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Motivated by availability of computers and computer

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Unformatted text preview: availability of computers and computer networks that allow us to gather and analyze large scale data. New Analytical Approach: Find statistical properties that characterize the structure of these networks and ways to measure them Create models of networks Predict behavior of networks on the basis of measured structural properties and models 5 Networks: Lecture 2 Graphs Graphs—1 We represent a network by a graph (N , g ), which consists of a set of nodes N = {1, . . . , n } and an n × n matrix g = [gij ]i ,j ∈N (referred to as an adjacency matrix), where gij ∈ {0, 1} represents the availability of an edge from node i to node j . The edge weight gij > 0 can also take on non-binary values, representing the intensity of the interaction, in which case we refer to (N , g ) as a weighted graph. We refer to a graph as a directed graph (or digraph) if gij �= gji and an undirected graph if gij = gji for all i , j ∈ N . 1 ⎡ 0 Example 1: ⎣ 0 1 1 0 0 ⎤ 0 1 ⎦ ⇒ 0 2 3 1 ⎡ 0 Example 2: ⎣ 1 1 1 0 1 1 ⎤ 1 1 ⎦⇒ 0 ⇒ 2 3 2 3 6 Networks: Lecture 2 Graphs Graphs—2 Another representation of a graph is given by (N , E ), where E is the set of...
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