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Diffstat (limited to 'doc/latex/latex/models/correlations')
-rw-r--r-- | doc/latex/latex/models/correlations/tune_qnn_adaptive.tex | 64 | ||||
-rw-r--r-- | doc/latex/latex/models/correlations/tune_rho.tex | 45 |
2 files changed, 109 insertions, 0 deletions
diff --git a/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex b/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex new file mode 100644 index 0000000..016b674 --- /dev/null +++ b/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex @@ -0,0 +1,64 @@ +\myprogram{{tune\_qnn\_adaptive}} + {Construct a multiplex with prescribed inter-layer correlations.} + {$<$degs1$>$ $<$degs2$>$ $<$mu$>$ $<$eps$>$ $<$beta$>$ [RND|NAT|INV]} + +\mydescription{This programs tunes the inter-layer degree correlation + exponent $\mu$. If we consider two layers of a multiplex, + and we denote by $k$ the degree of a node on the first layer + and by $q$ the degree of the same node on the second layers, + the inter-layer degree correlation function is defined as: + + \begin{equation*} + \overline{q}(k) = \sum_{q'} q' P(q'|k) + \end{equation*} + + where $\overline{q}(k)$ is the average degree on layer $2$ + of nodes having degree $k$ on layer $1$. + + The program assumes that we want to set the degree + correlation function such that: + + \begin{equation*} + \overline{q}(k) = a k^{\mu} + \end{equation*} + + where the exponent of the power-law function is given by + the user (it is indeed the parameter \textit{mu}), and + successively adjusts the pairing between nodes at the two + layers in order to obtain a correlation function as close + as possible to the desired one. The files \textit{degs1} + and \textit{degs2} contain, respectively, the degrees of + the nodes on the first layer and on the second layer. + + The parameter \textit{eps} is the accuracy of \textit{mu}. + For instance, if \textit{mu} is set equal to -0.25 + and \textit{eps} is equal to 0.0001, the program stops when + the configuration of node pairing corresponds to a value of + the exponent $\mu$ which differs from -0.25 by less than + 0.0001. + + The parameter \textit{beta} is the typical inverse + temperature of simulated annealing. + + If no other parameter is specified, or if the last parameter + is \texttt{RND}, the program starts from a random pairing of + nodes. If the last parameter is \texttt{NAT} then the + program assumes that the initial pairing is the natural one, + where the nodes have the same ID on both layers. Finally, + if \texttt{INV} is specified, the initial pairing is the + inverse pairing, i.e. the one where node 0 on layer 1 is + paired with node N-1 on layer 2, and so on. + + } + + +\myreturn{The program prints on \texttt{stdout} a pairing, i.e. a list +of lines in the format: + +\hspace{0.5cm} \textit{IDL1 IDL2} + +where \textit{IDL1} is the ID of the node on layer 1 and \textit{IDL2} +is the corresponding ID of the same node on layer 2. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/correlations/tune_rho.tex b/doc/latex/latex/models/correlations/tune_rho.tex new file mode 100644 index 0000000..27b6079 --- /dev/null +++ b/doc/latex/latex/models/correlations/tune_rho.tex @@ -0,0 +1,45 @@ +\myprogram{{tune\_rho}} + {Construct a multiplex with prescribed inter-layer correlations.} + {$<$rank1$>$ $<$rank2$>$ $<$rho$>$ $<$eps$>$ $<$beta$>$ [RND|NAT|INV]} + +\mydescription{This programs tunes the inter-layer degree correlation + coefficient $\rho$ (Spearman's rank correlation) of two + layers, by adjusting the inter-layer pairing of nodes. The + files \textit{rank1} and \textit{rank2} are the rankings of + nodes in the first and second layer, where the n-th line of + the file contains the rank of the n-th node (the highest + ranked node has rank equal to 1). + + The parameter \textit{rho} is the desired value of the + Spearman's rank correlation coefficient, while \textit{eps} + is the accuracy of \textit{rho}. For instance, + if \textit{rho} is set equal to -0.25 and \textit{eps} is + equal to 0.0001, the program stops when the configuration of + node pairing corresponds to a value of $\rho$ which differs + from -0.25 by less than 0.0001. + + The parameter \textit{beta} is the typical inverse + temperature of simulated annealing. + + If no other parameter is specified, or if the last parameter + is \texttt{RND}, the program starts from a random pairing of + nodes. If the last parameter is \texttt{NAT} then the + program assumes that the initial pairing is the natural one, + where the nodes have the same ID on both layers. Finally, + if \texttt{INV} is specified, the initial pairing is the + inverse pairing, i.e. the one where node 0 on layer 1 is + paired with node N-1 on layer 2, and so on. + + } + + +\myreturn{The program prints on \texttt{stdout} a pairing, i.e. a list +of lines in the format: + +\hspace{0.5cm} \textit{IDL1 IDL2} + +where \textit{IDL1} is the ID of the node on layer 1 and \textit{IDL2} +is the corresponding ID of the same node on layer 2. +} + +\myreference{\refcorrelations} |