A representation for the Mexican political networks

Social Networks 29 (2007) 81–92

A representation for the Mexican political networks夽 Philip A. Sinclair The British University in Egypt, El Sherouk City, Misr-Ismalia Desert Road, Postal No. 11837, P.O. Box 43, Egypt

Abstract A compressed graph representation for use with the Mexican political networks is introduced. Properties of these graphs are investigated. It is also explained how the Jorge–Schmidt power centrality index can be used to index the centrality of nodes in the original network from the compressed graph representation. © 2006 Elsevier B.V. All rights reserved. Keywords: Centrality; Political networks

1. Introduction At the Institute for Applied Mathematics at the National University of Mexico, researchers have developed the REDMEX database that contains information on politicians active in the political arena of Mexico from the 1920s through to the turn of the century. The database is implemented using Access and algorithms have been developed to allow the researchers to obtain information about parameters of the networks implicit in the database. One of the beautiful aspects of the networks is that they can be given a graphical representation which can reveal aspects of the network that are otherwise hidden. The famous representation of the political network of power of Mexico due to Gil and Schmidt (1996) clearly shows the unbroken reign of the ruling elite. Their analysis of this network shows that there are two main blocks, a military block in which the majority of the politicians are connected through the Mexican revolution and the military, and later a financier block in which relationships have been built through the financial sector. To help with their analysis Gil and Schmidt (1996) introduced an index of vertex centrality, the Gil and Schmidt power centrality index. The Gil–Schmidt power centrality index is a generalised degree centrality index that is comparable with the closeness 夽 The research for this paper was supported by IIMAS-UNAM. The opinions expressed herein are those of the author and not of the sponsoring agency.

E-mail address: [email protected] (P.A. Sinclair). 0378-8733/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.socnet.2006.03.001

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centrality index in that it uses distances from the indexed vertex to other vertices in the calculation of the index. Gil et al. (1997) found that the centrality values of the members of the Mexican network of power show an overall decrease during the century reflecting the fact that the network became less cohesive as the century progressed. In this paper a compressed network representation of the Mexican political system for a given year is introduced. The representation shows the clique/co-clique structure of the network and provides a convenient overview. It is explained how the Gil–Schmidt index for vertices in the original graph can be calculated from the compressed network. This can reduce computation time as the order of the compressed network is much reduced from the original political network. Another possible application for the compressed network representation is to obtain a centrality index for cliques. 2. Definitions All networks and graphs will be loopless and undirected. Let n(G) = |V (G)| be the number of vertices in the network or graph G. Let ρG (u) denote the degree of u ∈ V (G). For vertices ui and uj in the same component of G, let dG (ui , uj ) denote the distance between ui and uj in G. Let the eccentricity of ui ∈ V (G), be defined as eG (ui ) = maxuj ∈V (G) dG (ui , uj ). For 0 ≤ k ≤ eG (ui ), let Nk (ui ; G) = Nk (ui ) = {uj ∈ V (G)|dG (ui , uj ) = k}, be the kth-order neighbourhood of G with respect to ui , and let nk (ui ; G) = nk (ui ) = |Nk (ui ; G)| be the number of vertices in the kth-order neighbourhood. Then N0 (ui ) = {ui }, N1 (ui ) also written NG (ui ) is the set of neighbours of ui in G, and n1 (ui ) = ρ(ui ). If G is not connected then the above definitions apply to the component of G to which the vertex ui belongs. 3. The power index of centrality The Gil–Schmidt power centrality index, Gil and Schmidt (1996) and see also Gil et al. (1999), was developed by Gil and Schmidt to help in their analysis of the Mexican political networks. It is intuitively appealing because it is a generalised degree centrality index. Applied to a particular vertex v the index counts the degree of the vertex with weight 1, the order of the second-order neighbourhood with weight 21 , and in general, the order of the kth-order neighbourhood with weight 1k . Thus, for any particular actor in a network the most important other actors are the immediate neighbours and then in decreasing importance the actors of the second-order neighbourhood, third-order neighbourhood, etc. The sum obtained is then normalised by the number of actors in the same component as v. Formally, the Gil–Schmidt power centrality index is defined for all ui ∈ V (G) as I  (ui ; G) =

e(u ) ne(ui ) (ui ) 1 n2 (ui ) 1 i nj (ui ) + ··· + {n1 (ui ) + }= ti − 1 2 e(ui ) ti − 1 j

