Transcript Lecture 12 Network Analysis (3)

Lecture 12 Network Analysis (3)
12-1 Network Connectivity – C1Matrix
To measure the connectivity of a network, we
can use a Matrix set – called C matrices.
Connectivity matrices are useful in evaluating
network accessibility.
The first order connectivity matrix, defined as
the C1matrix, is based on the direct connection
between nodes.
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12-1 Network Connectivity –
Matrix (Cont.)
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12-1 Network Connectivity –
Matrix (Cont.)
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C
About C1 Matrix:
-Cell value 1 represents a direct connection between two
nodes (P and Q).
-Cell value 0 represents no direct connection between
two nodes.
-The last column shows the sum of values for each row.
The row sum indicates the number of links directly
connected to the corresponding node.
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12-2 Network Connectivity –
Matrix
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C
C1 matrix is based on direct connection.
Transportation systems always consist of both direct
and indirect connections. (for example, airline
passengers can choose their preferences – no
connection, one connection, or more.). So we will
need high-order C matrix:
C2= C1 X C1
Each cell in C2 matrix is the sum of the products of all
elements in the corresponding row and column.
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12-2 Network Connectivity –
Matrix (Cont.)
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C
Cell value in C2:
The row P, column Q cell in the C2 matrix is the sum of
the products of elements in row P and the
corresponding elements of column Q in the C1
matrix.
For example:
CP2 7,Q 3  C71,1 * C11,3  C71, 2 * C21,3  ...  C71,8 * C81,3
 0 *1  0 * 0  1* 0  0 * 0  1*1  0 *1  0 *1  1*1
2
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12-2 Network Connectivity –
Matrix (Cont.)
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12-2 Network Connectivity –
Matrix (Cont.)
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C
About C2 Matrix:
-Every element in the C2 matrix signifies the number of
unique ways that one could move from one node to
another through an indirect path of exactly two links.
-The row sum in the C2 matrix indicates the number of
different ways one could move from the corresponding
node to all other nodes in exactly two steps in the
system.
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12-3 Network Connectivity –
Matrix
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C
Indirect connectivity can be extended beyond the second order:
C3= C1 X C2
About C3 matrix:
-It is the connectivity matrix for indirect connection of exactly
three steps. Every element represents the number of ways to move
from one node to the other in exactly three steps.
-The row sum in the C3 matrix indicates the number of different
ways one could move from the corresponding node to all other
nodes in exactly three steps in the system.
-Matrix order larger than diameter value will contain redundant
information.
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12-3 Network Connectivity –
Matrix (Cont.)
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12-4 Network Accessibility
Accessibility matrix (T matrix):
- Sum of all meaningful C matrices from the first order
to the order equal to the network diameter. In the
above example, T= C1+C2+ C3
- A cell in the T matrix is the sum of values of the
corresponding row and column in the C1,C2 ,
C3 , ….matrices.
- A node with a higher row sum has greater
accessibility to the rest of the network.
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12-4 Network Accessibility (Cont.)
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In general, the
larger the
network
accessibility
value, the more
route choices are
available in the
system and the
better connected
the nodes.
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