graph/community/louvain.tex

% Copyright ©2015 The Gonum Authors. All rights reserved.
% Use of this source code is governed by a BSD-style
% license that can be found in the LICENSE file.

\documentclass{article}

\usepackage{amsmath,amsfonts}
\usepackage[margin=4cm]{geometry}

\title{Louvain algorithm for undirected and directed graphs}
\author{The {\tt gonum} Authors}

\begin{document}

\maketitle

The algorithm attempts to find communities (highly connected sub-graphs),
and it does this by minimising the modularity function
\begin{equation}
	Q(c) = \frac{1}{2m}\sum_i\sum_j\left[ A_{ij} - \gamma \frac{k_ik_j}{2m} \right] \delta_{ij}(c),
\end{equation}
where $c$ is a partition of nodes into subsets or communities,
$A_{ij}$ is the edge weight between nodes $i$ and $j$,
$\gamma$ is a tuning parameter,
\begin{equation}
m = \frac{1}{2}\sum_i\sum_jA_{ij},
\end{equation}
\begin{equation}
k_i = \sum_j{A_{ij}},
\end{equation}
and
\begin{equation}
	\delta_{ij}(c) = \left \{ \begin{array}{ll}
	1 & \text{if} \quad  c(i) = c(j) \\
	0 & \text{otherwise} \end{array} \right ..
\end{equation}
Here $c(i)$ denotes the community to which node $i$ belongs
in the partitioning $c$.

The algorithm finds a hierarchical community structure by iterating
between two phases:
\begin{enumerate}
	\item Find a set of communities that minimise $Q$.
	\item Construct a new graph, whose nodes are the communities
	found in the preceding phase one step.
\end{enumerate}
Each iteration of these two phases is called a `pass'.
In this way, the algorithm obtains a nested community structure,
where at each level $Q$ is minimised for the relevant graph.
We consider this process in more detail, in particular looking
at phase one first in the first pass, when each node is a single
node, and then how this generalises to later passes when each node
is a community.

\section{Undirected Graphs}

\subsection{Initial Pass}
\label{sec:initialPass}

The initial pass is simple as the initial pass uses the original graph,
and in all following passes graphs constructed in the previous pass's
phase two are used.
Here we will consider this initial simple formulation for phase one, and
in Section~\ref{sec:laterPasses} we consider how this generalises for
passes two and onwards.
Phase one works by initially allocating each node to a separate community,
and then iterating through each node $a$ and checking if moving it into
a different community $\beta$ will reduce $Q$.
If there are possible moves that will reduce $Q$, $a$ is moved into the
the community which will generate the largest reduction in $Q$.
This process is continued until there are no moves left to reduce $Q$
further, meaning a local minimum for $Q$ has been achieved.
Then the algorithm moves to phase two (constructing a new graph where
each node in the new graph is a community in the old graph).

Note that we assume the original graph to be simple and undirected.
First, we introduce some notation that will be useful:
Let $c(i)$ denote the community to which node $i$ belongs,
and let $\alpha$ be the community that the node $a$ mentioned above
belongs to, i.e., $\alpha = c_a$.
Then we define
\newcommand{\Stot}[1]{\Sigma_{\text{tot}}^{#1}}
\begin{equation}
	\Stot{\alpha} = \sum_{i \in \alpha}\sum_{j}A_{ij} = \sum_{i \in \alpha}k_i,
\end{equation}
\newcommand{\kin}[2]{k_{#1}^{#2}}
\begin{equation}
	\kin{i}{\alpha} = \sum_{j \in \alpha}A_{ij},
\end{equation}
and
\newcommand{\Sin}[1]{\Sigma_{\text{in}}^{#1}}
\begin{equation}
	\Sin{\alpha} = \sum_{i \in \alpha}\sum_{j \in \alpha}A_{ij} = \sum_{i \in \alpha}\kin{i}{\alpha}.
\end{equation}

