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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{amssymb}
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\usepackage{amsmath}
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\usetikzlibrary{calc,shapes,arrows}
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\usepackage{cleveref} 
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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{Node Policy}
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Nodes evaluate route \acp{ADV} with a function $\Gamma: R \to [0,100]\cap \mathbb{N}$, where $R$ is the set of routes.
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Given two routes $r_1,r_2$, a node prefers $r_1$ if $\Gamma(r_1)\geq \Gamma(r_2)$ and $r_2$ otherwise.
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As noted before, $r\in R$ is a couple $(d, \xi)$ of destination and attributes.
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The simplest form of policy returns the (possibly weighted) length of the \ac{AS} path indicated in the attributes $\xi$ (hop distance to $d$).
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Policies are implemented defining the $\Gamma_i$ function for each $i\in V$.
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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 them separately.
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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 consider them separately.
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Whenever node $i$ recives a route UPDATE for destination $d$ it updates its Adj-RIBs-In \footnote{the information structure pertaining route \ac{ADV}}, and if the new route $r$ obtain a newly best score $\Gamma(r)$, then it selects $\xi_m$ as the new preferred attribute for destination $d$, and its schedules a new \ac{ADV} to be sent according to MRAI.
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When the MRAi timer fires only the new best option is advertised.
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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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\section{Solution sketch No.2}
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Ideally, we would like to have increasing $T_{ij}$ along the propagation paths in $G^P$.
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We can propose an extension to the \ac{ADV} UPDATE message so to include sending node MRAI timer.
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\begin{enumerate}
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\item Node $i$ advertise route $r$;
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\item Node $i$ sends an UPDATE to $j\in N_i$ for $r$ with an MRAI of $T_{ij}$;
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\item If node $j$ accepts $r$ than $j$ sets its MRAI for its neighbour $z\in N_j$,  $T_{jz}=T_{ij}+c$.
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\item The procedure repeats from point 2 with $i=j$.
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\end{enumerate}
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$c$ is a reasonable enough value.
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\bibliographystyle{IEEEtran}
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\bibliography{biblio}
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\end{document}