756 lines
38 KiB
TeX
756 lines
38 KiB
TeX
% ! TeX program = lualatex
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\documentclass[a4paper,11pt]{article}
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% packages
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\usepackage{titlesec}
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\author{Andrew Hayes}
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\begin{document}
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\begin{titlepage}
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\begin{center}
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\hrule
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\vspace*{0.6cm}
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\Huge \textsc{cs4423}
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\vspace*{0.6cm}
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\hrule
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\LARGE
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\vspace{0.5cm}
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Networks
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\vspace{0.5cm}
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\begin{minipage}{0.495\textwidth}
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\vspace{0.4em}
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\raggedright
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\normalsize
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\begin{tabular}{@{}l l}
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Name: & Andrew Hayes \\
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Student ID: & 21321503 \\
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E-mail: & \href{mailto://a.hayes18@universityofgalway.ie}{a.hayes18@universityofgalway.ie} \\
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\end{tabular}
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\begin{minipage}{0.495\textwidth}
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\raggedleft
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\vspace*{0.8cm}
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\Large
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\today
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\vspace*{0.6cm}
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\end{minipage}
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\end{center}
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\end{titlepage}
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\pagenumbering{roman}
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\newpage
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\tableofcontents
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\newpage
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\setcounter{page}{1}
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\pagenumbering{arabic}
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\section{Introduction}
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\textbf{CS4423 Networks} is a Semester 2 module on \textbf{Network Science}.
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Modern societies are in many ways highly connected.
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Certain aspects of this phenomenon are frequently described as \textbf{networks}.
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CS4423 is an introduction to this emerging interdisciplinary subject.
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We'll cover several major topics in this module, including:
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\begin{itemize}
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\item Graphs \& Graph Theory, and how they relate to networks;
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\item Representations of networks, including as matrices;
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\item Computing with networks, using \mintinline{python}{networkx} in Python;
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\item Centrality measures;
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\item Random graphs;
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\item Small worlds;
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\item Models of growing graphs;
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\end{itemize}
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Lecture notes \& assignments will come in the form of Jupyter notebooks, which allows us to include interactive Python code with the text.
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\subsection{Lecturer Contact Information}
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\begin{itemize}
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\item Name: Dr Niall Madden.
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\item School of Mathematical \& Statistical Sciences, University of Galway.
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\item Office: Room ADB-1013, Arás de Brún.
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\item E-mail: \href{mailto://niall.madden@universityofgalway.ie}{niall.madden@universityofgalway.ie}.
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\item Website: \url{https://www.niallmadden.ie}
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\end{itemize}
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\subsection{Exam Information}
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First year lecturing, should be similar to old exam papers.
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Only looked at the past 2 years or so.
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\subsection{Schedule}
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Tentative schedule for labs / tutorials:
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\begin{itemize}
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\item Tuesday at 16:00 in AC215;
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\item Wednesday at 10:00 in CA116a.
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\end{itemize}
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There will be some practicals during the semester: Week 3 ``Introduction to Python \& Jupyter'' sessions, later weeks help with assignments, preparations for exam, etc.
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\subsection{Assessment}
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\begin{itemize}
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\item Two homework assignments.
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Tentative deadlines: Weeks 5 \& 10.
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Each contribute 10\% each to the final grade.
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\item One in-class test.
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Probably Week 7 (depending on FYP deadlines).
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Contributes 10\% to the final grade.
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\item Final exam: 70\%.
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\end{itemize}
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\subsection{Introduction to Networks}
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Newman (for example) broadly divides the most commonly studied real-world networks into four classes:
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\begin{enumerate}
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\item \textbf{Technological networks:} rely on physical infrastructure.
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In many cases, this infrastructure has been built over many decades and forms part of the backbone of modern societies, including roads \& other transportation networks, power grids, and communications networks.
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\item \textbf{Social networks:} the vertices of a social network are people (or, at leasts, User IDs), with edges representing some sort of \textbf{social interaction}.
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In sociology, the vertices are often called \textbf{actors}, and the edges are called \textbf{ties}.
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Social networks are not just online: sociologists have studied social networks long before people started exhibiting their relations to others online.
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Traditionally, data about the structure of social networks have been compiled by interviewing the people involved.
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\item \textbf{Information networks:} consist of \textbf{data items} which are linked to each other in some way.