(1)

j=1

where ti = n1 (ui ) + n2 (ui ) + · · · + ne(ui ) (ui ), that is ti is the order of the component of the network 1 is for the purpose of normalisation so that all centrality scores that contains ui . The factor ti −1 lie between zero and one; this makes comparisons between different networks easier. Different normalisations can be also used, e.g. for network density, and in much of the following une(u ) n (u ) normalised centrality is used for the sake of brevity. Let I(ui ; G) = I(ui ) = j=1i j j i be the centrality score of ui without normalisation. Then I  (ui ) =

I(ui ) ti −1 .

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4. Representing a political network The REDMEX database at UNAM contains information on over 5400 members of the Mexican government. For each of these people attributes are listed: personal information, education, political activities, membership of social groups, elected positions, congressional positions, government positions, professional activities, academic positions, publications, membership of professional groups, awards and decorations, international representations, and commissions. From certain attributes dyadic relations can be inferred. For example, in an analysis of the network surrounding the president Miguel Aleman Valdes, Gil et al. (1999) use four categories of relationship: military, family, friendship and political. For small networks identifying the relationships can be done by hand, however for large sets of politicians algorithms have been developed at UNAM to identify whether a relationship is present or not. If all actors recorded in the database and all relationships between pairs of actors were included a giant network of the Mexican political network would be created. The Mexican network of power described in Gil and Schmidt (1996) is a longitudal view of the elite of this of this network. The networks referred to in this paper were formed from data taken from the database at 5-yearly intervals and included all the politicians recorded in the database for a given year. In this paper neither the number nor type of relationship between pairs of actor is analysed, but only whether any kind of recorded relationship is present or not. Thus, the networks analysed are simple, that is they do not contain multiple edges. Another characteristic of these networks is that every politician belongs to a clique. A clique is a maximal subgraph of order at least three in which every pair of vertices is adjacent. Let the clique union subgraph J of G be the subgraph of G that is the union of all the cliques of G. If J = G then G is called a clique union network. Let C(G) = {C1 , . . . , Cp } be the set of all cliques of a network G. Let A(G) = {a1 , . . . , aq } be the set of all vertices that belong to more than one clique of C(G). Let the clique intersection network of G, ΓA , be the bipartite network with V (ΓA ) = C(G) ∪ A(G) and E(ΓA ) = {ak Ci iff ak ∈ V (Ci ) ∩ V (Cj ), Ci , Cj ∈ C(G)}. If G is a clique union network then V (G) = ∪Ci ∈C(G) V (Ci ) and E(G) = ∪Ci ∈C(G) E(Ci ). Let each Ci in V (ΓA ) be written ci . Let eΓA (u) be the eccentricity of u in ΓA , treating ΓA as an ordinary network. Some simple properties of the clique intersection network are now described and a method to calculate the Gil and Schmidt power centrality index of individual politicians from the clique intersection network is given. Note that by comparing the Gil and Schmidt power centrality index of the vertices ci in ΓA the centrality of different cliques in G can be compared. However, the numbers calculated will not typically be the same as that calculated for the clique centrality index in Gil et al. (1999). Theorem 1. Let G be a clique union network. Then   n(Ci ) − ρΓA (ai ) + |A(G)| n(G) = Ci ∈C(G)

=



Ci ∈C(G)

=



Ci ∈C(G)

ai ∈A(G)

n(Ci ) −



ρΓA (ci ) + |A(G)|

Ci ∈C(G)

(n(Ci ) − 1) − |E(ΓA (G))| + n(ΓA )

(2)

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Fig. 1. The clique intersection network of the political network of Mexico for 1987, Γ87 .

