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Table 2 Formal representation of graph measures

From: Characterization of protein-interaction networks in tumors

Name

Class

Definition

Description

Ref.

Closeness Centrality

size

C C i = 1 ∑ j d ( i , j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGdbWqdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaeGymaedabaWaaabuaeaacqWGKbazcqGGOaakcqWGPbqAcqGGSaalcqWGQbGAcqGGPaqkaSqaaiabdQgaQbqab0GaeyyeIuoaaaaaaa@3C7C@

d(i,j) is the length of the shortest path between vertices i and j. The sum of CC i over all vertices gives the total Closeness Centrality of a given subgraph.

[42]

Graph Diameter

size

G D = max ( d ( i , j ) ) N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrcqWGebarcqGH9aqpdaWcaaqaaiGbc2gaTjabcggaHjabcIha4jabcIcaOiabdsgaKjabcIcaOiabdMgaPjabcYcaSiabdQgaQjabcMcaPiabcMcaPaqaaiabd6eaobaaaaa@3D82@

d(i,j) is the length of the shortest path between vertices i and j. GD is computed for all pairs (i,j), and reflects the longest path identified.

[43]

Index of Aggregation

size

I o A = A B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGjbqscqWGVbWBcqWGbbqqcqGH9aqpdaWcaaqaaiabdgeabbqaaiabdkeacbaaaaa@3367@

A is the total number of vertices in the subgraph, and B is the total number of all given vertices in the graph.

[15]

Assortative Mixing Coefficient

distribution

k 1 and k 2 are the counts of edges of two vertices connected by a given edge. This measure reflects the edge-to-edge distribution over all edges of a graph.

[44]

Entropy of the distribution of edges

distribution

H = − ∑ k p ( k ) ln p ( k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibascqGH9aqpcqGHsisldaaeqbqaaiabdchaWjabcIcaOiabdUgaRjabcMcaPiGbcYgaSjabc6gaUjabdchaWjabcIcaOiabdUgaRjabcMcaPaWcbaGaem4AaSgabeqdcqGHris5aaaa@3EF4@

k is the count of edges of one vertex, and p(k) is the ratio of vertices that have k edges.

[45]

Betweenness

biological relevance

B = ∑ i ∈ V ∑ j , k σ ( j , i , k ) σ ( j , k ) N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGcbGqcqGH9aqpdaWcaaqaamaaqafabaWaaabuaeaadaWcaaqaaGGaciab=n8aZjabcIcaOiabdQgaQjabcYcaSiabdMgaPjabcYcaSiabdUgaRjabcMcaPaqaaiab=n8aZjabcIcaOiabdQgaQjabcYcaSiabdUgaRjabcMcaPaaaaSqaaiabdQgaQjabcYcaSiabdUgaRbqab0GaeyyeIuoaaSqaaiabdMgaPjabgIGiolabdAfawbqab0GaeyyeIuoaaOqaaiabd6eaobaaaaa@4C63@

σ(j,i,k) is the total number of shortest connections between vertices j and k, where each shortest connection has to pass vertex i, and σ(j,k) is the total number of shortest connections between j and k. We computed σ(j,i,k) and σ(j,k) for the entire OPHID graph, but then only used vertices also present in the subgraph generated on the basis of a given gene-expression data set.

[42]

Betweenness of all selected Vertices

biological relevance

 

As for Betweenness, but considering all selected vertices.

[42]

Stress Centrality

biological Relevance

S t C = ∑ i ∈ V ∑ j , k σ ( j , i , k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWucqWG0baDcqWGdbWqcqGH9aqpdaaeqbqaamaaqafabaacciGae83WdmNaeiikaGIaemOAaOMaeiilaWIaemyAaKMaeiilaWIaem4AaSMaeiykaKcaleaacqWGQbGAcqGGSaalcqWGRbWAaeqaniabggHiLdaaleaacqWGPbqAcqGHiiIZcqWGwbGvaeqaniabggHiLdaaaa@46AA@

σ(j,i,k) is the total number of shortest connections between vertices j and k, where each shortest connection has to pass vertex i.

