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Module (or community) structure detection, which has been successfully applied to many fields, is vital step for understand network dynamic and complex systems. Module structure not only is a crucial character of networks, but also is multiscale. Therefore, many multiscale module detection algorithms are proposed to resolve the problem. But a highly important issue for multiscale methods is that of how to select crucial partitions among multiscale network partitions so that these partitions can effectively help people to understand complex system. To solve the problem, we propose a novel partitionbased hierarchical clustering to select significant network partitions. Experiments on selection of benchmark and real networks demonstrate that the new method for selecting significant partitions is very effectively.
complex network, module structure, multiscale module detection, significant partition
Many complex systems, such as biology, mathematics, sociology and physics can be abstracted as networks that only include nodes and edges. The merit is simple, but can provide significant convenience to perceive and explain mysteries of complex systems. The assembling property of network, i.e. community, has triggered a big activity to this field [13], which is considered to be capable of revealing the network structure. Many methods have been put forward to detect network communities, such as modularity [4] and its variations [57], and hierarchical clustering methods based on node [8] or edge similarity [9].
Although much progress has been obtained, many issues challenges existing community detection algorithms greatly in traditional singlescale approaches. Singlescale community detection algorithm may encounter the socalled resolution limit problem [10] (or termed underpartition) when applied on the real complex networks. For instance, the modularity orientated optimization is a branch of the singlescale community mining methods, which have been widely investigated. However, even the most recently reported infomap method [11], which is powerful for detecting nonoverlapping communities, also suffers from this problem [12]. Other localbased singlescale community detection methods have overpartition problem since they have a tendency to group together those nodes with the strongest connections but leave out those with weaker connections, so that the divisions they generate may consist of smaller (even including several nodes) and dense cores [13]. One of the potential reasons for the underpartition and overpartition phenomenon in the singlescale community detection approaches is that the network itself has multiscale community structures. Also, measuring the node similarity solely according to the links in the original observed network in existing methods will potentially result in the inaccuracy of the solutions.
Another difficulty for existing community mining algorithms is that networks often have several levels organization [1415], leading to different relevant communities at various scales. For instance, in the PPI networks, the protein complexes are often organized hierarchically [1617]; the atomiclevel networks of protein 3D structures are characterized by multiple scales in time and space [1819]. Singlescale community detection methods are not suitable for the analysis of the multiscale networks in which there is not a single ‘best’ mesoscopic level of description, but rather multiple levels association with multiple scales in the network [20]. Some recent methods [2123] have been proposed to try to resolve the multiple scales of the complex system, but significant efforts are still needed. For instance, the stability method is a multiscale community mining algorithm based on extended modularity idea, its performance will be affected by the inherent weaknesses of modularity definition as some networks. Another multiscale map method is found to produce more groups without meaningful community structure at different scales. Apart from the two issues mentioned above, another most significant issue for people is that how to select crucial partitions among multiscale network partitions, and these partitions can effectively help one to understand complex system.
In this work, we propose a network partitionbased cluster method to select significant partitions based on ISIMB multiscale method [24]. The proposed method first calculates similarity (or distance) between two partitions using normalization variation of information, and a similarity matrix is generated, second, hierarchical clustering with a defined objective function is employed to select significant partitions. Experiments on selection of benchmark and real networks demonstrate that the new method for selecting significant partitions is very effectively. Particularly, we apply the multiscale concept to mine and select protein complexes in PPI networks and reveal their hierarchical organizations, which open a new way to understand and insight into protein complexes.
The ISIMB method has been proposed based on ISIM similarity. ISIM similarity first define transition probability from node $i$ to node $j$ at the $(t+1)th$ time is described below:
${{p}_{ij}}(t+1)=(1\alpha )\frac{1}{SP(i,j)}+\alpha \sum\limits_{k=1}^{\left {{U}_{i}} \right}{\frac{1}{{{d}_{i}}}}{{p}_{kj}}(t)$ (1)
where, $ α∈(0,1)$ parameter is a regulative factor, and $SP(i,j)$ is the shortest path between node $i$ and node $j$, $d_j$ is the degree of node $i$, and $U_j$ denotes the sets of neighbours of node $i$, and $k(k≠i)$ is a member of $U_j$. The equation (1) can be calculated by iterative (equation 2) or convergent (equation 3) equations on whole network, and the ISIM node similarity between node $i$ and node $j$ is defined by equation (4).
