Dynamic Social Network Modeling and Analysis Workshop Summary and Papers (2003) / Chapter Skim
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Accounting for Degree Distribution in Empirical Analysis of Network Dynamics
Accounting for Degree Distribution in Empirical Analysis of Network Dynamics
Pages 146-161
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From page 146... ...
Therefore empirical inference from observed network characteristics to the processes that could be responsible for network genesis and dynamics cannot be based only, or mainly, on the observed degree distribution. As an elaboration and practical implementation of this point, a statistical model for the dynamics of networks' expressed as digraphs with a fixed vertex set, is proposed in which the outUegree distribution is governed by parameters that are not connected to the parameters for the structural dynamics.
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derived various asymptotic properties of random undirected graphs with given degree sequences. They followed the mentioned literature in unclerlining the imporDYNAMIC SOCIAL N~TWORKHODELING AND ANALYSIS 147
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for network dynamics: first outHegrees, then network structure This paper is concerned with the statistical modeling of network evolution for data consisting of two or more repeated observations of a social network for a given fixed set of actors, represented by a directed graph with a given vertex set. When proposing statistical models for network evolution, theoretical credibility of the mode!
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These observed networks are regarded as M discrete observations on a stochastic process X(~) on the space of all digraphs on r1 vertices, evolving in continuous time.
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Mode/ definition: objective function Given that actor i makes a change at some moment t for which the current network is given by X(t) = x, the particular change made is assumed to be determined by the so-called objective function - which gives the numerical evaluation by the actor of the possible states of the network - together with a random element - accounting for 'unexplained changes', in other words, for the limitations of the model.
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is a random variable, indicating the part of the actor's preference not represented by the objective function, and assumed to be distributed according to the type l extreme value distribution with mean O and scale parameter 1 (Maddala, 1983~. Similarly, if at moment t actor i deletes a tie, then he chooses the other actor j, among those for which xij = 1, for which fi_~B,x(`i~j)
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pi_ for x Ill. Rate functions and stationary distributions In a simple model definition, the rates Ai+ and Ai- depend only on the outclegree of actor i.
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= 0. The network evolution model is not necessarily assumed to be stationary, and indeed it is likely that in empirical observations, networks often will be far from the stationary distributions of the corresponding evolution processes.
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D YNAMIC SOCIAL NETWORK MODELING AND Anal YSIS (15)
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IV. Parameter estimation For the estimation of parameters of this type of network evolution models for observations made at discrete moments to, .
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7-10. Space limitations prohibit (loin" justice to the richness of the original data and the social processes involved in the network dynamics.
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Figures 1 and 2 show, for these two models, the observed ranked outUegrees combined with simulated 90-~o intervals for the distributions of the ranked outbegrees. Figure 1 indicates a poor fit: the distributions of the 9 highest outUegrees in the fitted model are concentrated on too Tow values compared to the observed highest outUegrees; in the middle low range, the fitted distribution of the ranked DYNAMIC SOCIAL NE:TWORKMOI)
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In the network dynamics there is clear evidence for reciprocity of choices, for a network closure effect, and for a greater popularity of female actors. (A gender similarity effect also was tested, but 158 DYNAMIC SOCIAL NETWORK MODELING ED ANALYSIS
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More specifically, similarly to what is explained in Snipers (2002a) , the earlier model can be specified so that quite a good fit is obtained for the evolution over a limited time period of a network with, say, a rather low density and a relatively high amount of transitivity, but that the graph evolution process so defined will with probability one lead to an explosion' in the sense that at some moment, the graph density very DYNAMIC SOCIAL N~TWORKMOD~WG ED ISIS 159
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This implies that we can learn little about the processes leading to transitivity and related structural properties of graphs and digraphs by looking only at degree distributions, or by limiting attention to network evolution processes that are based only on degrees. REFERENCES Albert.
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2002a. Markov Chain Monte Cario Estimation of Expm nential Random Graph Models.
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