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Hi Elly,

The centrality values attached to the nodes are not independent from 
each other because a high centrality value of one node may imply a high 
centrality value of an adjacent node (see measures of degree 
assortativity etc.) -- or merely because the network has a given 
centralization and increasing the centrality of one node must decrease 
the centrality of another node.

If you estimate a regression model, you should therefore specify the 
channels of influence/dependence between the nodes as covariates. For 
example, for each node you can include the cumulated and/or average 
centrality of adjacent nodes (and possibly of indirect friends with path 
length 2). Depending on the centrality measure (e.g., eigenvector 
centrality), it may also make sense to include the aggregated centrality 
scores of structurally similar nodes because being connected to the same 
important other nodes makes both nodes in a dyad similarly important. 
There may be many other ways in which a node's centrality value depends 
on other nodes' values but also potentially just on features of the 
network or the local neighborhood of a node in the network.

Regression models where the dependencies are specified like this go by 
several names:

- If you estimate linear or generalized linear models and include only 
functions of direct neighbors, this is called a "spatial autocorrelation 
model" or a "network autocorrelation model". See the work of Patrick 
Doreian, for example. There is a nice article in Social Networks on 
"Specifying the weight matrix" for these models by Roger Leenders 
(http://dx.doi.org/10.1016/S0378-8733(01)00049-1). There is an R 
implementation for the linear case in the sna package by Carter Butts 
(the lnam function).

- If you include various other dependencies and estimate a binary (= 
logit or probit) model, this was termed "autologistic actor attribute 
model" (ALAAM) by Galina Daraganova and Garry Robins in their chapter 9 
of the ERGM book ("Exponential Random Graph Models for Social Networks") 
by Dean Lusher, Johan Koskinen and Garry Robins. I think there is an 
implementation in PNet or a related program, but others may know better 
than I do. This is kind of a special case of the network autocorrelation 
model but with additional dependencies.

- If you include only basic dependencies (= functions of direct 
neighbors and their attributes), this is called "multiparametric 
spatiotemporal autoregressive model" (m-STAR) in the spatial 
econometrics literature (see an article of Jude Hays, Aya Kachi and Rob 
Franzese here: http://dx.doi.org/10.1016/j.stamet.2009.11.005).

- Finally, I have created a generalized version of all of this. I call 
it "temporal network autocorrelation model" (TNAM) because it's a 
spatial *autocorrelation model* like the one specified above (see first 
bullet point), but it also includes various kinds of dependencies (hence 
the word *"network"*, see second bullet point), and it's possible to 
estimate it with *temporal* data/repeated observations of the network 
and/or the outcome variable (see bullet point 3). More generally, you 
can plug it into any model you would like, including tobit, survival, 
linear mixed models etc. I have implemented this in the tnam function in 
my R package xergm, along with a number of dependency terms to include. 
See here (http://rpackages.ianhowson.com/rforge/xergm/man/tnam.html) and 
here (http://rpackages.ianhowson.com/rforge/xergm/man/tnam-terms.html) 
for a description. A paper will be available in a few weeks.

There are three potential caveats here:

(1) It may be difficult to specify the dependencies appropriately if the 
attributes you are explaining are centralities. It may require you to 
think hard about what is causing them and to what extent you want them 
to be explained by network/influence terms vs. covariates, but I think 
technically it should be a subproblem of the more general models 
outlined above.

(2) A cross-sectional analysis does not really allow you to infer 
causality (this is tricky enough with longitudinal data). It's 
relatively certain that some of your independent variables/model terms 
will be partially caused by the centrality of the nodes.

And (3) centrality scores are usually fairly skewed and often also bound 
between 0 and 1, so it may be inappropriate to use a linear model. But 
this is a general statistical problem, not one that is specific to 
network analysis. You may want to consult the literature on beta 
regression, Box-Cox transformations etc. TNAM should be able to deal 
with this, but you have to find out what model (e.g., GLM with a beta 
distribution) would be appropriate for your data.

As an alternative, you may want to consider modeling the network using 
an exponential random graph model, rather than modeling centrality 
scores, which are merely a function of the network with a huge loss of 
information. By explaining the network, you basically explain the 
structure including who is central.

Best regards

Philip

Am 17.03.2015 um 02:16 schrieb Elly Power:
> ***** To join INSNA, visit http://www.insna.org *****
> Hello all,
>
> I was hoping I could get some advice on how (or if) I could use
> centrality measures (e.g., eigenvector centrality) as the /dependent/
> variable in some analyses.
>
> I know that we usually think of centrality as an independent variable,
> but it seems reasonable that we might want to predict centrality.
> Personally, I work on religious practice, and I want to understand if
> the nature of someone's religious practice might influence his/her
> centrality.
>
> The issue, of course, is that centrality measures are not independent.
> Does anyone know of any ways to deal with this? Is there anyone who has
> tried to look at this? Any direction would be very much appreciated.
>
> Thanks in advance for all of your suggestions.
>
> - Elly Power
>
> --
> Eleanor A. Power, PhD Candidate
> Department of Anthropology
> Stanford University
> 450 Serra Mall, Bldg 50
> Stanford, CA 94305
> www.stanford.edu/~epower <http://www.stanford.edu/%7Eepower>
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