Setting

We consider a clustering problem of nodes for undirected edge-weighted graphs G = (V, E) where V consists of n vertices {v₁, …, v_n}, and E is represented by the edge-weight matrix W_n, which is a n × n symmetric (real Hermitian) matrix with elements and w_i,i = 0 ( denotes a set of real numbers). Let us assume that there are K clusters of nodes, denoting them as c₁, …, c_K. We define a cluster block (k, k′) as a set of weights w_i,j such that nodes i and j belong to the cluster c_k and c_k′, respectively: v_i ∈ c_k and v_j ∈ c_k′ (1 ≤ k, k′ ≤ K). Here, we assume that each off-diagonal weight w_i,j is independently drawn from a certain distribution. With this assumption, we define a K-way community structure as characterized by different distributions in K × K cluster blocks. To elaborate this definition, we assume the following distribution for each cluster block: (1) where v_i ∈ c_k, v_j ∈ c_k′, and g is a certain probability distribution. This definition suggests that a pair of parameters (μ_k,k′, σ_k,k′) characterizes each cluster block, hence, community structure (Fig 2A). Note that in this definition we exclude the degenerate case in which μ_k,k′ = constant and σ_k,k′ = 0 such that variances become zero for the whole set of {w_i,j}.

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Fig 2. Setting of community structure and Tracy-Widom distribution.

Panel (A): Illustration of the setting of community structure in a matrix representation where nodes are arranged in the order of cluster labels. Each cluster block is characterized by mean μ and standard deviation σ with cluster block index (k, k′). Panel (B): The density function of the Tracy-Widom distribution for Gaussian orthogonal ensembles with β = 1 (the first derivative of F₁(x) in Eq (7)), generated by the function dtw in R-package {RMTstat}. The critical values at significance level α = 0.05 and α/4 = 0.0125 are 0.979 (green line) and 1.889 (red line), respectively.

https://doi.org/10.1371/journal.pone.0194079.g002

Since the community structure of interest is based on differences of weight distributions, it is translation- and scale-invariant for all weights. Hence, to simplify the problem, as a preprocess, we standardize off-diagonal elements of W_n using all off-diagonal weights w_i,j(i ≠ j) so that the mean is zero and the variance one. We denote as S the mapping that standardizes the edge-weight matrix in this way, transforming each element of the matrix as (2) where μ and σ are the mean and the standard deviation of the whole off-diagonal elements {w_i,j}. Practically, these mean and standard deviation may be replaced by the empirical counterparts μ_emp and σ_emp. For the standardized edge-weight matrix S(W_n), we assume that the mean and the standard deviation of g in Eq (1) are zero and one, respectively. In this setting, the mean and the standard deviation in cluster block (k, k′) are μ_k,k′ and σ_k,k′, respectively. The differences of these parameters distinguish between clusters in terms of the first and second moments, while controlling higher moments than two. Using this setting of community structure, we define no community case as a single community with K = 1 where μ_k,k′ = 0 and σ_k,k′ = 1 for S(W_n). Note that since g is arbitrary, including a mixture distribution of a certain distribution family, our definition of no community structure includes the case in which each weight is generated from a specific distribution in a list of distributions in random order. Importantly, when we shuffle the off-diagonal elements W_n at random (in element-wise manner), the community structure always disappears. Indeed, in such a case, each element of the shuffled matrix independently and identically follows the mixture distribution consisting of different components, i.e., ∑_k,k′ π_k,k′ g_k,k′ where π_k,k′ is the proportion of elements of cluster block (k, k′) for the original matrix W_n. We use this property for our statistical test as an alternative way to estimate confidence intervals.

