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\thispagestyle{empty} \vspace*{-0.75in} {\bssten MS\&E 337 \hfill
Lecture \#\arabic{lecture} Notes \\ Information Networks \hfill Spring 2010 \\
Prof. Amin Saberi \hfill Page 1 of \pageref{totalpag} \\}



\topic{Preferential Attachment} The preferential attachment model
was introduced by Barab\'asi and Albert   to explain the power-law
degree distribution  in complex real-world networks. Here is their
model, stated more formally.

Fix $m > 0$ constant.  Let $G_1, G_2, \ldots, G_t, \ldots$ be a
sequence of graphs such that $G_t$ is obtained from $G_{t-1}$ by
adding $v_t$ and $m$ edges to
$\left\{v_1,v_2,\ldots,v_{t-1}\right\}$ such that the probability of
adding a link from $v_t$ to $v_i$ is proportional to the degree of
$v_i$.  Specifically, we define intermediate graphs
$\Gamma_{t,1},\Gamma_{t,2}, \ldots, \Gamma_{t,m}$ such that
$\Gamma_{t,j}$ is obtained from $\Gamma_{t,j-1}$ by adding a link
between edge $v_t$ and $v_l$ with probability
$deg(v_l)/2(m(t-1)+j-1)$, and $G_t = \Gamma_{t-1,m} = \Gamma_{t,0}$.
We initialize the process by setting $G_1$ to be a singleton with
$m$ loops.

Barab\'asi and Albert ran  Monte Carlo simulations of the
preferential attachment process and estimated a power-law exponent
of $\alpha = 2.9\pm .1$.  This observation led to the following
heuristic ``mean-field'' argument for the true exponent to be
$\alpha = 3$:
\begin{enumerate}
\item Let $d_i(t)$ be the degree of node $i$ at time $t$:
\[d_i(t) = deg(v_i), v_i \in G_t.\]
If we think of this as a continuous process and take the derivative,
we get:
\begin{equation}
\frac{\delta d_i(t)}{\delta t} = \frac{m d_i(t)}{\sum_{j=1}^nd_j(t)}
= \frac{d_i(t)}{2t} \label{eq1}
\end{equation}
\item Let $t_i$ be the time that node $i$ arrives.  We can solve
(\ref{eq1}) to get:
\begin{equation} d_i(t) = m \sqrt \frac{t}{t_i} \end{equation}

\item Assuming that $t_i$ is distributed uniformly on $[0,t]$, then
\begin{equation}
\p\left(d_i(t) > k\right) = \p\left(m\sqrt t > k \sqrt t_i\right) =
\p\left(t_i < \frac{m^2t}{k^2} = \frac{m^2}{k^2}\right) \label{eq3}
\end{equation}
\item Differentiating (\ref{eq3}) gives
\begin{equation} \p(d_i(t) = k) = \frac{2m^2}{k^3}\end{equation}
\end{enumerate}

This gives a ``back of the envelope'' justification for the $\alpha
= 3$ power-law exponent.  We can prove this more rigorously.


Let $Z(k,t)$ be the random variable indicating the number of
vertices of degree $k$ at time $t$.  Let $N(k,t) = \E(Z(k,t))$.
Then
\begin{align}
N(k,t) - N(k,t-1) = \frac{m(k-1)N(k-1,t)}{2m(t-1)} -
\frac{mkN(k,t-1)}{2m(t-1)} + 1_{m=k} + \epsilon(k,t), \label{rec}
\end{align}
where $\epsilon(k,t)$ accounts for the possibility of multiple edges
and can be bounded by
\[|\epsilon(k,t)| =
O\left(\sum^m_{i=2}\frac{(k-i)^iN_{k-i}(t)}{(mt)^i}\right) =
O\left(\frac{k}{t}\right) = O\left(t^{-1/2}\right)\]

Solving the recurrence relationship (\ref{rec}) can be done by
algebraic manipulations (see Durret's Random Graph Dynamics, pp
92-93). For us, it is slightly easier.
%: define $p_k = \lim_{t\rightarrow \infty } N(k,t)/t$.
You can verify that
\[N(k,t) = \frac{(t-1)m(m+1)}{k(k+1)(k+2)} + O(t^{-1/2}),\]
by verifying it at $k = m$ and in the recurrence relation.

