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Functional units in complex networks: beyond cohesive modules
13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China
Functional units in complex networks: beyond cohesive modules Qing-Ju Jiao*. Xiao-Fan Wang*. Hong-Bin Shen*
*Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240 China (Tel: +86-21-34205320; e-mail: hbshen@ sjtu.edu.cn) Abstract: A large amount of networks are found to divide naturally into modules or communities within which the network connections are tight but between which they are sparser. However, it is still not clear that whether sparse modules in which the nodes are sparsely connected internally and densely connected with other sparse or cohesive modules possibly exist and have functional meaning in complex networks. In this study, we first test BinTree Seeking method that detects both cohesive and sparse modules in a network on 4 networks with known modular structure. And then, we further study modular structure of 23 complex networks of different types and scales, and promising results indeed reveal that the sparse modules composed by the cohesively isolated nodes widely co-exist with the cohesive modules. Furthermore, the two types of modules are found often spatially closely related with each other. Compared with the cohesive modules, sizes of sparse ones are generally smaller. Keywords: Complex network, module detection, cohesive, sparse, BTS.
modules in the same complex network regardless of its type. In order to answer this question, we first test BinTree Seeking (BTS) method proposed by authors on four networks with known modular structure. What’s more, we further apply BTS method to the collected 23 different types including social, computer software, technological and biological networks. The scales of these 23 networks vary from 105 to 7343 nodes and 168 to 16380 edges. As a result, we detect sparse modules in all 23 networks. Although it seems from the results that it is a general rule for a network has both cohesive and sparse modules, we find the preferences of sparse modules vary on different types of networks. We also illustrate some spatial organizations of these sparse modules that are found in the real networks, which show that sparse and cohesive modules sometimes are spatially correlated.
1. INTRODUCTION Networks that can describe diverse complex systems are successful tools to understand unknown domains in nature (Newman. 2006). Modular structure detection has received a considerable amount of attention in various fields because of its remarkable feature that the nodes in the same module have similar attributes. A cohesive module indicates its intravertices are densely connected, which at the same time are sparsely connected with the vertices in other modules (Newman et al. 2004a; Fortunato. 2010). Dominant efforts were devoted to mining cohesive modules of networks in different fields, including social networks (Girvan et al. 2002), technical networks of world wide web ( Newman et al. 2004a; Flake et al. 2002), biological networks of proteinprotein interaction network (Barabasi et al. 2004), metabolic network (Ravasz et al. 2002), and neural network (Park et al. 2004). In general, modules detection mentioned above are dominantly implemented based on the modular concept that the connections of nodes within the same group are denser than connections with the rest of the network. But, are sparse modules in which the nodes are sparsely connected internally and densely connected with other sparse or cohesive modules possible in complex networks? Fortunately, this issue has attracted attentions, and some studies have been reported in biological networks (Pinkert et al. 2010; Jiao et al. 2011). In these studies, both cohesive and sparse modules were proved to be functionally related. These finding reveal that we need to extend our traditional concept about the functional unit in a complex network by paying more attention to the sparse modules.
2. METHOD 2.1 BinTree Seeking method Unlike previous approaches that mainly extract functional units by identifying cohesive modules, we have recently developed an algorithm called BinTree Seeking (BTS) method that can find unorthodox structures of sparse modules in protein-protein interaction networks (Jiao et al. 2011). The members in a sparse module are defined as sparsely connected internally and densely connected with other sparse or cohesive modules at the same time. In order to distinguish and search cohesive and sparse modules, Edge Density of Module (EDM), Link Density (LD), Edge Density of Bridge Matrix (EDBM), and Edge Density for a Network (EDN) are defined below:
Based on the results obtained in biological networks, the motivation of this paper is to study whether it is a general principle for the sparse modules co-existing with cohesive 978-3-902823-39-7/2013 © IFAC
Given a module r, its EDM is defined as: 94
10.3182/20130708-3-CN-2036.00040
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
n
EDM
r
n
¦¦r
ij
lr n u ( n 1) / 2
(1)
i 1 j 1 2
n n
Where lr is the actual number of edges in the module r, n is the total nodes in r. The Link Density (LD) between a node and module r is as follows: n
LD
¦e
ai
E (W , B )
(2)
/n
i 1
Where eai=1 if there is a link between a and the i-th node in r, otherwise eai=0. Edge Density of Bridge Matrix (EDBM) between modules r1 and r2, and the Edge Density for a Network R (EDN) are defined respectively as: n1
EDBM
EDN
n2
1 / 2 ¦ ¦ bm ij
(3)
i 1 j 1
n1 u n 2
LR N u ( N 1) / 2
N
N
i
j
¦¦R
ij
N2 N
(4)
Where bmij
cohesive and sparse modules in protein-protein interaction (PPI) networks. For more details about BTS, the readers are referred to ref. (Jiao et al. 2011), and BTS software is available at: http://www.csbio.sjtu.edu.cn/bioinf/BTS/. 1 N ¦ ( Aij BW iW j )( wij Pij ) M iz j
(6)
To effectively detect cohesive and sparse modules in various networks, three thresholds of (a1, a2, a3) are introduced in BTS method, which play significant roles. a1 is the lower limit of the link density of cohesive module, a2 is the upper limit of link density of sparse module and a3 is the lower limit of edge density of bridge matrix required to confirm the existence of bridge matrix. In (Jiao et al. 2011), authors discuss the choice of three thresholds in detail on PPI networks. In order to apply BTS method to various types of networks and detect modules successfully, except for the classical values of three thresholds (a1=a30.7, a2=a31.5, a3=EDN1, write as (0.7, 1.5, 1 ) in a simple form), we also provide one with other five groups of three thresholds: (0.5, 1.3, 1), (0.5, 1.7, 1), (0.85, 1.3, 1), (0.8, 1.4, 1) and (0.5, 1.5, 1). 2.2 Comparison to other methods
1L ri , j r2 ,if verticesiand jareconnected, ® ¯0L ri , j r2 ,if verticesiand jarenotconnected.