[42]

Connectivity

density

C = A B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqGH9aqpdaWcaaqaaiabdgeabbqaaiabdkeacbaaaaa@30E9@

A is the total number of edges realized in a given graph, and B is the maximum number of edges possible.

[43]

Clustering Coefficient

density

C L U S T i = A B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGmbatcqWGvbqvcqWGtbWucqWGubavdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaemyqaeeabaGaemOqaieaaaaa@372E@

A is the total number of edges between the nearest neighbors of vertex i, and B is the maximum number of possible edges between the nearest neighbors of vertex i. The sum of CLUST i over all vertices gives the total Clustering Coefficient of a given subgraph.

[46]

Number of edges divided by the number of vertices

density

N e N v = A B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtcqWGLbqzcqWGobGtcqWG2bGDcqGH9aqpdaWcaaqaaiabdgeabbqaaiabdkeacbaaaaa@34EC@

A is the total number of edges in a given graph, and B is the number of selected vertices in a given graph.

-

Community

density

C o m m = A B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqWGVbWBcqWGTbqBcqWGTbqBcqGH9aqpdaWcaaqaaiabdgeabbqaaiabdkeacbaaaaa@3516@

A is the total number of edges, where both connected vertices are in the given subgraph, and B is the total number of edges, where one connected vertex is in the subgraph and the other vertex is outside it.

[47]

Entropy

density

H ( G ) = ∑ v ∈ V , i ( v ) > = 2 ( i ( v ) − 1 ) ∗ log ( | E | − | V | + 1 i ( v ) − 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibascqGGOaakcqWGhbWrcqGGPaqkcqGH9aqpdaaeqaqaaiabcIcaOiabdMgaPjabcIcaOiabdAha2jabcMcaPiabgkHiTiabigdaXiabcMcaPiabgEHiQiGbcYgaSjabc+gaVjabcEgaNjabcIcaOmaalaaabaGaeiiFaWNaemyrauKaeiiFaWNaeyOeI0IaeiiFaWNaemOvayLaeiiFaWNaey4kaSIaeGymaedabaGaemyAaKMaeiikaGIaemODayNaeiykaKIaeyOeI0IaeGymaedaaaWcbaGaemODayNaeyicI4SaemOvayLaeiilaWIaemyAaKMaeiikaGIaemODayNaeiykaKIaeyOpa4Jaeyypa0JaeGOmaidabeqdcqGHris5aOGaeiykaKcaaa@6057@

where |E| is the total number of edges, |V| is the total number of vertices, and i(v) is the number of edges of vertex v.

[48]

Graph Centrality

density

G C i = 1 max ( d ( i , j ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrcqWGdbWqdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaeGymaedabaGagiyBa0MaeiyyaeMaeiiEaGNaeiikaGIaemizaqMaeiikaGIaemyAaKMaeiilaWIaemOAaOMaeiykaKIaeiykaKcaaaaa@3EDC@

max(d(i,j)) is the length of the shortest path between vertices i and j for a given vertex i.

[42]

Number of walks of length n

density

N W = ∑ N W i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtcqWGxbWvcqGH9aqpdaaeabqaaiabd6eaojabdEfaxnaaBaaaleaacqWGPbqAaeqaaaqabeqaniabggHiLdaaaa@35FA@

NW i is one walk with a length of n edges in the subgraph.

[43]

Sum of the Wiener Number

density

W i = 1 2 ∗ ∑ i , j d ( i , j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGxbWvdaWgaaWcbaGaemyAaKgabeaakiabg2da9maalaaabaGaeGymaedabaGaeGOmaidaaiabgEHiQmaaqafabaGaemizaqMaeiikaGIaemyAaKMaeiilaWIaemOAaOMaeiykaKcaleaacqWGPbqAcqGGSaalcqWGQbGAaeqaniabggHiLdaaaa@3FB1@

d(i,j) is the length of the shortest path between vertices i and j. We computed the Sum of the Wiener Number for each vertex.

