Other authors refer to network concepts as network statistics or network indices. Network concepts include connectivity, mean connectivity, density, variance of the connectivity (related to the heterogeneity) etc. Network concepts can be used as descriptive statistics for networks. While some network concepts (e.g. connectivity) have found important uses in biology and genetics, other network concepts (e.g. network centralization) appear less interesting to biologists. Before attempting to understand why some concepts are more interesting than others, it is important to understand how network concepts relate to each other in biologically interesting networks. As a step toward this goal, we explore the meaning of network concepts in module networks, which are defined below.

In the following, we review fundamental network concepts. Further details on the definitions and notations can be found in the Methods section.

The node connectivity is given by

C o n n e c t i v i t y i = k i = ∑ j ≠ i a i j . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGVbWBcqWGUbGBcqWGUbGBcqWGLbqzcqWGJbWycqWG0baDcqWGPbqAcqWG2bGDcqWGPbqAcqWG0baDcqWG5bqEdaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdUgaRnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0ZaaabuaeaacqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAcqGHGjsUcqWGPbqAaeqaniabggHiLdGccqGGUaGlaaa@4F57@

In unweighted networks, the connectivity k_i of node i equals the number of directly linked neighbors. In weighted networks, the connectivity equals the sum of connection weights with all other nodes. Highly connected 'hub' genes are thought to play an important role in organizing the behavior of biological networks 6789. Connectivity has been found to be an important complementary gene screening variable for finding biologically significant genes in cancer 1011 and primate brain development 12.

The line density 13 is defined as the mean off-diagonal adjacency and is closely related to the mean connectivity.

D e n s i t y = ∑ i ∑ j ≠ i a i j n ( n − 1 ) = S 1 ( k ) n ( n − 1 ) = m e a n ( k ) n − 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebarcqWGLbqzcqWGUbGBcqWGZbWCcqWGPbqAcqWG0baDcqWG5bqEcqGH9aqpdaWcaaqaamaaqababaWaaabeaeaacqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAcqGHGjsUcqWGPbqAaeqaniabggHiLdaaleaacqWGPbqAaeqaniabggHiLdaakeaacqWGUbGBcqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkaaGaeyypa0ZaaSaaaeaacqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOGqadiab=TgaRjabcMcaPaqaaiabd6gaUjabcIcaOiabd6gaUjabgkHiTiabigdaXiabcMcaPaaacqGH9aqpdaWcaaqaaiabd2gaTjabdwgaLjabdggaHjabd6gaUjabcIcaOiab=TgaRjabcMcaPaqaaiabd6gaUjabgkHiTiabigdaXaaacqGGSaalaaa@65F0@

where the function S_p(·) is defined for a vector v as S_p(v) = ∑_ivip MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaqhaaWcbaGaemyAaKgabaGaemiCaahaaaaa@3112@ = (v^p)^τ 1.

The normalized connectivity centralization (also known as degree centralization) is a simple and widely used index of the connectivity distribution. By definition 14, the normalized connectivity centralization is given by

C e n t r a l i z a t i o n = n n − 2 ( max ⁡ ( k ) n − 1 − D e n s i t y ) ≈ max ⁡ ( k ) n − D e n s i t y . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@71A6@

A frequent question of social network analysis concerns the causes and consequences of centralization in network structure, i.e. the extent to which certain nodes are far more central than others within the network in question. The centralization index has been used to describe structural differences of metabolic networks 15.

Many measures of network heterogeneity are based on the variance of the connectivity, and authors differ on how to scale the variance 13. Our definition of the network heterogeneity equals the coefficient of variation of the connectivity distribution, i.e.

H e t e r o g e n e i t y = v a r i a n c e ( k ) m e a n ( k ) = n S 2 ( k ) S 1 ( k ) 2 − 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6745@

This heterogeneity measure is scale invariant with respect to multiplying the connectivity by a scalar. Biological networks tend to be very heterogeneous: while some 'hub' nodes are highly connected, the majority of nodes tend to have very few connections. Describing the heterogeneity (inhomogeneity) of the connectivity (degree) distribution has been the focus of considerable research in recent years 6161718.

The clustering coefficient of node i is a density measure of local connections, or 'cliquishness' 1920. Specifically,

C l u s t e r C o e f i = n i π i = ∑ l ≠ i ∑ m ≠ i , l a i l a l m a m i { ( ∑ l ≠ i a i l ) 2 − ∑ l ≠ i a i l 2 } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGSbaBcqWG1bqDcqWGZbWCcqWG0baDcqWGLbqzcqWGYbGCcqWGdbWqcqWGVbWBcqWGLbqzcqWGMbGzdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaemOBa42aaSbaaSqaaiabdMgaPbqabaaakeaaiiGacqWFapaCdaWgaaWcbaGaemyAaKgabeaaaaGccqGH9aqpdaWcaaqaamaaqababaWaaabeaeaacqWGHbqydaWgaaWcbaGaemyAaKMaemiBaWgabeaakiabdggaHnaaBaaaleaacqWGSbaBcqWGTbqBaeqaaOGaemyyae2aaSbaaSqaaiabd2gaTjabdMgaPbqabaaabaGaemyBa0MaeyiyIKRaemyAaKMaeiilaWIaemiBaWgabeqdcqGHris5aaWcbaGaemiBaWMaeyiyIKRaemyAaKgabeqdcqGHris5aaGcbaWaaiWabeaadaqadaqaamaaqababaacbiGae4xyae2aaSbaaSqaaiab+LgaPjab+XgaSbqabaaabaGae4hBaWMaeyiyIKRae4xAaKgabeqdcqGHris5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGHsisldaaeqaqaaiabdggaHnaaDaaaleaacqWGPbqAcqWGSbaBaeaacqaIYaGmaaaabaGaemiBaWMaeyiyIKRaemyAaKgabeqdcqGHris5aaGccaGL7bGaayzFaaaaaiabc6caUaaa@7DE4@

In unweighted networks, n_i equals twice the number of direct connections among the nodes connected to node i, and π_i equals twice the maximum possible number of direct connections among the nodes connected to node i. Consequently, ClusterCoef_i equals 1 if and only if all neighbors of i are also connected to each other. For general weighted networks with 0 ≤ a_ij ≤ 1, one can prove 0 ≤ ClusterCoef_i ≤ 1 21. The relationship between the clustering coefficient and modular structure has been investigated by several authors 20222324.

The topological overlap between nodes i and j reflects their relative interconnectedness 2025. It is defined by

T o p O v e r l a p i j = l i j + a i j min ⁡ { k i , k j } + 1 − a i j , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGubavcqWGVbWBcqWGWbaCcqWGpbWtcqWG2bGDcqWGLbqzcqWGYbGCcqWGSbaBcqWGHbqycqWGWbaCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9maalaaabaGaemiBaW2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGHRaWkcqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaaaOqaaiGbc2gaTjabcMgaPjabc6gaUnaacmqabaGaem4AaS2aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqWGRbWAdaWgaaWcbaGaemOAaOgabeaaaOGaay5Eaiaaw2haaiabgUcaRiabigdaXiabgkHiTiabdggaHnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaaakiabcYcaSaaa@5C7A@

where l_ij = ∑_u≠i,ja_iua_uj. In an unweighted network, l_ij equals the number of nodes to which both i and j are connected. In this case, TopOverlap_ij = 1 if the node with fewer connections satisfies two conditions: (a) all of its neighbors are also neighbors of the other node, and (b) it is connected to the other node. In contrast, TopOverlap_ij = 0 if i and j are un-connected and the two nodes do not share any neighbors. By convention, TopOverlap_ii = 1. One can prove that 0 ≤ a_ij ≤ 1 implies 0 ≤ TopOverlap_ij ≤ 1 21.

