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Diffstat (limited to 'doc/latex/latex/models/growth')
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diff --git a/doc/latex/latex/models/growth/nibilab_linear_delay.tex b/doc/latex/latex/models/growth/nibilab_linear_delay.tex new file mode 100644 index 0000000..afbf083 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delay.tex @@ -0,0 +1,67 @@ +\myprogram{{nibilab\_linear\_delay}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives first on layer $1$, and its replica on + the layer $2$ appears after a time delay $\tau$ sampled from + the power-law function: + + \begin{equation*} + P(\tau) \sim \tau^{-\beta} + \end{equation*} + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \item \textbf{beta} the exponent of the power-law delay + function which determines the arrival of replicas on layer $2$ + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex b/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex new file mode 100644 index 0000000..5863951 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex @@ -0,0 +1,68 @@ +\myprogram{{nibilab\_linear\_delay\_mix}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival and randomly selected first layer.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives on one of the two layers, chosen + uniformly at random, and its replica on the other layer + appears after a time delay $\tau$ sampled from the power-law + function: + + \begin{equation*} + P(\tau) \sim \tau^{-\beta} + \end{equation*} + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \item \textbf{beta} the exponent of the power-law delay + function which determines the arrival of replicas on layer $2$ + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_delta.tex b/doc/latex/latex/models/growth/nibilab_linear_delta.tex new file mode 100644 index 0000000..27ebbd0 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delta.tex @@ -0,0 +1,57 @@ +\myprogram{{nibilab\_linear\_delta}} + {Multiplex linear preferential attachment model -- + Synchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ $<$c$>$ $<$d$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives at the same time on both layers. + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_random_times.tex b/doc/latex/latex/models/growth/nibilab_linear_random_times.tex new file mode 100644 index 0000000..5f1c0fe --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_random_times.tex @@ -0,0 +1,63 @@ +\myprogram{{nibilab\_linear\_random\_times}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival with randomly sampled arrival times on + layer 2.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ } + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives on layer $1$, but its replica on the + other layer appears at a uniformly chosen random time in + $[m0+1; N]$. + + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_nonlinear.tex b/doc/latex/latex/models/growth/nibilab_nonlinear.tex new file mode 100644 index 0000000..2e48a18 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_nonlinear.tex @@ -0,0 +1,60 @@ +\myprogram{{nibilab\_nonlinear}} + {Multiplex non-linear preferential attachment model -- + Synchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$alpha$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex non-linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto \frac{\left(k\lay{1}_j\right)^{\alpha}} + {\left(k\lay{2}_j\right)^{\beta}} + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto \frac{\left(k\lay{2}_j\right)^{\alpha}} + {\left(k\lay{1}_j\right)^{\beta}} + \end{equation*} + + Each node arrives simultaneously on both layers. + + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{alpha, beta} exponents of of the attaching probability + function + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/node_deg_over_time.tex b/doc/latex/latex/models/growth/node_deg_over_time.tex new file mode 100644 index 0000000..351045e --- /dev/null +++ b/doc/latex/latex/models/growth/node_deg_over_time.tex @@ -0,0 +1,50 @@ +\myprogram{{node\_deg\_over\_time.py}} + {Time evolution of the degree of a node in a growing graph.} + {$<$layer$>$ $<$arrival\_times$>$ $<$node\_id$>$ + [$<$node\_id$>$ ...]} + +\mydescription{Compute the degree $k_{i}(t)$ of node $i$ in a growing + network as a function of time. The file \textit{layer} + contains the edge list of the final network. Each line of + the file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{arrival\_times} is a list of node arrival times, in + the format: + + \hspace{0.5cm} \textit{time\_i node\_i} + + where \textit{time\_i} is the time at which \textit{node\_i} arrived + in the graph. Notice that \textit{time\_i} must be an integer in the + range [0, N-1], where N is the total number of nodes in the final + graph. + + The third parameter \textit{node\_id} is the ID of the node whose + degree over time will be printed on output. If more than + one \textit{node\_id} is provided, the degrees over time of all the + corresponding nodes are printed on output. + } + + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{t kit} + + where \textit{kit} is the degree of node \textit{i} at + time \textit{t}. The first line of output is in the format: + + \hspace{0.5cm} \textit{\#\#\#\# node\_id} + + where \textit{node\_id} is the ID of node \textit{i}. + + If more than one \textit{node\_id}s is provided as input, the program + prints the degree over time of all of them, sequentially. + +} + +\myreference{\refgrowth} |