We are interested in how $Q$ will change if we move a node $a$ from its
current community $\alpha$, to a new community $\beta$.
This will have two effects, it will remove the terms from $Q$
related to $a$ in $\alpha$, which we will call $Q^-$ and it will add terms
related to $a$ in $\beta$, which we will call $Q^+$.
The total change in $Q$ caused by the movement of $a$ from $\alpha$ to $\beta$ is
\begin{equation}
	\Delta Q = Q^{+} - Q^{-},
\end{equation}
where
\begin{align*}
Q^- &= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2\sum_{i \in \alpha, \, i \neq a} \left( A_{ia} - \gamma \frac{k_ik_a}{2m} \right) \right] \\
	&= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2 \left( \kin{a}{\alpha} -A_{aa}\right) - \gamma \frac{2k_a}{2m}\sum_{i \in \alpha, \, i \neq a} k_i \right] \\
	&= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2 \left( \kin{a}{\alpha} -A_{aa}\right) - \gamma \frac{2k_a}{2m}\left( \Stot{\alpha} - k_a \right)  \right], \\
\end{align*}
and
\begin{align*}
Q^+ &= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2\sum_{i \in \beta} \left( A_{ia} - \gamma \frac{k_ik_a}{2m} \right) \right] \\
		&= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2\kin{a}{\beta} - \gamma \frac{2k_a}{2m}\sum_{i \in \beta} k_i \right] \\
		&= \frac{1}{2m}\left[ \left( A_{aa} - \gamma \frac{k_a^2}{2m} \right)
+ 2\kin{a}{\beta} - \gamma \frac{2k_a\Stot{\beta}}{2m} \right]. \\
\end{align*}
The first term in both these expressions ($Q^-$ and $Q^+$) is the same, and so cancels:
\begin{equation}
\Delta Q = \frac{1}{2m}\left[ \left( 2\kin{a}{\beta} - \gamma \frac{2k_a\Stot{\beta}}{2m} \right)
		- \left( 2 \left( \kin{a}{\alpha} -A_{aa}\right) - \gamma \frac{2k_a}{2m}\left( \Stot{\alpha} - k_a \right) \right) \right].
\end{equation}

\subsection{Later Passes}
\label{sec:laterPasses}

In phase two a `meta-graph' is constructed where nodes correspond to
the communities found in the preceding phase one step, and edge weight
between two such communities (nodes, in the meta-graph)
$\alpha$ and $\beta$ are defined to be
\begin{equation}
	A_{\alpha \beta}^* = \sum_{i \in \alpha}\sum_{j \in \beta}A_{ij}.
	\label{eqn:Aij*}
\end{equation}
Note that $i$ and $j$ refer to nodes in the original graph, not nodes
in the previous graph, and so holds any meta-graph, not just the first.
Also note that this definition of $A^*_{\alpha \beta}$ allows for
$A^*_{\alpha \alpha}$ to be non-zero as
\begin{equation}
A_{\alpha \alpha}^* = \sum_{i \in \alpha}\sum_{j \in \alpha}A_{ij} = \Sin{\alpha}.
\end{equation}

In this newly constructed graph, $\alpha$ and $\beta$ are nodes, but
also refer to communities (sets of nodes) in the original graph, and I
use these two interpretations interchangeably.
This should be the only ambiguous bit of notation in this document, I hope.

The results of Section~\ref{sec:initialPass} generalise to these meta-graphs,
and the generalised results mirror those of Section~\ref{sec:initialPass} closely
-- I distinguish the new results from those of Section~\ref{sec:initialPass} by a
superscript $*$.
I use $i$ and $j$ to denote nodes of the original graph as in Section~\ref{sec:initialPass},
and use $z$ and $w$ to denote nodes of the meta-graph (communities of the original).
I use analogous notation to Section~\ref{sec:initialPass}, $c^*(z)$, to
denote the community to which node $z$ of the meta-graph belongs,
and let $\mathfrak{a}$ be the community that the node $\alpha$ belongs to
($c^*(\alpha) = \mathfrak{a}$), i.e.
\begin{equation}
	\mathfrak{a} = \{z | c^*(z) = c^*(\alpha) \}.
\end{equation}