All that is left is to show that $Z(k,t)$ is concentrated around its
expectation. To prove this, we define a martingale process $Z_i(k,t)
= \E(Z(k,t)|Y_1,Y_2,\ldots,Y_{i-1}),$ where the $Y_i$'s are the
random choices made at time $i$.  We can then evoke the
Azuma-Hoeffding inequality:
\begin{lemma} (Azuma-Hoeffding inequality). Let $(X_t)^n_{t=0}$ be a
martingale with $|X_{t+1} - X_t| \leq c$ for $t = 0,\ldots,n-1$.
Then
\[\p(|X_n-X_0| \geq x) \leq \exp\left(-\frac{x^2}{2c^2n}\right).\]
\end{lemma}
If we note that $|Z_i(k,t) - Z_{i-1}(k,t)| \leq 2m$, then we are
done.



\topic{Connections to P\'olya Urn Scheme}

We give an equivalent description of the preferential attachment
process as a combination of several P\'olya urn processes. The
P\'olya urn model is proposed and analyzed much earlier in the
beautiful work of P\'olya and Eggenberger in the early twentieth
century.
%{\bf Preferential Attachment model}


{\bf P\'olya urn scheme}

In early twentieth century, P\'olya proposed and analyzed the
following model known as the P\'olya urn model. Suppose we have an
urn with $r$ red balls and $b$ blue balls. At each step $i$, we pick
a ball uniformly at random from the urn and replace it with two
balls of the same color.

P\'olya showed that this model is equivalent to another process as
follows. Choose a parameter (like "strength" or "attractiveness")
$p$, and at each step, {\em independently} of the decision in
previous steps, toss a coin with bias $p$ and put a blue or red ball
in the urn depending on the outcome. P\'olya specified the
distribution of $p$ for which this mimics the urn model and showed
that it is a $\beta$-distribution with appropriate parameters.

For a complete treatment of P\'olya urn scheme we need to define
exchangeability of random variables and cover de Finnetti's Theorem.
You can find those in Durrett's book Probability: Theory and
Examples. Here we only sketch the main idea:

Let
\[X_t = \left\{ \begin{array}{ll}1 &: \mbox{ ball chosen at time $t$
is blue}\\0 &: \mbox{ otherwise}\end{array} \right.\] Then
\[\p(X_1 = 1, X_2 =0, X_3 = 1) =
\frac{b}{r+b}\frac{r}{r+b+1}\frac{b+1}{r+b+2} = \p(\mbox{``rbb'''})
= \p(\mbox{``bbr'''}).\] The property that the ordering of events
doesn't affect their cumulative probability, only the number of
events of each type, is known as \emph{exchangeability}. The
probability of seeing $n_1$ red balls and $n_2$ blue balls after
$n=n_1+n_2$ steps is
\begin{align*}
\p(\mbox{Number of red balls is } n_1)&=
\frac{n!}{n_1!n_2!}\frac{r(r+1)\ldots(r+n_1-1)b(b+1)\ldots(b+n_2-1)}{(r+b)\ldots(r+b+n-1)}\\
&=
\frac{n!}{n_1!n_2!}\frac{(r+b-1)!}{(r-1)!(b-1)!}\frac{(r+n_1-1)!(b+n_2-1)}{(r+b+n-1)!}
\end{align*}
Suppose $n_1/n = x$, and taking the limit for each term,
\begin{align*}
&\lim_{n\rightarrow \infty} (r+n_1-1)!/n_1! \rightarrow n_1^{r-1} \\
&\lim_{n\rightarrow \infty} (b+n_2-1)!/n_2! \rightarrow n_2^{b-1} \\
&\lim_{n\rightarrow \infty} (r+b+n-1)!/n! \rightarrow n^{b+r-1},
\end{align*}
the probability of seeing $n_1$ red balls converges to
\[\lim_{n\rightarrow \infty} \p(\mbox{Number of red balls is } n_1)
\rightarrow \frac{(r+b-1)!}{(r-1)!(b-1)!} \frac{1}{n}
x^{r-1}(1-x)^{b-1},\]

which is a random variable with distribution $\beta(r,b)$.