$\mathbf{P}(t+1)=(1\alpha )\mathbf{SPV}+\alpha \mathbf{WP}(t)$ (2)
$\mathbf{P}=(1\alpha ){{(\mathbf{I}\alpha \mathbf{W})}^{1}}\mathbf{SPV}$ (3)
${{S}_{ij}}=\frac{{{\mathbf{P}}_{ij}}+{{\mathbf{P}}_{ji}}}{2}$ (4)
For a network, ISIMB method first set some values (such as 60) for parameter $α$, and then, a series of similarity matrix of the network are obtained according ISIM, third, the number of modules are selected via the number of eigenvalues of similarity matrixes that larger than 0.2, at last, we employ hierarchical clustering with similarity matrix and preset number of communities to mine multiscale community structure.
Using ISIMB method, we get a series of partitions of a network at different scales. However, one expects that the related or significant partitions can be identified from multiscale community structures easily. Before identifying the significant scales, we should give a definition of it. A related or significant scale in a series of states is stable and it is more persistent than others when changing the resolution scale of the topology because they are usually more important to understand the system. In this work, a significant partition with the same number of communities and modular structure should span more parameters ($α$) than other scales. Here, we present a new method to detect related scales. The method uses hierarchical clustering to group different partitions revealed by the ISIMB method. The similarity between two partitions are measured by normalization variation of information (NVI) [25] which has all the properties of a proper distance measure.
$\begin{align}& V(X,Y)=H(X)+H(Y)2I(X;Y) \\& \text{ }=H(XY)+H(YX) \\ & \text{ }=\sum\limits_{xy}{P(x,y)\log \frac{P(x,y)}{P(y)}\sum\limits_{xy}{P(x,y)\log \frac{P(x,y)}{P(x)}}} \\\end{align}$ (5)
where, X and Y are two network partitions, H(X) is the entropy of X and H(XY) is the conditional entropy, i.e., the additional information needed to describe X once we know Y. I(X:Y) is defined by equation 6.
$I(X;Y)=\sum\limits_{x=1}^{K}{\sum\limits_{y=1}^{{{K}^{'}}}{P(x,y)\log \frac{P(x,y)}{P(x)P(y)}}}$ (6)
where, K and K' are the number of network partitions X and Y respectively.
First, the distances among 60 partitions (a partition is similar with a node in traditional way) which are labeled by a series of digits from 1 to 60 in order are computed and a distance matrix is generated. Second, the hierarchical clustering is performed and the number of clusters (nc) are set from 1 to 60. Third, an objective function F is defined based on the average of intracluster distance and average intercluster distance.
\(F=\frac{\frac{1}{nc}\sum\limits_{nc=1}^{60}{(\sum\limits_{X\in {{C}_{nc}}}{\sum\limits_{Y\in {{C}_{nc}},Y\ne X}{NVI(X,Y)}})}}{\frac{2}{nc\times (nc1)}\sum\limits_{{{C}_{p}},{{C}_{q}}}^{{}}{(\frac{1}{{{n}_{p}}\times {{n}_{q}}}\sum\limits_{X\in {{C}_{p}}}{\sum\limits_{Y\in {{C}_{q}}}{NVI(X,Y)}})}}\) (7)
where, X and Y are two network partitions, nc is the number of clusters, $n_p$ and $n_q$ are the number of partitions in clusters $C_p$ and $C_q$ respectively. In contrast to traditional cluster methods, the cluster methods used here is constrained clustering because the partitions in one cluster need to be ordinal. Therefore, for the final clustering result, it not only satisfies the minimum of the objective function F but also makes the following equation 0.
$\sum\limits_{i=1}^{{{C}_{number}}1}{({{L}_{i+1}}{{L}_{i}})}({{C}_{number}}1)$ (8)
where, C_{number} is the number of partitions in a cluster, L_{i} (1, 2, …, 60) is the label of partition i. At last, we select a partition to represent the cluster. For a given cluster, we compute the average NVI of each partition among other partitions, and pick the partition that has the minimum average of the NVI to represent the given cluster.