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Examples include relational databases.
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Sets of information (like scientific publications) have been linking to each other (e.g., through citations) long before computers were invented, although links in digital form are easier to follow.
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\\\\
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The \textbf{WWW} is probably the most widespread \& best-known example of an information network.
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Its nodes are \textbf{web pages} containing information in form of text \& pictures, and its edges are the \textbf{hyperlinks}, allowing us to surf or navigate from page to page.
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Hyperlinks run in one direction only, from the page that contains the hyperlink to the page that is referenced.
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Therefore, the WWW is a \textbf{directed network}, a graph where each edge has a direction.
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\item \textbf{Biological networks:}
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\begin{itemize}
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\item \textbf{Biochemical networks} represent molecular-level patterns of interaction \& control mechanisms in the biological cell, including metabolic networks, protein-protein interaction networks, \& genetic regulatory networks.
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\item A \textbf{neural network} can be represented as a set of vertices, the neurons, connected by two types of directed edges, one for excitatory inputs and one for inhibitory inputs.
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(Not to be confused with an artificial neural network).
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\item \textbf{Ecological networks} are networks of ecological interactions between species.
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\end{itemize}
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\end{enumerate}
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In each case, a network connects parts of a system (\textbf{nodes}) by some means (\textbf{links}).
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Different techniques are used to display, discover, \& measure the structure in each example.
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\\\\
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In its simplest form, a \textbf{network} is just a collection of points (called \textbf{vertices} or \textbf{nodes}), some of which are joined in pairs (called \textbf{edges} or \textbf{links}).
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Many systems of interest are composed of individual parts that are in some way linked together: such systems can be regarded as networks, and thinking about them in this way can often lead to new \& useful insights.
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\\\\
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\textbf{Network science} studies the patterns of connections between the components of a system.
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Naturally, the structure of the networks can have a big impact on the behaviour of the system.
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A \textbf{network} is a simplified representation of a complex system by vertices \& edges.
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The scientific study of networks is an interdisciplinary undertaking that combines ideas from mathematics, computer science, physics, the social sciences, \& biology.
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Between these scientific fields, many tools have been developed for analysing, modeling, \& understanding networks.
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\subsubsection{Network Measures}
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\textbf{Centrality} is an example of a useful \& important type of network measure; it is concerned with the question of how important a particular vertex or edge is in a networked system.
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Different concepts have been proposed to capture mathematically what it means to be central.
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For example, a simple measure of the centrality of a vertex is its \textbf{degree}, that is, the number of edges it is part of (or, equivalently, the number of vertices it is adjacent to).
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Applications of centrality include determining which entities in a social network have the most influence, or which links in a power grid are most vulnerable.
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\\\\
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Which measurements \& calculations give meaningful answers for a particular system depends of course on the specific nature of the system and the questions one wants to ask.
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\subsubsection{Network Concepts}
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Another interesting network concept is the \textbf{small-world effect}, which is concerned with the question of how far apart two randomly chosen points in a network typically are.
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Here, \textbf{distance} is usually measured by the number of edges one would need to cross over when travelling along a \textbf{path} from one vertex to another.
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In real-world social networks, the distance between people tends to be rather small.
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\section{Graphs}
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A \textbf{graph} can serve as a mathematical model of a network.
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Later, we will use the \mintinline{python}{networkx} package to work with examples of graphs \& networks.
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\subsection{Example: The Internet (circa 1970)}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.7\textwidth]{./images/f7dec1970.jpg}
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\caption{
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The Internet (more precisely, ARPANET) in December 1970.
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Nodes are computers, connected by a link if they can directly communicate with each other.
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At the time, only 13 computers participated in that network.
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}
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\end{figure}
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\begin{code}
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\begin{minted}[linenos, breaklines, frame=single]{text}
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UCSB SRI UCLA
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SRI UCLA STAN UTAH
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UCLA STAN RAND
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UTAH SDC MIT
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RAND SDC BBN
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MIT BBN LINC
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BBN HARV
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LINC CASE
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HARV CARN
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CASE CARN
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\end{minted}
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\caption{\texttt{arpa.adj}}
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\end{code}
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The following \textbf{diagram}, built from the adjacencies in \verb|arpa.adj|, contains the same information as in the above figure, without the distracting details of US geography;
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this is actually an important point, as networks only reflect the \textbf{topology} of the object being studied.