(5)
And n1, n2 are the number of nodes in modules r1 and r2 respectively, LR is the actual number of edges a network R, N is the total number of nodes in network R. Starting from an adjacency matrix of a network, BTS will first randomly select a seed node, and then try to build its cohesive modules by adding the nodes whose link density values are larger than a1, and build its sparse modules by adding the nodes whose link density values are smaller than a2. As a result, we will obtain a Binary Tree (BinTree) with a root and two leaves, i.e., left subtree represents the adjacency matrix that includes a cohesive modules and the other represents the adjacency matrix that includes a sparse module. Second, the bridge matrixes of the cohesive modules are built if its edge density of bridge matrix value larger than a3. Likewise, for the sparse modules obtained by the previous step, its bridge matrixes are also built. Third, remove the nodes whose link density values do not meet the threshold a1 in cohesive module and regulate the sparse module based on its bridge matrixes. Finally, repeat these steps, until all nodes are processed. When BTS finally converges, it will generate a binary tree, where each leaf represents a state of possible divisions composed of both cohesive and sparse modules. Then, we can use a kind of error function to evaluate the qualities of the leaf states so that we can pick up the best formulation, e.g. the leaf corresponding to the lowest error function E (6) (Pinkert et al. 2010). Another merit of BTS that makes it especially suitable for unsupervised module detection is that the number of modules in a network can be automatically determined in the BTS approach. In (Jiao et al. 2011), we have shown that BTS can find both meaningful 95
Although our aim is not to show good performance of BTS method, we also compare BTS method with other three methods using four networks with known modular structure, which are listed in Table 1. The first is a synthetic network that is composed of 72 nodes and 448 edges is constructed, one of which is cohesive module and the other two form sparse modules. The role of generation is similar to the algorithm that generates SB Benchmark proposed by Subelj (Subelj et al. 2011a), but not identical. The synthetic network comprises three communities of 16, 32 and 24 nodes, respectively. The first two are sparse modules with average degree 16 and 8 respectively, while the third corresponds to cohesive module and its average degree is fixed to 16. 12 out of 16 edges in module 1 are linked to module 2 and the other edges are connected to module 3. Likewise, except for those edges that are connected to module 1, the rest of edges in module 2 are placed between module 2 and 3. For the cohesive module, most of edges are connected to intramodule except links that are connected to modules 1 and 2. The other three networks are Davis’s southern woman (Subelj et al. 2011a), Scottish corpor. Interlocks (Subelj et al. 2011a) and Jung networks (Subelj et al. 2011b) respectively. The Davis network that is a well-known bipartite network describes the relationship of social collaborations between women in Natchez Mississippi. The other bipartite network is Scottish that supports corporate interlocks in Scotland between 1904 and 1905. The last network is technological network, where nodes represent software classes and edges correspond to different types of dependencies among them. Different types of modules in these four networks are detected by BTS method. In addition, three methods that deal with the detection of block structures in networks are also employed to apply to these networks. Among them are the mixture model method (NL method) (Newman et al. 2007)
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
Table 2. Descriptions of 23 networks
proposed by Newman et al, and the module detection method (Infomod method) (Rosvall et al. 2007) by optimizing the function of minimum description length principle. The third method is an alternative approach (Pinkert method) (Pinkert et al. 2010) different from prior definitions of what actually constitutes a “module” to mine modules in networks. Detected module results by four methods are first compared with known metadata, and then the compared outcomes are measured by normalized mutual information (NMI) (Danon et al. 2005). The results are presented in Table 1. Note that the parameter of number of modules is set true figure that is in accord with real number of modules for NL and Pinkert method. Table 1. The performance test of four methods Network
Nodes /Edges
BTS
NL
Infomod
Pinkert
NMI Systhesis Davis Scottish Jung
72 /448 32 /89 228 /358 398 /943
0.646
0.423
0.533
0.275
0.666
0.818
0.669
0.665
0.565
0.275
0.122
0.1536
0.588
0.591
0.537
0.451
The results in Table 1 report the performance of various methods on tested networks. From the results we can find that BTS method can get better performance than other two methods on Systhntic, Scottish and Jung networks including cohesive and (or) sparse networks. While in Davis network, Infomod method slightly wins BTS method, and NL method performs significantly better than other methods, which could be attributed to a known number of modules. Using the Pinkert method, we get poor result on the tested networks because the code is implemented by authors. So, the code may be distinct from original code that is not obtained from the paper or author’s website.
Network Name
Nodes
Edges
Ref
Csphd (S) Erdos (S)
1384 492
1703 1417
Nooy et al. 2005 Nooy et al. 2005
Football (S)
115
615
Girvan et al. 2002
Isle_of_Man (S) Jazz (S) Science (S) Collaboration (S) Roget (S) Geom (S)
675 198 1589 5242 1022 7343
2007 2742 2742 14490 5075 11898
Nooy et al. 2005 Duch et al. 2005 Newman. 2001 Leskovec et al. 2007 Kunth. 1993 Nooy et al. 2005
Java (C)
1538
7817
Nooy et al. 2005
A00 (C)
352
384
Nooy et al. 2005
A96 (C) C98 (C) E-mail (T)
1096 112 1133
1677 168 5451
Nooy et al. 2005 Nooy et al. 2005 Guimera et al. 2003
Odlis (T) SmallW (T) Polbook (T)
2909 396 105
16380 994 441
Reitz. 2010 Garfield. 2001 Krebs. 2004
Power (T) Usair (T) Yeast PIN (B)
4941 332 2361
6594 2126 6646
KPI (B)
887
1844
DIP yeast (B)
2147
4275
Watts et al. 1998 Nooy et al. 2005 Bu et al. 2003 Breitkreutz et al. 2010 Salwinski et al. 2004
BIND human (B)
3724
8748
Brown et al. 2005
3.2 Sparse modules co-exist with cohesive ones in complex networks To better understand coexistence of cohesive and sparse modules in various complex networks. We apply BTS approach to 23 networks. It is revealed by Table 3 that sparse modules are not occasional observations in some specific complex networks, e.g. protein-protein interaction networks in (Jiao et al. 2011), and they have been found in all tested 23 networks. These results suggest that modules of both cohesive and sparse are remarkable features in complex networks. The nodes’ similar functions in a cohesive module are reflected by the direct links among them, while functions in sparse modules are exhibited by indirect linking and depending on other modules. In considering of this, the relationship between different nodes should be evaluated by both the two types of modules.