Statistical test

In this section, we develop a statistical test for the existence of community structure defined in the previous section (i.e., K = 1 vs. K > 1). We base our test on the asymptotic behavior of the eigenvalues of S(W_n) (n goes to ∞) when there is no community structure. A useful result of Random Matrix Theory in this context is that if the elements of an infinite dimensional symmetric matrix X independently follow a certain distribution with mean zero and variance one, then the empirical (random) distribution of the eigenvalue λ of , where X_n is the principal submatrix of X for the first n rows and columns, converges almost surely to a Wigner semicircular distribution as n goes to ∞ (semicircular law). Note that this law holds for any generative distribution of the elements in matrix X (as long as independently drawn), which is referred to as the universality property of the law. Also, this law holds even if we replace diagonal elements with zero’s, as in our case. Further, strong Bai-Yin theorem suggests that with the additional condition of the distribution of each element (namely, the existence of a fourth moment), the largest magnitude of eigenvalues is almost certainly bounded by 2. These two theorems imply that the largest magnitude of eigenvalues almost surely converges to two ([24], p.136).

In order to apply this property to our context, we consider a normalization mapping of edge-weight matrix W_n, transforming each element of the matrix as (3) where S is the standardization mapping in Eq (2). Now, let us assume that the elements in an edge-weight matrix W_n are generated as in Eq (1). In this setting, if the largest magnitude of eigenvalues of T(W_n) does not converge to two, then, there should be some K-way community structure in the graph (K > 1) because of our assumption in Eq (1) (Note that without the assumption in Eq (1), this property does not hold. For instance, one can make a scale-free graph where the eigenvalues do not follow the semicircular law [25]). However, the converse argument does not necessarily hold. That is to say, the fact that the convergence of the largest magnitude of eigenvalues to two does not imply that there is no community structure (i.e., K = 1). A simple counter example is given as follows (proof in S1 Appendix).

Example 1. Let W_n be a n × n symmetric edge-weight matrix that has K-way community structure with the same cluster size (n/K) as defined in the previous section. Suppose that μ_k,k′ = 0 for ∀k, k′, for k ≠ k′, and . Then, the largest magnitude of eigenvalues of T(W_n) almost surely converges to two as n goes to ∞.

Nonetheless, in our setting, we can show that an additional condition on the eigenvalue distribution for an exponentially mapped edge-weight matrix ensures that the converse argument also holds. For this purpose, we introduce the exponential mapping Exp that transforms each element of W_n as (4) where is a tuning parameter (we do not explicitly denote the dependence of Exp on t because of cluttering). Subsequently, we define the normalization mapping T_e for the exponentially transformed matrix as (5)

Now, the following theorem provides a necessary and sufficient condition for the existence of community structure (proof in S2 Appendix).

Theorem 1. Let W_n be a n × n weight matrix defined in the previous section with the fixed proportion of cluster sizes (r₁, …, r_K) and the pairs of fixed parameters {(μ_k,k′, σ_k,k′)}(k, k′ = 1, …, K). Suppose that there exists the moment-generating function M(t) in an open interval containing zero for g (g is defined in Eq (1)). Then, the following statements (C1) and (C2) are equivalent:

1. (C1) There is no community structure (i.e., K = 1)
2. (C2) Each of the largest magnitudes of eigenvalues of T(W_n) and T_e(W_n) for any non-zero real value t₀ ≠ 0 almost surely converges to two, as n goes to ∞.

Theorem 1 motivates us to use the largest magnitude of eigenvalues of edge-weight matrix to establish a statistical test on the null hypothesis H₀: (6) Practically, to test the null hypothesis H₀, we focus on positive and negative extreme values of eigenvalues. The largest eigenvalue may deviate positively from two, while the smallest eigenvalue may deviated negatively from -2.

Comments.