 Proof. When we make the sum Ci ∈C(G) n(Ci ) we count i ∈ A(G), ρΓA (ai ) times,  each vertex a once for each clique that ai belongs to. Thus, the sum Ci  n(C ) − i ∈C(G) ai ∈A(G)  ρΓA (ai ) counts ) − n(C the number of vertices of G that do not belong to A(G) and i Ci ∈C(G) ai ∈A(G) ρΓA (ai ) +   |A(G)| = n(G). Since ΓA (G) is bipartite, ai ∈A(G) ρ (a ) = ρ (c ) and the second line Γ i Γ i A Ci ∈C  A of (2) is true. Finally, because ΓA (G) is bipartite, Ci ∈C(G) ρΓA (ci ) = |E(ΓA (G))| and so the final line of Eq. (2) is true. 

Order dΓ87 (ci )

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

66 13

8 8

62 10

8 1

17 1

43 1

27 2

5 0

25 2

15 0

14 4

17 3

31 1

25 1

The 14 cliques and their respective orders.

Let S ⊆ V (G) be such that S = ∅. Suppose that u, v ∈ V (G) belong to the same component of G, but to different components of G − S, then S separates u from v in G and S is called a separating set of G. For example, in the Fig. 1 the set {x, y, z} is a separating set in Γ87 . If |S| = 1 then S is said to contain a separating vertex. A connected network G for which n(G) ≥ 2, is k-connected if either there exists a spanning subgraph H of G and H is a clique on p vertices, for p ≥ k + 1, or if there exist at least two non-adjacent vertices in G and there does not exist a separating set S ⊆ V (G) such that |S| ≤ k − 1. Let ΓA be a clique union network then ΓA is said to be k-connected, for k ≥ 1, if ΓA is connected, |C(ΓA )| > 1 and there does not exist a separating set S for any S ⊆ A(G) and |S| < k. Separating sets in a network are important because they represent sets of actors within which a change can have a strong effect on the network. In the extreme case the loss of a separating vertex disconnects a component in the network. If more than one actor belongs to a separating set then the loss of one actor can effect the importance of the remaining actors. For example, if the politician x left the network of Fig. 1 then the network would become less cohesive and we might argue that the importance of politician y to the network would increase. This change for y is not reflected by the change in centrality of y and in this sense the analysis of separating sets can be seen as complementary to the analysis obtained using the Gil-Schmidt power centrality index.