3. RESULTS 3.1 Applying BTS method to 23 networks Total 23 complex networks are studied in this paper, which are summarized in Table 2. These networks have following features: (1) different types of networks, where the 23 networks can be divided into 4 categories of social (S), computer software (C), technological (T) and biological (B) networks, (2) the scales of the tested networks vary significantly from 105 to 7,343 nodes, and from 168 to 16,380 edges, (3) different network topological organizations reflected by the edge densities. In general, a network can be represented as a graph where a node corresponds to an individual or object and edge to a special relationship or interaction. Problems now we are facing is how to divide the whole nodes in a graph into meaningful groups.
In order to verify the rationality of existence of sparse modules, we compare the outputs from BTS and those from the state-of-the-art cohesive-specific module detection approach of Newman-fast algorithm (Newman. 2004b) because we do not know modular structure of these networks. From the literature of ref. (Leskovec et al. 2008), we know that cohesive modules with less than 10 nodes possibly lack 96
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comparing these outputs, we find that there is an intersection of 98 nodes between the small modules from Newman-fast approach and the sparse modules from BTS algorithm, and an intersection of 202 nodes between the large modules from Newman-fast approach and the cohesive modules from BTS algorithm. It is also worth mentioning that although there are a large number of overlapped nodes between the small modules from Newman-fast method and the sparse modules from BTS, their node component organizations are quite different. The reason for the differences is that the small modules are subjective to the cohesive definitions in the Newman-fast method, while in the BTS, sparse modules are totally different from the cohesive definition. These findings indicate the following: (1) the core set of nodes in the cohesive modules can be well identified by both the cohesive-specific Newman-fast algorithm and the BTS method that can mine both dense and sparse modules; (2) although the modules with less than 10 nodes from the Newman-fast approach possibly lack obvious cohesive modular significance (Leskovec et al. 2008), these nodes can be potentially re-organized into sparse modules of important functional units, which should be investigated in a different way from the traditional cohesive-specific approach.
obvious modular significance. However, in order to provide a full network based vision of the modules, both cohesive modules of larger and smaller than 10-node sizes obtained from Newman-fast algorithm are performed statistics, which are defined as large and small modules respectively. We then can compare the relationship between the outputs from BTS and Newman method to reveal the inherent mechanism of the sparse module. Table 3. List of Modules detected by BTS in 23 complex networks Network
Cohesive
Sparse
Total
Csphd
5
13
18
Erdos
8
16
24
Football
15
6
21
Isle_of_Man
1
9
10
Jazz
5
12
17
Science
7
17
24
Collaboration
6
19
25
Roget
16
10
26
Geom
4
16
20
Java
9
19
28
A00
8
6
14
A96
7
17
24
C98
3
7
10
E-mail
7
17
24
Odlis
6
16
22
SmallW
3
2
5
Polbook
7
8
15
Power
5
19
24
Usair
5
9
14
Yeast PIN
5
9
14
KPI
8
21
29
DIP yeast
26
33
59
BIND human
26
39
65
Fig. 1. The distributions of nodes in A00 network mined by BTS method and Newman-fast algorithm As a second example, let’s analyze modular structures of science network, which describes the cooperative relationship among scientists working on network theory and experiment. This network includes 1461 linked nodes and 2742 edges (128 isolated nodes are not considered). As a result, 24 modules and 275 modules are mined by BTS method and Newman-fast algorithm respectively. According to BTS, 1140 nodes are divided into the sparse modules, and 321 nodes belong to the cohesive modules. In the Newman-fast algorithm, total 887 nodes belong to small cohesive modules, and the remaining 574 nodes compose large cohesive modules respectively. By comparing these results, we get a large intersection composed of 836 nodes between the 887 nodes of small modules from Newman-fast algorithm and 1140 nodes in sparse modules from BTS approach. In addition, 270 nodes are overlap between the 574 nodes in large modules from Newman-fast algorithm and 321 nodes in cohesive modules from BTS.
We firstly compared the results on the A00 network that has 352 nodes and 384 edges. After A00 network is analyzed by two methods of BTS and Newman-fast, Fig. 1 shows the nodes distributions in different types of modules. In Newman-fast algorithm, all the detected modules are considered cohesive, where the differences are the module sizes (large v.s. small). While in the BTS’s outputs, we will obtain both cohesive and sparse modules. As shown in Fig. 1, the total number of nodes in the small modules from the Newman-fast approach is 110 and the remaining 242 nodes are located in the large modules. At the same time, we find 138 and 214 nodes in the sparse and cohesive modules respectively from the BTS algorithm. By systematically
All these results above show that besides the cohesive modules that can be discovered by both the Newman-fast algorithm and BTS approach, we cannot ignore the remaining 97
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
nodes because they can form functional units in the type of sparse modules. These sparse modules as revealed by refs. (Pinkert et al. 2010; Jiao et al. 2011) also hold important functions. According to BTS, sparse modules widely exist in the tested 23 different networks. 3.3 Preferences of sparse structures Which type of networks do sparse modules prefer to locate? It is difficult to answer this question accurately and we will study this problem in a simple way. The proportions of nodes in sparse and cohesive modules from BTS are computed (Fig. 2). From Fig. 2, we can see the differences between different types of networks. In general, technological and computer software networks tested in this study have lower proportion of nodes in sparse modules according to BTS. For biological networks, high proportions of nodes in sparse modules are found in Fig. 2, meaning that biological networks more possibly possess sparse structures and could have weak significant cohesive modules, which are consistent with other studies (Pinkert et al. 2010; Jiao et al. 2011). Significant differences are found in the tested 9 social networks, where 4 of them show obvious sparse structures (Csphd, Is, Sc, and Geom), and 3 of them illustrate obvious cohesive features (Fo, Jazz, and Roget). These results reveal that the modular structures in social networks are varying very much according to the different relationships represented.