1. Strictly speaking, the independent assumption on weights is broken if we transform them by T or T_e using an empirical mean and standard deviation μ_emp and σ_emp. For simplicity, however, we ignore such an effect in the present paper.
2. Our method is not applicable to a directed graph, because in that case an edge-weight matrix becomes non-symmetric; hence, Theorem 1, which is based on properties of eigenvalues of symmetric matrices, does not hold.
3. The spectral method in [19–21] takes a slightly different approach from our method. In the context of unweighted graph, they consider a symmetric matrix representing an expected fraction of edges between two communities (hence, the summation of entries in a row is one). Further, the size of the matrix in their approach is K × K while that of our method is n × n. Moreover, in their approach, the second largest eigenvalue is considered because the largest eigenvalue is constant (always one). As other community detection methods, it is not trivial to generalize their method to real-valued graphs.
4. Theorem 1 implies that if there is no community structure (i.e., edges are i.i.d. generated), the ratio of the second largest eigenvalue to the largest eigenvalue converges to one. This is a general property to a symmetric matrix. This result is contrasted with Brody’s conjecture that for a positive non-symmetric matrix in which all edges are i.i.d. generated, the ratio of the second largest eigenvalue to the largest eigenvalue goes to zero [26–28].
5. The exponentially transformed matrix in Theorem 1 does not replace non-backtracking matrix in [11]. Rather, the exponentially transformed matrix serves to capture differences of variances in generative models.
6. In the current setting, means and variances in generative models are fixed. One may wonder how much this condition can be relaxed. From the argument about unweighted graphs by [12, 29], we speculate that one may be able to relax the condition such that the difference of means may be larger than Ω(log n/n). Indeed, assuming that this condition holds after standardizing the edge-weight matrix, Eq (3) in S2 Appendix becomes λ₁(M_n) ≥ A(log n)². Hence, in this relaxed condition as well as in non-relaxed condition, it holds that if there is community structure with equal variances the largest magnitude of eigenvalues does not converge to two (the first part of the proof of Theorem 1). Moreover, using the relaxed condition, it can be shown that for the unstandardized matrix, mean differences should be also larger than Ω(log n/n). Note that from the prime number theorem [30], the reciprocal of log n/n denotes the number of prime numbers less than n. This observation provides us the following interpretation of the results. If we assume that the community size is the same across different communities, the number of prime numbers denotes the number of irreducible topologies of communities (for example, four-community structure may be reduced to two-community structure by paring two communities while three-community structure is not reducible). We speculate that such an irreducible community structure is easier to detect than the remainder of structures and that if such structures are more available, the detection of community structure becomes easier. This interpretation suggests that the number of prime numbers may be inversely related to detectable differences of means, which is consistent with the lower bound Ω(log n/n).
7. One may wish to tune the value of t₀ as follows. We consider two-community structure, assuming that edges within communities are generated by while edges between communities by . The first moment (mean) of the exponentially transformed variable generated from N(μ, σ) is given by exp(μt + σ²t²/2) while the second moment is exp(2μt + 2σ²t²). Using these results, we can analytically evaluate the difference in means of the exponentially transformed variables between and normalized by the square root of the average of variances. It can be shown that in this case μ is irrelevant for the normalized difference. So, one may choose t₀ that maximizes the normalized difference for a given σ₁ and σ₂, or, simply the ratio . From Fig 3, one may choose t₀ between 0.5 and 1.

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Fig 3. Normalized differences of exponentially transformed variables between normal distributions N(μ, σ₁) and N(μ, σ₂).

The X-axis denotes t₀ in Theorem 1 and the Y-axis normalized difference. We set to 1.1, 1.6, 2.1, 2.6, and 3.1.

https://doi.org/10.1371/journal.pone.0194079.g003

The behavior of the largest eigenvalue has been well studied in the literature when elements of the edge-weight matrix W_n are independently generated by certain symmetric distributions g (typically Gaussian, otherwise, its density function may be even with less heavier tails than Gaussian distributions) with mean zero and variance one for non-diagonal elements and with mean zero and variance two for diagonal elements. In this setting, the largest eigenvalue λ_max asymptotically follows the Tracy-Widom distribution for Gaussian orthogonal ensembles with parameter β = 1: (7) where with where q(x) is the solution of Painlevé II equation d²q/dx² = xq + 2q³ with the boundary condition q(x) ∼ Ai(x) as x → ∞ [31, 32]. Note that the Tracy-Widom distribution is for the maximum eigenvalue of a specific type of symmetric matrix (e.g., Gaussian ensembles) while the semicircular law holds for the distribution of eigenvalues in a general type of symmetric matrix (Wigner ensembles). Moreover, in our framework, the diagonal elements are all zero, which is a slightly different situation than the conventional assumption for the Tracy-Widom distribution. Nevertheless, because of the universality property of the Tracy-Widom distribution ([33], Theorem 21.4.3), we can safely apply Eq (7) to our context (obviously, our context satisfies the condition of universality that the diagonal part should be symmetric with a sub-Gaussian tail).

