
Figure 3: Accuracy obtained on 1-dimensional GBM as a function of n, when r in = 0.08 and r out = 0.05, for Algorithm 1 and Algorithm 2. Results are averaged over 100 realisations, and error bars show the standard error.

Figure 4 :
Figure 4: Evolution of accuracy (blue curve) with respect to r in , for a GBM with n = 3000 and r out = 0.06.Results are averaged over 5 realisations.By the red curve we show the index of the ideal eigenvector, again with respect to r in .

Figure 5 :Figure 6 :
Figure5: Accuracy obtained on 1-dimensional GBM for different clustering methods.Motif Counting 1 corresponds to the algorithm described in[8] and Motif Counting 2 to the algorithm described in[9].Results are averaged over 50 realisations, and error bars show the standard error.