The performance of ISIMB method is tested on both synthetic and realworld benchmark networks with other multiscale methods, Stability, Map and RB [26] methods. The stability method detects module by optimizing a function ‘Stability’. The modular structure generated by stability method significantly depends on a time scale (Markov time). Using different time scales, one can obtain various community structures with different scales. Hence the Markov time acts effectively as a resolution regulator that reveals multiscale community structures from fine to coarser as the Makov time grows. In particular, some standard partitioning measures, such as modularity, correspond to ‘Stability’ with different time scales. Likewise, the Map method reveals multiscale community structure also using Markov time. In the Map method, the community detection problem is interpreted as compression of the description length of a random walk. In order to reveal multiscale community structure, Map method introduces dynamic Markov time into the process of map encoding. The RB method finds community structures in networks by minimizing the energy of the spin glass, and reveals modules at different scales using a resolution regulator in the energy. Different from other three methods, the resolution regulator in RB method varied from big to small: as one decreases the resolution regulator, communities in the network are merged, and more big communities are generated. When the resolution regulator is 1, the function energy in the RB method is the standard modularity quality function. In addition, the community results detected by four multiscale methods are compared with real partitions (or real community structure) using Normalized Mutual Information (NMI, see equation 9). Given two partitions $χ=(X_1,X_2,⋯,X_(n_X ))$ and $γ=(Y_1,Y_2,⋯,Y_(n_Y ))$ of a network, with $n_X$ and $n_Y$ communities, respectively:
$NMI(\chi ,\gamma )=\frac{2\sum\limits_{i=1}^{{{n}_{X}}}{\sum\limits_{j=1}^{{{n}_{Y}}}{n_{ij}^{XY}\log (\frac{n_{ij}^{XY}\cdot N}{n_{i}^{X}\cdot n_{j}^{Y}})}}}{\sum\limits_{i=1}^{{{n}_{X}}}{n_{i}^{X}\log (\frac{n_{i}^{X}}{N})+\sum\limits_{j=1}^{{{n}_{Y}}}{n_{j}^{Y}\log (\frac{n_{j}^{Y}}{N})}}}$ (9)
where, N is the number of nodes in a network, $n_i^X$ and $n_j^Y$ are the number of nodes in the communities $X_i$ and $Y_j$ respectively, $n_{ij}^{XY}$ is the number of nodes shared by the communities $X_i$ and $Y_j$: $n_{ij}^{XY}=X_i∩Y_j $. The larger the NMI value, the better the community partition.
Figure 1. H 152 network structure (A) and two hierarchical community structures (B)
The first synthetic network is the H 152 network, which describes a homogeneous in degree network with two predefined hierarchical levels. It contains 256 nodes in total, and 15 edges are placed in the most internal community with 16 nodes. 2 edges are linked among the most internal communities and form the most external community including 64 nodes, and 1 edge is linked with any other node at random in the network. The H 152 network (see Figure 1A) has hierarchical community structures (see Figure 1B): the first level includes 16 communities and each community has 16 nodes, and the second level includes 4 communities and each community has 64 nodes. Form the definition of community (or module) in which group of vertices within which connections are dense, but between which connection are sparse, we can infer that 16way partition is more stable than 4way partition because the nodes in 16way partition are more densely connected than the nodes in 4way partition. Therefore, a better multiscale community detection method should find two phenomenons. The one is that the method should find hierarchical community structures (both 16way and 4way partitions in H 152 network), and the other one is that the multiscale community detection should find more stable network partition, that is the 16way partition should span more parameters (the parameters are resolution regulators in ISIMB and RB methods, and are Markov time in Stability and Map methods. see Figure 2) than 4way partition.
Figure 2 illustrates the multiscale community structures discovered by four multiscale community detection methods of ISIMB, Stability, Map and RB methods. We can see that ISIMB and Map methods detect both 16way and 4way partitions successfully by varying $α$ in ISIMB and the Markov time in Map. Likewise, For the Stability and RB algorithms, they also successfully find the two partitions. These results mean that all four methods show good performance on the first phenomenon mentioned above. For the second phenomenon, the 16way partition found by ISIMB persists longest in four methods which indicates that the performance of ISIMB is best. Furthermore, for the Stability and RB algorithms, although they also successfully find the two partitions, the 4way partition is a more stable partition than the 16way partition in the two method. In fact, Stability and RB methods are extension of Modularity, so they may suffer from the same resolution limit problem.
Figure 2. The multiscale community structures of H 152 network discovered by four multiscale community detection methods
The ISIMB method is then applied to another large synthetic network named LFR benchmark proposed by Lancichinetti et al. In the LFR benchmark [27], some parameters should be decided by a user: (1) the number of nodes N, the average degree $<k>$ and maximum degree $<maxk>$. (2) Minimum for the micro community sizes $<minc>$ and maximum for the micro community sizes $<maxc>$. (4) The number of overlapping nodes (micro communities only) $<on>$, the number of memberships of the overlapping nodes (micro communities only) $<om>$. (5) Minimum for the macro community size $<minc>$, maximum for the macro community size $<maxC>$. (6) Mixing parameter for the macro communities $<mμ 1>$, mixing parameter for the micro communities $<mμ 2>$. The parameters used to generate the synthetic network are: N=1000, $k=maxk=16$, $minc=maxc=10$, $on=om=0$, $minC=maxC=50$, $mμ 1=0.03$ and $mμ 2=0.08$. Then, according to the LFR, the generated synthetic network will have two hierarchical community structures with 40 communities and 20 communities respectively. The first level will have stronger community structure than the second level because its communities have stronger local features set by the parameters.