Our main interest lies in (sub-)networks comprised of nodes that form a module inside a larger network. Since a particular module network may encode a pathway or a protein complex, these special types of networks have great practical importance. Similar to the term 'cluster', no consensus on the meaning of the term 'module' seems to exist in the literature. In our applications, we use a clustering procedure to identify modules (clusters) of nodes with high topological overlap. We follow the suggestion of 20 to turn the topological overlap matrix TopOverlap into a dissimilarity measure by subtracting it from 1, i.e. dissTopOverlap_ij = 1 - TopOverlap_ij.

We use dissTopOverlap_ij as input of average linkage hierarchical clustering to arrive at a dendrogram (clustering tree) 27. Modules are defined as the branches of the dendrogram. For example, in Figure 1 we show the dendrograms of our network applications. Genes or proteins of proper modules are assigned a color (e.g. turquoise, blue etc). Genes outside any proper module are colored grey. Our module definition depends on how the branches are cut off the dendrogram. Several methods and criteria for identifying branches in a dendrogram have been proposed, see e.g. 202128. In practice, it is advisable to study how robust the results are with respect to alternative module detection methods. In our online R software tutorial, we show that our findings are highly robust with respect to alternative module definitions. In addition, we use a functional enrichment analysis of the resulting modules to provide indirect evidence that the modules are biologically meaningful. Our module detection approach has led to biologically meaningful modules in several applications 9101220282930 but we make no claim that it is optimal. Our theoretical results will apply to all module detection methods that result in approximately factorizable networks.

We define an adjacency matrix A to be exactly factorizable if, and only if, there exists a vector CF with non-negative elements such that

a_ij = CF_iCF_j    for all   i ≠ j

If the non-negative solution of equation (7) is unique, it is referred to as conformity vector CF and CF_i is the conformity of node i. One can easily show that the vector CF is not unique if the network contains only n = 2 nodes. However, for n > 2 it is unique for a weighted network, see our derivations surrounding equation (20).

We also define the concept of conformity for a general, non-factorizable network. The idea is to find an exactly factorizable adjacency matrix A_CF = CF CF^τ - diag(CF²) + I that best approximates A. Note that the diagonal elements of A_CF and A equal 1.

In the appendix, we define the conformity as a maximizer of the factorizability function FA(v)=1−∑i∑j≠i(aij−vivj)2∑i∑j≠i(aij)2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaWgaaWcbaGaemyqaeeabeaakiabcIcaOGqadiab=zha2jabcMcaPiabg2da9iabigdaXiabgkHiTmaalaaabaWaaabeaeaadaaeqaqaaiabcIcaOiabdggaHnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeyOeI0IaemODay3aaSbaaSqaaiabdMgaPbqabaGccqWG2bGDdaWgaaWcbaGaemOAaOgabeaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaaqaaiabdQgaQjabgcMi5kabdMgaPbqab0GaeyyeIuoaaSqaaiabdMgaPbqab0GaeyyeIuoaaOqaamaaqababaWaaabeaeaacqGGOaakcqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaaqaaiabdQgaQjabgcMi5kabdMgaPbqab0GaeyyeIuoaaSqaaiabdMgaPbqab0GaeyyeIuoaaaaaaa@5D67@. Alternative methods of decomposing an adjacency matrix are briefly discussed below.

In equation (43), we define a measure of network factorizability as follows

F ( A ) = 1 − ‖ ( A − I ) − ( A C F − I ) ‖ F 2 ‖ A − I ‖ F 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrcqGGOaakcqWGbbqqcqGGPaqkcqGH9aqpcqaIXaqmcqGHsisldaWcaaqaamaafmaabaGaeiikaGIaemyqaeKaeyOeI0IaemysaKKaeiykaKIaeyOeI0IaeiikaGIaemyqae0aaSbaaSqaaiabdoeadjabdAeagbqabaGccqGHsislcqWGjbqscqGGPaqkaiaawMa7caGLkWoadaqhaaWcbaGaemOrayeabaGaeGOmaidaaaGcbaWaauWaaeaacqWGbbqqcqGHsislcqWGjbqsaiaawMa7caGLkWoadaqhaaWcbaGaemOrayeabaGaeGOmaidaaaaakiabc6caUaaa@4F03@

The factorizability F(A) is normalized to take on values in the unit interval [0, 1]. The higher F(A), the better A_CF - I approximates A - I.

Approximate factorizability is a very strong structural assumption on an adjacency matrix. It certainly does not hold for general networks. However, we provide empirical evidence that many clusters (modules) of genes or proteins in real networks are approximately factorizable. Table 1 reports the mean values of F(A) for the applications considered in this paper. For example in the Drosophila PPI network, the mean factorizability F(A) is 0.82 across 'proper' modules defined as clusters in the network. In contrast, the factorizability of the subnetwork comprised of non-module nodes is only 0.17. In the yeast PPI network, the mean factorizability of proper modules is 0.85 while it equals only 0.20 for the grey module. In the weighted yeast gene co-expression network, the mean factorizability of proper modules equals 0.73 while it is only 0.18 for the improper module. Similarly in the unweighted yeast gene co-expression network, the mean factorizability of proper modules equals 0.62 while it is only 0.11 for the improper module. A more detailed table presenting network concepts in each module is also provided [see Additional file 1].

Our empirical results support the following

Observation 1 For many modules defined with a clustering procedure, the subnetwork comprised of the module nodes is approximately factorizable.

This observation motivates us to study network concepts in approximately factorizable networks.

We refer to the standard network concepts known from the literature as fundamental network concepts. In general, fundamental network concepts are functions of the off-diagonal elements of the adjacency matrix A. More precisely, we use network concept functions to define different types of network concepts depending on the input matrix (see Table 2 and equation (21)). For example, when inputting an adjacency matrix with its diagonal elements replaced by 0, one arrives at fundamental network concepts (see Definition 5 in the Methods section). When inputting the conformity-based (CF-based) adjacency matrix A_CF with its diagonal elements replaced by 0, one arrives at CF-based network concepts (see Definition 6 in the Methods section). The conformity vector can be used to define the approximate CF-based matrix

A_CF,app = CF CF^τ = [CF_iCF_j].

Note that the i-th diagonal element of A_CF,app equals CFi2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@314A@. When A_CF,app is used as input of a network concept function, one arrives at an approximate CF-based concept (see Definition 7 in the Methods section).

We will demonstrate that approximate CF-based concepts satisfy simple relationships. Below, we show that these simple relationships carry over to fundamental network concepts in approximately factorizable networks.

In Definition 7, we provide a formula for calculating approximate CF-based analogs of the fundamental network concepts. Specifically, we find

k C F , a p p , i = C F i S 1 ( C F ) , D e n s i t y C F , a p p = S 1 ( C F ) 2 n ( n − 1 ) ≈ ( S 1 ( C F ) n ) 2 , C e n t r a l i z a t i o n C F , a p p = n S 1 ( C F ) ( n − 1 ) ( n − 2 ) ( max ⁡ ( C F ) − S 1 ( C F ) n ) ≈ S 1 ( C F ) n ( max ⁡ ( C F ) − S 1 ( C F ) n ) , H e t e r o g e n e i t y C F , a p p = n S 2 ( C F ) ( S 1 ( C F ) ) 2 − 1 , C l u s t e r C o e f C F , a p p , i = ( S 2 ( C F ) S 1 ( C F ) ) 2 , T o p O v e r l a p C F , a p p , i j = C F i C F j ( S 2 ( C F ) + 1 ) min ⁡ ( C F i , C F j ) S 1 ( C F ) + 1 − C F i C F j , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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neadjab=zeagjabcMcaPaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83qamKae8NrayKaeiykaKcaaaGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaOGaeiilaWcabaGaemivaqLaem4Ba8MaemiCaaNaem4ta8KaemODayNaemyzauMaemOCaiNaemiBaWMaemyyaeMaemiCaa3aaSbaaSqaaiabdoeadjabdAeagjabcYcaSiabdggaHjabdchaWjabdchaWjabcYcaSiabdMgaPjabdQgaQbqabaaakeaacqGH9aqpaeaadaWcaaqaaiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaem4qamKaemOray0aaSbaaSqaaiabdQgaQbqabaGccqGGOaakcqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOiab=neadjab=zeagjabcMcaPiabgUcaRiabigdaXiabcMcaPaqaaiGbc2gaTjabcMgaPjabc6gaUjabcIcaOiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaem4qamKaemOray0aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=neadjab=zeagjabcMcaPiabgUcaRiabigdaXiabgkHiTiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaem4qamKaemOray0aaSbaaSqaaiabdQgaQbqabaaaaOGaeiilaWcaaaaa@6B92@

where S_p(CF) = ∑_i(CF_i)^p. Note that the approximate CF-based clustering coefficient does not depend on the i-index. This is why we sometimes omit this index and simply write ClusterCoef_CF,app.