[43]

Total number of triangles of a subgraph and its dilation

Modularity

 

Given a subgraph g of graph G, the complement of g, denoted as g, is the subgraph implied by the set of vertices

N(g) = N(G)\N(g)

The dilation of g is the subgraph δ(g) implied by the vertices in g plus the vertices directly connected to a vertex in g. The coat of nearest neighbors of the subgraph is defined as

DN(g) = δ(g)\N(g)

The set of all valid triangles for g is defined as

VT(g) = {x,y,z | (x,y,z ∈ N(δ(g)) ^ (x,y),(y,z),(z,x) ∈ E(δ(g))) ∩ (x ∈ N(g) ^ z ∈ DN(g))}

where N is the number of vertices and E is the number of edges in the graph. The result for a subgraph g is the total number of elements in VT(g).

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Localized Modularity

modularity

L M = | E inside | | E within the  ( direct )  neighbors | ∗ | E inside | ∗ | E to the outside | | E within the  ( direct )  neighbors | 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B47C@

where |E| is the total number of edges.

[49]

modified Vertex Distance Number

modularity

m V D = ∑ i , j ∈ V , i ≠ j V 1 d ( i , j ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGTbqBcqWGwbGvcqWGebarcqGH9aqpdaaeWbqaamaalaaabaGaeGymaedabaGaemizaqMaeiikaGIaemyAaKMaeiilaWIaemOAaOMaeiykaKYaaWbaaSqabeaacqaIYaGmaaaaaaqaaiabdMgaPjabcYcaSiabdQgaQjabgIGiolabdAfawjabcYcaSiabdMgaPjabgcMi5kabdQgaQbqaaiabdAfawbqdcqGHris5aaaa@4931@

d(i,j) is the length of the shortest path between vertices i and j. For this measure, i and j are all selected from V.

-

Eigenvalues

cycles

E V = ∑ j | E R j | 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrcqWGwbGvcqGH9aqpdaaeqbqaaiabcYha8jabdweafjabdkfasnaaBaaaleaacqWGQbGAaeqaaOGaeiiFaW3aaWbaaSqabeaacqaIYaGmaaaabaGaemOAaOgabeqdcqGHris5aaaa@3B61@

ER j is the real part of the j-th Eigenvalue for the adjacency matrix of the given subgraph.

[50]

Subgraph Centrality

cycles

S C = 1 N ∑ i = 1 N ∑ k = 1 ∞ ( A k ) i i k ! MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWucqWGdbWqcqGH9aqpdaWcaaqaaiabigdaXaqaaiabd6eaobaadaaeWbqaamaaqahabaWaaSaaaeaacqGGOaakcqWGbbqqdaahaaWcbeqaaiabdUgaRbaakiabcMcaPiabdMgaPjabdMgaPbqaaiabdUgaRjabcgcaHaaaaSqaaiabdUgaRjabg2da9iabigdaXaqaaiabg6HiLcqdcqGHris5aaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaaaa@4917@

A is the adjacency matrix. We computed SC for k [1,99].

[42]

Cyclic Coefficient

cycles

θ ( i ) = 2 k i ∗ ( k i − 1 ) ∗ ∑ j , k 1 S i ( j , k ) θ = 1 / N ∗ θ ( i ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaacciGae8hUdeNaeiikaGIaemyAaKMaeiykaKIaeyypa0ZaaSaaaeaacqaIYaGmaeaacqWGRbWAdaWgaaWcbaGaemyAaKgabeaakiabgEHiQiabcIcaOiabdUgaRnaaBaaaleaacqWGPbqAaeqaaOGaeyOeI0IaeGymaeJaeiykaKcaaiabgEHiQmaaqafabaWaaSaaaeaacqaIXaqmaeaacqWGtbWudaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdQgaQjabcYcaSiabdUgaRjabcMcaPaaaaSqaaiabdQgaQjabcYcaSiabdUgaRbqab0GaeyyeIuoaaOqaaiab=H7aXjabg2da9iabigdaXiabc+caViabd6eaojabgEHiQiab=H7aXjabcIcaOiabdMgaPjabcMcaPaaaaaa@590D@

S i is the smallest possible cycle of vertex i and two of its neighboring vertices k. The total Cyclic Coefficient for all vertices N is then given as θ

[42]

  1. Name, formal representation, and short description of graph measures computed for the categories of size, distribution, biological relevance, density, modularity, and cycles.