【グラフ理論とネットワーク理論】Extended Complete Graph
Connectivity and Damage Index
Connectivity
Connectivity express the reliability of the graph
- Node Connectivity Cv(G) : Minimum number of nodes to be removed to make a graph unconnected.
- Branch Connectivity Ce(G) : Minimum number of branches to be removed to make a graph unconnected.
- Node connectivity = d → d - 1 node failures OK
- Branch connectivity = d → d - 1 branch failures OK
Theorem1
Cv(G) ≦ Ce(G) ≦ δ(G) : δ = degree (the number of branches of one node)
Damage index
Damage index is an index of the increase of hops to connect in case of failure.
α-Node(Branch) Damage Index Iα(G) (Jα(G))
is defined as blow
Maximum diameter of a graph without α nodes(branches)
-Diameter of the original graph G
Design Method of Extended Complete Graph
Extended Complete Graph : Complete graph Gd → Line conversion, m times → Extended complete graph.
Lm(Gd) = L(L(…(L(G))…)) (m times)
Ex. 1
Number of nodes | Number of branched | |
---|---|---|
Gd | d+1 | (d+1)d |
L(Gd) | (d+1)d | (d+1)d2 |
L2(Gd) | (d+1)d2 | (d+1)d3 |
Comparison
Complete graph | Extended Complete Graph | |
---|---|---|
正則性 Regularity | d-Regular | d-Regular |
Number of nodes V |
d + 1 | (d + 1)dm |
Number of branches E |
(d + 1)d | (d+1)dm+1 |
直径 Diameter k(G) | 1 | m+1 |
Node connectivity Cv(G) | d | d |
Branch connectivity Ce(G) | d | d |
次数 Degree δ(G) | d | d |
節点罹障度 Node damage index Iα(G) | 0 | ≦ m |
枝罹障度 Branch damage index Jα(G) | 1 | ≦ m+1 |
Throughput ratio in case of failure
k(G)/(Jα(G) + k(G)) → 0.5 for both Gd and Lm(Gd)