Here we demonstrate a major advantage of approximate CF-based network concepts: they exhibit simple relationships. Using the fact that S₁(k_CF,app) = S₁(CF)², and the approximation n/(n - 1) ≈ 1, equations (8) imply the following relationship

H e t e r o g e n e i t y C F , a p p ≈ C l u s t e r C o e f C F , a p p D e n s i t y C F , a p p − 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7131@

or equivalently,

C l u s t e r C o e f C F , a p p , i ≈ ( 1 + H e t e r o g e n e i t y C F , a p p 2 ) 2 × D e n s i t y C F , a p p . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7905@

Further, it is straightforward to derive a simple relationship between approximate CF-based topological overlap, connectivity and heterogeneity under the following mild assumptions: 1S2(CF)≈0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaiabdofatnaaBaaaleaacqaIYaGmaeqaaOGaeiikaGccbmGae83qamKae8NrayKaeiykaKcaaiabgIKi7kabicdaWaaa@367C@ and 1−CFiCFjmin⁡(CFi,CFj)S1(CF)≈0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXiabgkHiTiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaem4qamKaemOray0aaSbaaSqaaiabdQgaQbqabaaakeaacyGGTbqBcqGGPbqAcqGGUbGBcqGGOaakcqWGdbWqcqWGgbGrdaWgaaWcbaGaemyAaKgabeaakiabdoeadjabdAeagnaaBaaaleaacqWGQbGAaeqaaOGaeiykaKIaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakieWacqWFdbWqcqWFgbGrcqGGPaqkaaGaeyisISRaeGimaadaaa@4C13@. Specifically, we find

T o p O v e r l a p C F , a p p , i j ≈ max ⁡ ( C F i , C F j ) S 2 ( C F ) S 1 ( C F ) = max ⁡ ( C F i S 1 ( C F ) , C F j S 1 ( C F ) ) n n S 2 ( C F ) S 1 ( C F ) 2 ≈ max ⁡ ( k C F , a p p , i , k C F , a p p , j ) n ( 1 + H e t e r o g e n e i t y C F , a p p 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaGaemivaqLaem4Ba8MaemiCaaNaem4ta8KaemODayNaemyzauMaemOCaiNaemiBaWMaemyyaeMaemiCaa3aaSbaaSqaaiabdoeadjabdAeagjabcYcaSiabdggaHjabdchaWjabdchaWjabcYcaSiabdMgaPjabdQgaQbqabaaakeaacqGHijYUaeaacyGGTbqBcqGGHbqycqGG4baEcqGGOaakcqWGdbWqcqWGgbGrdaWgaaWcbaGaemyAaKgabeaakiabdoeadjabdAeagnaaBaaaleaacqWGQbGAaeqaaOGaeiykaKYaaSaaaeaacqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOGqadiab=neadjab=zeagjabcMcaPaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83qamKae8NrayKaeiykaKcaaiabg2da9maalaaabaGagiyBa0MaeiyyaeMaeiiEaGNaeiikaGIaem4qamKaemOray0aaSbaaSqaaiabdMgaPbqabaGccqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=neadjab=zeagjabcMcaPiabcYcaSiabdoeadjabdAeagnaaBaaaleaacqWGQbGAaeqaaOGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkcqGGPaqkaeaacqWGUbGBaaWaaSaaaeaacqWGUbGBcqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOiab=neadjab=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@CA64@

In the following subsection, we outline the conditions when equations (9) and (10) hold approximately for fundamental network concepts in approximately factorizable module networks.

In the Methods section, we provide a heuristic argument for the following

Observation 2 In approximately factorizable networks, fundamental network concepts are approximately equal to their approximate CF-based analogs,

FundamentalNetworkConcept ≈ NetworkConcept_CF,app.

The observation implies that in approximately factorizable networks, Connectivity ≈ Connectivity_CF,app and Density ≈ Density_CF,app, etc. Observation 2 is illustrated for network density, centralization, heterogeneity, and clustering coefficients in Figure 2 (Drosophila PPI network), Figure 3 (yeast PPI network), and Figure 4 (weighted and unweighted yeast gene co-expression networks; density is not included due to limited space). A consequence of this observation is that the simple relationships satisfied by approximate CF-based network concepts also apply to their corresponding fundamental network concepts in approximately factorizable networks. In particular, equations (9) and (10) imply the following

Observation 3 In approximately factorizable networks, the following relationships hold among fundamental network concepts

mean(ClusterCoef) ≈ (1 + Heterogeneity²)² × Density,

and

T o p O v e r l a p i j ≈ max ⁡ ( k i , k j ) n ( 1 + H e t e r o g e n e i t y 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGubavcqWGVbWBcqWGWbaCcqWGpbWtcqWG2bGDcqWGLbqzcqWGYbGCcqWGSbaBcqWGHbqycqWGWbaCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabgIKi7oaalaaabaGagiyBa0MaeiyyaeMaeiiEaGNaeiikaGIaem4AaS2aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqWGRbWAdaWgaaWcbaGaemOAaOgabeaakiabcMcaPaqaaiabd6gaUbaacqGGOaakcqaIXaqmcqGHRaWkcqWGibascqWGLbqzcqWG0baDcqWGLbqzcqWGYbGCcqWGVbWBcqWGNbWzcqWGLbqzcqWGUbGBcqWGLbqzcqWGPbqAcqWG0baDcqWG5bqEdaahaaWcbeqaaiabikdaYaaakiabcMcaPiabc6caUaaa@640A@

Observation 3 is important since it highlights the fact that seemingly disparate network concepts satisfy simple and intuitive relationships in approximately factorizable networks. Equations (11) and (12) are illustrated in Figure 5 (Drosophila PPI network), Figure 6 (yeast PPI network), and Figure 7 (weighted and unweighted yeast gene co-expression networks; TOM plots are not included due to limited space). Equation (12) has several important consequences. To begin with, it illustrates that the topological overlap between the most highly connected node and all other nodes is approximately constant. Specifically, if we denote the index of the most highly connected node by [1] and its connectivity by k_[1] = max(k), then

T o p O v e r l a p [ 1 ] j ≈ k [ 1 ] n ( 1 + H e t e r o g e n e i t y 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGubavcqWGVbWBcqWGWbaCcqWGpbWtcqWG2bGDcqWGLbqzcqWGYbGCcqWGSbaBcqWGHbqycqWGWbaCdaWgaaWcbaGaei4waSLaeGymaeJaeiyxa0LaemOAaOgabeaakiabgIKi7oaalaaabaGaem4AaS2aaSbaaSqaaiabcUfaBjabigdaXiabc2faDbqabaaakeaacqWGUbGBaaGaeiikaGIaeGymaeJaey4kaSIaemisaGKaemyzauMaemiDaqNaemyzauMaemOCaiNaem4Ba8Maem4zaCMaemyzauMaemOBa4MaemyzauMaemyAaKMaemiDaqNaemyEaK3aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGGUaGlaaa@5E8A@

As an aside, we briefly mention that TopOverlap_[1]j has a simple interpretation in terms of the hierarchical clustering dendrogram that results from using dissTopOverlap_ij = 1 -TopOverlap_ij as input. In this case, TopOverlap_[1]j is related to the longest branch length in the dendrogram.