Figure 3 compares the results from the four multiscale algorithms. We also use the two phenomenons mentioned above to measure the four multiscale algorithms. From this figure, the four algorithms can discover the two hierarchical community structures which are perfectly consistent with the predefined community structures (NMI is equal to 1). But, only ISIMB method correctly reveals the more importance of the 40way partition by a longer time of persisted $α$ compared to the 20way partition. Although the Stability, Map and RB methods also successfully discover the predefined twolayer hierarchical community structures, they regard the 20way partition as more stable. Overall, comparing with other three method, ISIMB method can find stable and significant community structure from networks more effectively.
Figure 3. The multiscale community structures in the LFR synthetic network reveled by four multiscale methods
The third benchmark network is the Karate network with 34 nodes and 78 edges [28]. The network is widely used to test various module detection methods, and is naturally divided into 2 clusters. Figure 4 shows the multiscale modular structure detected by four methods. From this figure, we can see that ISIMB captures two stable phases with 4 and 2 communities respectively. The 2way partition (Figure 5A) agrees exactly with the benchmark (NMI is equal to 1). Figure 5B shows the stable structure of the 4way partition which has large modularity found by the ISIMB on the Karate network. Apart from the large modularity, the 4way partition also has higher partition density than 2way partition. Compared to ISIMB, the Stability method discovers only one stable partition with 2 communities. But the Map does not detect any stable partition, even 2way partition. The RB method discovers both 4way and 2way partition, but the 2way partition is not consistent with the real partition (NMI is equal to 0.8372).
Figure 4. Multiscale modular structures of Karate network detected by four methods
Figure 5. The community structures of 2way (A) and 4way (B) partition detected by ISIMB method
3.2 Selection of significant community structure on benchmark networks
The network partitionbased method is first applied to detect significant community structure on three benchmark networks mentioned above: H 152 network, LFR synthetic network and Karate network, and the results are shown in Figure 6. From the figure we can see that the two significant modular structure with 16 and 4 communities in H 152 network are discovered by our method. Likeness, the significant community structure in LFR synthetic network are also revealed by our method. In the Karate network, the plot of the result is the same as the plot in ISIMB in the figure 4, which means that the significant community structure is also detected by our method.
Figure 6. The significant community structure on three benchmark networks
3.3 Selection of significant modular structure on PPI network
Our method is then applied to detect hierarchical protein complexes in proteinprotein interaction (PPI) network. The network includes 334 proteins and 1,879 interactions, and the 334 proteins are composed by 8 yeast protein root complexes and 70 protein subcomplexes. In PPI networks, protein complexes generally correspond to module (or community) structures. Protein complexes not only exhibit community structure, but also display multiple scales because they are organized hierarchically. First, we employ ISIMB method to reveal multiscale of protein complexes. But, a most significant issue for biologists is that how to select crucial partitions among multiscale network partitions, and these partitions can effectively cover proteins complexes in biological networks. The multiscale of the PPI network is analyzed by ISIMB method and the results are shown in Figure 7.
Figure 7. The multiscale and significant community structure of the PPI network
From the Figure 7, we can see that the original multiscale structure (blue line) does not reveal significant community structure because they do not have more persistent phenomena. After significant community selection, more stable community structure is discovered (red line in Figure 7), such as, 65way partition, 58way partition, and so on. Apart from, the significant partitions (or states) of the PPI network are measured using the real protein complexes, and 4 evaluation indexes [29], $Rec all_c$ (see equation 11), $Pre cision_c$ (see equation 12), $Rec all_p$ (see equation 13) and $Pre cision_p$ (see equation 14) are computed. The $Rec all_c$ and $Pre cision_c$ are defined at complex level and the other two are defined at complexprotein pair level. Let S be a cluster, C be a reference complex, V_{s} be the set of proteins contained in S and V_{c} be the set of proteins contained in C. The match score is defined by equation 10.
