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other applications, the interconnectedness of a network is of critical impor- sance. A network can be expressed visually with 'edges' connecting 'nodes', as in the example shown below. Mathematically, this information can be stored in the associated 'adjacency natrix' A = (a¿¿) where, for each pair of nodes (i, j), we have aij 1, 0, if an edge connects (i, j); otherwise. Remarkably: the number of paths of length a from node i to node j is equal to the (i, j) entry of the matrix power Aª. (a) Compute the 4 × 4 adjacency matrix A for the network displayed below. (Note: the diagonal entries až = 0 since there are no edges connecting a node to itself.) (b) Compute A² and A³ and hence complete the displayed tables.

other applications, the interconnectedness of a network is of critical impor- sance. A network can be expressed visually with 'edges' connecting 'nodes', as in the example shown below. Mathematically, this information can be stored in the associated 'adjacency natrix' A = (a¿¿) where, for each pair of nodes (i, j), we have aij 1, 0, if an edge connects (i, j); otherwise. Remarkably: the number of paths of length a from node i to node j is equal to the (i, j) entry of the matrix power Aª. (a) Compute the 4 × 4 adjacency matrix A for the network displayed below. (Note: the diagonal entries až = 0 since there are no edges connecting a node to itself.) (b) Compute A² and A³ and hence complete the displayed tables.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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In epidemiology, genetics, communications, transportation, logistics and many other applications, the interconnectedness of a network is of critical impor- tance. A network can be expressed visually with 'edges' connecting ‘nodes’, as in the example shown below. Mathematically, this information can be stored in the associated 'adjacency matrix' A = (ažj) where, for each pair of nodes (i, j), we have Aij = { 1, 0, if an edge connects (i, j); otherwise. Remarkably: the number of paths of length q from node i to node j is equal to the (i, j) entry of the matrix power A¹. (a) Compute the 4 × 4 adjacency matrix A for the network displayed below. (Note: the diagonal entries a = 0 since there are no edges connecting a node to itself.) (b) Compute A² and A³ and hence complete the displayed tables.
Transcribed Image Text:In epidemiology, genetics, communications, transportation, logistics and many other applications, the interconnectedness of a network is of critical impor- tance. A network can be expressed visually with 'edges' connecting ‘nodes’, as in the example shown below. Mathematically, this information can be stored in the associated 'adjacency matrix' A = (ažj) where, for each pair of nodes (i, j), we have Aij = { 1, 0, if an edge connects (i, j); otherwise. Remarkably: the number of paths of length q from node i to node j is equal to the (i, j) entry of the matrix power A¹. (a) Compute the 4 × 4 adjacency matrix A for the network displayed below. (Note: the diagonal entries a = 0 since there are no edges connecting a node to itself.) (b) Compute A² and A³ and hence complete the displayed tables.
3 २ 14 (# of length 1 paths 1-1) = 0 (# of length 1 paths 2-2) = ? (# of length 1 paths 1-2) = ? (# of length 1 paths 1→3) = ? (# of length 2 paths 1-2) = 0 (# of length 2 paths 1-3) = 1 (# of length 2 paths 2→2) = ? (# of length 2 paths 3-1) = ? (# of length 2 paths 3-3) = 2 (# of length 3 paths 1-1) = ? (# of length 3 paths 1-2) = ? (# of length 3 paths 3-2) = 4 (# of length 3 paths 4→2) = ? (# of length 3 paths 4-4) = ?
Transcribed Image Text:3 २ 14 (# of length 1 paths 1-1) = 0 (# of length 1 paths 2-2) = ? (# of length 1 paths 1-2) = ? (# of length 1 paths 1→3) = ? (# of length 2 paths 1-2) = 0 (# of length 2 paths 1-3) = 1 (# of length 2 paths 2→2) = ? (# of length 2 paths 3-1) = ? (# of length 2 paths 3-3) = 2 (# of length 3 paths 1-1) = ? (# of length 3 paths 1-2) = ? (# of length 3 paths 3-2) = 4 (# of length 3 paths 4→2) = ? (# of length 3 paths 4-4) = ?
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