Lecture 11 Network Analysis (2)11-1 Evaluation of Network Structure – γ Index (Cont.)Slide 311-2 Evaluation of Network Structure – α Index11-2 Evaluation of Network Structure – α Index (Cont.)Slide 611-3 Network Diameter19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC11Lecture 11 Network Analysis (2)11-1 Evaluation of network structure – γ IndexRelative complexity and connectivity – can be evaluated in several ways (γ and α).The γ index is defined as the ratio of the actual number of links to the maximum possible number of links in a network.Left example: Actual links – 4 Maximum possible number of links - 619/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC2211-1 Evaluation of Network Structure 11-1 Evaluation of Network Structure – – γ Index (Cont.)Index (Cont.)A graph is planar if it can be drawn in a plane without graph edges crossing. In a planar graph, the maximum number of links is always equal to 3(n-2). n: number of nodes.network. ain links ofnumber :)2(3maxlnlllThe value of the γ index ranges between 0 and 1. A value closer to 0 indicates? Closer to 1?19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC3311-1 Evaluation of Network Structure 11-1 Evaluation of Network Structure – – γ Index (Cont.)Index (Cont.)For non-planar graphs (such as airline transportation), the maximum number of links is equal to n(n-1)/2.ABCDEFABCDEF19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC4411-2 Evaluation of Network Structure 11-2 Evaluation of Network Structure – – αα IndexIndexThe α index is defined as the ratio of the actual number of circuits to the maximum number of circuits in the network.Circuit – A circuit is a loop in the network.When a loop exists, each node has two ways to reach any other node in the network.52maxnccc19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC5511-2 Evaluation of Network Structure 11-2 Evaluation of Network Structure – – αα Index (Cont.)Index (Cont.)ABCDGraph (network) ABCD has 4 nodes. Maximum loop number is 3. αα value is 1/3. Question: How many ways you can go from A to C?19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC6611-2 Evaluation of Network Structure 11-2 Evaluation of Network Structure – – αα Index (Cont.)Index (Cont.)The αα value evaluates network structure in terms of the number of ways you can proceed from one node to another.In general:A better developed transportation network<->higher values on both αα and γ <->higher levels of complexity and connectivity.19/1/1419/1/14Jun Liang, Geography @ UNCJun Liang, Geography @ UNC7711-3 Network Diameter11-3 Network DiameterThe diameter of a network represents the maximum number of steps required to move from any node to any other node through the shortest possible routes within a connected network.12345678Diameter of the left graph is