$match\_score(S,C)=\frac{\left {{V}_{S}}\bigcap {{V}_{C}} \right}{\left {{V}_{S}}\bigcup V \right}$ (10)
If $match\_score(S,C)≥0.5$, we say S and C match each other. Given a set of reference complexes $C={C_1,C_2,⋯,C_n}$ and a set of predicted complexes $ρ={S_1,S_2,⋯,S_m}$, the 4 evaluation indexes are defined as follows:
$\operatorname{Re}cal{{l}_{c}}=\frac{\left \{{{C}_{i}}{{C}_{i}}\in C\wedge \exists {{S}_{j}}\in \rho ,{{S}_{j}}\text{ }matches\text{ }{{C}_{i}}\} \right}{\left C \right}$ (11)
$\Pr ecisio{{n}_{c}}=\frac{\left \{{{S}_{j}}{{S}_{j}}\in \rho \wedge \exists {{C}_{i}}\in C,{{C}_{i}}\text{ }matches\text{ }{{\text{S}}_{j}}\} \right}{\left \rho \right}$ (12)
$\operatorname{Re}cal{{l}_{p}}=\frac{\sum\nolimits_{i=1}^{n}{\max \{overlap({{C}_{i}},{{S}_{j}})\forall {{S}_{j}}\text{ }that\text{ }mathes\text{ }{{C}_{i}}\}}}{\sum\nolimits_{i=1}^{n}{\left {{C}_{i}} \right}}$ (13)
$\Pr ecisio{{n}_{p}}=\frac{\sum\nolimits_{j=1}^{m}{\max \{overlap({{S}_{j}},{{C}_{i}})\forall {{C}_{i}}\text{ }that\text{ }mathes\text{ }{{S}_{j}}\}}}{\sum\nolimits_{j=1}^{m}{\left {{S}_{j}} \right}}$ (14)
where, $overlap(C_i,S_j)=V_{C_i }∩V_{S_j} $.
In order to verify the performance of our method, four singlescale methods (such as ClusterONE, CFinder, CMC and MCODE) are employed to discover protein complexes. Comparing these methods, the 2 stable states are also the winner in the 4 indexes. For example, the values of indexes of ClusterONE, CFinder [30], CMC and MCODE [31] are 1.6, 1.0899, 0.8868 and 1.3037 respectively (See Table 1). These results indicate that the new method for selecting significant partitions is very effectively because the 2 stable partitions are consistent with real protein complexes, and also that the methods for revealing multiscale community is successful.
Table 1. The performance on 4 evaluation indexes in various of methods
Methods 
No. communities 
Recall_{c} 
Precision_{c} 
Recall_{p} 
Precision_{c} 
Total 
ISIM 
65 
0.5190 
0.5538 
0.2193 
0.5539 
1.8460 
58 
0.4937 
0.5690 
0.2415 
0.5629 
1.8670 

ClusterONE 
52 
0.4231 
0.5577 
0.2108 
0.4048 
1.6000 
CFinder 
504 
0.5128 
0.1667 
0.2246 
0.1858 
1.0899 
CMC 
45 
0.1923 
0.3111 
0.1472 
0.2361 
0.8868 
MCODE 
17 
0.1795 
0.6471 
0.1070 
0.3702 
1.3037 
Many complex systems in nature can be described by networks, and then be resolved easily. One of significant characterization in complex networks is community or modular structure. Comparing with singlescale community structure, detecting multiscale community structure is important for understanding the structure and dynamics of complex network. Selection of crucial partitions among multiscale network partitions can be applied in many fields. For example, multiscale module structure in PPI network can reveal evolution of protein complex or proteins with similar function, and selection of crucial partitions among multiscale community structure in PPI networks can find more stable protein complexes or groups of proteins with similar functions. Furthermore, take social network as an example, multiscale community structure should be able to reveal evolution of people in friendship, collaboration or others, and selection of crucial partitions among multiscale community structure in social network can find more stable friendship or collaboration. Although some multiscale community structure detection methods have been proposed, few algorithms for selecting significant network partition are designed. In the work, we introduce a network partitionbased cluster method for selecting significant community structure which can help one to understand complex system. Comparing with other singlescale and multiscale methods, our method can select significant network partitions effectively.
Although the proposed algorithm is efficient, it has problems for further discussion. For example, the performance of ISIMB is measured on both synthetic networks and realworld networks, but the measured method is qualitative (by two phenomenons), not is quantitative. So, a quantitative method should be proposed to measure different multiscale community structure detection algorithms. The second problems are that the proposed method in this work relies on ISIMB method totally. Therefore, the proposed method may inherit some drawbacks from ISIMB method. Such as, the proposed method cannot select crucial partitions from very sparse networks.
The authors acknowledge the supports from the National Language Committee scientific research projects of China (No. YB13550), the National Natural Science Foundation of China (No. 61806007) and Science and Technology Development Plan Project of Henan (No. 182102310920).
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