Given this notation, we can observe that
\begin{equation}
m^* = \frac{1}{2}\sum_{z}\sum_{w}{A_{zw}^*} = \frac{1}{2}\sum_{z}\sum_{w}{\sum_{i \in z}\sum_{j \in w}A_{ij}} = \frac{1}{2}\sum_i\sum_jA_{ij} = m,
\end{equation}
\begin{equation}
k_{z}^* = \sum_{w}{A_{zw}^*} =  \sum_{w}{\sum_{i \in z}\sum_{j \in w}A_{ij}} = \sum_{i \in z}\sum_{j}A_{ij} = \Stot{z},
\end{equation}
\begin{equation}
	\Stot{\mathfrak{a} *} = \sum_{z \in \mathfrak{a}}\sum_{w}A_{zw}^* = \sum_{z \in \mathfrak{a}}k_z^* = \sum_{z \in \mathfrak{a}}\Stot{z},
\end{equation}
\begin{equation}
	\kin{z}{\mathfrak{a} *} = \sum_{w \in \mathfrak{a}}{A_{zw}^*} = \sum_{w \in \mathfrak{a}}{\sum_{i \in z}\sum_{j \in w}A_{ij}},
\end{equation}
and
\begin{equation}
\Sin{\mathfrak{a} *} = \sum_{z \in \mathfrak{a}}\sum_{w \in \mathfrak{a}}A_{zw}^* = \sum_{z \in \mathfrak{a}}\kin{z}{\mathfrak{a} *} = \sum_{z \in \mathfrak{a}}\sum_{w \in \mathfrak{a}}{\sum_{i \in z}\sum_{j \in w}A_{ij}}.
	%\label{eqn:Sin}
\end{equation}

If we let $\mathfrak{b}$ denote the community to which we are considering moving $\alpha$,
then the expression for $\Delta Q$ from Section~\ref{sec:initialPass} trivially generalises to
\begin{equation}
\Delta Q = \frac{1}{2m}\left[ \left( 2 \kin{\alpha}{\mathfrak{b} *} - \gamma \frac{2k_{\alpha}^*\Stot{\mathfrak{b} *}}{2m} \right)
		- \left( 2\left( \kin{\alpha}{\mathfrak{a} *} - A_{\alpha \alpha}^* \right) - \gamma \frac{2k_{\alpha}^*}{2m} \left( \Stot{\mathfrak{a} *} - k_{\alpha}^* \right ) \right) \right] \\
\end{equation}

\section{Directed Graphs}
\label{sec:directedGraphs}

It is of interest to consider how this generalises to directed graphs.
If we are to treat incoming and outgoing nodes equally, there are several
thoughts on how to extend the algorithm to directed graphs, of which we
will explore three:
\begin{itemize}
	\item Construct an undirected graph first, and then use the undirected case.
	\item Generalise the expressions from the undirected case to the directed case,
	we will consider two different suggestions for such generalisations.
\end{itemize}
We will show that one of the two `generalisation of expressions' approaches is
equivalent to constructing an undirected graph, and the other is not.

\subsection{Construction of an undirected graph}
A simple approach to generalising to directed graphs is to construct
an undirected graph with edge weights
\begin{equation}
A_{ij} = B_{ij} + B_{ji},
\label{eqn:undirectedAB}
\end{equation}
and simply use the undirected algorithm.
Another suggestion is to average the directed edges to make
an undirected graph, i.e. to use a directed graph with edge weights
\begin{equation}
A_{ij} = \frac{B_{ij} + B_{ji}}{2}.
\end{equation}
This raises an important question: does scaling all edge weights across
the entire graph by a constant affect the results of the algorithm?
Hopefully not, but worth checking.
We can follow this through the results for the undirected graph by
substituting $A_{ij}^{(1)} = pA_{ij}$, $p \in \mathbb{R}$, and
distinguishing the new expressions by a superscript ${(1)}$. These
new expressions are:
\begin{equation}
m^{(1)} = \frac{1}{2}\sum_i\sum_jpA_{ij}  = p\frac{1}{2}\sum_i\sum_j A_{ij} = pm ,
\end{equation}
\begin{equation}
k_i^{(1)} = \sum_j{pA_{ij}} = p\sum_j{A_{ij}} = pk_i,
\end{equation}
and so
\begin{align*}
	Q^{(1)}(c) &= \frac{1}{2pm}\sum_i\sum_j\left[ pA_{ij} - \gamma \frac{pk_ipk_j}{2pm} \right] \delta_{ij}(c) \\
	&= \frac{1}{2m}\sum_i\sum_j\left[ A_{ij} - \gamma \frac{k_ik_j}{2m} \right] \delta_{ij}(c) \\
	&= Q(c)
\end{align*}
Note that as we have shown $Q^{(1)} = Q$ there is no need to go into the remainder of the terms
involved in the algorithm, as they all derive from $Q$.