Fig. 3. The average sizes of sparse and cohesive modules in various networks. 3.5 Possible organizing structures of sparse modules We have shown above that the sparse and cohesive modules can simultaneously exist in the same network, but the spatial organization structures of the sparse modules are still not clear. Here we report two possible organizing structures for sparse modules in complex networks observed from the results on the 23 networks. Fig. 4A.1 illustrates the bridge sparse module that links one or more other cohesive modules, and Fig. 4A.2 shows a real example from the tested science network. It is also worth pointing out that these bridge modules were also found in the biological networks. Another common organization of the sparse modules we observed is that they can interact with each other as shown in Fig. 4B.1, where Fig. 4B.2 illustrates a real example from the Geom network.
Fig. 2. The proportions of nodes in different networks from sparse and cohesive modules detected by BTS method. 3.4 Sizes of sparse modules We further study the sizes of sparse modules in various networks. Fig. 3 compares the average sizes of the mined sparse and cohesive modules in tested 23 networks by BTS. Generally speaking, the sizes of sparse modules are smaller than cohesive modules that exist in the same network, although 3 exceptions are observed (Csphd, Sc, and KPI). This phenomenon further motivates us to go further investigating what kinds of spatial organizations of these sparse modules, which will be discussed in the next section.
Fig. 4. Schematic illustration to show two possible organizations of sparse modules. 4. CONCLUSIONS Complex network is a set of people, proteins, webpages, or other elements with some pattern of contacts or interactions between them. Most of elements in complex networks exhibit modular features which are studied deeply by sociologists, mathematicians, and biologists. Based on the principle that elements in the same cohesive module have similar attributes, 98
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Leskovec, J., Kleinberg, J., and Other, C. (2007). Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD), 1, 2. Leskovec, J., Lang, K.J., and Other, A. (2008). Statistical properties of community structure in large social and information networks. In Proceedings of the 17th International Conference on World Wide Web, 695-704. Newman, M.E.J. (2006). Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103, 8577-8582. Newman, M.E.J. and Girvan M. (2004a). Finding and evaluating community structure in networks. Physical Review E, 69, 026113. Newman, M.E.J. and Leicht E. (2007). Mixture models and exploratory analysis in networks. Proceedings of the National Academy of Sciences, 104, 9564-9569. Newman, M.E.J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 98, 404-409. Newman, M.E.J. (2004b). Fast algorithm for detecting community structure in networks. Physical Review E, 69, 066133. Nooy, D.W., Mrvar, A., and Other, V. (2011). Exploratory social network analysis with Pajek. Cambridge University Press, New York. Park, Y.S., Chon, T.S., and Other, I.S. (2004). Hierarchical community classification and assessment of aquatic ecosystems using artificial neural networks. Science of the Total Environment, 327, 105-122. Pinkert, S., Schultz, J., and Other, J. (2010). Protein interaction networks—more than mere modules. Plos Computational Biology, 6, e1000659. Ravasz, E., Somera, A.L., and Other, D.A. (2002). Hierarchical organization of modularity in metabolic networks. Science, 297. 1551-1555. Reitz J.M. (2010). Online dictionary for library and information science. Westport, CT: Libraries Unlimited. Rosvall, M. and Bergstrom, C.T. (2007). An informationtheoretic framework for resolving community structure in complex networks. Proceedings of the National Academy of Sciences, 104, 7327-7331. Salwinski, L., Miller, C.S., and Other, A.J. (2004) The database of interacting proteins: 2004 update. Nucleic Acids Research, 32, D449-D451. Subelj, L. and Bajec, M. (2011a). Generalized network community detection. In Proceedings of the ECML PKDD Workshop on Finding Patterns of Human Behaviors in Network and Mobility Data, 66-84. Subelj, L. and Bajec, M. (2011b). Community structure of complex software systems: Analysis and applications. Physica A: Statistical Mechanics and its Applications, 390, 2968-2975. Subelj, L. and Bajec, M. (2012). Ubiquitousness of linkdensity and link-pattern communities in real-world networks. The European Physical Journal B, 85, 1-11. Watts, D. and Strogatz, S. (1998). Collective dynamics of small world networks. Nature, 393, 440-442.
dominant efforts have been devoted to mining cohesive modules and significant progress has been achieved. Recent studies reveal that networks comprise even more sophisticated modules than traditional cohesive modules (Newman et al. 2007; Subelj et al. 2012). For example, sparse modules in PPI networks were verified existence and the nodes in them have similar functions in recent literatures (Pinkert et al. 2010; Jiao et al. 2011). Using (or extended) BTS method that can successfully mine both cohesive and sparse clusters in various types of networks, we analyze the meanings of cohesive and sparse in particular modules detected from four types (social, computer software, technological and biological networks) of networks. Furthermore, in order to better show the Ubiquitousness of the coexistence of cohesive and sparse modules in complex networks, the modular structure of 23 different types of networks are also investigated. Our results suggest that sparse modules commonly exist with cohesive modules, indicating both of the two different types of modules should be analyzed simultaneously in order to reveal the functions of the whole network. REFERENCES Barabasi, A.L. and Oltvai, Z.N. (2004). Network biology: Understanding the cell's functional organization. Nature Reviews Genetics, 5, 101-U115. Breitkreutz, A., Choi, N., and Other, J.G. (2010). A global protein kinase and phosphatase interaction network in yeast. Science Signalling, 328, 1043. Brown, K.R. and Jurisica, I. (2005). Online predicted human interaction database. Bioinformatics, 21, 2076-2082. Bu, D., Zhao, Y., and Other, L. (2003). Topological structure analysis of the protein–protein interaction network in budding yeast. Nucleic Acids Research, 31, 2443-2450. Danon, L., Díaz-Guilera, A., and Other, J. (2005). Comparing community structure identification. Journal of Statistical Mechanics: Theory and Experiment, 2005, P09008. Duch, J. and Arenas, A. (2005). Community detection in complex networks using extremal optimization. Physical Review E, 72, 027104. Fortunato, S. (2010) Community detection in graphs. Physics Reports, 486, 75-174. Garfield, E. (2001). From computational linguistics to algorithmic historiography. paper presented at the Symposium in Honor of Casimir Borkowski at the University of Pittsburgh School of Information Sciences. Girvan, M. and Newman, M.E.J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99, 7821-7826. Guimera, R., Danon, L., and Other A. (2003). Self-similar community structure in a network of human interactions. Physical Review E, 68, 065103. Jiao, Q. J., Zhang, Y.K., and Other, L.N. (2011). Bintree seeking: a novel approach to mine both bi-sparse and cohesive modules in protein interaction networks. Plos One, 6, e27646. Krebs, V. (2004). Web site: http://www.orgnet.com/. Knuth, D.E. (1993). The Stanford GraphBase: a platform for combinatorial computing. AcM Press.