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Theorem 2. Suppose that G is a clique union network. If ΓA is k-connected then G is k-connected. If G is connected then ΓA (G) is connected. Proof. We first show that if ΓA is k-connected then G is k-connected and start by proving directly that if ΓA is connected then G is connected. So suppose that ΓA is connected. Let u ∈ V (Ci ) and v ∈ V (Cj ), for Ci , Cj ∈ C(G). Suppose that P = x1 , . . . , xt , x1 = ci , xt = cj is a path in ΓA . Because ΓA is bipartite, the vertices of P alternate between vertices ck ∈ C(G) and vertices ak ∈ A(G). Thus, the subsequence x2 , x4 , . . . , xt−1 of vertices of A(G) in P defines a path in G since x2r and x2(r+1) belong to the same clique, for 1 ≤ r ≤ t−3 2 . Hence, u, x2 , x4 , . . . , xt−1 , v is a path in G and as u and v have been chosen arbitrarily we see that any pair of vertices in G are connected by a path. Thus, G is connected. The rest of the proof is by contradiction; suppose that ΓA is k-connected, but that G is not k-connected, for k ≥ 2. By definition, |V (ΓA )| > 1, therefore |C(G)| > 1 and G is not a clique. Thus, we can choose u, v ∈ V (G) such that there exists a uv-separating set S ⊆ V (G), with |S| < k. Let S be chosen such that |S| is minimum, that is so that there is no smaller set for which G − S is disconnected. Because ΓA is k-connected, S ⊆ A(G), for otherwise S would be a separating set in ΓA . Let v ∈ S\A(G) such that v ∈ V (Ck ) for Ck ∈ C(G). Let S  = S − {v}. By minimality, G − S  is connected and v is a separating vertex of G − (S  \{v}). Let H1 and H2 be two components of G − S such that there exists h1 ∈ V (H1 ) ∩ NG (v) and h2 ∈ V (H2 ) ∩ NG (v). Since G is a clique union graph and vh1 , vh2 ∈ E(G) both v and h1 belong to the same clique and both v and h2 belong to the same clique. Now however, because h1 h2 ∈ E(G), h1 and h2 belong to different cliques of G and v ∈ A(G), a contradiction to the choice of v. Now we prove directly that if G is connected then ΓA is connected. We show that any pair of vertices in ΓA are connected by a path. Choose any two arbitrary distinct vertices of ΓA , say xi and xj . Since each vertex of G belongs to a clique it follows that there is a vertex vi ∈ V (G) such that either vi = xi ∈ A(G) or vi ∈ V (Ci ) and ci = xi . Similarly, there is a vertex vj ∈ V (G) such that vj = xj ∈ A(G) or vj ∈ V (Ci ) and cj = xj . Since G is connected there exists a vi vj -path in G and it is easy to see how this induces an xi xj -path in ΓA .  It is not true that if G is k-connected then Γ (G) is k-connected, see for example the networks of Fig. 2. Note that the example of Fig. 2 relies on the existence of cliques of low order, as the vertices of the separating set form a clique. The following two theorems describe how the Gil and Schmidt power centrality index of a politician can be calculated from the clique intersection network, they can be proved using an inductive argument (see Appendix A).

Fig. 2. G is 4-connected, ΓA (G) is not 4-connected.

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Theorem 3. Let G be a clique union network suppose that there exists a u ∈ V (C1 )\A(G), for C1 ∈ C(G). Let ΓA be the clique intersection network of G. If eΓA (c1 ) is odd then  m  dΓA (c1 ,cj )=2i {n(Cj ) − ρΓA (cj )} + n2i+1 (c1 ) I(u) = n(C1 ) − 1 + (3) i+1 i=1

eΓA (c1 )−1 . 2

where m = If eΓA (c1 ) is even then I(u) = n(C1 ) − 1 + + where m =

2



eΓA (c1 ) 2

m 



dΓA (c1 ,cj )=2i {n(Cj ) − ρΓA (cj )} + n2i+1 (c1 )

i=1

i+1

dΓA (c1 ,cj )=eΓA (c1 ) {n(Cj ) − ρΓA (cj )}

eΓA (c1 ) + 2

− 1.

Theorem 4. Let G be a clique union network. Let ΓA be the clique intersection network of G and let v ∈ A(G). If eΓA (v) is even then  m  dΓA (v,cj )=2i−1 {n(Cj ) − ρΓA (cj )} + n2i (v; ΓA ) I(v) = i i=1

e

(v)

where m = ΓA2 . If eΓA (v) is odd then  m  dΓA (v,cj )=2i−1 {n(Cj ) − ρΓA (cj )} + n2i (v; ΓA ) I(v) = i i=1  2 dΓ (v,cj )=eΓ (v) {n(Cj ) − ρΓA (cj )} A A + eΓA (v) + 1 where m =

(4)

eΓA (v)−1 . 2

Example 1. Let z be the vertex shown in the network of Fig. 1. At distance 1 lie the vertices: c1 and c9 , at distance 3 lie the vertices c2 , c3 , c4 , c6 , c7 , c11 , c12 and c13 , and at distance 5 lie the vertices c5 and c14 . For i = 1, the sum in Eq. (4) of Theorem 4 is  25 + 66 − 15 + 13 dΓA (z,cj )=1 {n(Cj ) − ρΓA (cj )} + n2 (z; ΓA ) = = 89 1 1 For i = 2, the sum in Eq. (4) is  310 − 30 + 5 185 1 dΓA (z,cj )=3 {n(Cj ) − ρΓA (cj )} + n4 (z; ΓA ) = = = 92 2 2 2 2 For i = 3, the sum in Eq. (4) is  1 40 dΓA (z,cj )=5 {n(Cj ) − ρΓA (cj )} = 13 = 3 3 3

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Fig. 3. A part of the clique intersection network of the political network of 1985, Γ85 .