In the following, we relate TopOverlap_[1]j to the fundamental network concept Centralization. According to equation (3), max⁡(k)n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiGbc2gaTjabcggaHjabcIha4jabcIcaOGqadiab=TgaRjabcMcaPaqaaiabd6gaUbaaaaa@3560@ ≈ Centralization + Density. Substituting this expression in equation (13) implies

TopOverlap_[1]j ≈ (Centralization + Density)(1 + Heterogeneity²).

Equation (14) is illustrated in Figure 5 (Drosophila PPI network), Figure 6 (yeast PPI network), and Figure 7 (weighted and unweighted yeast gene co-expression networks).

Here we demonstrate another advantage of approximate CF-based network concepts. They allow one to relate fundamental network concepts to simple functions of the connectivity. Toward this end, note the following simple relationship between the conformity CF and the approximate CF-based connectivity k_CF,app:

C F i = k C F , a p p , i S 1 ( k C F , a p p ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaem4AaS2aaSbaaSqaaiabdoeadjabdAeagjabcYcaSiabdggaHjabdchaWjabdchaWjabcYcaSiabdMgaPbqabaaakeaadaGcaaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGccbmGae83AaS2aaSbaaSqaaiabdoeadjabdAeagjabcYcaSiabdggaHjabdchaWjabdchaWbqabaGccqGGPaqkaSqabaaaaOGaeiOla4caaa@4A36@

Since in approximately factorizable networks k_CF,i ≈ k_i, we find that the conformity CF is approximately given by the scaled connectivity, i.e.

C F ≈ k S 1 ( k ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFdbWqcqWFgbGrcqGHijYUdaWcaaqaaiab=TgaRbqaamaakaaabaGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFRbWAcqGGPaqkaSqabaaaaOGaeiOla4caaa@3859@

This equation shows that conformity can be interpreted as a scaled connectivity in approximately factorizable networks. Since approximate CF-based network concepts are simple functions of the conformity, substituting kS1(k) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGqadiab=TgaRbqaamaakaaabaGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFRbWAcqGGPaqkaSqabaaaaaaa@33A0@ for CF implies that approximate CF-based concepts can be approximated by simple functions of the connectivity. For example, we find the following simple expressions for the cluster coefficient and the topological overlap.

Observation 4

C l u s t e r C o e f i ≈ ( S 2 ( k ) ) 2 ( S 1 ( k ) ) 3 , T o p O v e r l a p i j ≈ k i k j ( S 2 ( k ) + S 1 ( k ) ) min ⁡ ( k i , k j ) S 1 ( k ) + S 1 ( k ) − k i k j ≈ m a x ( k i , k j ) n S 2 ( k ) S 1 ( k ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaGaem4qamKaemiBaWMaemyDauNaem4CamNaemiDaqNaemyzauMaemOCaiNaem4qamKaem4Ba8MaemyzauMaemOzay2aaSbaaSqaaiabdMgaPbqabaaakeaacqGHijYUaeaadaWcaaqaaiabcIcaOiabdofatnaaBaaaleaacqaIYaGmaeqaaOGaeiikaGccbeGae83AaSMaeiykaKIaeiykaKYaaWbaaSqabeaacqaIYaGmaaaakeaacqGGOaakcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=TgaRjabcMcaPiabcMcaPmaaCaaaleqabaGaeG4mamdaaaaakiabcYcaSaqaaiabdsfaujabd+gaVjabdchaWjabd+eapjabdAha2jabdwgaLjabdkhaYjabdYgaSjabdggaHjabdchaWnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaGcbaGaeyisISlabaWaaSaaaeaacqWGRbWAdaWgaaWcbaGaemyAaKgabeaakiabdUgaRnaaBaaaleaacqWGQbGAaeqaaOGaeiikaGIaem4uam1aaSbaaSqaaiabikdaYaqabaGccqGGOaakcqWFRbWAcqGGPaqkcqGHRaWkcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=TgaRjabcMcaPiabcMcaPaqaaiGbc2gaTjabcMgaPjabc6gaUjabcIcaOiabdUgaRnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaem4AaS2aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=TgaRjabcMcaPiabgUcaRiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83AaSMaeiykaKIaeyOeI0Iaem4AaS2aaSbaaSqaaiabdMgaPbqabaGccqWGRbWAdaWgaaWcbaGaemOAaOgabeaaaaGccqGHijYUdaWcaaqaaGqaciab+1gaTjab+fgaHjab+Hha4jabcIcaOiabdUgaRnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaem4AaS2aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkaeaacqWGUbGBaaWaaSaaaeaacqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOiab=TgaRjabcMcaPaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83AaSMaeiykaKcaaiabcYcaSaaaaaa@AFF4@

where the last approximation assumes S 1 ( k ) S 2 ( k ) ≈ 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGccbeGae83AaSMaeiykaKcabaGaem4uam1aaSbaaSqaaiabikdaYaqabaGccqGGOaakcqWFRbWAcqGGPaqkaaGaeyisISRaeGimaadaaa@3A2B@ and S 1 ( k ) − k i k j min ⁡ ( k i , k j ) S 1 ( k ) ≈ 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGccbeGae83AaSMaeiykaKIaeyOeI0Iaem4AaS2aaSbaaSqaaiabdMgaPbqabaGccqWGRbWAdaWgaaWcbaGaemOAaOgabeaaaOqaaiGbc2gaTjabcMgaPjabc6gaUjabcIcaOiabdUgaRnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaem4AaS2aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=TgaRjabcMcaPaaacqGHijYUcqaIWaamaaa@4D8E@

Since the number of genes inside a module (module size) varies greatly among the modules, it is natural to wonder whether the reported relationships between network concepts are due to the underlying module sizes. We find that the relationship between fundamental network concepts and their approximate CF-based analogs remains highly significant even after correcting for module sizes [see Additional file 2]. The same holds for the relationships between network concepts. Thus, none of the reported relationships is trivially due to module sizes. But we find that many network concepts depend on the underlying module size. We find that large modules are less factorizable than small modules: there is a strong negative correlation between module factorizability F(A) and module size. We also find that fundamental network concepts (e.g. density) depend on module size in our applications. For the factorizability, density, centralization, heterogeneity and mean clustering coefficient, the correlation coefficients with module size are -0.84, -0.46, -0.17, 0.26, and -0.36 in Drosophila PPI module networks; they are -0.55, -0.52, 0.05, 0.5, and -0.44 in yeast PPI module networks; they are -0.93, -0.52, -0.82, 0.27, and -0.55 in weighted yeast gene co-expression module networks; they are -0.86, -0.77, -0.56, 0.87, and -0.85 in unweighted yeast gene co-expression module networks. A more detailed analysis is presented in the Additional files [see Additional file 2].

A simple, exactly factorizable network is given by an adjacency matrix A with constant adjacencies (a_ij = b, b ∈ (0, 1]). The adjacency matrix is exactly factorizable since a_ij = CF_iCF_j where CF_i = b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabdkgaIbWcbeaaaaa@2E14@. This network can be interpreted as the expected adjacency matrix of an Erdös-Rényi network 37. One can easily derive the following expressions for the fundamental network concepts: Connectivity_i = (n - 1)b, Density = b, Centralization = 0, Heterogeneity = 0, ClusterCoef_i = b and TopOverlap_ij = b.