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Functional units in complex networks: beyond cohesive modules Qing-Ju Jiao*. Xiao-Fan Wang*. Hong-Bin Shen*
*Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240 China (Tel: +86-21-34205320; e-mail: hbshen@ sjtu.edu.cn) Abstract: A large amount of networks are found to divide naturally into modules or communities within which the network connections are tight but between which they are sparser. However, it is still not clear that whether sparse modules in which the nodes are sparsely connected internally and densely connected with other sparse or cohesive modules possibly exist and have functional meaning in complex networks. In this study, we first test BinTree Seeking method that detects both cohesive and sparse modules in a network on 4 networks with known modular structure. And then, we further study modular structure of 23 complex networks of different types and scales, and promising results indeed reveal that the sparse modules composed by the cohesively isolated nodes widely co-exist with the cohesive modules. Furthermore, the two types of modules are found often spatially closely related with each other. Compared with the cohesive modules, sizes of sparse ones are generally smaller. Keywords: Complex network, module detection, cohesive, sparse, BTS.
modules in the same complex network regardless of its type. In order to answer this question, we first test BinTree Seeking (BTS) method proposed by authors on four networks with known modular structure. What’s more, we further apply BTS method to the collected 23 different types including social, computer software, technological and biological networks. The scales of these 23 networks vary from 105 to 7343 nodes and 168 to 16380 edges. As a result, we detect sparse modules in all 23 networks. Although it seems from the results that it is a general rule for a network has both cohesive and sparse modules, we find the preferences of sparse modules vary on different types of networks. We also illustrate some spatial organizations of these sparse modules that are found in the real networks, which show that sparse and cohesive modules sometimes are spatially correlated.
1. INTRODUCTION Networks that can describe diverse complex systems are successful tools to understand unknown domains in nature (Newman. 2006). Modular structure detection has received a considerable amount of attention in various fields because of its remarkable feature that the nodes in the same module have similar attributes. A cohesive module indicates its intravertices are densely connected, which at the same time are sparsely connected with the vertices in other modules (Newman et al. 2004a; Fortunato. 2010). Dominant efforts were devoted to mining cohesive modules of networks in different fields, including social networks (Girvan et al. 2002), technical networks of world wide web ( Newman et al. 2004a; Flake et al. 2002), biological networks of proteinprotein interaction network (Barabasi et al. 2004), metabolic network (Ravasz et al. 2002), and neural network (Park et al. 2004). In general, modules detection mentioned above are dominantly implemented based on the modular concept that the connections of nodes within the same group are denser than connections with the rest of the network. But, are sparse modules in which the nodes are sparsely connected internally and densely connected with other sparse or cohesive modules possible in complex networks? Fortunately, this issue has attracted attentions, and some studies have been reported in biological networks (Pinkert et al. 2010; Jiao et al. 2011). In these studies, both cohesive and sparse modules were proved to be functionally related. These finding reveal that we need to extend our traditional concept about the functional unit in a complex network by paying more attention to the sparse modules.
2. METHOD 2.1 BinTree Seeking method Unlike previous approaches that mainly extract functional units by identifying cohesive modules, we have recently developed an algorithm called BinTree Seeking (BTS) method that can find unorthodox structures of sparse modules in protein-protein interaction networks (Jiao et al. 2011). The members in a sparse module are defined as sparsely connected internally and densely connected with other sparse or cohesive modules at the same time. In order to distinguish and search cohesive and sparse modules, Edge Density of Module (EDM), Link Density (LD), Edge Density of Bridge Matrix (EDBM), and Edge Density for a Network (EDN) are defined below:
Based on the results obtained in biological networks, the motivation of this paper is to study whether it is a general principle for the sparse modules co-existing with cohesive 978-3-902823-39-7/2013 © IFAC
Given a module r, its EDM is defined as: 94
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IFAC LSS 2013 July 7-10, 2013. Shanghai, China
n
EDM
r
n
¦¦r
ij
lr n u ( n 1) / 2
(1)
i 1 j 1 2
n n
Where lr is the actual number of edges in the module r, n is the total nodes in r. The Link Density (LD) between a node and module r is as follows: n
LD
¦e
ai
E (W , B )
(2)
/n
i 1
Where eai=1 if there is a link between a and the i-th node in r, otherwise eai=0. Edge Density of Bridge Matrix (EDBM) between modules r1 and r2, and the Edge Density for a Network R (EDN) are defined respectively as: n1
EDBM
EDN
n2
1 / 2 ¦ ¦ bm ij
(3)
i 1 j 1
n1 u n 2
LR N u ( N 1) / 2
N
N
i
j
¦¦R
ij
N2 N
(4)
Where bmij
cohesive and sparse modules in protein-protein interaction (PPI) networks. For more details about BTS, the readers are referred to ref. (Jiao et al. 2011), and BTS software is available at: http://www.csbio.sjtu.edu.cn/bioinf/BTS/. 1 N ¦ ( Aij BW iW j )( wij Pij ) M iz j
(6)
To effectively detect cohesive and sparse modules in various networks, three thresholds of (a1, a2, a3) are introduced in BTS method, which play significant roles. a1 is the lower limit of the link density of cohesive module, a2 is the upper limit of link density of sparse module and a3 is the lower limit of edge density of bridge matrix required to confirm the existence of bridge matrix. In (Jiao et al. 2011), authors discuss the choice of three thresholds in detail on PPI networks. In order to apply BTS method to various types of networks and detect modules successfully, except for the classical values of three thresholds (a1=a30.7, a2=a31.5, a3=EDN1, write as (0.7, 1.5, 1 ) in a simple form), we also provide one with other five groups of three thresholds: (0.5, 1.3, 1), (0.5, 1.7, 1), (0.85, 1.3, 1), (0.8, 1.4, 1) and (0.5, 1.5, 1). 2.2 Comparison to other methods
1L ri , j r2 ,if verticesiand jareconnected, ® ¯0L ri , j r2 ,if verticesiand jarenotconnected.