{\bf Preferential attachment revisited}
%\begin{thm}
%The preferential attachment model is equivalent to the following: \\
%Let
%\begin{align}
%\Psi_k = \beta(m,(2k-3)m) \\
%\Phi_k = \Psi_k \prod_{i=k+1}^n (1-\Psi_i).
%\end{align}
%Then the degree of node $k$ is proportional to $\Phi_k$.


It is not hard to see that there is a close connection between the
the preferential attachment model of Barab\'asi and Albert and the
P\'olya urn model in the following sense: every new connection that
a vertex gains can be represented by a new ball added in the urn
corresponding to that vertex. We use this idea to give an equivalent
description of the scale-free graph which is easier to analyze. We
will see throughout the paper the properties of this description
that make it useful for understanding the graph.

To derive this representation, let us consider first a two urn
model, with the number of balls in one urn representing the degree
of a particular vertex $k$, and the number of balls in the other
representing the sum of the degrees of the vertices $1,\dots, k-1$.
We will start this process at the point when $n=k$ and $k$ has
connected to precisely $m$ vertices in $\{1,\dots, k-1\}$.  Note
that at this point, the urn representing the degree of $k$ has $m$
balls, while the other one has $(2k-3)m$ balls.


Taking into account that the two urns start with $m$ and $(2k-3)m$
balls, respectively, we see that the evolution of the two bins is a
P\'olya urn with strengths $\psi_k$ and $1-\psi_k$, where
\begin{equation}
\label{psik-dis} \psi_k\sim\beta\left(m , (2k-3)m
%\frac{(k-1)^2}{k}
\right).
\end{equation}
%
%\err{(Note that this differs from Noam's version, which
%claimed that
%\[
%\psi_k\sim\beta\left(m+2mu\frac{k-1}{k},(2k-3)m+2\frac{(k-1)^2}{k}mu\right).
%\]
%}
%%


Having the above insight, we construct an alternative description of
the preferential attachment model. Let $\psi_1=1$, and for every
$2\leq k\leq n$, we take
 $\psi_k$ to be distributed
according to $\beta(m,(2k-3)m)$. For $1\leq k\leq n$, we take
\[
\phi_k=\psi_k\prod_{j=k+1}^n(1-\psi_j).
\]
It is easy to see that $\sum_{k=1}^n\phi_k=1$. Let
\[
l_k=\sum_{j=1}^k\phi_k.
\]
For every $a\in[0,1]$, we define $\kappa(a)=\min\{k:l_k\geq a\}$.
Let $\{U_{i,k}\}_{1\leq i\leq m,1\leq k\leq n}$ be independent
random variables, uniform on $[0,1]$. For $k>j$, we draw an edge
between $k$ and $j$ if for some $1\leq i\leq m$ we have
\begin{equation}\label{eq:kappa}
j=\kappa(U_{i,k}l_{k-1}).
\end{equation}
%We allow multiple edges --- the number of edges connecting $k$ to
%$j$ is the number of values of $i$ such that  (\ref{eq:kappa}) is
%satisfied.
\begin{thm}\label{lem:polbar}
The random graph described above has the same distribution as the
$n$-vertex preferential attachment model with parameter $m$.
\end{thm}

The proof of the above Theorem follows from the theory of P\'olya
urns. For the details of the proof, you can see [BBCS07]. The main
advantage of this new representation is that it contains much more
independence and is therefore much simpler to analyze. For example
you can see the analysis of joint degree distributions, the limiting
distribution of subgraphs in balls of any given radius $k$ around a
random vertex in the preferential attachment graph in [BBCS07].


\label{totalpag}
\end{document}