Hence, by Theorem 4 5 185 40 + = 194 2 3 6   By Theorem 1, n(G) = i n(Ci ) − ρ(ci ) + |A(G)| = 363 − 47 + 19 = 335. The normalisation is done with the order of the component containing z which is 335 − 15 − 5 − 1 = 314. Then 5 1169 1 I  (z) = × 194 = = 0.620 . . . 314 6 1884 I(z) = 89 +

Order dΓ85 (ci )

C1

C2

C3

C6

C10

C12

67 12

20 5

18 11

13 3

39 3

4 3

Let G be the network of Fig. 3. The data for the network is taken from the political network of Mexico for 1985. Let u be a vertex of V (C1 )\A(G). Then e(c1 ) = 5 and by equation (3) of Theorem 3, n(C2 ) + n(C3 ) − ρΓA (c2 ) − ρΓA (c3 ) + n3 (c1 ) 2 n(C6 ) + n(C10 ) + n(C12 ) − ρΓA (c6 ) − ρΓA (c10 ) − ρΓA (c12 ) + n5 (c1 ) + 3 20 + 18 − 5 − 11 + 4 13 + 39 + 4 − 3 − 3 − 3 + 1 + = 95 = 66 + 2 3 By Theorem 1, n(G) = 141 and so I(u) = n(C1 ) − 1 +

95 = 0.674 . . . 141 The clique union networks are an effective way to give a simple representation for the political networks because the majority of vertices in a political network for a given year belong to a I  (u) =

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Fig. 4. A part of the clique intersection network of the political network of 1985, Γ85 with the cliques C1 and C2 replaced with a 2-clique.

relatively small number of large cliques. It can also be the case that a large number of politicians belong to two cliques that contain many politician in common. In this case a body of politicians can be represented using a 2-clique, see for example Fig. 4. By doing this we simplify the representation and we can still see where separating sets of low order in the network are. The t-clique was introduced, by Luce (1950). A t-clique H is a maximal subgraph H of G in which dG (ui , uj ) ≤ t, for every ui , uj ∈ V (H) (there is no subgraph H  of G such that dG (ui , uj ) ≤ t for every ui , uj ∈ V (H  ) and H is a subgraph of H  ). The minimum degree in a t-clique is the most important factor in determining the lower bound for the value of the centrality index in that clique. In a clique network I  (u) = 1 for all vertices in the clique; this is not true for a 2-clique and Theorems 3 and 4 should not be used. Theorem 5 gives a lower bound for the value of I(u). The lower bound for a vertex u is obtained because each kth-neighbourhood for k < e(u) forms a separating set. Theorem 5. Let G be a k-connected t-clique with minimum degree δ. Then for u ∈ V (G): 1 n − 1 + δ(t − 1) − k(2t − 1) +k i t t

I(u) ≥

i=1

Proof. Since G is k-connected, and each Ni (u) is a separating set for 1 ≤ i ≤ eG (u) − 1, ni (u) ≥ k for each 1 ≤ i ≤ eG (u) − 1. It follows that I(u) ≥ δ +

k k n − 1 − du − k(eG (u) − 2) + ··· + + 2 eG (u) − 1 eG (u)

= δ−k+k

e G (u) i=1

=

1 n − 1 − δ − k(eG (u) − 1) + i eG (u)

e G (u) n − 1 + δ(eG (u) − 1) − k(2eG (u) − 1) 1 +k . eG (u) i i=1



P.A. Sinclair / Social Networks 29 (2007) 81–92

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Example 2 (T). he minimum value of a vertex in a 2-clique with c vertices and minimum degree δ is I(u) =

c+δ−1 2

With the normalisation I  (u) = If δ =

c−1 2

1 c−1

c+δ−1 2(c − 1)

then I  (u) = 0.75 and if δ =

c−1 5

then I  (u) = 0.6.

