## Revision a6b51dd0

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.gitignore
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*.aux

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*.bbl

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*.blg

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*.fdb_latexmk

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*.synctex.gz

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*.fls

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# Notes and Specs

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This repo is about insights, models and specification for our analysis and code.

bgp_model/bgp_model.tex
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\documentclass[twocolumn, 10pt]{article}

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\usepackage{acronym}

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\usepackage{tikz}

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\usepackage{cleveref}

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\usetikzlibrary{calc,shapes,arrows}

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\title{BGP model and other oddities}

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\begin{document}

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\maketitle

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\acrodef{ADV}{advertisement}

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\acrodef{AS}{autonomous system}

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\acrodef{MiCe}{Milani Centrality}

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We model a eBGP speaker graph; in the following \ac{AS}, speaker and node are used interchangeably.

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\section{Preliminaries}

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Nodes form a communication network to propagate BGP \acp{ADV}, the BGP messages.

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\acp{ADV} can be of several types, the two related to prefix/destinations are

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\begin{itemize}

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\item UPDATE;

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\item WITHDRAWAL.

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\end{itemize}

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We are not interested in the others, devoted to connection management.

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We use the terms prefix and destination interchangeably and they indicate an IP subnet or a collection of IP subnets.

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A route is a pair $r=(d,\xi)$ of destination and attributes.

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$\xi$ contains information such as:

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\begin{itemize}

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\item path, which is a list of \acp{AS};

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\item originator id, which is the IP address for the entry point of the originating \ac{AS}.

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\end{itemize}

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\section{BGP graph}

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Nodes form a strongly connected bidirectional graph $G(V,E)$.

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We indicate as $G^k(V^k, E^k)$ the fully connected subgraph of tier-one BGP nodes, such that, $V^k\subset V,\ E^K = V^k\times V^k$.

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Links can be either peer-to-peer or customer-provider depending on the role involved nodes are playing.

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We indicate with $\Lambda=\{\pi,c,s\}$ the labels indicating peer, consumer and provider respectively.

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The function $\lambda:E\to \Lambda$ assigns to a node edge $(i,j)$ the role $i$ has with respect to $j$.

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Hence, $\lambda(i,j) = \pi \iff \lambda(j,i)=\pi$ and $\lambda(i,j) = c \iff \lambda(j,i)=s$

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Note that, $\lambda(i,j) = \pi\ \forall i,j\in V^k$.

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We call $N_i = \{j\in V: \exists (i,j)\in E\}$ the set of neighbours for node $i$ and $C_i=\{j\in N_i: \lambda(i,j)=s\}$ the set of customers of node $i$.

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\section{Advertisement propagation}

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When a node $i\in V$ generates an \ac{ADV} for the route $r$, it is propagated to all nodes using the links of $G(V,E)$.

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Propagation considers:

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\begin{itemize}

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\item $\lambda()$;

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\item node policies blocking \ac{ADV} forwarding.

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\end{itemize}

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In the following we neglect blocking policies for the ease of analysis but it should not change it too much.

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\begin{figure}

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\centering

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\begin{tikzpicture}

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\tikzset{edge/.style = {->,> = latex',line width=0.2mm}}

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\node[rectangle,text width=5em,align=center,draw=black] (b1) {$j$ receives $r$ from $i$};

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\node[diamond,text width=4em,align=center,draw=black,below of=b1,node distance=7em] (b2) {$\lambda(i,j)==c?$};

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\node[rectangle,text width=5em,align=center,draw=black] at ($(b2.west)+(-1.0,-1.5)$) (b3) {$j$ propagates to $C_j$};

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\node[rectangle,text width=5em,align=center,draw=black] at ($(b2.east)+(+1.0,-1.5)$) (b4) {$j$ propagates to $N_j\setminus \{i\}$};

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\draw[edge] (b1)->(b2);

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\draw[edge] (b2)-|(b3) node[midway, draw=none, above] {n} ;

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\draw[edge] (b2)-|(b4) node[midway, draw=none, above] {y} ;

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\end{tikzpicture}

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\caption{Flow chart for \ac{ADV} forwarding.}

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\label{fig:adv_forwarding}

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\end{figure}

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\cref{fig:adv_forwarding} represents the deciding policy for \ac{ADV} forwarding using the standard policy of \textit{no-valley and prefer-customer}~\cite{elmokashfi2010scalability}.