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\begin{code}
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\begin{minted}[linenos, breaklines, frame=single]{python}
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H = nx.read_adjlist("../data/arpa.adj")
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opts = { "with_labels": True, "node_color": 'y' }
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nx.draw(H, **opts)
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\end{minted}
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\caption{\texttt{arpa.adj}}
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\end{code}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.7\textwidth]{./images/qwe_download.png}
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\caption{ The ARPA Network as a Graph }
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\end{figure}
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\subsection{Simple Graphs}
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A \textbf{simple graph} is a pair $G = (X,E)$ consisting of a finite set $X$ of objects called \textit{nodes}, \textit{vertices}, or \textit{points} and a set of \textit{links} or \textit{edges} $E$ which are each a set of two different vertices.
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\begin{itemize}
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\item We can also write $E \subseteq \binom{X}{2}$, where $\binom{X}{2}$ ($X$ \textit{choose} 2) is the set of all $2$-element subsets of $X$.
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\item The \textbf{order} of the graph $G$ is denoted as $n = |X|$, where $n$ is the number of vertices in the graph.
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\item The \textbf{size} of the graph is denoted as $m = |E|$, where $m$ is the number of edges in the graph.
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Naturally, $m \leq \binom{n}{2}$.
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\end{itemize}
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\subsection{Subgraphs \& Induced Subgraphs}
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Given $G = (X,E)$, a \textbf{subgraph} of $G$ is $H=(Y, E_H)$ with $Y \subseteq X$ and $E_H \subseteq E \cap \binom{Y}{s}$;
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therefore, all the nodes in $H$ are also in $G$ and any edge in $H$ was also in $G$, and is incident only to vertices in $Y$.
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\\\\
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One of the most important subgraphs of $G$ is the \textbf{induced subgraph} on $Y \subseteq X$: $H = (Y, E \cap \binom{Y}{2})$;
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that is, given a subset $Y$ of $X$, we include all possible edges from the original graph $G$ too.
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Each node has a list of \textbf{neighbours} which are the nodes it is directly connected to by an edge of the graph.
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\subsection{Important Graphs}
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The \textbf{complete graph} on a vertex set $X$ is the graph with edge set $\binom{X}{2}$.
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For example, if $X = \{0,1,2,3 \}$, then $E = \{01,02,03,12,13,23\}$
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\\\\
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The \textbf{Petersen graph} is a graph on 10 vertices with 15 edges.
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It can be constructed as the complement of the line graph of the complete graph $K_5$, that is, as the graph with the vertex set $X = \binom{ \{0,1,2,3,4\} }{2}$ (the edge set of $K_5$) and with an edge between $x,y \in X$ whenever $x \cap y = \emptyset$.
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\\\\
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A graph is \textbf{bipartite} if we can divide the node set $X$ into two subsets $X_1$ and $X_2$ such that:
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\begin{itemize}
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\item $X_1 \cap X_2 = \emptyset$ (the sets have no edge in common);
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\item $X_1 \cup X_2 = X$.
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\end{itemize}
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For any edge $(u_1, u_2)$, we have $u_1 \in X_1$ and $u_2 \in X_2$; that is, we only ever have edges between nodes from different sets.
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Such graphs are very common in Network Science, where nodes in the network represent two different types of entities; for example, we might have a graph wherein nodes represent students and modules, with edges between students and modules they were enrolled in, often called an \textbf{affiliation network}.
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\\\\
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A \textbf{complete bipartite graph} is a particular bipartite graph wherein there is an edge between every node in $X_1$ and every node in $X_2$.
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Such graphs are denoted $K_{m,n}$, where $|X_1| = m$ and $|X_2|=n$.
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\\\\
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The \textbf{path graph} with $n$ nodes, denoted $P_n$, is a graph where two nodes have degree 1, and the other $n-2$ have degree 2.
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\\\\
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The \textbf{cycle graph} on $n \geq 3$ nodes, denoted $C_n$ (slightly informally) is formed by adding an edge between the two nodes of degree 1 in a path graph.
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\subsection{New Graphs from Old}
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The \textbf{complement} of a graph $G$ is a graph $H$ with the same nodes as $G$ but each pair of nodes in $H$ are adjacent if and only if they are \textit{not adjacent} in $G$.
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The complement of a complete graph is an empty graph.