5. Conclusion In this paper a representation that can be used to compresses big networks has been introduced. The representation is especially useful when the original network has a small number of large cliques. It is shown how the Gil–Schmidt power centrality index of vertices in the original network can be calculated with the compressed networks. This can lead to savings in computation time as the order of the compressed network is greatly reduced. The technique of reducing cliques to form a compressed network can be generalised to include other dense subgraphs. It may perhaps, in simple cases, be possible to develop a formula for the Gil–Schmidt power centrality index with such compressed networks. In the above analysis, the differences between different politicians occur because they belong to different cliques or to the intersection of cliques. Due to the nature of politics, politicians tend to be grouped in cliques and because of this it is of interest for a coarse view of the network to consider the clique structure. Much of the structure in a political network for a given year actually occurs inside the cliques as described above, because politicians hold different offices, and also because there are multiple and varied relationships. Gil et al. (1999) proposed a model composed of many networks induced by the relations contained in the database. Thus, here can be a network induced by a relationship ‘friends at school’, and one by the relationship ‘belonged to the same government committee’, and by the relationship ‘belongs to the same family’, and so on. In this way a politician can be seen as belonging to many different networks. There can be several different approaches to indexing the centrality politicians in such sets of networks. Acknowledgments The author would like to express his thanks to Prof. Jorge Gil, Alejandro Ruiz, and an anonymous referee for their help. Appendix A For subsets A and B of V (G) let E(A, B) denote the set of edges with one end in A and one end in B and let |E(A, B)| be the number of these edges. Proof of Theorem 3. By choice of u, N1 (u; G) = V (C1 )\{u}. Thus, n1 (u) = n(C1 ) − 1. The vertices of C1 are represented by the vertices of N1 (u) in ΓA . We claim that for 1 ≤ i < eG (u) − 1:

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ni+1 (u; G) =

n(Cj ) − |EΓA (N2i (c1 ), N2i (c1 ))| + n2i+1 (c1 )

dΓA (c1 ,cj )=2i



=

{n(Cj ) − ρΓA (cj )} + n2i+1 (c1 )

(5)

dΓA (c1 ,cj )=2i

The vertices that are in N2 (u; G) are the vertices that are in the cliques Cj of G for which dΓA (c1 , cj ) = 2, but that are not vertices of N1 (c1 ; ΓA ). In the sum  n(Cj ) (6) dΓA (c1 ,cj )=2

the vertices of N1 (c1 ; ΓA ) are counted |EΓA (N1 (c1 ), N2 (c1 ))| times; each vertex of N1 (c1 ; ΓA ) is counted once for each clique represented by a vertex of N2 (c1 ; ΓA ) that the vertex belongs to. Furthermore, some of the vertices of N3 (c1 ; ΓA ), may belong to more than one clique in the sum of Eq. (6). The vertices in N3 (c1 ; ΓA ) are counted |EΓA (N2 (c1 ), N3 (c1 ))| times in the sum of Eq. (6). Hence  n(Cj ) − |EΓA (N1 (c1 ), N2 (c1 ))| n2 (u; G) = dΓA (c1 ,cj )=2

− |EΓA (N2 (c1 ), N3 (c1 ))| + n3 (c1 ; ΓA )  = n(Cj ) − |EΓA (N2 (c1 ), N2 (c1 ))| + n3 (c1 ; ΓA ) dΓA (c1 ,cj )=2