Since A is exactly factorizable, the fundamental network concepts equal their CF-based analogs. However, the approximate CF-based concepts are different from their exact counterparts, see Table 3. For reasonably large values of n, the fundamental network concepts are very close to their approximate CF-based analogs. This illustrates Observation 2. With the results in Table 3, one can easily verify Observation 3 and equation (16) in this example.

In the following, we will consider a block diagonal adjacency matrix where each block has constant adjacencies, i.e.

A = ( 1 b 1 ⋯ b 1 0 0 ⋯ 0 b 1 1 ⋯ b 1 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ b 1 b 1 ⋯ 1 0 0 ⋯ 0 0 0 ⋯ 0 1 b 2 ⋯ b 2 0 0 ⋯ 0 b 2 1 ⋯ b 2 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 0 b 2 b 2 ⋯ 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9B6A@

We assume that the first and second blocks have dimensions n₁ × n₁ and n₂ × n₂, respectively. Such a block diagonal matrix can be interpreted as a network with two distinct modules. Setting n₂ = 0 results in the simple constant adjacency matrix, which we considered before.

We denote by f₁ = (1, 1,..., 1, 0, 0, ..., 0) the vector whose first n₁ components equal 1 and the remaining components equal 0. Similarly, we define f₂ = (0, 0, ...,0, 1, 1, ..., 1) = 1 - f₁. To simplify the calculation of the conformity, we further assume that

n 2 ( n 2 − 1 ) b 2 2 n 1 ( n 1 − 1 ) b 1 2 < 1. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabd6gaUnaaBaaaleaacqaIYaGmaeqaaOGaeiikaGIaemOBa42aaSbaaSqaaiabikdaYaqabaGccqGHsislcqaIXaqmcqGGPaqkcqWGIbGydaqhaaWcbaGaeGOmaidabaGaeGOmaidaaaGcbaGaemOBa42aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWGUbGBdaWgaaWcbaGaeGymaedabeaakiabgkHiTiabigdaXiabcMcaPiabdkgaInaaDaaaleaacqaIXaqmaeaacqaIYaGmaaaaaOGaeyipaWJaeGymaeJaeiOla4caaa@47B0@

Then the conformity is uniquely defined by

C F = b 1 f 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFdbWqcqWFgbGrcqGH9aqpdaGcaaqaaiabdkgaInaaBaaaleaacqaIXaqmaeqaaaqabaGccqWFMbGzdaWgaaWcbaGaeGymaedabeaakiabcYcaSaaa@35B4@

as one can show using equations (36) and (37) in the appendix. Further, using Proposition 10 in the appendix, one can show that the factorizability is given by

F ( A ) = n 1 ( n 1 − 1 ) b 1 2 n 1 ( n 1 − 1 ) b 1 2 + n 2 ( n 2 − 1 ) b 2 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5781@

In particular, if n₁ ≈ n₂ and b₁ = b₂, i.e. if the adjacency matrix is comprised of two nearly identical blocks, the factorizability is F(A) ≈ 1/2. Similarly, one can show that if the matrix A is comprised of B identical blocks, then F(A) ≈ 1/B.

This block diagonal network allows one to arrive at explicit formulas for fundamental-, CF-based-, and approximate CF-based network concepts, see Table 4.

In the following, we study the relationship between fundamental network concepts and their approximate CF-based analogs in the limit when the block diagonal network becomes approximately factorizable. Specifically, we calculate network concepts in the limit b₂ → 0 when n₁, n₂ and b₁ are kept fixed. Under this assumption, b₂→ 0 is equivalent to F(A) → 1. Then, one can easily show that

lim ⁡ F ( A ) → 1 C o n n e c t i v i t y i = n 1 − 1 n 1 C o n n e c t i v i t y C F , a p p , i , lim ⁡ F ( A ) → 1 D e n s i t y = n 1 − 1 n 1 D e n s i t y C F , a p p , lim ⁡ F ( A ) → 1 C e n t r a l i z a t i o n = n 1 − 1 n 1 C e n t r a l i z a t i o n C F , a p p , lim ⁡ F ( A ) → 1 H e t e r o g e n e i t y = H e t e r o g e n e i t y C F , a p p , lim ⁡ F ( A ) → 1 T o p O v e r l a p i j = ( n 1 − 1 ) b 1 + 1 n 1 b 1 + 1 T o p O v e r l a p C F , a p p , i j , lim ⁡ F ( A ) → 1 C l u s t e r C o e f i = C l u s t e r C o e f C F , a p p , i . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9070@

For reasonably large values of n₁ (say n₁ > 20), these limits illustrate Observation 2. Similarly, one can easily verify Observation 3 and equation (16) in the case when the factorizability F(A) is close to 1 and n₁ is reasonably large.

We study the properties of an adjacency matrix (network) A that satisfies the following three conditions:

(A.1) A is symmetric and has dimension n × n.

(A.2) The entries of A are bounded within [0, 1], that is, 0 ≤ a_ij ≤ 1 for all 1 ≤ i,j ≤ n.

(A.3) The diagonal elements of A are all 1, that is, a_ii = 1 for all 1 ≤ i ≤ n.

One can easily show that the vector CF is not unique if an exactly factorizable network contains only n = 2 nodes. However, for n > 2 the conformity is uniquely defined when dealing with a weighted network where a_ij > 0.

Specifically, we prove the following statement. If A is an n × n (n ≥ 3) dimensional adjacency matrix with positive entries (a_ij > 0), then the system of equations in (7) has at most one solution CF with positive entries. If the solution exists, it is given by

C F i = ( p i ( ∏ m = 1 n p m ) 1 / ( 2 ( n − 1 ) ) ) 1 n − 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaWgaaWcbaGaemyAaKgabeaakiabg2da9maabmaabaWaaSaaaeaacqWGWbaCdaWgaaWcbaGaemyAaKgabeaaaOqaamaabmaabaWaaebmaeaacqWGWbaCdaWgaaWcbaGaemyBa0gabeaaaeaacqWGTbqBcqGH9aqpcqaIXaqmaeaacqWGUbGBa0Gaey4dIunaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGymaeJaei4la8IaeiikaGIaeGOmaiJaeiikaGIaemOBa4MaeyOeI0IaeGymaeJaeiykaKIaeiykaKcaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaWaaSaaaeaacqaIXaqmaeaacqWGUbGBcqGHsislcqaIYaGmaaaaaOGaeiilaWcaaa@5029@

where pi=∏j=1naij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaradabaGaemyyae2aaSbaaSqaaiabdMgaPjabdQgaQbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemOBa4ganiabg+Givdaaaa@3B79@ denotes the 'product connectivity' of the i-th node.

Proof: by assumption, we have a_ij = CF_iCF_j for a positive vector CF and n ≥ 3. Multiplying both sides of equation (7) yields ∏m∏l≠malm=∏m∏l≠mCFlCFm=(∏l=1nCFl)2(n−1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5F7C@. Since ∏l=1nCFl MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqeWaqaaiabdoeadjabdAeagnaaBaaaleaacqWGSbaBaeqaaaqaaiabdYgaSjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHpis1aaaa@36FF@ is positive, we find ∏l=1nCFl=(∏m∏l≠malm)12(n−1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqeWaqaaiabdoeadjabdAeagnaaBaaaleaacqWGSbaBaeqaaaqaaiabdYgaSjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHpis1aOGaeyypa0ZaaeWaaeaadaqeqaqaamaarababaGaemyyae2aaSbaaSqaaiabdYgaSjabd2gaTbqabaaabaGaemiBaWMaeyiyIKRaemyBa0gabeqdcqGHpis1aaWcbaGaemyBa0gabeqdcqGHpis1aaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWcaaqaaiabigdaXaqaaiabikdaYiabcIcaOiabd6gaUjabgkHiTiabigdaXiabcMcaPaaaaaaaaa@4E77@. Similarly, eliminating the i-th row and column from A yields ∏l≠iCFl=(∏m≠i∏l≠m,ialm)12(n−1)=(∏m∏l≠malm/(∏l≠iali)2)12(n−1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7843@. Since CFi=∏l=1nCFl/∏l≠iCFl MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalyaabaWaaebmaeaacqWGdbWqcqWGgbGrdaWgaaWcbaGaemiBaWgabeaaaeaacqWGSbaBcqGH9aqpcqaIXaqmaeaacqWGUbGBa0Gaey4dIunaaOqaamaarababaGaem4qamKaemOray0aaSbaaSqaaiabdYgaSbqabaaabaGaemiBaWMaeyiyIKRaemyAaKgabeqdcqGHpis1aaaaaaa@45D5@, we conclude that CF_i is uniquely defined by