(5)
And n1, n2 are the number of nodes in modules r1 and r2 respectively, LR is the actual number of edges a network R, N is the total number of nodes in network R. Starting from an adjacency matrix of a network, BTS will first randomly select a seed node, and then try to build its cohesive modules by adding the nodes whose link density values are larger than a1, and build its sparse modules by adding the nodes whose link density values are smaller than a2. As a result, we will obtain a Binary Tree (BinTree) with a root and two leaves, i.e., left subtree represents the adjacency matrix that includes a cohesive modules and the other represents the adjacency matrix that includes a sparse module. Second, the bridge matrixes of the cohesive modules are built if its edge density of bridge matrix value larger than a3. Likewise, for the sparse modules obtained by the previous step, its bridge matrixes are also built. Third, remove the nodes whose link density values do not meet the threshold a1 in cohesive module and regulate the sparse module based on its bridge matrixes. Finally, repeat these steps, until all nodes are processed. When BTS finally converges, it will generate a binary tree, where each leaf represents a state of possible divisions composed of both cohesive and sparse modules. Then, we can use a kind of error function to evaluate the qualities of the leaf states so that we can pick up the best formulation, e.g. the leaf corresponding to the lowest error function E (6) (Pinkert et al. 2010). Another merit of BTS that makes it especially suitable for unsupervised module detection is that the number of modules in a network can be automatically determined in the BTS approach. In (Jiao et al. 2011), we have shown that BTS can find both meaningful 95
Although our aim is not to show good performance of BTS method, we also compare BTS method with other three methods using four networks with known modular structure, which are listed in Table 1. The first is a synthetic network that is composed of 72 nodes and 448 edges is constructed, one of which is cohesive module and the other two form sparse modules. The role of generation is similar to the algorithm that generates SB Benchmark proposed by Subelj (Subelj et al. 2011a), but not identical. The synthetic network comprises three communities of 16, 32 and 24 nodes, respectively. The first two are sparse modules with average degree 16 and 8 respectively, while the third corresponds to cohesive module and its average degree is fixed to 16. 12 out of 16 edges in module 1 are linked to module 2 and the other edges are connected to module 3. Likewise, except for those edges that are connected to module 1, the rest of edges in module 2 are placed between module 2 and 3. For the cohesive module, most of edges are connected to intramodule except links that are connected to modules 1 and 2. The other three networks are Davis’s southern woman (Subelj et al. 2011a), Scottish corpor. Interlocks (Subelj et al. 2011a) and Jung networks (Subelj et al. 2011b) respectively. The Davis network that is a well-known bipartite network describes the relationship of social collaborations between women in Natchez Mississippi. The other bipartite network is Scottish that supports corporate interlocks in Scotland between 1904 and 1905. The last network is technological network, where nodes represent software classes and edges correspond to different types of dependencies among them. Different types of modules in these four networks are detected by BTS method. In addition, three methods that deal with the detection of block structures in networks are also employed to apply to these networks. Among them are the mixture model method (NL method) (Newman et al. 2007)
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
Table 2. Descriptions of 23 networks
proposed by Newman et al, and the module detection method (Infomod method) (Rosvall et al. 2007) by optimizing the function of minimum description length principle. The third method is an alternative approach (Pinkert method) (Pinkert et al. 2010) different from prior definitions of what actually constitutes a “module” to mine modules in networks. Detected module results by four methods are first compared with known metadata, and then the compared outcomes are measured by normalized mutual information (NMI) (Danon et al. 2005). The results are presented in Table 1. Note that the parameter of number of modules is set true figure that is in accord with real number of modules for NL and Pinkert method. Table 1. The performance test of four methods Network
Nodes /Edges
BTS
NL
Infomod
Pinkert
NMI Systhesis Davis Scottish Jung
72 /448 32 /89 228 /358 398 /943
0.646
0.423
0.533
0.275
0.666
0.818
0.669
0.665
0.565
0.275
0.122
0.1536
0.588
0.591
0.537
0.451
The results in Table 1 report the performance of various methods on tested networks. From the results we can find that BTS method can get better performance than other two methods on Systhntic, Scottish and Jung networks including cohesive and (or) sparse networks. While in Davis network, Infomod method slightly wins BTS method, and NL method performs significantly better than other methods, which could be attributed to a known number of modules. Using the Pinkert method, we get poor result on the tested networks because the code is implemented by authors. So, the code may be distinct from original code that is not obtained from the paper or author’s website.