Using the Tracy-Widom distribution in Eq (7), we set confidence intervals for our statistical test as follows. For the normalized edge-weight matrix T(W_n), we set the confidence interval CI_max of the largest eigenvalue λ_max at level α. Since the violation of the semicircular law occurs as the positive deviation from the expected value, we consider the one-sided confidence interval as (−∞, q) where q is a critical value at significant level α, i.e., P(λ_max ≥ q|H₀) = α, which is estimated by F₁(x) in Eq (7) (refer to the shape of its first derivative in Fig 2B). If the generative distribution g is not symmetric or is heavy-tailed, one may evaluate the distribution of the largest eigenvalues by means of a permutation test for T(W_n). Though the permutation test may provide an accurate confidence interval, it is not computationally efficient because we need to compute eigenvalues a large number of times. Therefore, when the number of nodes is large, one may opt for the Tracy-Widom distribution to efficiently obtain confidence intervals. In addition to the largest eigenvalue, we also test the smallest eigenvalue λ_min, which may violate the semicircular law (what matters is indeed the largest magnitude of eigenvalue). In this case, the confidence interval CI_min is given by (−q, ∞). In similar fashion, we test the largest and the smallest eigenvalue of the exponentially normalized weight matrix. We first standardize the data and then apply the mapping T_e where we set t₀ to 1/2 as default. This results in the transformed matrix T_e(S(W_n)) (we denote the confidence intervals as and , respectively). Since this procedure involves a series of four statistical tests, we set the level of significance to α/4 for each test, taking into account the Bonferroni correction (Algorithm 1; is an indicator function: 1 for correct a; 0 otherwise).

Algorithm 1. Testing the existence of community structure

Input: Edge-weight matrix W, confidence intervals CI_max, CI_min, and at level α/4.

s ← 0

(max. eigenvalue of T(W) ∈ CI_max)

(min. eigenvalues of T(W) ∈ CI_min)

(max. eigenvalue of T_e(S(W)) ∈ )

(min. eigenvalue of T_e(S(W)) ∈ )

if s = 4 then

 Accept H₀

else

 Reject H₀

end if

Data generation

For the data structure in this simulation study, we adopted that in [34], setting the number of clusters to five and cluster size to (10s, 20s, 30s, 40s, 50s), where we manipulated integer s. In this setting, we have 5 × 5 = 25 cluster blocks. In each cluster block, weights were independently drawn from a Gaussian distribution N(μ_k,k′, σ_k,k′) where μ_k,k′ and are the mean and the variance for a cluster block (k, k′). We generated 100 datasets for each setting.

Results

When the number of nodes ranges from 150 to 1500, the distribution function F₁(x) in Eq (7) provides a good approximation of the critical value at a significance level of α = 0.05 under the null hypothesis H₀ (Fig 4A). Since the function F₁(x) provides the asymptotic probability distribution, this result suggests that the function F₁(x) also provides a good approximation of the critical value when the number of nodes exceeds this range. In regard to statistical power, it is implied that our method can readily detect the existence of community structure when means μ_k,k′ in each block differ by at most 0.3 (3 × 0.05 + 3 × 0.05) when σ_k,k′ = 1 with the number of nodes being 750 (Fig 4B). On the other hand, the power may not be sufficient when differences among cluster blocks are characterized by variances (Fig 4C). However, the application of our method to the exponentially transformed matrix by Exp considerably improves the power (Fig 4D). All these results suggest good performance of our method in testing for the existence of community structure in a graph.