C F i = ( ∏ m ∏ l ≠ m a l m ) 1 2 ( n − 1 ) ( ∏ m ≠ i ∏ l ≠ m , i a l m ) 1 2 ( n − 1 ) = ( ∏ m ∏ l ≠ m a l m ) 1 2 ( n − 1 ) ( ∏ m ∏ l ≠ m a l m ) 1 2 ( n − 2 ) ( ∏ m = 1 n a i m ) 1 n − 2 = ( p i ( ∏ m = 1 n p m ) 1 / ( 2 ( n − 1 ) ) ) 1 n − 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@C2B6@

In general, we define a network concept function to be a tensor valued function (e.g. the connectivity vector) that takes a square matrix (e.g. the network adjacency matrix) as input.

Denote by M = [m_ij] a general n × n matrix. Then we will study the following network concept functions:

C o n n e c t i v i t y i ( M ) = ∑ j m i j = e i τ M 1 , D e n s i t y ( M ) = ∑ i ∑ j m i j n ( n − 1 ) , C e n t r a l i z a t i o n ( M ) = n n − 2 ( max ⁡ ( M 1 ) n − 1 − D e n s i t y ( M ) ) , H e t e r o g e n e i t y ( M ) = n ( 1 τ M M 1 ) ( 1 τ M 1 ) 2 − 1 , T o p O v e r l a p i j ( M ) = e i τ M M e j + e i τ M e j min ⁡ { e i τ M 1 , e j τ M 1 } + 1 − e i τ M e j , C l u s t e r C o e f i ( M ) = e i τ M M M e i e i τ M B M M e i , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGbdaaaaeaacqWGdbWqcqWGVbWBcqWGUbGBcqWGUbGBcqWGLbqzcqWGJbWycqWG0baDcqWGPbqAcqWG2bGDcqWGPbqAcqWG0baDcqWG5bqEdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabd2eanjabcMcaPaqaaiabg2da9aqaamaaqafabaGaemyBa02aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpieWacqWFLbqzdaqhaaWcbaGaemyAaKgabaacciGae4hXdqhaaOGaemyta0Kae8xmaeJaeiilaWcaleaacqWGQbGAaeqaniabggHiLdaakeaacqWGebarcqWGLbqzcqWGUbGBcqWGZbWCcqWGPbqAcqWG0baDcqWG5bqEcqGGOaakcqWGnbqtcqGGPaqkaeaacqGH9aqpaeaadaWcaaqaamaaqababaWaaabeaeaacqWGTbqBdaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAaeqaniabggHiLdaaleaacqWGPbqAaeqaniabggHiLdaakeaacqWGUbGBcqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkaaGaeiilaWcabaGaem4qamKaemyzauMaemOBa4MaemiDaqNaemOCaiNaemyyaeMaemiBaWMaemyAaKMaemOEaONaemyyaeMaemiDaqNaemyAaKMaem4Ba8MaemOBa4MaeiikaGIaemyta0KaeiykaKcabaGaeyypa0dabaWaaSaaaeaacqWGUbGBaeaacqWGUbGBcqGHsislcqaIYaGmaaWaaeWaaeaadaWcaaqaaiGbc2gaTjabcggaHjabcIha4jabcIcaOiabd2eanjab=fdaXiabcMcaPaqaaiabd6gaUjabgkHiTiabigdaXaaacqGHsislcqWGebarcqWGLbqzcqWGUbGBcqWGZbWCcqWGPbqAcqWG0baDcqWG5bqEcqGGOaakcqWGnbqtcqGGPaqkaiaawIcacaGLPaaacqGGSaalaeaacqWGibascqWGLbqzcqWG0baDcqWGLbqzcqWGYbGCcqWGVbWBcqWGNbWzcqWGLbqzcqWGUbGBcqWGLbqzcqWGPbqAcqWG0baDcqWG5bqEcqGGOaakcqWGnbqtcqGGPaqkaeaacqGH9aqpaeaadaGcaaqaamaalaaabaGaemOBa4MaeiikaGIae8xmaeZaaWbaaSqabeaacqGFepaDaaGccqWGnbqtcqWGnbqtcqWFXaqmcqGGPaqkaeaacqGGOaakcqWFXaqmdaahaaWcbeqaaiab+r8a0baakiabd2eanjab=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fdaXiabc2ha9jabgUcaRiabigdaXiabgkHiTiab=vgaLnaaDaaaleaacqWGPbqAaeaacqGFepaDaaGccqWGnbqtcqWFLbqzdaWgaaWcbaGaemOAaOgabeaaaaGccqGGSaalaeaacqWGdbWqcqWGSbaBcqWG1bqDcqWGZbWCcqWG0baDcqWGLbqzcqWGYbGCcqWGdbWqcqWGVbWBcqWGLbqzcqWGMbGzdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabd2eanjabcMcaPaqaaiabg2da9aqaamaalaaabaGae8xzau2aa0baaSqaaiabdMgaPbqaaiab+r8a0baakiabd2eanjabd2eanjabd2eanjab=vgaLnaaBaaaleaacqWGPbqAaeqaaaGcbaGae8xzau2aa0baaSqaaiabdMgaPbqaaiab+r8a0baakiabd2eanjabdkeacnaaBaaaleaacqWGnbqtaeqaaOGaemyta0Kae8xzau2aaSbaaSqaaiabdMgaPbqabaaaaOGaeiilaWcaaaaa@47B4@

where the components of matrix B_M in the denominator of the clustering coefficient function are given by b_ij = 1 if i ≠ j and b_ii = Ind(m_ii > 0). Here the indicator function Ind(·) takes on the value 1 if the condition is satisfied and 0 otherwise.

For the sake of brevity, we study only a limited selection of network concept functions and do not claim that these are more important than others studied in the literature. Our general formalism for relating fundamental network concepts to their approximate CF-based analogs should allow the reader to adapt our derivations to alternative concepts as well.

Now we are ready to define the fundamental network concepts that are studied in this article.

Definition 5 (Fundamental Network Concept) The fundamental network concepts of a network A are defined by evaluating the network functions (equation (21)) on A - I, i.e.

FundamentalNetworkConcept = NetworkConcept(A - I).

As special cases of this definition, we find the following concepts. The connectivity (also known as degree) of the i-th node is given by

k i = C o n n e c t i v i t y i ( A − I ) = ∑ j ≠ i a i j . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdoeadjabd+gaVjabd6gaUjabd6gaUjabdwgaLjabdogaJjabdsha0jabdMgaPjabdAha2jabdMgaPjabdsha0jabdMha5naaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemyqaeKaeyOeI0IaemysaKKaeiykaKIaeyypa0ZaaabuaeaacqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAcqGHGjsUcqWGPbqAaeqaniabggHiLdGccqGGUaGlaaa@541C@

The line density 13 equals the mean adjacency, i.e

D e n s i t y ( A − I ) = ∑ i ∑ j ≠ i a i j n ( n − 1 ) = S 1 ( k ) n ( n − 1 ) = m e a n ( k ) n − 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebarcqWGLbqzcqWGUbGBcqWGZbWCcqWGPbqAcqWG0baDcqWG5bqEcqGGOaakcqWGbbqqcqGHsislcqWGjbqscqGGPaqkcqGH9aqpdaWcaaqaamaaqababaWaaabeaeaacqWGHbqydaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAcqGHGjsUcqWGPbqAaeqaniabggHiLdaaleaacqWGPbqAaeqaniabggHiLdaakeaacqWGUbGBcqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkaaGaeyypa0ZaaSaaaeaacqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOGqadiab=TgaRjabcMcaPaqaaiabd6gaUjabcIcaOiabd6gaUjabgkHiTiabigdaXiabcMcaPaaacqGH9aqpdaWcaaqaaiabd2gaTjabdwgaLjabdggaHjabd6gaUjabcIcaOiab=TgaRjabcMcaPaqaaiabd6gaUjabgkHiTiabigdaXaaacqGGUaGlaaa@6AB9@

For notational convenience, we sometimes omit the reference to the adjacency matrix and simply write Density to denote the fundamental network concepts.