Network Name
Nodes
Edges
Ref
Csphd (S) Erdos (S)
1384 492
1703 1417
Nooy et al. 2005 Nooy et al. 2005
Football (S)
115
615
Girvan et al. 2002
Isle_of_Man (S) Jazz (S) Science (S) Collaboration (S) Roget (S) Geom (S)
675 198 1589 5242 1022 7343
2007 2742 2742 14490 5075 11898
Nooy et al. 2005 Duch et al. 2005 Newman. 2001 Leskovec et al. 2007 Kunth. 1993 Nooy et al. 2005
Java (C)
1538
7817
Nooy et al. 2005
A00 (C)
352
384
Nooy et al. 2005
A96 (C) C98 (C) E-mail (T)
1096 112 1133
1677 168 5451
Nooy et al. 2005 Nooy et al. 2005 Guimera et al. 2003
Odlis (T) SmallW (T) Polbook (T)
2909 396 105
16380 994 441
Reitz. 2010 Garfield. 2001 Krebs. 2004
Power (T) Usair (T) Yeast PIN (B)
4941 332 2361
6594 2126 6646
KPI (B)
887
1844
DIP yeast (B)
2147
4275
Watts et al. 1998 Nooy et al. 2005 Bu et al. 2003 Breitkreutz et al. 2010 Salwinski et al. 2004
BIND human (B)
3724
8748
Brown et al. 2005
3.2 Sparse modules co-exist with cohesive ones in complex networks To better understand coexistence of cohesive and sparse modules in various complex networks. We apply BTS approach to 23 networks. It is revealed by Table 3 that sparse modules are not occasional observations in some specific complex networks, e.g. protein-protein interaction networks in (Jiao et al. 2011), and they have been found in all tested 23 networks. These results suggest that modules of both cohesive and sparse are remarkable features in complex networks. The nodes’ similar functions in a cohesive module are reflected by the direct links among them, while functions in sparse modules are exhibited by indirect linking and depending on other modules. In considering of this, the relationship between different nodes should be evaluated by both the two types of modules.
3. RESULTS 3.1 Applying BTS method to 23 networks Total 23 complex networks are studied in this paper, which are summarized in Table 2. These networks have following features: (1) different types of networks, where the 23 networks can be divided into 4 categories of social (S), computer software (C), technological (T) and biological (B) networks, (2) the scales of the tested networks vary significantly from 105 to 7,343 nodes, and from 168 to 16,380 edges, (3) different network topological organizations reflected by the edge densities. In general, a network can be represented as a graph where a node corresponds to an individual or object and edge to a special relationship or interaction. Problems now we are facing is how to divide the whole nodes in a graph into meaningful groups.
In order to verify the rationality of existence of sparse modules, we compare the outputs from BTS and those from the state-of-the-art cohesive-specific module detection approach of Newman-fast algorithm (Newman. 2004b) because we do not know modular structure of these networks. From the literature of ref. (Leskovec et al. 2008), we know that cohesive modules with less than 10 nodes possibly lack 96
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
comparing these outputs, we find that there is an intersection of 98 nodes between the small modules from Newman-fast approach and the sparse modules from BTS algorithm, and an intersection of 202 nodes between the large modules from Newman-fast approach and the cohesive modules from BTS algorithm. It is also worth mentioning that although there are a large number of overlapped nodes between the small modules from Newman-fast method and the sparse modules from BTS, their node component organizations are quite different. The reason for the differences is that the small modules are subjective to the cohesive definitions in the Newman-fast method, while in the BTS, sparse modules are totally different from the cohesive definition. These findings indicate the following: (1) the core set of nodes in the cohesive modules can be well identified by both the cohesive-specific Newman-fast algorithm and the BTS method that can mine both dense and sparse modules; (2) although the modules with less than 10 nodes from the Newman-fast approach possibly lack obvious cohesive modular significance (Leskovec et al. 2008), these nodes can be potentially re-organized into sparse modules of important functional units, which should be investigated in a different way from the traditional cohesive-specific approach.
obvious modular significance. However, in order to provide a full network based vision of the modules, both cohesive modules of larger and smaller than 10-node sizes obtained from Newman-fast algorithm are performed statistics, which are defined as large and small modules respectively. We then can compare the relationship between the outputs from BTS and Newman method to reveal the inherent mechanism of the sparse module. Table 3. List of Modules detected by BTS in 23 complex networks Network
Cohesive
Sparse
Total
Csphd
5
13
18
Erdos
8
16
24
Football
15
6
21
Isle_of_Man
1
9
10
Jazz
5
12
17
Science
7
17
24
Collaboration
6
19
25
Roget
16
10
26
Geom
4
16
20
Java
9
19
28
A00
8
6
14
A96
7
17
24
C98
3
7
10
7
17
24
Odlis
6
16
22
SmallW
3
2
5
Polbook
7
8
15
Power
5
19
24
Usair
5
9
14
Yeast PIN
5
9
14
KPI
8
21
29
DIP yeast
26
33
59
BIND human
26
39
65
Fig. 1. The distributions of nodes in A00 network mined by BTS method and Newman-fast algorithm As a second example, let’s analyze modular structures of science network, which describes the cooperative relationship among scientists working on network theory and experiment. This network includes 1461 linked nodes and 2742 edges (128 isolated nodes are not considered). As a result, 24 modules and 275 modules are mined by BTS method and Newman-fast algorithm respectively. According to BTS, 1140 nodes are divided into the sparse modules, and 321 nodes belong to the cohesive modules. In the Newman-fast algorithm, total 887 nodes belong to small cohesive modules, and the remaining 574 nodes compose large cohesive modules respectively. By comparing these results, we get a large intersection composed of 836 nodes between the 887 nodes of small modules from Newman-fast algorithm and 1140 nodes in sparse modules from BTS approach. In addition, 270 nodes are overlap between the 574 nodes in large modules from Newman-fast algorithm and 321 nodes in cohesive modules from BTS.