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Fig 4. Boxplots represent distributions of the largest eigenvalues for various settings.

Panel (A): No-community case (K = 1) of Gaussian ensembles for different number of nodes from 150 to 1500 in x-axis. Panel (B): Five-way community case with the number of nodes 750 and cluster size (50, 100, 150, 200, 250). Each cluster block is characterized by means of a Gaussian distribution (while fixing variance = 1), which is randomly chosen from {−μ, μ} with equal probabilities. The value of μ is manipulated from 0 to 0.5 of width 0.1 in x-axis. Panel (C): A five-way community case characterized by variance (while fixing mean = 0), which is randomly chosen from {1, σ²} with equal probabilities. The value of σ is manipulated from 1 to 6 of width 1 in x-axis. Panel (D): A five-way community case in the same setting as in (C), but each edge-weight matrix is transformed by the exponential mapping Exp in Eq (4) with t₀ = 1/2. In all panels, the green line denotes the 95 percentile of the largest eigenvalue under the null hypothesis H₀ in (6).

https://doi.org/10.1371/journal.pone.0194079.g004

Lastly, we compare the performance of our method with the remaining methods. We applied our method as outlined in Algorithm 1 to synthetic data, setting α to 0.05 (hence, α/4 = 0.0125). When the community structure is characterized by mean differences, the performance of our method is comparable with the clustering entropy method with signed spectral clustering (CE + SignedSpec), while it outperforms the clustering entropy method with conventional spectral clustering (CE + ConvSpec) (Fig 5A). On the other hand, when the community structure is characterized by scale differences, our method considerably outperforms other methods (Fig 5B).

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Fig 5. Comparison of the power of the test for three different methods.

Our method, the clustering entropy method for the resulting cluster solution using the signed spectral clustering method (CE + SignedSpec), and the clustering entropy method using conventional spectral clustering (CE + ConvSpec). The true community structure is set as follows: cluster size (50, 100, 150, 200, 250); means and variances are manipulated in x-axis of Panel (A) and (B) as in Fig 4B and 4C, respectively.

https://doi.org/10.1371/journal.pone.0194079.g005

Data

First, we applied our method to the following benchmark graph datasets: Karate club, Karate [35]; co-authorships in network science, Co-authours [36]; Tribal relationships in highland New Guinea, Gahuku-Gama [37]. The datasets of Karate and Co-authours are binary (i.e., {0, 1}), while the edges in the dataset of Gahuku-Gama take discrete signed values, {−1, 0, 1}. The number of nodes for these datasets are 34, 1589, and 16, respectively. These datasets have been well studied in terms of detecting community structure [7].

Second, we applied our method to a real-valued edge-weighted graph: resting state functional MRI (fMRI) data [38]. The original dataset consists of the level of BOLD (Blood-Oxygen-Level Dependent) signals at short intervals, which reflects neural activity at tiny regions of the brain, called ‘voxel’ (4949 voxels in this dataset). We pre-processed this dataset by evaluating temporal correlations among these voxels and carrying out Fisher’s z-transformation for them, which results in a 4949 edge-weight matrix W. The objective in this dataset is to test our method on a real-valued, edge-weight matrix and to draw useful inferences from the analysis.

Results

For the first group of real datasets, our method finds some community structure (i.e., K > 1), whether we estimate critical values using the Tracy-Widom distribution or a permutation test (Fig 6). Note that in the binary case, we always obtain the same results for the original matrix and for the exponentially transposed matrix, because T(W) = T_e(S(W)). So, we tested only T(W) in Karate and Co-authors datasets, setting the significance level to α/2.

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Fig 6. Results of application of our method to real datasets.