The normalized connectivity centralization (also known as degree centralization) 14 is given by

C e n t r a l i z a t i o n ( A − I ) = n n − 2 ( max ⁡ ( k ) n − 1 − D e n s i t y ) = n ( n − 2 ) ( n − 1 ) ( max ⁡ ( k ) − m e a n ( k ) ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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TgaRjabcMcaPiabgkHiTiabd2gaTjabdwgaLjabdggaHjabd6gaUjabcIcaOiab=TgaRjabcMcaPiabcMcaPiabc6caUaaa@8050@

Our definition of the network heterogeneity equals the coefficient of variation of the connectivity distribution, i.e.

H e t e r o g e n e i t y ( A − I ) = v a r i a n c e ( k ) m e a n ( k ) = n S 2 ( k ) S 1 ( k ) 2 − 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6C0A@

Note that Heterogeneity(b * M) = Heterogeneity(M) for a scalar b ≠ 0.

The clustering coefficient of node i is a density measure of local connections, or 'cliquishness' 1920. Specifically,

C l u s t e r C o e f i = C l u s t e r C o e f i ( A − I ) = n i π i = ∑ l ≠ i ∑ m ≠ i , l a i l a l m a m i { ( ∑ l ≠ i a i l ) 2 − ∑ l ≠ i a i l 2 } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGSbaBcqWG1bqDcqWGZbWCcqWG0baDcqWGLbqzcqWGYbGCcqWGdbWqcqWGVbWBcqWGLbqzcqWGMbGzdaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdoeadjabdYgaSjabdwha1jabdohaZjabdsha0jabdwgaLjabdkhaYjabdoeadjabd+gaVjabdwgaLjabdAgaMnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemyqaeKaeyOeI0IaemysaKKaeiykaKIaeyypa0ZaaSaaaeaacqWGUbGBdaWgaaWcbaGaemyAaKgabeaaaOqaaGGaciab=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@93EF@

The topological overlap between nodes i and j reflects their relative interconnectedness. It is defined by

T o p O v e r l a p i j = T o p O v e r l a p i j ( A − I ) = l i j + a i j min ⁡ { k i , k j } + 1 − a i j , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGubavcqWGVbWBcqWGWbaCcqWGpbWtcqWG2bGDcqWGLbqzcqWGYbGCcqWGSbaBcqWGHbqycqWGWbaCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9iabdsfaujabd+gaVjabdchaWjabd+eapjabdAha2jabdwgaLjabdkhaYjabdYgaSjabdggaHjabdchaWnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiikaGIaemyqaeKaeyOeI0IaemysaKKaeiykaKIaeyypa0ZaaSaaaeaacqWGSbaBdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabgUcaRiabdggaHnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaGcbaGagiyBa0MaeiyAaKMaeiOBa4Maei4EaSNaem4AaS2aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqWGRbWAdaWgaaWcbaGaemOAaOgabeaakiabc2ha9jabgUcaRiabigdaXiabgkHiTiabdggaHnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaaakiabcYcaSaaa@7373@

where l_ij = ∑_u≠i,ja_iua_uj.

In the following, we will present explicit formulas for the fundamental network concepts in Definition 5 when the adjacency matrix A is exactly factorizable, i.e. if a_ij = CF_iCF_j. We define the CF-based adjacency matrix as follows

A_CF := CF CF^τ - diag(CF²) + I,

where diag(CF²) denotes the diagonal matrix with diagonal elements CFi2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@314A@, i = 1 ...n. Then one can easily show that for exactly factorizable networks

A = A C F , N e t w o r k C o n c e p t ( A − I ) = N e t w o r k C o n c e p t ( A C F − I ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGadaaabaGaemyqaeeabaGaeyypa0dabaGaemyqae0aaSbaaSqaaiabdoeadjabdAeagjabcYcaSaqabaaakeaacqWGobGtcqWGLbqzcqWG0baDcqWG3bWDcqWGVbWBcqWGYbGCcqWGRbWAcqWGdbWqcqWGVbWBcqWGUbGBcqWGJbWycqWGLbqzcqWGWbaCcqWG0baDcqGGOaakcqWGbbqqcqGHsislcqWGjbqscqGGPaqkaeaacqGH9aqpaeaacqWGobGtcqWGLbqzcqWG0baDcqWG3bWDcqWGVbWBcqWGYbGCcqWGRbWAcqWGdbWqcqWGVbWBcqWGUbGBcqWGJbWycqWGLbqzcqWGWbaCcqWG0baDcqGGOaakcqWGbbqqdaWgaaWcbaGaem4qamKaemOrayeabeaakiabgkHiTiabdMeajjabcMcaPiabc6caUaaaaaa@66B9@

Using our definition of network concept functions in equations (21), one can easily derive the following formulas for NetworkConcept(A_CF - I) in terms of the quantities S_p(CF) = ∑_iCFip MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGgbGrdaqhaaWcbaGaemyAaKgabaGaemiCaahaaaaa@31C1@.

C o n n e c t i v i t y i ( A C F − I ) = C F i S 1 ( C F ) − C F i 2 , D e n s i t y ( A C F − I ) = S 1 ( C F ) 2 − S 2 ( C F ) n ( n − 1 ) , C e n t r a l i z a t i o n ( A C F − I ) = n n − 2 ( max ⁡ ( C o n n e c t i v i t y ( A C F − I ) ) n − 1 − D e n s i t y ( A C F − I ) ) , H e t e r o g e n e i t y ( A C F − I ) = n ( S 2 ( C F ) S 1 ( C F ) 2 − 2 S 3 ( C F ) S 1 ( C F ) + S 4 ( C F ) ) ( S 1 ( C F ) 2 − S 2 ( C F ) ) 2 − 1 , C l u s t e r C o e f i ( A C F − I ) = ( S 2 ( C F ) − C F i 2 ) 2 − ( S 4 ( C F ) − C F i 4 ) ( S 1 ( C F ) − C F i ) 2 − ( S 2 ( C F ) − C F i 2 ) 2 . T o p O v e r l a p i j ( A C F − I ) = C F i C F j ( S 2 ( C F ) − C F i 2 − C F j 2 ) + C F i C F j min ⁡ ( C F i ( S 1 ( C F ) − C F i ) , C F j ( S 1 ( C F ) − C F j ) ) + 1 − C F i C F j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGbdaaaaeaacqWGdbWqcqWGVbWBcqWGUbGBcqWGUbGBcqWGLbqzcqWGJbWycqWG0baDcqWGPbqAcqWG2bGDcqWGPbqAcqWG0baDcqWG5bqEdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdgeabnaaBaaaleaacqWGdbWqcqWGgbGraeqaaOGaeyOeI0IaemysaKKaeiykaKcabaGaeyypa0dabaGaem4qamKaemOray0aaSbaaSqaaiabdMgaPbqabaGccqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOGqadiab=neadjab=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neadjab=zeagjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeyOeI0IaeGOmaiJaem4uam1aaSbaaSqaaiabiodaZaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=neadjab=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neadjab=zeagjabcMcaPiabgkHiTiabdoeadjabdAeagnaaDaaaleaacqWGPbqAaeaacqaI0aanaaGccqGGPaqkaeaacqGGOaakcqWGtbWudaWgaaWcbaGaeGymaedabeaakiabcIcaOiab=neadjab=zeagjabcMcaPiabgkHiTiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqGHsislcqGGOaakcqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOiab=neadjab=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@E872@

When A_CF - I is used as input of a network concept function, it gives rise to a CF-based network concept as detailed in the following

Definition 6 (CF-based Network Concepts) Assume that the conformity vector CF can be defined for a general adjacency matrix A. Then the CF-based network concepts are defined by evaluating the network concept functions on A_CF - I = CF CF^τ - diag(CF²), i.e.