\subsection{First generalisation of expressions approach}

One suggested extension to directed graphs is to modify the expressions
involved by adding the `from' case and the `to' case for each term.
If we let $B_{ij}$ be the edge weight between nodes $i$ and $j$ in
the directed graph, and distinguishing these extended expressions by
a superscript $(2)$, the extended expressions become:
\begin{equation}
m^{(2)} = \frac{1}{2}\left ( \sum_i\sum_jB_{ij} + \sum_i\sum_jB_{ji}\right)  = \frac{1}{2}\sum_i\sum_j \left( B_{ij} + B_{ji} \right) ,
\end{equation}
\begin{equation}
k_i^{(2)} = \sum_jB_{ij} + \sum_jB_{ji} = \sum_j{\left( B_{ij} + B_{ji} \right)},
\end{equation}
and similarly
\begin{equation}
	Q^{(2)}(c) = \frac{1}{2m}\sum_i\sum_j\left[ \left( B_{ij} + B_{ji} \right) - \gamma \frac{k_i^{(2)}k_j^{(2)}}{2m} \right] \delta_{ij}(c).
\end{equation}

Note how this is equivalent to the construction of an undirected graph as
per Equation~(\ref{eqn:undirectedAB}). Similarly to above,
there is no need to go into the remainder of the terms
involved in the algorithm, as they all derive from $Q$.


\subsection{Second generalisation of expressions approach}

Another approach to generalising the expressions to the
directed case, that still treats incoming and outgoing edges
as equally important, is to propose an alternative modularity
expression:
\newcommand{\dkin}[1]{k_{#1}^{\text{in}}}
\newcommand{\dkout}[1]{k_{#1}^{\text{out}}}
\begin{equation}
Q^{(3)}(c) = \frac{1}{2m}\sum_i\sum_j\left[ 2B_{ij} - 2\gamma \frac{\dkin{i}\dkout{j}}{2m} \right] \delta_{ij}(c), \\
\end{equation}
where
\begin{equation}
\dkout{i} = \sum_j{B_{ij}}
\quad \quad \text{and} \quad \quad
\dkin{i} = \sum_j{B_{ji}},
\end{equation}
so $k_i^{(2)} = \dkin{i} + \dkout{i}$.
Note I leave the factor of two in the expression for $Q^{(3)}$ so that it
remains as comparable to that for $Q^{(2)}$ as possible.
There is no need for alternative $m$, as it will still be the same as above.
$Q^{(3)}$ will differ from $Q^{(2)}$ in two ways.
Firstly, as $k_i^{(2)} = \dkin{i} + \dkout{i}$,
\begin{align*}
\sum_i\sum_j k_i^{(2)} k_j^{(2)} \delta_{ij}(c)  &= \sum_i\sum_j  (\dkin{i} + \dkout{i}) (\dkin{j} + \dkout{j}) \delta_{ij}(c) \\
 &= \sum_i\sum_j \left[ (\dkin{i}\dkin{j} + \dkout{i}\dkout{j}) + (\dkin{i}\dkout{j} + \dkin{j}\dkout{i}) \right] \delta_{ij}(c). \\
 &= \sum_i\sum_j \left[ (\dkin{i}\dkin{j} + \dkout{i}\dkout{j}) + 2\dkin{i}\dkout{j} \right] \delta_{ij}(c), \\
\end{align*}
and similarly,
\begin{equation}
\sum_i\sum_j  \left( B_{ij} + B_{ji} \right) \delta_{ij}(c) = 2\sum_i\sum_j B_{ij} \delta_{ij}(c).
\end{equation}
From these two expressions, we can see that
\begin{equation}
Q^{(3)} - Q^{(2)} = \frac{1}{2m}\sum_i\sum_j \gamma \frac{\dkin{i}\dkin{j} + \dkout{i}\dkout{j}}{2m} \delta_{ij}(c).
\end{equation}