Karate, Co-authors, and Gahuku-Gama from left to right panels. A star denotes the maximum or minimum eigenvalues of the normalized matrix T(W), while a cross denotes those of the exponentially normalized matrix T_e(S(W)). The top or bottom edges of boxes denote critical values of these eigenvalues at significance level α/2 with α = 0.05 for Karate and Co-authors datasets, and α/4 for Gahuku-Gama dataset. These critical values resulted from a permutation test with 1000 randomized realizations. In contrast, red dashed lines denote critical values derived from the Tracy-Widom distribution F₁(x).

https://doi.org/10.1371/journal.pone.0194079.g006

It is observed in Fig 6 that confidence intervals largely match between the Tracy-Widom distribution and the permutation test for the Karate and Co-authors datasets. On the other hand, there is some discrepancy between these for Gahuku-Gama data. A possible explanation for this is due to the small number of nodes in the dataset: the Tracy-Widom distribution describes the asymptotic behavior of the eigenvalue when n goes to ∞.

For fMRI dataset, our test rejected the null hypothesis H₀, yielding the maximum and minimum eigenvalues as 31.0 and -7.2 for T(W), and 31.8 and -10.9 for T_e(S(W)), which provides strong evidence that community structure exists in this graph. Furthermore, we carried out our test for subsets of voxels in brain regions that are anatomically predefined, where the number of voxels ranges from 13 to 498. Our test results suggest that community structure may exist in each region (except for brain region 16) (Fig 7). This result supports the conjecture on heterogeneity of brain activities in anatomically defined brain regions, discussed in the neuroscience literature [39].

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Fig 7. Results of application of our method to the fMRI dataset.

Stars denote the maximum or minimum eigenvalues λ for normalized weight matrices by mapping T in various brains regions with an edge-weight matrix W_k,k, indexed by the brain region k in the x-axis. Crosses denote counterparts for exponentially normalized weight matrices by the mapping T_e. Horizontal lines denote lines y = −2 and y = 2, which correspond to values at which the minimum and maximum eigenvalues asymptotically converge.

https://doi.org/10.1371/journal.pone.0194079.g007

Relevant methods

One of the most popular approaches to detection of communities in an unweighted graph is based on ‘modularity’ [40, 41], which is defined as (8) where e_k,k′ is one-half of the fractions of edges between cluster k and k′, and a_k = ∑_k′ e_k,k′. The modularity Q denotes deviation of the number of edges from possible random configurations, hence, serving as an objective function for finding a community structure. The algorithm of optimizing Q is to start with node-community (a community consisting of a single node) and to aggregate communities to increase Q in a similar fashion to a hierarchical clustering algorithm [42]. We use an algorithm of this kind proposed by [41], which is referred to as ‘Newman’. On the other hand, Louvain methods [43–45] are a variant of the modularity-based methods, which optimizes Q (or, different type of Q) by means of iterating the following two steps. The first step is to optimize Q by aggregating communities in the aforementioned manner. The second step is to re-parameterize each community as a single node. These steps are alternatively carried out until no further increment in Q is possible. Here, we use one of the most popular methods by [43], referred to as ‘Louvain’. For a threshold of detecting community structure, we use an analytical approximation of modularity for an Erdős-Rényi random graph with n nodes and probability p of connecting two nodes, which is given as by [46]. Note that in a sparse graph, even without any community structure, modularity Q can take a large value. The analytical approximation captures this point, providing a useful criterion of community detection, though the confidence interval is not readily available. For another relevant method to modularity, we consider an approach based on eigenvalues of a modularity matrix by [47]. In a similar line to the graph Laplacian [48], this method partitions nodes based on the eigenvector of the modularity matrix corresponding to the largest positive eigenvalue. By repeatedly evaluating such an eigenvector, we continue to partition nodes until no further positive eigenvalue is obtained. Here, we use this method (referred to as ‘Split’) for the first partition of nodes, evaluating the largest eigenvalue of the modularity matrix. Furthermore, we consider a versatile approach: a Bayesian clustering method for communities in a graph by [49]. This method explicitly models community memberships as probabilistic parameters, which are optimized in a Bayesian manner (referred to as ‘Bayesian’). Lastly, we include a bootstrap method by [16], which is combined with the community detecting method ‘Newman’, setting the proportion of disturbance to 5% (referred to as ‘Bootstrap’). In this method, for simplicity, we evaluate concordances of community structure between the original graph and bootstrapped graphs by means of Adjusted Rand Index [50] in the same spirit as [51].