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\\\\
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A graph $G$ can be thought of as being made from ``things'' that have connection to each other: the ``things'' are nodes, and their connections are represented by an edge.
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However, we can also think of edges as ``things'' that are connected to any other edge with which they share a vertex in common.
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This leads to the idea of a line graph:
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the \textbf{line graph} of a graph $G$, denoted $L(G)$ is the graph where every node in $L(G)$ corresponds to an edge in $G$, and for every pair of edges in $G$ that share a node, $L(G)$ has an edge between their corresponding nodes.
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\section{Matrices of Graphs}
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There are various was to represent a graph, including the node set, the edge set, or a drawing of the graph;
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one of the most useful representations of a graph for computational purposes is as a \textbf{matrix}; the three most important matrix representations are:
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\begin{itemize}
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\item The \textbf{adjacency matrix} (most important);
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\item The \textbf{incidence matrix} (has its uses);
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\item The \textbf{graph Laplacian} (the coolest).
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\end{itemize}
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\subsection{Adjacency Matrices}
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The \textbf{adjacency matrix} of a graph $G$ of order $n$ is a square $n \times n$ matrix $A = (a_{i,j})$ with rows \& columns corresponding to the nodes of the graph, that is, we number the nodes $1, 2, \dots, n$.
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Then, $A$ is given by:
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\begin{align*}
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a_{i,j} =
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\begin{cases}
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1 & \text{if nodes } i \text{ and } j \text{ are joined by an edge,} \\
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0 & \text{otherwise}
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\end{cases}
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\end{align*}
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Put another way, $a_{i,j}$ is the number of edges between node $i$ and node $j$.
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Properties of adjacency matrices include:
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\begin{itemize}
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\item $\sum^N_{i=1} \sum^N_{j=1} a_{i,j} = \sum_{u \in X}\text{deg}(u)$ where $\text{deg}(u)$ is the degree of $u$.
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\item All graphs that we've seen hitherto are \textit{undirected}: for all such graphs, $A$ is symmetric.
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$A = A^T$ and, equivalently, $a_{i,j} = a{j,i}$.
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\item $a_{i,i} = 0$ for all $i$.
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\item In real-world examples, $A$ is usually \textbf{sparse} which means that $\sum^N_{i=1} \sum^N_{j=1} a_{i,j} \ll n^2$, that is, the vast majority of the entries are zero.
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Sparse matrices have huge importance in computational linear algebra: an important idea is that is much more efficient to just store the location of the non-zero entities in a sparse matrix.
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\end{itemize}
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Any matrix $M = (m_{i,j})$ with the properties that all entries are zero or one and that the diagonal entries are zero (i.e., $m_{i,j}=0$) is an adjacency matrix of \textit{some} graph (as long as we don't mind too much about node labels).
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In a sense, every square matrix defines a graph if:
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\begin{itemize}
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\item We allow loops (an edge between a node and itself).
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\item Every edge has a weight: this is equivalent to the case for our more typical graphs that every potential edge is weighted 0 (is not in the edge set) or 1 (is in the edge set).
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\item There are two edges between each node (one in each direction) and they can have different weights.
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\end{itemize}
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\subsubsection{Examples of Adjacency Matrices}
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Let $G = G(X,E)$ be the graph with $X = \{a,b,c,d,e\}$ nodes and edges $\{a \leftrightarrow b, b \leftrightarrow c, b \leftrightarrow d, c \leftrightarrow d, d \leftrightarrow e \}$.
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Then:
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\begin{align*}
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A =
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\begin{pmatrix}
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0 & 1 & 0 & 0 & 0 \\
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1 & 0 & 1 & 1 & 0 \\
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0 & 1 & 0 & 1 & 0 \\
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0 & 1 & 1 & 0 & 1 \\
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0 & 0 & 0 & 1 & 0 \\
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\end{pmatrix}
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\end{align*}
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The adjacency matrix of $K_4$ is:
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\begin{align*}
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A =
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\begin{pmatrix}
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0 & 1 & 1 & 1 \\
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1 & 0 & 1 & 1 \\
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1 & 1 & 0 & 1 \\
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1 & 1 & 1 & 0 \\
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\end{pmatrix}
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\end{align*}
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\subsection{Degree}
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The \textbf{degree} of a node in a simple graph is the number of nodes to which it is adjacent, i.e., its number of neighbours.
