Tively simple model can easily lead to cases where

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tively simple model can easily lead to cases where exact inference procedures are intractable. 1.3. Model Specification and Inference Fields such as probability, graph theory, and computer science have each posited specific models which can be applied to network data; however, Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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6 B. P. Olding and P. J. Wolfe when appealing to the existing literature, it is often the case that neither the models nor the analysis tools put forward in these contexts have been developed specifically for inference. In this section, we introduce two basic network models and relate them to classical statistics. The first such model consists of nodes whose degrees are identically distributed, whereas the sec- ond implies group structure via latent categorical covariates. Inferring rela- tionships amongst groups of nodes from data in turn requires the standard tools of statistics, including parameter estimation and hypothesis testing. We provide examples of such procedures below, illustrating their compu- tational complexity, and introduce corresponding notions of approximate inference. 1.3.1. Erd¨ os–R´ enyi: A first illustrative example We begin by considering one of the simplest possible models from random graph theory, attributed to Erd¨ os and R´ enyi (1959) and Gilbert (1959), and consisting of pairwise links that are generated independently with proba- bility p . Under this model, all nodes have identically distributed degrees; it is hence appropriate to describe instances in which no group structure (by way of categorical covariates) is present. In turn, we shall contrast this with an explicit model for structure below. Adapted to the task of modeling undirected network data, the Erd¨ os– enyi model may be expressed as a sequence of ( n 2 ) Bernoulli trials corre- sponding to off-diagonal elements of the adjacency matrix A . Definition 1.1 (Erd¨ os–R´ enyi Model). Let n > 1 be integral and fix some p [0 , 1] . The Erd¨ os–R´ enyi random graph model corresponds to matri- ces A ∈ { 0 , 1 } n × n defined element-wise as i, j ∈ { 1 , 2 , . . . , n } : i < j, A ij iid Bernoulli( p ); A ji = A ij , A ii = 0 . Erd¨ os–R´ enyi thus provides a one-parameter model yielding indepen- dent and identically distributed binary random variables representing the absence or presence of pairwise links between nodes; as this binary relation is symmetric, we take A ji = A ij . The additional stipulation A ii = 0 for all i implies that our relation is also irreflexive; in the language of graph theory, the corresponding (undirected, unweighted) graph is said to be simple , as it exhibits neither multiple edges nor self-loops. The event i j is thus a Bernoulli( p ) random variable for all i = j
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  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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