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Difficulty: MediumTransportation Modes and Network Connectivity

A regional transport planning board is evaluating four different transport network graph topologies that each connect five key industrial nodes (v=5v = 5). Arrange the following network topologies in order of their topological connectivity as measured by the Beta Index (β\beta), from the LOWEST connectivity level to the HIGHEST connectivity level.

  1. 1Linear tree network consisting of 4 transport links (e=4e = 4)
  2. 2Simple ring circuit network consisting of 5 transport links (e=5e = 5)
  3. 3Delta network consisting of 7 transport links (e=7e = 7)
  4. 4Complete mesh network consisting of 10 transport links (e=10e = 10)

Answer

Linear tree network (e=4,β=0.80e = 4, \beta = 0.80) ��� Simple ring circuit network (e=5,β=1.00e = 5, \beta = 1.00) → Delta network (e=7,β=1.40e = 7, \beta = 1.40) → Complete mesh network (e=10,β=2.00e = 10, \beta = 2.00)
Topological connectivity in network analysis is measured using the Beta Index formula β=ev\beta = \frac{e}{v}, where ee represents the number of links (edges) and vv represents the number of nodes (vertices). For a fixed set of v=5v = 5 nodes, the connectivity value increases as more links are added. Evaluating each option yields: Linear tree network (β=45=0.80\beta = \frac{4}{5} = 0.80), Simple ring circuit (β=55=1.00\beta = \frac{5}{5} = 1.00), Delta network (β=75=1.40\beta = \frac{7}{5} = 1.40), and Complete mesh network (β=105=2.00\beta = \frac{10}{5} = 2.00). Arranging these values in ascending order places the linear tree network first, followed by the simple ring circuit, the delta network, and finally the complete mesh network.

Step-by-Step Solution

1
Identify the formula for the Beta Index in network graph analysis
The Beta Index is defined as β=ev\beta = \frac{e}{v}, where ee is the number of edges (routes or links) and vv is the number of vertices (nodes or settlements).
The Beta Index quantifies the degree of connectivity of a network graph.
2
Calculate the Beta Index for each given network topology with v=5v = 5
Linear tree network: β=45=0.80\beta = \frac{4}{5} = 0.80; Simple ring circuit: β=55=1.00\beta = \frac{5}{5} = 1.00; Delta network: β=75=1.40\beta = \frac{7}{5} = 1.40; Complete mesh network: β=105=2.00\beta = \frac{10}{5} = 2.00.
Substituting the specific edge counts (ee) and vertex count (v=5v = 5) yields the exact numerical connectivity value for each network.
3
Sequence the topologies in ascending order of their Beta Index values
Rank order: 0.80<1.00<1.40<2.000.80 < 1.00 < 1.40 < 2.00.
The question specifies arranging from the lowest connectivity level to the highest connectivity level.

Key Concept

Topological Network Connectivity and Beta Index Analysis
Estimated Time:1m 30s
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Topics

Transportation Modes and Network Connectivity

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