\section{Directed Graphs in more detail}
\label{sec:directedGraphsDetail}

In Section \ref{sec:directedGraphs} we essentially showed three
things:
\begin{itemize}
	\item How an undirected graph could be constructed from a directed
	graph, thereby allowing the undirected algorithm to be used for
	directed graphs.
	\item How scaling all edge weights by a non-zero constant would not
	affect the modularity function.
	\item An alternative approach to extending the algorithm to
	directed graphs that is not equivalent to first reducing it
	to an undirected graph.
\end{itemize}
It is this third point that we will explore here.
Analogously to Sections \ref{sec:initialPass} and \ref{sec:laterPasses} we will
break this up into the initial pass and the later passes.

\subsection{Initial pass}
\label{sec:initialPassDirected}

Continuing with the notation of Section \ref{sec:initialPass}, in which
$c(i)$ denotes the community to which node $i$ belongs,
and $\alpha = c(a)$, we define
\newcommand{\dinStot}[1]{\Sigma_{\text{tot}}^{\text{in}(#1)}}
\newcommand{\doutStot}[1]{\Sigma_{\text{tot}}^{\text{out}(#1)}}
\begin{equation}
	\doutStot{\alpha} = \sum_{i \in \alpha}\sum_{j}B_{ij} = \sum_{i \in \alpha}\dkout{i}
	\quad \quad \text{and} \quad \quad
	\dinStot{\alpha} = \sum_{i \in \alpha}\sum_{j}B_{ji} = \sum_{i \in \alpha}\dkin{i},
\end{equation}
\newcommand{\dinkin}[2]{k_{#1}^{\text{in}(#2)}}
\newcommand{\doutkin}[2]{k_{#1}^{\text{out}(#2)}}
\begin{equation}
	\doutkin{i}{\alpha} = \sum_{j \in \alpha}B_{ij}
	\quad \quad \text{and} \quad \quad
	\dinkin{i}{\alpha} = \sum_{j \in \alpha}B_{ji},
\end{equation}
and we will entertain one more ambiguous notation choice:
%\newcommand{\Sin}[1]{\Sigma_{\text{in}}^{#1}}
\begin{equation}
	\Sin{\alpha} = \sum_{i \in \alpha}\sum_{j \in \alpha}B_{ij} = \sum_{i \in \alpha}\doutkin{i}{\alpha} = \sum_{i \in \alpha}\dinkin{i}{\alpha}.
\end{equation}

