0 200 400 600 0
2 4 6 8 10 12 14
Network Size
AverageShortestPathLength
Maximum
Random
Minimum
Figure 3.4: Tight bounds on the average distance of a graph with N = 40 vertices and 39≤m≤780 edges. These bounds have been computed analytically. The average shortest path length for random graphs has been estimated by finding the mean shortest path length of 104randomly generated graphs of the same order and size. The expected average distance of a random graph is very close to the minimum, even for relatively sparse networks. For graphs with edge density ρ >0.25, it is virtually identical to the minimum one.
the same order and size. An example that shows the tight upper and lower bounds of the average distance of a graph with N = 40 and 39 ≤m ≤780 vertices is shown in Figure 3.4.
more generally signal propagation) among various nodes of a network, since it indi- cates how important each vertex or edge is for the function of such a network, and how robust it is with respect to vertex or edge removal [28]. The vertex betweenness centrality is defined as
B(u) = ∑
(s,t)∈V2(G) s̸=u̸=t
σst(u)
σst , (3.46)
where σst is the number of shortest paths between vertices s and t and σst(u) is the number of shortest paths between s and t that go through vertex u. Equation (3.46) computes the total number of shortest paths of all the pairs of vertices in the graph that go through a given vertex u. If there is more than one such path, we divide by their total number σst, since they are assumed to be equally important. The betweenness centrality of a vertex is sometimes normalized by the total number of all vertex pairs that we took into account for computing it, which is equal to (N−1
2
).
Bnorm(u) = 1 (N−1
2
) ∑
(s,t)∈V2(G) s̸=u̸=t
σst(u)
σst . (3.47)
The vertex betweenness is always nonnegative. The only vertices with betweenness centrality equal to zero are the ones with degree equal to 1. In order to assess the betweenness centrality of a network, we find the average of all vertices:
Bv(G) = 1 N
∑
u∈V(G)
B(u). (3.48)
Networks with a large betweenness centrality usually have few vertices that play a major role in the communications among every other vertex. Conversely, small betweenness centrality indicates that the vertices of the network tend to be equally important or that there are many different shortest paths among the various parts of the network.
The edge betweenness centrality is similarly defined as the sum of the fraction of
shortest paths of all vertex pairs in the network that go through a given edge:
B(f) = ∑
(s,t)∈V2(G) s̸=t
σst(f) σst
(3.49)
where in this case σst(f) is the number of shortest paths between s and t that go through edge f. The edge betweenness centrality of the network is defined in the same manner as before:
Be(G) = 1 m
∑
f∈E(G)
B(f). (3.50)
The betweenness of an edge is always positive for a connected network.
The betweenness centrality of a graph is an important proxy of how robust the network is to random vertex or edge removals. Removing a vertex or an edge with large betweenness centrality means that the communication among many vertex pairs will be affected, since they will now be forced to exchange information through al- ternative, possibly longer paths. Graphs which include nodes or edges with large betweenness centralities are sensitive to random removal of that set of vertices or edges. The vertex or edge betweenness centrality of a graph does not give any in- formation about the centralities of individual vertices or edges, which may largely vary from edge to edge. For networks with the same betweenness centrality, large variations among vertices or edges reveal a sensitivity to targeted attacks, since re- moving the most central vertices may significantly disrupt the network function. In this section we show that the betweenness centrality of a graph is inherently related to its average shortest path length.
Theorem 3. The average betweenness centrality of a network G(N, m) is a linear function of its average distance,
B(G) = (N−1)( ¯D(G)−1)
2 . (3.51)
Proof.
B(G) = 1 N
∑
u∈V(G)
B(u) = 1 N
∑
u∈V(G)
∑
(s,t)∈V2(G) s̸=u̸=t
σst(u) σst
= 1 2N
∑
u∈V(G)
∑
s∈V(G) s̸=u
∑
t∈V(G) t̸=u t̸=s
σst(u) σst
= 1 2N
∑
s∈V(G)
∑
t∈V(G) t̸=s
1 σst
∑
u∈V(G) u̸=s u̸=t
σst(u)
= 1 2N
∑
s∈V(G)
∑
t∈V(G) t̸=s
1
σstσst(|P(s, t)| −1) = 1 2N
∑
s∈V(G)
∑
t∈V(G) t̸=s
(d(s, t)−1)
= 1 2N
[ 2
(N 2
)
D(¯ G)−2 (N
2 )]
.
(3.52)
Simplifying the last equation, the average betweenness centrality of a graph becomes the one stated in the theorem.
It is worth mentioning that the average betweenness centrality of the network is only dependent on its size indirectly, through the average distance of the graph. For a fixed order, the average betweenness centrality of a network decreases as we add new edges (Lemma 1).
Corollary 10. A network has minimum (maximum) average betweenness centrality if and only if it has minimum (maximum) average distance. The minimum possible average betweenness centrality of a graph of order N and size m is equal to
Bmin(G) = N −1
2 − m
N (3.53)
and the maximum possible average betweenness centrality of such a graph is Bmax(G) =
(C
2
)+C(P+1
2
)+P(C−α) +(P+1
3
)
N − N −1
2 (3.54)
where C, P, and α are defined in equations (3.32), (3.33), and (3.34), respectively.
Proof. The networks with the smallest or largest average betweenness centrality are the graphs with the smallest or largest average distance respectively. Replacing them from equations (3.3) and (3.31), the bounds for the average betweenness centrality of graphs follow.
Corollary 11. The minimum sum of betweenness centralities of all the vertices of a network is equal to the number of vertices that are not neighbors.
Proof. From equations (3.48) and (3.53), we see that
G∈Cmin(N,m)
[ ∑
u∈V(G)
B(u) ]
=N · Bmin(G) = (N
2 )
−m. (3.55)
Theorem 4. The average edge betweenness centrality of a network is directly propor- tional to the average distance of the network, equal to
Be(G) = 1 m
(N 2
)
D(¯ G). (3.56)
Furthermore, the minimum and maximum average edge betweenness centrality of a network of order N, and size m are, respectively
Bemin(N, m) = N(N −1)
m −1 (3.57)
and
Bmaxe (N, m) = (C
2
)+C(P+1
2
)+P(C−α) +(P+1
3
)
m (3.58)
where C, P and α are the same as in equations (3.32), (3.33), and (3.34).
Proof. We follow the same method as in the proof of the vertex betweenness centrality:
Be(G) = 1 m
∑
e∈E(G)
B(e) = 1 m
∑
e∈E(G)
∑
(s,t)∈V2(G) s̸=t
σst(e) σst
= 1 m
∑
(s,t)∈V2(G)
∑
e∈E(G)
σst(e) σst = 1
m
∑
(s,t)∈V2(G)
1 σst
∑
e∈E(G)
σst(e)
= 1 m
∑
(s,t)∈V2(G)
1
σstσstd(s, t)
= 1 m
(N 2
) D(¯ G).
(3.59)
Replacing the average distance by its minimum and maximum bounds, we get equa- tions (3.57) and (3.58), respectively.