For meaningful comparison of computation time among different methods, we ran these methods in the same programming language, Matlab, using publicly available codes for Newman in [52], Louvain in [53] and Bayesian in [54]. For our method, Split method, and Bootstrap method, we ourselves programmed corresponding Matlab codes.

Data

We consider the following real datasets: Social networks in Indian Villages [55, 56] with 203 nodes and 523 edges (referred to as ‘IndianVillage’); Protein-protein interactions in budding yeast [57] with 2361 nodes and 6600 edges (referred to as ‘Yeast’); Word associations based on empirical studies [58] with 10617 nodes and 63000 edges (referred to as ‘FreeAssoc’; we transformed the original data into an undirected graph by adding edges if there is a connection between nodes in either direction). In addition, we also consider inverted graphs in which the status of an edge is inverted in these datasets (i.e., if there is an edge, it is removed; otherwise, it is added). We expected that this would clarify differences of performance among the methods in question. Finally, we generate weighted versions of these datasets as follows. If there is an edge, a weight is randomly generated from N(0, 1), otherwise from N(0, 0.01). We apply our method and Bayesian method to these datasets (the remainder of methods are not applicable to a weighted graph).

Results

For the original datasets, the performance of our method is comparable to other relevant methods, because the existence of community structures is well detected (Table 1). On the other hand, our method suggests the existence of community structures for the inverted graphs as well. Bayesian method and Split method yielded similar results. However, the modularity-based methods Newman and Louvain suggest no community structures while the performance of Bootstrap method is in-between. These differences arise from different (implicit) assumptions in the methods. Our method and Bayesian method focus on differences of patterns in occurrence of edges in communities, while the modularity-based approaches focus only on high density of edges in communists. For practical usage, this implies that one should carefully choose a method, depending on what type of community structure one aims to detect. In case of weighted datasets, both our method and Bayesian method yield similar results on detection of community structures. Lastly, as regards computation time, our method outperforms the remainder of the methods. This is possibly due to that these methods go through a procedure to search for community memberships including the number of communities, while our method does not include such a procedure.

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Table 1. Results of application to unweighted graphs of real datasets: IndianVillage, Yeast and FreeAssoc.

In the column of ‘Type’ in the table, ‘Ori’ denotes the original graph while ‘Inv’ the inverted graph. Further, ‘Ori.w’ denotes the weighted original graph while ‘Inv.w’ denotes the weighted inverted graph. For each cell in the table, computation time and a corresponding statistic to detect community structure are displayed. A star marker in digits denotes that the result supports the existence of community structure. These statistics and critical values are given as follows. For our method, the maximum magnitude of eigenvalues λ is used. The critical value is given by the Tracy-Widom distribution in Eq (7). For Newman and Louvain methods, modularity Q is used with the critical value 0.45, 0.48, and 0.29 for IndianVillage, Yeast and FreeAssoc, respectively, based on the analytical approximation of modularity for an Erdős-Rényi random graph. For Split method, a positive largest eigenvalue of modularity matrix λ′ suggests community structure while a negative largest eigenvalue λ′ non-community structure. For Bayesian method, the difference of marginal log-likelihood for K = 1 and K = 2 (‘Dif’; subtraction of K = 1 case from K = 2 case) is used. A positive difference suggests community structure while a negative difference non-community structure. For Bootstrap method, we evaluate stability of community structure by means of Adjusted Rand Index (ARI) between the targeted graph and bootstrapped graphs (the number of replicates is set to 100). We compare the median of ARI (mARI) with the distribution of ARI when the targeted graph is randomized. If mARI falls within the 95% confidence interval, it suggests that there is no community structure. Seemingly, this method is not computationally efficient. We were not able to obtain the results for FreeAssoc within 72 hours.

https://doi.org/10.1371/journal.pone.0194079.t001