Analogously to Section \ref{sec:initialPass}, we are interested in how
$Q^{(3)}$ will change if we move a node $a$ from its
current community $\alpha$, to a new community $\beta$,
and analogously this will have two effects -- it will remove the terms
from $Q^{(3)}$ related to $a$ in $\alpha$, which we will call $Q^{-(3)}$
and it will add terms related to $a$ in $\beta$, which we will call $Q^{+(3)}$.
The total change in $Q^{(3)}$ caused by the movement of $a$ from $\alpha$ to $\beta$ is
\begin{equation}
	\Delta Q^{(3)} = Q^{+(3)} - Q^{-(3)},
\end{equation}
where
\begin{align*}
Q^{-(3)} &= \frac{1}{2m}\left[ \left( 2B_{aa} - 2\gamma \frac{\dkin{a}\dkout{a}}{2m} \right)
+ \sum_{i \in \alpha, \, i \neq a} \left( 2B_{ia} + 2B_{ai} - 2\gamma \frac{\dkin{i}\dkout{a}}{2m} - 2\gamma \frac{\dkin{a}\dkout{i}}{2m} \right) \right] \\
				 &= \frac{1}{2m}\left[ \left( 2B_{aa} - 2\gamma \frac{\dkin{a}\dkout{a}}{2m} \right)
+ 2(\dinkin{a}{\alpha} - B_{aa}) + 2(\doutkin{a}{\alpha} - B_{aa}) \hdots \right . \\
 & \quad \quad \quad \quad \quad \quad \left .
- \frac{2\gamma\dkout{a}}{2m} (\dinStot{\alpha} - \dkin{a}) - \frac{2\gamma\dkin{a}}{2m} (\doutStot{\alpha} - \dkout{a}) \right] \\
\end{align*}
and
\begin{align*}
Q^{+(3)} &= \frac{1}{2m}\left[ \left( 2B_{aa} - 2\gamma \frac{\dkin{a}\dkout{a}}{2m} \right)
+ \sum_{i \in \beta} \left( 2B_{ia} + 2B_{ai} - 2\gamma \frac{\dkin{i}\dkout{a}}{2m} - 2\gamma \frac{\dkin{a}\dkout{i}}{2m} \right) \right] \\
				 &= \frac{1}{2m}\left[ \left( 2B_{aa} - 2\gamma \frac{\dkin{a}\dkout{a}}{2m} \right)
+ 2\dinkin{a}{\beta} + 2\doutkin{a}{\beta} - \frac{2\gamma\dkout{a}}{2m} \dinStot{\beta}  - \frac{2\gamma\dkin{a}}{2m} \doutStot{\beta} \right] \\
\end{align*}
Similarly to Section \ref{sec:initialPass}, the first term in both these expressions is the same, and so cancels, leaving:
\begin{align*}
\Delta Q^{(3)} &= \frac{2}{2m}\left[
\left( \dinkin{a}{\beta} + \doutkin{a}{\beta} - \frac{\gamma\dkout{a}}{2m} \dinStot{\beta}  - \frac{\gamma\dkin{a}}{2m} \doutStot{\beta} \right) \right. \\
& \hspace{-1cm}
- \left. \left( (\dinkin{a}{\alpha} - B_{aa}) + (\doutkin{a}{\alpha} - B_{aa}) - \frac{\gamma\dkout{a}}{2m} (\dinStot{\alpha} - \dkin{a}) - \frac{\gamma\dkin{a}}{2m} (\doutStot{\alpha} - \dkout{a})  \right) \right] \\
 &= \frac{2}{2m}\left[ (\dinkin{a}{\beta}-\dinkin{a}{\alpha}) + (\doutkin{a}{\beta}-\doutkin{a}{\alpha}) + 2B_{aa} \right. \\
& \hspace{-1cm} \left.
- \frac{\gamma\dkout{a}}{2m} (\dinStot{\beta}-\dinStot{\alpha}) - \frac{\gamma\dkin{a}}{2m} (\doutStot{\beta} - \doutStot{\alpha}) - \frac{2\gamma\dkin{a}\dkout{a}}{2m} \right]
\end{align*}


\subsection{Later passes}
\label{sec:laterPassesDirected}

In phase two a `meta-graph' is constructed where nodes correspond to
the communities found in the preceding phase one step, and edge weight
between two such communities (nodes, in the meta-graph)
$\alpha$ and $\beta$ are defined to be
\begin{equation}
	B_{\alpha \beta}^* = \sum_{i \in \alpha}\sum_{j \in \beta}B_{ij}.
	\label{eqn:Bij*}
\end{equation}
Note that $i$ and $j$ refer to nodes in the original graph, not nodes
in the previous graph, and so holds any meta-graph, not just the first.
Also note that this definition of $B^*_{\alpha \beta}$ allows for
$B^*_{\alpha \alpha}$ to be non-zero, in fact
\begin{equation}
B_{\alpha \alpha}^* = \sum_{i \in \alpha}\sum_{j \in \alpha}B_{ij} = \Sin{\alpha}.
\end{equation}

In this newly constructed graph, $\alpha$ and $\beta$ are nodes, but
also refer to communities (sets of nodes) in the original graph, and I
use these two interpretations interchangeably, completely analogously to
Section \ref{sec:laterPasses}.

The results of Section~\ref{sec:initialPassDirected} generalise to these meta-graphs,
and the generalised results mirror those of Section~\ref{sec:initialPassDirected} closely
-- I distinguish the new results from those of Section~\ref{sec:initialPassDirected} by a
superscript $*$.
I use $i$ and $j$ to denote nodes of the original graph as in Sections~\ref{sec:initialPass}
and \ref{sec:initialPassDirected},
and use $z$ and $w$ to denote nodes of the meta-graph (communities of the original).
I use analogous notation to Section~\ref{sec:initialPass}, $c^*(z)$, to
denote the community to which node $z$ of the meta-graph belongs,
and let $\mathfrak{a}$ be the community that the node $\alpha$ belongs to,
i.e., $\mathfrak{a} = c^*(\alpha) $.