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Propagation policies and $G$ structure (it includes a full meshed graph $G^k$ of nodes with peer links), implies a very specific pattern of route propagation from one node $i\in V\setminus V^k$ to the rest of the network.

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We model this pattern by sub-dividing the graph $G$ in three components as the propagation happens in three phases:

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\begin{itemize}

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\item Phase 1: node $i\in V\setminus V^k$ generates an \ac{ADV} for route $r$ and this spreads toward a node $z\in V^k$; we call the resulting sub-graph $G^{(1)}(V^{(1)},E^{(1)})$;

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\item Phase 2: \ac{ADV} propagates toward tier one nodes $V^k$; the resulting sub-graph is $G^{(2)}=G^k$;

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\item Phase 3: \ac{ADV} propagates from $z\in V^k$ to all the nodes $j\in V\setminus V^k\setminus V^{(1)}$; we call the resulting sub-graph $G^{(3)}(V^{(3)},E^{(3)})$.

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\end{itemize}

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\subsection{Phase 1 graph}

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Let $i$ be the \ac{ADV} originator and $j\in V$ another node in $G$.

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$j\in V^{(1)} \iff \exists p_{ij}$ a path in $G$ between $i$ and $j$, with $p_{ij}=(i,k_0)\dots (k_l,j)$ such that:

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\begin{itemize}

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\item $k_x\notin V^k,\ \forall x=0,\dots,l$;

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\item we have one of the following conditions:

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	\begin{itemize}

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	\item $\lambda(\rho,t)=c\ \forall (\rho,t)\in p_{ij}$;

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	\item $\exists z\leq l: \lambda(\rho,t)=c\ \forall (\rho,t)\in p_{ik_z}, \lambda(k_z,k_{z+1})=\pi, \lambda(\rho,t)=s\ \forall (\rho,t)\in p_{k_{z+1},j}$

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	\end{itemize}

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\end{itemize}

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\subsection{Phase 3 graph}

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Phase 3 graph is the one connecting the nodes $V^k$ to all the nodes $V^{(3)}=V\setminus V^k\setminus V^{(1)}$.

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$j\in V^{(3)} \iff \exists p_{ij}$ a path in $G$ between $i\in V^k$ and $j$, with $p_{ij}=(i,k_0)\dots (k_l,j)$ such that:

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\begin{itemize}

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	\item $j,k_x\notin V^k\cup V^{(1)},\ \forall x=0,\dots,l$;

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	\item $\lambda(\rho,t)=s\ \forall (\rho,t)\in p_{ij}$;

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\end{itemize}

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\subsection{Propagation graph}

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Given an originator $i\in V\setminus V^k$, we call propagation graph $G^P=(V^{(1)}\cup V^{(2)}\cup V^{(3)},E^{(1)}\cup E^{(2)}\cup E^{(3)})$ the graph comprising all the feasible links for \ac{ADV} propagation.

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\section{MRAI timers}

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For each neighbour $j$, every node $i$ consider an MRAI timer value $T_{ij}$.

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The typical value for it is 30 seconds.

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Each node will consider route advertisements separately.

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Routes can differ for the destination $d$ or for attributes $\xi$.

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If two routes share the same destination $d$ but they include different attributes $\xi_1\neq \xi_2$ then node $i$ will process two different UPDATES messages.

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With respect to the destination $d$, node $i$ will not send more than one message every $T_{ij}$ seconds.

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If multiple route messages $r_1,\dots,r_m$ arrive to $i$ for the same destination $d$ but with different attributes $\xi_1,\dots,\xi_m$, node $i$ will propagate them separately, waiting $T_{ij}$ seconds between one transmission and the other\footnote{Note: this behaviour is mandated by the RFC, section 9.2.1.1 which foresees anyway some overhead issues in actual implementation. Solution to overhead is left to implementers.}.

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For each destination $d$, node $i$ has a queue of route advertisements $r_1,\dots,r_m$, and, for each $j\in N_i$ it keeps a function $\tau(i,j,d): V\times V\times D\to [0,T_{ij}]$ expressing the minimum remaining time for another transmission.

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\subsection{Fabrikant issue}

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In the Fabrikant et al. paper~\cite{fabrikant2011something} it is describe a possible issue related to timers and path exploration.

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This issue happens when there are multipaths in the propagation graph $G^P$ and a specific timer allocation $T_{ij}$.

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In the following we suppose that all nodes $i$ have $\tau(i,j,d)=0,\ \forall j\in N_i$ (ready to fire).

