\begin{algorithm} 
\caption{Generation of realizable degree sequence from a prescribed distribution}
\label{alg: Seq}
Consider size $N$ and the prescribed distribution given by $\mathbb{P}(D=d), d\in \mathbb{Z}_{>0}$

\begin{enumerate}
\item Following the algorithm described by \cite{Newman:2010}, we first draw a set of $N$ positive integer from the prescribed distribution.

\item If the sum of all $N$ integers in Step 1 is even, proceed the sequence to Step 3. 
If there is an odd total Degree, randomly chose one of the vertices and changed its degree by drawing again from the prescribed distribution until the total number of stubs is even, then proceed to the next step.

\item Since both the order of vertices and the degree sequence have no impact when generating random graphs, we re-order and sort the sequence in descending order, as degree sequence $\textbf{d}$.

\item Check if the sequence $\textbf{d}$ satisfies Theorem~\ref{thm: Realizable}.
If satisfied, take $\textbf{d}$ as degree sequence.
Otherwise, restart from Step 1.
\end{enumerate}
\end{algorithm}

\begin{algorithm}
\caption{Generation of realizable degree sequence from a prescribed distribution}
Consider size $N$ and the prescribed distribution given by $\mathbb{P}(D=d), d\in \mathbb{Z}_{>0}$
\begin{enumerate}
\item Following the algorithm described by \cite{Newman:2010}, we first draw a set of $N$ positive integer from the prescribed distribution.
\item If the sum of all $N$ integers in Step 1 is even, proceed the sequence to Step 3.
If there is an odd total Degree, randomly chose one of the vertices and changed its degree by drawing again from the prescribed distribution until the total number of stubs is even, then proceed to the next step.
\item Since both the order of vertices and the degree sequence have no impact when generating random graphs, we re-order and sort the sequence in descending order, as degree sequence $\textbf{d}$.
\item Check if the sequence $\textbf{d}$ satisfies Theorem~\ref{thm: Realizable}.
If satisfied, take $\textbf{d}$ as degree sequence.
Otherwise, restart from Step 1.
\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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