 Clearly, |EΓA (N2 (c1 ), N2 (c1 ))| = dΓ (c1 ,cj )=2 ρΓA (cj ). Hence, Eq. (5) holds when i = 1. A Since ΓA is bipartite, vertices at an odd distance from c1 in ΓA belong to A(G) and vertices at an even distance from c1 in ΓA represent cliques of G. It follows by induction, that for i < eG (u) − 1, the vertices at distance i + 1 from u in G belong to the cliques represented by the vertices in N2i (c1 ; ΓA ), and that Eq. (5) holds. Now consider the case i = eG (u) − 1. If eΓA (c1 ) is odd, say eΓA (c1 ) = 2p + 1, then the (2p + 1)th distance class contains one or more vertices from A(G) and Eq. (5) holds, for i = p. e (c ) Otherwise, if eΓA (c1 ) is even, say eΓA (c1 ) = 2p, then for i = p = ΓA2 1 :  n(Cj ) − |EΓA (N2p−1 (c1 ), N2p (c1 ))| np+1 (u; G) = dΓA (c1 ,cj )=2p



=

{n(Cj ) − ρΓA (cj )}

dΓA (c1 ,cj )=2p

Then 2 np+1 (u; G) = p+1 =

2



dΓA (c1 ,cj )=2p n(Cj ) − 2|EΓA (N2p−1 (c1 ), N2p (c1 ))|



eΓA (c1 ) + 2

dΓA (c1 ,cj )=2p {n(Cj ) − ρΓA (cj )}

eΓA (c1 ) + 2

The result now follows from the definition of I(u; G).



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Proof of Theorem 4. The vertices of N1 (v; G) are those that belong to the cliques represented by the vertices of N1 (v; ΓA ). The vertex v is counted ρΓA (v) times in the sum 

n(Cj )

(7)

cj ∈N1 (v,ΓA )

once for each clique that v belongs to. The vertices of N2 (v; ΓA ) also belong to these cliques. They are either vertices that are common to more than one of the cliques represented by N1 (v; ΓA ), or that are common to the cliques represented by N1 (v; ΓA ) and those represented by N3 (v; ΓA ), or both. The vertices of N2 (v; ΓA ) are counted |EΓA (N1 (v), N2 (v))| times in the sum of Eq. (7). Hence 

n1 (v; G) =

n(Cj ) − ρΓA (v) − |EΓA (N1 (v), N2 (v))| + n2 (v; ΓA )

cj ∈N1 (v,ΓA )



=

{n(Cj ) − ρΓA (cj )} + n2 (v; ΓA )

cj ∈N1 (v,ΓA )

Since ΓA is bipartite, the vertices at odd distance from v in ΓA represent cliques of G and vertices at even distance from v in ΓA belong to A(G). By induction, the vertices  at distance i in G belong to the cliques represented by the vertices in N2i−1 (v; ΓA ). In the sum cj ∈N2i−1 (v) n(Cj ) the vertices that belong to more than one clique, those of N2i−2 (v; ΓA ) and those of N2i (v; ΓA ), are counted |EΓA (N2i−1 (v), N2i−1 (v))| times. The vertices in N2i (v; ΓA ) are distance i from v and those in N2i−2 (v; ΓA ) are distance i − 1 from v in G. e (v)−1 Hence, for 1 ≤ i ≤  ΓA 2 : 

ni (v; G) =

n(Cj ) − |EΓA (N2i−1 (v), N2i−1 (v))| + n2i (v; ΓA )

dΓA (v,cj )=2i−1



=

{n(Cj ) − ρΓA (cj )} + n2i (v; ΓA )

(8)

dΓA (v,cj )=2i−1

If eΓA (v) is even, say eΓA (v) = 2p, then the vertices of the (2p)th distance class of ΓA (G) are in A(G) and Eq. (8) holds for i = p. Otherwise, if eΓA (v) is odd, say eΓA (v) = 2p + 1, then for e (v)+1 i = p + 1 = ΓA 2 : np+1 (v; G) =



n(Cj ) − |EΓA (N2p (v), N2p+1 (v))|

dΓA (v,cj )=2p+1

=



dΓA (v,cj )=2p+1

Thus, result is as required.



{n(Cj ) − ρΓA (cj )}

92

P.A. Sinclair / Social Networks 29 (2007) 81–92

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