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\textbf{Duplicate ADV.} Let $i\in V$ originate a route advertisement $r$ and let $j\in V$ receives it at certain time $t_0$.

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$j$ will process the \ac{ADV} and propagate it at time $t_0+\epsilon$; if $j$ finds more advantageous, given the new advertisement, to reach $d$ through another node, let's say $z\in V$, it will implement that in its routing table and will propagate the \ac{ADV} with the modified $\xi$.

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However, if $z$ also receives the \ac{ADV} from $i$ at $t_0$, it has to change its routing table and propagate the \ac{ADV} accordingly but, if it does that at time greater than $t_0+\epsilon$, it happens that $j$ indeed receives the updated route from $z$ which may change again its routing table but it cannot propagate the new information as $\tau(j,k,d)>0,\ \forall k\in N_j$.

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This behaviour makes $j$ send two \acp{ADV} for the same destination $d$, the first of which was computed on outdated information.

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Worse than that, the updated \ac{ADV} cannot be propagated before $T_{ik}$ seconds to the neighbours $k\in N_j$.

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\textbf{Duplicate duplication.} Suppose that $k\in N_j$ receives the \ac{ADV} at time $t_0+\epsilon$.

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$k$ generates hence an \ac{ADV} and propagates the wrong information.

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The duplication can happen again and $k$ receives at $t_1$ an \ac{ADV} from a node $h$ in a path from $k$ and $j$.

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So $k$ has to wait $\tau(j,x,d)$ seconds before sending the updated information to its neighbour $x\in N_k$.

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Now, if $T_{kx} > T_{jk}$ then $k$ will receive the update from $j$ and send only one duplicate \ac{ADV}.

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But, if $T_{kx} < T_{jk}$ it means that the forementioned step can be repeated twice and $k$ will send a total of 4 \ac{ADV} messages.

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\textbf{Exponential explosion.} In general and in presence of multipaths, if $p_{ij}=(i,k_0),\dots,(k_l,j)$ is a propagation path (a path in $G^P$), then the number of \ac{ADV} messages (and route oscillation) is exponential on the overall number of decreasing timers on the path.

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\section{Solution sketch}

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Ideally, we would like to have increasing $T_{ij}$ along the propagation paths in $G^P$.

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Because of path aggregation\footnote{It is an important feature but Bird does not perform it.}, $\xi$ does not describe the exact path but just a sequence of node subsets.

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Hence, it is difficult for a node to select its timer wrt its neighbours.

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Instead we can use \ac{MiCe} to infer the position of a node\footnote{We would need the position on the path but \ac{MiCe} is computed graph-wide.} and tune the timers $T_{ij}$ accordingly.

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Since, two nodes $j_1,j_2$ at the periphery, hence with similar centrality, could be one in $G^{(1)}$ and the other in $G^{(3)}$ respectively, we need to consider their case separately.

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Considering a graph-wide maximum timer $T=30$ and \ac{MiCe} centrality $c_i\in [0,1]$ for node $i$, one strategy could be to assign:

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\begin{itemize}

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	\item $T_{ij}=\frac{Tc_i}{2},\ \forall i\in V^{(1)}$

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	\item $T_{ij}=\frac{T}{2},\ \forall i\in V^k$

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	\item $T_{ij}=\frac{T(1-c_i)}{2}+\frac{T}{2}=\frac{T(2-c_i)}{2},\ \forall i\in V^{(3)}$

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\end{itemize}

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\textbf{Note:} we are not sure about \ac{MiCe} normalization in the interval $[0,1]$ applicability.

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\bibliographystyle{IEEEtran}

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\bibliography{biblio}

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\end{document}

bgp_model/biblio.bib
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@ARTICLE{elmokashfi2010scalability,

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author={A. {Elmokashfi} and A. {Kvalbein} and C. {Dovrolis}},

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journal={IEEE Journal on Selected Areas in Communications},

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title={On the Scalability of BGP: The Role of Topology Growth},

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year={2010},

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volume={28},

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number={8},

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pages={1250-1261},

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keywords={Internet;routing protocols;telecommunication network reliability;telecommunication network topology;BGP scalability;topology growth;BGP routing;Internet;multihoming;peering;routing protocol;Peer to peer computing;Topology;Routing;Internet topology;Scalability;Network topology;Internetworking;Routing;Topology;BGP},

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doi={10.1109/JSAC.2010.101003},

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ISSN={0733-8716},

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month={October},}

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@article{fabrikant2011something,

