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=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@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=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWbdaaaaeaacqWGRbWAdaWgaaWcbaGaem4qamKaemOrayKaeiilaWIaemyyaeMaemiCaaNaemiCaaNaeiilaWIaemyAaKgabeaaaOqaaiabg2da9aqaaiabdoeadjabdAeagnaaBaaaleaacqWGPbqAaeqaaOGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakieWacqWFdbWqcqWFgbGrcqGGPaqkcqGGSaalaeaacqWGebarcqWGLbqzcqWGUbGBcqWGZbWCcqWGPbqAcqWG0baDcqWG5bqEdaWgaaWcbaGaem4qamKaemOrayKaeiilaWIaemyyaeMaemiCaaNaemiCaahabeaaaOqaaiabg2da9aqaamaalaaabaGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaOqaaiabd6gaUjabcIcaOiabd6gaUjabgkHiTiabigdaXiabcMcaPaaacqGHijYUdaqadaqaamaalaaabaGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkaeaacqWGUbGBaaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGGSaalaeaacqWGdbWqcqWGLbqzcqWGUbGBcqWG0baDcqWGYbGCcqWGHbqycqWGSbaBcqWGPbqAcqWG6bGEcqWGHbqycqWG0baDcqWGPbqAcqWGVbWBcqWGUbGBdaWgaaWcbaGaem4qamKaemOrayKaeiilaWIaemyyaeMaemiCaaNaemiCaahabeaaaOqaaiabg2da9aqaamaalaaabaGaemOBa4Maem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkaeaacqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkcqGGOaakcqWGUbGBcqGHsislcqaIYaGmcqGGPaqkaaWaaeWaaeaacyGGTbqBcqGGHbqycqGG4baEcqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkcqGHsisldaWcaaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83qamKae8NrayKaeiykaKcabaGaemOBa4gaaaGaayjkaiaawMcaaaqaaaqaaiabgIKi7cqaamaalaaabaGaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkaeaacqWGUbGBaaWaaeWaaeaacyGGTbqBcqGGHbqycqGG4baEcqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkcqGHsisldaWcaaqaaiabdofatnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae83qamKae8NrayKaeiykaKcabaGaemOBa4gaaaGaayjkaiaawMcaaiabcYcaSaqaaiabdIeaijabdwgaLjabdsha0jabdwgaLjabdkhaYjabd+gaVjabdEgaNjabdwgaLjabd6gaUjabdwgaLjabdMgaPjabdsha0jabdMha5naaBaaaleaacqWGdbWqcqWGgbGrcqGGSaalcqWGHbqycqWGWbaCcqWGWbaCaeqaaaGcbaGaeyypa0dabaWaaOaaaeaadaWcaaqaaiabd6gaUjabdofatnaaBaaaleaacqaIYaGmaeqaaOGaeiikaGIae83qamKae8NrayKaeiykaKcabaGaeiikaGIaem4uam1aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWFdbWqcqWFgbGrcqGGPaqkcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaaGccqGHsislcqaIXaqmaSqabaGccqGGSaalaeaacqWGdbWqcqWGSbaBcqWG1bqDcqWGZbWCcqWG0baDcqWGLbqzcqWGYbGCcqWGdbWqcqWGVbWBcqWGLbqzcqWGMbGzdaWgaaWcbaGaem4qamKaemOrayKaeiilaWIaemyyaeMaemiCaaNaemiCaaNaeiilaWIaemyAaKgabeaaaOqaaiabg2da9aqaamaabmaabaWaaSaaaeaacqWGtbWudaWgaaWcbaGaeGOmaidabeaakiabcIcaOiab=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 )