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For a node $v$ we denote this number $\text{deg}(v)$.
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The degree of a node can serve as a (simple) measure of the importance of a node in a network.
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Recall that one of the basic properties of an adjacency matrix is $\sum^n_{i=1} \sum^n_{j=1} a_{i,j} = \sum_{u \in X} \text{deg}(u)$, where $\text{deg}(u)$ is the degree of $u$ and $n$ is the order of the graph;
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this relates to a (crude) measure of how connected a network is: the \textbf{average degree}:
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\begin{align*}
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\text{Average degree} = \frac{1}{n} \sum_{u \in X} \text{deg}(u) = \frac{1}{n}\sum^n_{i,j} a_{i,j}
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\end{align*}
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However, if the size of the network (the number of edges) is $m$, then the total sum of degrees is $2m$ (since each edge contributes to the degree count of two nodes), meaning that the average degree is $\frac{2m}{n}$.
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\subsection{Walks}
|
|
A \textbf{walk} in a graph is a series of edges (perhaps with some repeated) $\{ u_1 \leftrightarrow v_1, u_2 \leftrightarrow u_2, \dots, u_p \leftrightarrow v_p\}$ with the property that $v_i = u_{i+1}$.
|
|
If $v_p = u_1$, then it is a \textbf{closed walk}.
|
|
The \textbf{length} of a walk is the number of edges in it.
|
|
\\\\
|
|
Adjacency matrices can be used to enumerate the number of walks of a given length between a pair of vertices.
|
|
Obviously, $a_{i,j}$ is the number of walks of length 1 between node $i$ and node $j$.
|
|
We can extract that information for node $j$ by computing the product of $A$ and $e_j$ (column $j$of the identity matrix).
|
|
|
|
\section{Connectivity \& Permutations}
|
|
\subsection{Notation}
|
|
To start, let's decide on our notation:
|
|
\begin{itemize}
|
|
\item If we write $A = (a_{i,j})$, we mean that $A$ is a matrix and $a_{i,j}$ is its entry row $i$, column $j$.
|
|
\item We also write such entries as $(A)_{i,j}$;
|
|
the reason for this slightly different notation is to allow us to write, for example, $(A^2)_{i,j}$ is the entry in row $i$, column $j$ of $B = A^2$.
|
|
\item The \textbf{trace} of a matrix is the sum of its diagonal entries, that is, $\text{tr}(A) = \sum^n_{i=1}a_{i,i}$. (Very standard).
|
|
\item When we write $A > 0$, we mean that all entries of $A$ are positive.
|
|
\end{itemize}
|
|
|
|
\subsection{Counting Walks}
|
|
Recall that the \textbf{adjacency matrix} of a graph $G$ of order $N$ is a square $n \times n$ matrix $A = (a_{i,j})$ with rows and columns corresponding to the nodes of the graph.
|
|
$a_{i,j}$ is set to be the number of edges between nodes $i$ and $j$.
|
|
We learned previously that:
|
|
\begin{itemize}
|
|
\item If $e_j$ is the $j^\text{th}$ column of the identity matrix $I_n$, then $(Ae_j)_i$ is the number of walks of length 1 from node $i$ to node $j$.
|
|
Also, it is the same as $a_{i,j}$.
|
|
|
|
\item Moreover, $(A(Ae_j))_i = (A^2e_j)$ is the number of walks of length 2 from node $i$ to node $j$.
|
|
We can conclude that, if $B=A^2$, then $b_{i,j}$ is the number of walks of length 2 between nodes $i$ and $j$.
|
|
Note that $b_{i,i}$ is the degree of node $i$.
|
|
|
|
\item In fact, if $B=A^k$, then $b_{i,j}$ is the number of walks of length $k$ between nodes $i$ and $j$.
|
|
\end{itemize}
|
|
|
|
\subsection{Paths}
|
|
A \textbf{trail} is walk with no repeated edges.
|
|
A \textbf{cycle} is a trail in which the first and last nodes are the same, but no other node is repeated;
|
|
a \textbf{triangle} is a cycle of length 3.
|
|
A \textbf{path} is a walk in which no nodes (and so no edges) are repeated.
|
|
(The idea of a path is hugely important in network theory, and we will return to it often).
|
|
\\\\
|
|
The \textbf{length} of a path is the number of edges in that path.