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  title={There's something about MRAI: Timing diversity can exponentially worsen BGP convergence},

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  author={Alex Fabrikant and Umar Syed and Jennifer Rexford},

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  journal={2011 Proceedings IEEE INFOCOM},

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  year={2011},

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  pages={2975-2983}

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}

specs/bgp_graphml.tex
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\documentclass[twocolumn, 10pt]{article}

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\usepackage{acronym}

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\usepackage{tikz}

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\usepackage{cleveref}

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\usetikzlibrary{calc,shapes,arrows}

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\title{BGP graphml specification}

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\begin{document}

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\maketitle

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\newcommand{\mandatory}{\textcolor{red}}

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\acrodef{AS}{autonomous system}

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\acrodef{ADV}{advertisement}

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This document describes the graphml representation of a BGP network.

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Graphml is a standard for representing graphs enabling attribute embedding.

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A BGP network is a simple undirected graph $G(V, E)$ (no self-loops, no multi-edges) modeling BGP \ac{ADV} communication.

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\begin{itemize}

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\item \textbf{A node} is an eBGP speaker node representing an \ac{AS};

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\item \textbf{An edge} represents a BGP \ac{ADV} communication link (hence undirected).

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\end{itemize}

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\section{Attributes}

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We indicate with \textit{destination\_string} a concatenation of IP subnets, in the form of: {\footnotesize{\verb[<ip_address>/<mask_bits>,<ip_address>/<mask_bits>,...[ }}

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\\

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Node attributes (mandatory in red):

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\begin{itemize}

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\item \mandatory{type}: \{T,M,C,CP\}, which respectively indicates if a node is tier-1, mid-level, customer or content provider;

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\item destinations: [destination\_string], it indicates the destinations exposed by the node.

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\end{itemize}

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Edge attributes (mandatory in red):

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\begin{itemize}

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\item \mandatory{type}: \{transit, peer\}, which for a couple $(i, j)\in V\times V$, indicates whether there is a customer-provider or peer-to-peer relationship among $i$ and $j$.

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\item \mandatory{customer}: $z\in V$, in case the edge $(i,j)$ type is \textit{transit}, $z$ identifies the customer among $i,j$ (the relationship provider is hence the other node). It is undefined in the case the edge type is not \textit{transit};

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\item node\_a: $i\in (i, j)$;

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\item node\_b: $j\in (i, j)$;

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\item preference\_a: [integer], it indicates preference of node\_a to pick edge $(i,j)$ among node\_a neighbouring links;

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\item preference\_b: [integer], it indicates preference of node\_b to pick edge $(i,j)$ among node\_b neighbouring links;

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\item mrai: [interval] in seconds, it indicates the per link $(i,j)$ minimum route \ac{ADV} interval. Note, node-based mrai setup (Fabrikant gadget) can be described by setting all its mrai edge intervals to the same value.

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\end{itemize}

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\section{Example}

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The following is a basic graphml example with mandatory attributes only.

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\footnotesize

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\begin{verbatim}

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<?xml version='1.0' encoding='utf-8'?>

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<graphml xmlns="http://graphml.graphdrawing.org/xmlns"

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xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"

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xsi:schemaLocation="http://graphml.graphdrawing.org/xmlns

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http://graphml.graphdrawing.org/xmlns/1.0/graphml.xsd">

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<key attr.name="type" attr.type="string" for="edge" id="d2"/>

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<key attr.name="customer" attr.type="long" for="edge" id="d1"/>

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<key attr.name="type" attr.type="string" for="node" id="d0"/>

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<graph edgedefault="undirected">

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<node id="1">

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  <data key="d0">T</data>

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</node>

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<node id="2">

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  <data key="d0">M</data>

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</node>

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<node id="3">

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  <data key="d0">C</data>

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</node>

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<node id="4">

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  <data key="d0">CP</data>

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</node>

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<edge source="2" target="1">

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  <data key="d2">transit</data>

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  <data key="d1">2</data>

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</edge>

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<edge source="3" target="2">

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  <data key="d2">transit</data>

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  <data key="d1">3</data>

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</edge>

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<edge source="4" target="2">

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  <data key="d2">transit</data>

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  <data key="d1">4</data>

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</edge>

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<edge source="3" target="4">

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  <data key="d2">peer</data>

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</edge>

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</graph></graphml>

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\end{verbatim}

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%\bibliographystyle{IEEEtran}

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%\bibliography{biblio}

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\end{document}


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