Abstract
The paper proposes a model for the formation of economic and social networks by integrating an empirical component with a theoretical and an applicative one. The empirical network is derived from the analysis of the peer group conducted with the DEA methodology by solving a DEA-VRS model. The theoretical part of the model is based on an equation of utility behavior on the part of the agents and forms the basis for the assumption of dependence in the formation of network bonds. Finally, the applicative part estimates an ERGM model that reproduces the structural characteristics of the empirical network of the empirical component of the training model. In conclusion, the model proposed here enriches the class of models of formation of economic and social networks and represents a starting point for real applications.
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Notes
To give an example, suppose that an economic agent decides to form a certain number of bonds for each of which there are training and maintenance costs but that for the formation of the network he has a limited budget of financial resources equal to 100.Therefore, if the network will consist of 3 links (whether direct or indirect) and that for the formation of one it is necessary to bear a cost of 50 while for the formation of a second of 20, it is evident that the formation of the third depends on the costs of two previous ones and by the budget constraint. In fact, in this example the resources that he can allocate for the formation of the third link is only of 100 (budget) − 50 (cost of first link) − 20 (cost of second link) = 30.
Christakis et al. (2010) take this path as a solution to the presence of multiple balances in the presence of externalities (Bala and Goyal 2000; Jackson and Wolinsky 1996; Currarini et al. 2009, 2010). Other solutions to multiple balances can be found in (Chandrasekhar and Jackson 2014; Sheng 2012) using the under network as a unit of analysis, or that of modelling the formation of the network as an imperfect information game used in (Leung 2014b), or adopt restrictions on the type of externalities (Miyauchi 2016), in addition to those concerning utility functions.
If this is possible it could mean that the DEA-based network also converges towards an exponential distribution.
In truth, this which is only an assumption could become a real possibility if the DEA methodology and therefore the formation model of DEA-based networks were made stochastic.
We avoid using the term “observed” because we believe that this term should only be used for those networks which have actually been established and which the researcher keeps track of.
We believe that introducing this assumption not only serves to overcome the internal inconsistency of the model (Pinto 2020) but which also serves to represent a truth in which the formation of bonds is dependent on them.
This original way of forming links, or using the DEA, introduced it (Pinto 2020).
In other words, in our opinion, the meeting technology represents the sociological substrate of the peer analysis conducted in the DEA.
Consider that agent 4 is the owner of the damaged car and agents 2 and 5 are the repairer and the body builder of note 5 respectively.
Condition \(\sum \lambda =1\) is is made in the DEA BBC models in the form of the envelope and note 5. (Cooper et al. 2007) pp.gg. 91-93
The direct consequence of this is that it would be necessary to modify the left side of the ERGM model, in other words it is necessary to adjust the adjacency matrix of the excellent lambdas in accordance with the evidence coming from the strategic model and then proceed with the new matrix to estimate the ergm model.
The marginal utility that is achieved by passing from the absence of a bond (a = 0) to the formation of the bond (a = 1).
These variables that we call binding variables can be collected in a stochastic matrix \(X = \left[ {X_{IJ} } \right]\) wherein the row i and column j inputs indicate a link between i and j. The space of all possible adjacency matrices is indicated with X, while an embodiment of X is indicated with x = [xij].
When this relationship deduced from our DEA model does not exist then formalizing the dependence could be more complicated. And this for example happens for the DEA-CCR models [(Cooper et al. 2007) chap. 3].
We have used this but the DEA modeling is wide and therefore the possible network formation models based on the DEA methodology that can be built can be many, for this reason we can talk about the class of network formation models based on the DEA modeling our model maintains for DEA-VRS models with both measurements at input and output.
The ideal would be to have many empirical matrices.
It should be noted that we use the term “empirical” to indicate what is constructed from the DEA (Pinto 2020) methodology, and we distinguish it from the term “observed” which we reserve for what could be produced in reality.
In other words, having assumed that the usefulness of the agents in forming a bond depends on the attackers, it is necessary to stably know how the bonds are formed as a result of these attributes.
Generally, however, the interpretation is at the bond level and therefore the estimated parameters associated with each configuration of an ERGM model indicate whether the probability of a new bond forming is positive (positive sign of the estimated parameter) when a further specific configuration occurs decreases (negative sign of the estimated parameter) (Handcock et al. 2003a, 2003b, 2019).
All the models have been estimated with the “Stepping” method provided by the ergm function of the ergm package with an MCMC sample size of 2000 and 10 steps. This choice appeared the best both from the point of view of the estimate and from that of the simulation with respect to the default method which is the MCMLE. The other possible ones are: "Robbins-Monro" and "Stochastic-Approximation".
We are not sure that this model also includes the constraint that the sum of the optimal lambdas for each proposed agent must be equal to 1. Epliciting in the term "edgecov" of the ergm function of the statnet package in R the term form = "sum" the package does not recognize this input.
Il goodness-of-fit for model statistics are in the following.xlsx files:..\..\file data e analysis\gofmodel1.xlsx.,..\..\file data e analysis\gofmodel2.xlsx,..\..\file data e analysis\gofmodel3.xlsx.
This was made clear in the text when it was claimed to consider null and void in the formation and maintenance of the bonds. It is therefore clear that the presence of costs changes this economic assessment and therefore the incentive to form and maintain bonds.
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Pinto, C. The dependence of bonds in the formation models of DEA-based networks. Soc. Netw. Anal. Min. 12, 18 (2022). https://doi.org/10.1007/s13278-021-00849-6
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DOI: https://doi.org/10.1007/s13278-021-00849-6