|
|
A path from node $u$ to node $v$ is a \textbf{shortest path} if there is no path between them that is shorter (although there could be other paths of the same length).
|
|
Finding shortest paths in a network is a major topic that we will return to at another time.
|
|
\begin{itemize}
|
|
\item Every path is also a walk.
|
|
\item If a particular walk is the shortest walk between two nodes then it is also the shortest path between two nodes.
|
|
\item If $k$ is the smallest natural number of which $(A^k)_{i,j} \neq 0$, then the shortest walk from node $i$ to node $j$ is of length $k$.
|
|
\item It follows that $k$ is also the length of the shortest path from node $i$ to node $j$.
|
|
\end{itemize}
|
|
|
|
For example, consider the following adjacency matrix and its powers:
|
|
\begin{align*}
|
|
A =
|
|
\begin{pmatrix}
|
|
0 & 1 & 0 & 0 & 0 \\
|
|
1 & 0 & 1 & 0 & 0 \\
|
|
0 & 1 & 0 & 1 & 1 \\
|
|
0 & 0 & 1 & 0 & 1 \\
|
|
0 & 0 & 1 & 1 & 0
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
|
|
\begin{align*}
|
|
A^2 =
|
|
\begin{pmatrix}
|
|
1 & 0 & 1 & 0 & 0 \\
|
|
0 & 2 & 0 & 1 & 1 \\
|
|
1 & 0 & 3 & 0 & 0 \\
|
|
0 & 1 & 1 & 2 & 1 \\
|
|
0 & 1 & 1 & 1 & 2
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
|
|
\begin{align*}
|
|
A^3 =
|
|
\begin{pmatrix}
|
|
0 & 2 & 0 & 1 & 1 \\
|
|
2 & 0 & 4 & 1 & 1 \\
|
|
0 & 4 & 2 & 4 & 4 \\
|
|
1 & 1 & 4 & 2 & 3 \\
|
|
1 & 1 & 4 & 3 & 2
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
|
|
We can observe that, where $A$ is the adjacency matrix of the graph $G$:
|
|
\begin{itemize}
|
|
\item $(A^2)_{i,i}$ is the degree of node $i$.
|
|
\item $\text{tr}(A^2)$ is the degree sum of the nodes in $G$.
|
|
\item $(A^3)_{i,i} \neq 0 $ if node $i$ is in a triangle.
|
|
\item $\frac{\text{tr}(A^3)}{6}$ is the number of triangles in $G$.
|
|
\item If $G$ is bipartite, then $(A^3)_{i,j} = 0$ for all $i,j$.
|
|
\end{itemize}
|
|
|
|
\subsection{Connectivity}
|
|
Let $G$ be a graph and $A$ its adjacency matrix:
|
|
in $G$, node $i$ can be \textbf{reached} from node $j$ if there is a path between them.
|
|
If node $i$ is reachable from node $j$, then $(A^k)_{i,j} \neq 0$ for some $k$.
|
|
Also, note that $k \leq n$.
|
|
Equivalently, since each power of $A$ is non-negative, we can say that $(I + A + A^2 + A^3 + \cdots + A^k) > 0$.
|
|
\\\\
|
|
A graph/network is \textbf{connected} if there is a path between every pair of nodes.
|
|
That is, every node is reachable from every other node.
|
|
If a graph is not connected, we say that it is \textbf{disconnected}.
|
|
Determining if a graph is connected or not is important; we'll see later that this is especially important with directed graphs.
|
|
A graph $G$ of order $n$ is connected if and only if, for each $i,j$, there is some $K \leq n$ for which $(A^k)_{i,j} \neq 0$.
|
|
|
|
\subsection{Permutation Matrices}
|
|
We know that the structure of a network is not changed by labelling its nodes.
|
|
Sometimes, it is useful to re-label the nodes in order to expose certain properties, such as connectivity.
|
|
Since we think of the nodes as all being numbered from 1 to $n$, this is the same as \textbf{permuting} the numbers of some subset of the nodes.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/permexampe.png}
|
|
\caption{ Example wherein nodes are re-labelled to expose certain properties of the graph }
|
|
\end{figure}
|
|
|
|
When working with the adjacency matrix of a graph, such a permutation is expressed in terms of a \textbf{permutation matrix} $P$;
|
|
this is a $0$-$1$ matrix (also known as a Boolean or a binary matrix) where this is a single $1$ in every row \& column.