We firstly compared the results on the A00 network that has 352 nodes and 384 edges. After A00 network is analyzed by two methods of BTS and Newman-fast, Fig. 1 shows the nodes distributions in different types of modules. In Newman-fast algorithm, all the detected modules are considered cohesive, where the differences are the module sizes (large v.s. small). While in the BTS’s outputs, we will obtain both cohesive and sparse modules. As shown in Fig. 1, the total number of nodes in the small modules from the Newman-fast approach is 110 and the remaining 242 nodes are located in the large modules. At the same time, we find 138 and 214 nodes in the sparse and cohesive modules respectively from the BTS algorithm. By systematically
All these results above show that besides the cohesive modules that can be discovered by both the Newman-fast algorithm and BTS approach, we cannot ignore the remaining 97
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
nodes because they can form functional units in the type of sparse modules. These sparse modules as revealed by refs. (Pinkert et al. 2010; Jiao et al. 2011) also hold important functions. According to BTS, sparse modules widely exist in the tested 23 different networks. 3.3 Preferences of sparse structures Which type of networks do sparse modules prefer to locate? It is difficult to answer this question accurately and we will study this problem in a simple way. The proportions of nodes in sparse and cohesive modules from BTS are computed (Fig. 2). From Fig. 2, we can see the differences between different types of networks. In general, technological and computer software networks tested in this study have lower proportion of nodes in sparse modules according to BTS. For biological networks, high proportions of nodes in sparse modules are found in Fig. 2, meaning that biological networks more possibly possess sparse structures and could have weak significant cohesive modules, which are consistent with other studies (Pinkert et al. 2010; Jiao et al. 2011). Significant differences are found in the tested 9 social networks, where 4 of them show obvious sparse structures (Csphd, Is, Sc, and Geom), and 3 of them illustrate obvious cohesive features (Fo, Jazz, and Roget). These results reveal that the modular structures in social networks are varying very much according to the different relationships represented.
Fig. 3. The average sizes of sparse and cohesive modules in various networks. 3.5 Possible organizing structures of sparse modules We have shown above that the sparse and cohesive modules can simultaneously exist in the same network, but the spatial organization structures of the sparse modules are still not clear. Here we report two possible organizing structures for sparse modules in complex networks observed from the results on the 23 networks. Fig. 4A.1 illustrates the bridge sparse module that links one or more other cohesive modules, and Fig. 4A.2 shows a real example from the tested science network. It is also worth pointing out that these bridge modules were also found in the biological networks. Another common organization of the sparse modules we observed is that they can interact with each other as shown in Fig. 4B.1, where Fig. 4B.2 illustrates a real example from the Geom network.
Fig. 2. The proportions of nodes in different networks from sparse and cohesive modules detected by BTS method. 3.4 Sizes of sparse modules We further study the sizes of sparse modules in various networks. Fig. 3 compares the average sizes of the mined sparse and cohesive modules in tested 23 networks by BTS. Generally speaking, the sizes of sparse modules are smaller than cohesive modules that exist in the same network, although 3 exceptions are observed (Csphd, Sc, and KPI). This phenomenon further motivates us to go further investigating what kinds of spatial organizations of these sparse modules, which will be discussed in the next section.
Fig. 4. Schematic illustration to show two possible organizations of sparse modules. 4. CONCLUSIONS Complex network is a set of people, proteins, webpages, or other elements with some pattern of contacts or interactions between them. Most of elements in complex networks exhibit modular features which are studied deeply by sociologists, mathematicians, and biologists. Based on the principle that elements in the same cohesive module have similar attributes, 98
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dominant efforts have been devoted to mining cohesive modules and significant progress has been achieved. Recent studies reveal that networks comprise even more sophisticated modules than traditional cohesive modules (Newman et al. 2007; Subelj et al. 2012). For example, sparse modules in PPI networks were verified existence and the nodes in them have similar functions in recent literatures (Pinkert et al. 2010; Jiao et al. 2011). Using (or extended) BTS method that can successfully mine both cohesive and sparse clusters in various types of networks, we analyze the meanings of cohesive and sparse in particular modules detected from four types (social, computer software, technological and biological networks) of networks. Furthermore, in order to better show the Ubiquitousness of the coexistence of cohesive and sparse modules in complex networks, the modular structure of 23 different types of networks are also investigated. Our results suggest that sparse modules commonly exist with cohesive modules, indicating both of the two different types of modules should be analyzed simultaneously in order to reveal the functions of the whole network. REFERENCES Barabasi, A.L. and Oltvai, Z.N. (2004). Network biology: Understanding the cell's functional organization. Nature Reviews Genetics, 5, 101-U115. Breitkreutz, A., Choi, N., and Other, J.G. (2010). A global protein kinase and phosphatase interaction network in yeast. Science Signalling, 328, 1043. Brown, K.R. and Jurisica, I. (2005). Online predicted human interaction database. Bioinformatics, 21, 2076-2082. Bu, D., Zhao, Y., and Other, L. (2003). Topological structure analysis of the protein–protein interaction network in budding yeast. Nucleic Acids Research, 31, 2443-2450. Danon, L., Díaz-Guilera, A., and Other, J. (2005). Comparing community structure identification. Journal of Statistical Mechanics: Theory and Experiment, 2005, P09008. Duch, J. and Arenas, A. (2005). Community detection in complex networks using extremal optimization. Physical Review E, 72, 027104. Fortunato, S. (2010) Community detection in graphs. Physics Reports, 486, 75-174. Garfield, E. (2001). From computational linguistics to algorithmic historiography. paper presented at the Symposium in Honor of Casimir Borkowski at the University of Pittsburgh School of Information Sciences. Girvan, M. and Newman, M.E.J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99, 7821-7826. Guimera, R., Danon, L., and Other A. (2003). Self-similar community structure in a network of human interactions. Physical Review E, 68, 065103. Jiao, Q. J., Zhang, Y.K., and Other, L.N. (2011). Bintree seeking: a novel approach to mine both bi-sparse and cohesive modules in protein interaction networks. Plos One, 6, e27646. Krebs, V. (2004). Web site: http://www.orgnet.com/. Knuth, D.E. (1993). The Stanford GraphBase: a platform for combinatorial computing. AcM Press.
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