Given this notation, we get all the same results as in \ref{sec:laterPasses}, but
each split into two cases `out' and `in', separating by direction, essentially, so
\newcommand{\dkinStar}[1]{k_{#1}^{\text{in} *}}
\newcommand{\dkoutStar}[1]{k_{#1}^{\text{out} *}}
\begin{equation}
\dkoutStar{z} = \sum_w{B_{zw}^*} = \sum_w\sum_{i \in z}\sum_{j \in w}B_{ij} = \sum_{i \in z}\sum_jB_{ij} = \doutStot{z},
\end{equation}
\begin{equation}
\dkinStar{z}  = \sum_w{B_{wz}^*} = \sum_w\sum_{i \in z}\sum_{j \in w}B_{ji} = \sum_{i \in z}\sum_jB_{ji} = \dinStot{z},
\end{equation}
\newcommand{\dinStotStar}[1]{\Sigma_{\text{tot}}^{\text{in}(#1) *}}
\newcommand{\doutStotStar}[1]{\Sigma_{\text{tot}}^{\text{out}(#1) *}}
\begin{equation}
	\doutStotStar{\mathfrak{a}} = \sum_{z \in \mathfrak{a}}\sum_{w}B_{zw}^* = \sum_{z \in \mathfrak{a}}\dkoutStar{z} = \sum_{z \in \mathfrak{a}}\doutStot{z},
\end{equation}
\begin{equation}
	\dinStotStar{\mathfrak{a}}  = \sum_{z \in \mathfrak{a}}\sum_{w}B_{wz}^* = \sum_{z \in \mathfrak{a}}\dkinStar{z}  = \sum_{z \in \mathfrak{a}}\dinStot{z},
\end{equation}
\newcommand{\dinkinStar}[2]{k_{#1}^{\text{in}(#2) *}}
\newcommand{\doutkinStar}[2]{k_{#1}^{\text{out}(#2) *}}
\begin{equation}
	\doutkinStar{z}{\mathfrak{a}} = \sum_{w \in \mathfrak{a}}{B_{zw}^*} = \sum_{w \in \mathfrak{a}}{\sum_{i \in z}\sum_{j \in w}B_{ij}},
\end{equation}
\begin{equation}
	\dinkinStar{z}{\mathfrak{a}}  = \sum_{w \in \mathfrak{a}}{B_{wz}^*} = \sum_{w \in \mathfrak{a}}{\sum_{i \in z}\sum_{j \in w}B_{ji}},
\end{equation}
and
\begin{equation}
\Sin{\mathfrak{a} *} = \sum_{z \in \mathfrak{a}}\sum_{w \in \mathfrak{a}}A_{zw}^* = \sum_{z \in \mathfrak{a}}\kin{z}{\mathfrak{a} *} = \sum_{z \in \mathfrak{a}}\sum_{w \in \mathfrak{a}}{\sum_{i \in z}\sum_{j \in w}A_{ij}}.
	%\label{eqn:Sin}
\end{equation}

If we let $\mathfrak{b}$ denote the community to which we are considering moving $\alpha$,
then the expression for $\Delta Q$ from Section~\ref{sec:initialPassDirected} simply generalises as
\begin{align*}
\Delta Q^{(3)} &= \frac{2}{2m}\left[ (\dinkinStar{\alpha}{\mathfrak{b}}-\dinkinStar{\alpha}{\mathfrak{a}}) + (\doutkinStar{\alpha}{\mathfrak{b}}-\doutkinStar{\alpha}{\mathfrak{a}}) + 2B_{\alpha\alpha}^* \right. \\
& \hspace{-1cm} \left.
- \frac{\gamma\dkoutStar{\alpha}}{2m} (\dinStotStar{\mathfrak{b}}-\dinStotStar{\mathfrak{a}}) - \frac{\gamma\dkinStar{\alpha}}{2m} (\doutStotStar{\mathfrak{b}} - \doutStotStar{\mathfrak{a}}) - \frac{2\gamma\dkinStar{\alpha}\dkoutStar{\alpha}}{2m} \right]
\end{align*}


\end{document}