|
|
If the nodes of a graph $G$ (with adjacency matrix $A$) are listed as entries in a vector ${q}$, then:
|
|
\begin{itemize}
|
|
\item $Pq$ is a permutation of the nodes.
|
|
\item $PAP^T$ is the adjacency matrix of the graph with that node permutation applied.
|
|
\end{itemize}
|
|
|
|
In many examples, we will have a symmetric $P$ for the sake of simplicity, but in general, $P \ neq P^T$.
|
|
However, $P^TAP = PAP^T$and $P^T = P^{-1}$ so $PAP^T = PAP^{-1}$.
|
|
\\\\
|
|
A graph with adjacency matrix $A$ is \textbf{disconnected} if and only if there is a permutation matrix $P$ such that
|
|
\begin{align*}
|
|
A &= P \begin{pmatrix} X & O \\ O^T & Y \end{pmatrix} &
|
|
PAP^T &= P \begin{pmatrix} X & O \\ O & Y \end{pmatrix}
|
|
\end{align*}
|
|
where $O$ represents the zero matrix with the same number of rows as $X$ and the same number of columns as $Y$.
|
|
|
|
\section{Permutations \& Bipartite Networks}
|
|
\subsection{Graph Connectivity}
|
|
Recall that a graph is \textbf{connected} if there is a path between every pair of nodes.
|
|
If the graph is not connected, we say that it is \textbf{disconnected}.
|
|
We now know how to check if a graph is connected by looking at powers of its adjacency matrix.
|
|
However, that is not very practical for large networks.
|
|
Instead, we can determine if a graph is connected by just looking at the adjacency matrix, provided that we have ordered the nodes properly.
|
|
|
|
\subsection{Connected Components}
|
|
If a network is not connected, then we can divide it into \textbf{components} which \textit{are} connected.
|
|
The number of connected components is the number of blocks in the permuted adjacency matrix.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/connectedcomponents.png}
|
|
\caption{ Connected components example }
|
|
\end{figure}
|
|
|
|
\section{Bipartite Networks: Colours \& Computations}
|
|
\subsection{Class Survey Example}
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/suverydata.png}
|
|
\caption{ Final survey data }
|
|
\end{figure}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/suverygaaph.png}
|
|
\caption{ Final survey graph, with order 39 and size 87 }
|
|
\end{figure}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/surveysubgrap.png}
|
|
\caption{ Subgraph of the survey network based on 7 randomly chosen people, with order 16 and size 24 }
|
|
\end{figure}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/surveymatrix.png}
|
|
\caption{ Adjacency matrix where the nodes for people are listed first }
|
|
\end{figure}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/surveymatrix2.png}
|
|
\caption{ $B = A^2$ }
|
|
\end{figure}
|
|
|
|
Since we know from before that $(A^k)_{i,j}$ is the number of walks of length $k$between nodes $i$ and $j$, we can see that in this context:
|
|
\begin{itemize}
|
|
\item For the first 7 rows \& columns, $b_{i,j}$ is the number of programmes in common between person $i$ and person $j$.
|
|
(This even works for $i=j$, but the number of programmes a person has in common with themselves is just the number they watch).
|
|
\item For the last 9 rows \& columns, $b_{i,j}$ is the number of people who watch both programmes $i$ and $j$.
|
|
\end{itemize}
|
|
|
|
\subsection{Projections}
|
|
Given a bipartite graph $G$ whose node set $V$ has parts $V_1$ \& $V_2$, and \textbf{projection} of $G$ onto (for example) $V_1$ is the graph with:
|
|
\begin{itemize}
|
|
\item Node set $V_1$;
|
|
\item An edge between a pair of nodes in $V_1$ if they share a common neighbour in $G$.
|
|
\end{itemize}
|
|
|
|
In the context of our survey example, a projection onto $V_1$ (people/actors) gives us the graph of people who share a common programme.
|
|
To make such a graph:
|
|
\begin{itemize}
|
|
\item Let $A$ be the adjacency matrix of $G$.
|
|
\item Let $B$ be the submatrix of $A^2$ associated with the nodes in $V_1$.