NetworkConcept_CF := NetworkConcept(A_CF - I).

By definition, fundamental network concepts are equal to their CF-based analogs if A is exactly factorizable.

In the following, we define approximate CF-based analogs of the fundamental network concepts. The theoretical advantage of these approximate CF-based concepts is that they satisfy simple relationships. Define the approximate CF-based adjacency matrix as follows

A_CF,app = CF CF^τ.

Note that only the diagonal elements differ between A_CF,app and A_CF . We define the approximate CF-based network concepts by using AC_F,app as input of the network concept functions as detailed in the following

Definition 7 (Approximate CF-based Network Concepts) The approximate CF-based network concepts of a network A with conformity CF are defined by evaluating the network functions (equations (21)) on A_CF,app = CF CF^τ, i.e.

NetworkConcept_CF,app := NetworkConcept(A_CF,app).

Here we will provide a heuristic derivation of Observation 2. Since the components of CF are positive, one can easily show that S₄(CF) ≤ S₂(CF)². For many large, exactly factorizable networks, the ratio S₄(CF)/S₂(CF)² is close to 0. Since S₄(CF)/S₂(CF)² = ‖(ACF−I)−ACF,app‖F2/‖ACF,app‖F2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcgaqaamaafmaabaGaeiikaGIaemyqae0aaSbaaSqaaiabdoeadjabdAeagbqabaGccqGHsislcqWGjbqscqGGPaqkcqGHsislcqWGbbqqdaWgaaWcbaGaem4qamKaemOrayKaeiilaWIaemyyaeMaemiCaaNaemiCaahabeaaaOGaayzcSlaawQa7amaaDaaaleaacqWGgbGraeaacqaIYaGmaaaakeaadaqbdaqaaiabdgeabnaaBaaaleaacqWGdbWqcqWGgbGrcqGGSaalcqWGHbqycqWGWbaCcqWGWbaCaeqaaaGccaGLjWUaayPcSdWaa0baaSqaaiabdAeagbqaaiabikdaYaaaaaaaaa@5052@, this implies that A_CF - I ≈ A_CF,app. Since the network concept functions are continuous functions, this implies NetworkConcept(A_CF - I) ≈ NetworkConcept(A_CF,app). These derivations are summarized in the following

Observation 8 (Approximate Formulas for CF-based Concepts) If S₄(CF)/S₂(CF)² ≈ 0, then

NetworkConcept(A_CF - I) ≈ NetworkConcept(A_CF,app).

In particular, for exactly factorizable networks (i.e. A - I = A_CF - I), this means that the fundamental network concepts can be approximated by their approximate CF-based analogs.

In our real data applications, we show empirically that equation (31) holds even in networks that satisfy the assumptions of Observation 8 only approximately.

In the appendix (equation (43)), we define a measure of network factorizability as follows

F ( A ) = 1 − ‖ ( A − I ) − ( A C F − I ) ‖ F 2 ‖ A − I ‖ F 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrcqGGOaakcqWGbbqqcqGGPaqkcqGH9aqpcqaIXaqmcqGHsisldaWcaaqaamaafmaabaGaeiikaGIaemyqaeKaeyOeI0IaemysaKKaeiykaKIaeyOeI0IaeiikaGIaemyqae0aaSbaaSqaaiabdoeadjabdAeagbqabaGccqGHsislcqWGjbqscqGGPaqkaiaawMa7caGLkWoadaqhaaWcbaGaemOrayeabaGaeGOmaidaaaGcbaWaauWaaeaacqWGbbqqcqGHsislcqWGjbqsaiaawMa7caGLkWoadaqhaaWcbaGaemOrayeabaGaeGOmaidaaaaakiabcYcaSaaa@4EFF@

Thus, in approximately factorizable networks (i.e. F(A) close to 1), A - I can be approximated by A_CF - I. For a continuous network functions, this implies

NetworkConcept(A -I) ≈ NetworkConcept(A_CF- I),

i.e. the fundamental network concepts are approximately equal to their CF-based analogs in approximately factorizable networks. Observation 8 states that

NetworkConcept(A_CF -I) ≈ NetworkConcept(A_CF,app).

Combining the last two equations leads to NetworkConcept(A - I) ≈ NetworkConcept(A_CF,app). These derivations are summarized as follows.

In approximately factorizable networks, the fundamental network concepts are approximately equal to their approximate CF-based analogs, i.e.

FundamentalNetworkConcept ≈ NetworkConcept_CF,app.

Gene co-expression networks are constructed from microarray data that measures the transcriptional response of cells to changing conditions. We consider the case of n genes with gene expression profiles across m microarray samples. Thus, the gene expression profiles are given by an n × m matrix

X = [x_ij] = (x₁ x₂ ... x_n)^τ, i = 1, ..., n; j = 1, ..., m,

where the i-th row xiτ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWF4baEdaqhaaWcbaGaemyAaKgabaacciGae4hXdqhaaaaa@3180@ is the transcriptional responses of the i-th gene.

Recently, several groups have suggested thresholding the pairwise Pearson correlation coefficient cor(x_i, x_j) in order to arrive at gene co-expression networks, which are sometimes referred to as 'relevance' networks 1132. In these networks, a node corresponds to the gene expression profile of a given gene. The corresponding adjacency matrix is determined from a measure of co-expression between the genes. In the examples below, we will use the absolute value of the Pearson correlation coefficient between the gene expression profiles to measure co-expression.

To transform the co-expression measure into an adjacency, one can make use of an adjacency function. The choice of the adjacency function determines whether the resulting network will be weighted (soft-thresholding) or unweighted (hard-thresholding). The adjacency function is a monotonically increasing function that maps the interval [0, 1] into [0, 1]. A widely used adjacency function is the signum function which implements 'hard' thresholding involving the threshold parameter τ. Specifically,

a_ij = Signum(|cor(x_i, x_j)|, τ) = Ind(|cor(x_i, x_j)| ≥ τ),

where the indicator function Ind(·) takes on the value 1 if the condition is satisfied and 0 otherwise. Hard thresholding using the signum function leads to intuitive network concepts (e.g., the node connectivity equals the number of direct neighbors), but it may lead to a loss of information: if τ has been set to 0.8, there will be no connection between two nodes if their similarity equals 0.79.

To avoid the disadvantages of hard thresholding, we proposed a 'soft' thresholding approach that raises the absolute value of the correlation to the power β ≥ 1 21, i.e.

a_ij = Power(|cor(x_i, x_j)|, β) = |cor(x_i, x_j)|^β.

In our yeast cell cycle gene co-expression network analysis, we followed the analysis steps described in 21. Briefly, we used the 2001 most varying and connected genes. Next, we used the power adjacency function with β = 7 (equation (35)) to construct a weighted gene co-expression network and the signum adjacency function with τ = 0.65 (equation (34)) to construct an unweighted network.

Using our R software tutorial, the reader can easily verify that our conclusions are highly robust with respect to a) different ways of constructing co-expression networks and b) different ways of constructing modules.