\begin{algorithm}
\caption{Construction of a random network from a degree sequence with configuration model}
\label{alg: Config}
Suppose we want to generate a realization of a finite graph with $N$ vertices and a given degree sequence $\textbf{d}=(d_1,d_2,...,d_N)$. 

\begin{enumerate}
\item Following the algorithm described by \cite{Newman:2010}, we first create $N$ vertices labeled from $i=1$ to $N$, and attach $d_i$ \textbf{stubs} (half edges) to the $i$-th vertex.

\item Then we randomly pair any of the two distinct stubs to form edges by the configuration model: 
\begin{enumerate}
\item Vertices are treated as a partition of all stubs, which each vertex labeled by $i$ is isomorphic to a set $w_i$ of the partition contains all the stubs attach to it, so the set $w_i$ has $d_i$ element.

\item Randomly picking any two of the distinct stubs among all stubs and form a pair, even they come from the same partition set or there are already other pairs between the same two sets.
Once two stubs are paired, they will be recorded and removed from later paring candidates.

\item Repeat Step (b) until no unpaired stubs are left.

\item Map the paired stubs and partition sets back to graph, as edges and vertices, accordingly.

\item If there are multi-edges and loops in the result graph reject the result and restart from (a), until a simple graph is attained.
\end{enumerate}

\item Output the simple graph as final result.
\end{enumerate}
\end{algorithm}

\begin{algorithm}
\caption{Construction of a random network from a degree sequence with configuration model}
Suppose we want to generate a realization of a finite graph with $N$ vertices and a given degree sequence $\textbf{d}=(d_1,d_2,...,d_N)$.
\begin{enumerate}
\item Following the algorithm described by \cite{Newman:2010}, we first create $N$ vertices labeled from $i=1$ to $N$, and attach $d_i$ \textbf{stubs} (half edges) to the $i$-th vertex.
\item Then we randomly pair any of the two distinct stubs to form edges by the configuration model:
\begin{enumerate}
\item Vertices are treated as a partition of all stubs, which each vertex labeled by $i$ is isomorphic to a set $w_i$ of the partition contains all the stubs attach to it, so the set $w_i$ has $d_i$ element.
\item Randomly picking any two of the distinct stubs among all stubs and form a pair, even they come from the same partition set or there are already other pairs between the same two sets.
Once two stubs are paired, they will be recorded and removed from later paring candidates.
\item Repeat Step (b) until no unpaired stubs are left.
\item Map the paired stubs and partition sets back to graph, as edges and vertices, accordingly.
\item If there are multi-edges and loops in the result graph reject the result and restart from (a), until a simple graph is attained.
\end{enumerate}
\item Output the simple graph as final result.
\end{enumerate}
\end{algorithm}
{
    "title": "disease transmission on random graphs using edge-based percolation",
    "id": "2401.06872",
    "abstract": "edge-based percolation methods can be used to analyze disease transmission on complex social networks. this allows us to include complex social heterogeneity in our models while maintaining tractability. here we review the seminal works on this field by newman et al (2001); newman (2002, 2003), and miller et al (2012). we present a systematic discussion of the theoretical background behind these models, including an extensive derivation of the major results. we also connect these results relate back to the classical literature in random graph theory molloy and reed (1995, 1998). finally, we also present an accompanying r package that takes epidemic and network parameters as input and generates estimates of the epidemic trajectory and final size. this manuscript and the r package was developed to help researchers easily understand and use network models to investigate the interaction between different community structures and disease transmission.",
    "categories": "cs.si math.ds q-bio.pe",
    "doi": "",
    "created": "2024-01-12",
    "updated": "",
    "authors": [
        "s. zhao",
        "f. m. g. magpantay"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.06872"
}
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