|
|
\item Let $C$ be the adjacency matrix with the property:
|
|
\begin{align*}
|
|
c_{i,j} =
|
|
\begin{cases}
|
|
1 & b_{i,j} > 0 \text{ and } i \neq j \\
|
|
0 & \text{otherwise}
|
|
\end{cases}
|
|
\end{align*}
|
|
That is, $b_{i,j} = 0 $ or $i=j$.
|
|
\item Let $G_{V_1}$ be the graph on $V_1$ with adjacency matrix $C$.
|
|
Then, $G_{V_1}$ is the \textbf{projection of $G$ onto $V_1$}.
|
|
\end{itemize}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/surveygv1.png}
|
|
\caption{ $G_{V_1}$ computed for our survey data }
|
|
\end{figure}
|
|
|
|
\subsection{Colouring}
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/colouredsurvey.png}
|
|
\caption{ The original survey graph is more easily digestible if coloured }
|
|
\end{figure}
|
|
|
|
For any bipartite graph, we can think of the nodes in the two sets as \textbf{coloured} with different colours.
|
|
For instance, we can think of nodes in $X_1$ as white nodes and those in $X_2$ as black nodes.
|
|
A \textbf{vertex-colouring} of a graph $G$is an assignment of (finitely many) colours to the nodes of $G$ such that any two nodes which are connected by an edge have different colours.
|
|
A graph is called \textbf{$N$-colourable} if it has a vertex colouring with at most $N$ colours.
|
|
The \textbf{chromatic number} of a graph $G$ is the \textit{smallest $N$} for which a graph $G$ is $N$-colourable.
|
|
The following statements about a graph $G$ are equivalent:
|
|
\begin{itemize}
|
|
\item $G$ is bipartite;
|
|
\item $G$ is 2-colourable;
|
|
\item Each cycle in $G$ has even length.
|
|
\end{itemize}
|
|
|
|
\section{Trees}
|
|
A \textbf{cycle} in a simple graph provides, for any two nodes on that cycle, at least two different paths from node $a$ to node $b$.
|
|
It can be useful to provide alternative routes for connectivity in case one of the edges should fail, e.g., in an electrical network.
|
|
\\\\
|
|
A graph is called \textbf{acyclic} if it does not contain any cycles.
|
|
A \textbf{tree} is a simple graph that is \textit{connected} \& \textit{acyclic}.
|
|
In other words, between any two vertices in a tree there is exactly one simple path.
|
|
Trees can be characterised in many different ways.
|
|
\\\\
|
|
\textbf{Theorem:} Let $G=(X,E)$ be a (simple) graph of order $n=|X|$ and size $m=|E|$.
|
|
Then, the following are equivalent:
|
|
\begin{itemize}
|
|
\item $G$ is a tree (i.e., acyclic \& connected);
|
|
\item $G$ is connected and $m=n-1$.
|
|
\item $G$ is a minimally connected graph (i.e., removing any edge will disconnect $G$).
|
|
\item $G$ is acyclic and $m=n-1$.
|
|
\item $G$ is a maximally acyclic graph (i.e., adding any edge will introduce a cycle in $G$);
|
|
\item There is a unique path between each pair of nodes in $G$.
|
|
\end{itemize}
|
|
|
|
All trees are \textbf{bipartite}:
|
|
there are a few ways of thinking about this;
|
|
one is that a graph is bipartite if it has no cycles of odd length -- since a tree has no cycles, it must be bipartite.
|
|
|
|
\subsection{Cayley's Formula}
|
|
\textbf{Theorem:} there are exactly $n^{n-2}$ distinct (labelled) trees on the $n$-element vertex set $X=\{0,1,2, \dots, n-1\}$ if $n>1$.
|
|
|
|
\subsubsection{Pr\"ufer Codes}
|
|
The \textbf{Pr\"fer code} of a tree can be determined (destructively) as follows:
|
|
\begin{enumerate}
|
|
\item Start with a tree $T$ with nodes labelled $0, 1, \dots, n-1$ and an empty list $a$.
|
|
\item Find the \textbf{leaf node} $x$ with the smallest label (with a ``leaf node'' being a node of degree 1. Every tree must have at least two leaf nodes).
|
|
\item Append the label of its unique neighbour $y$to the list $a$.
|
|
\item Remove $x$ (and the edge $x \leftrightarrow y$) from $T$.
|
|
\item Repeat steps 2-3 until $T$ has only two ndoes left.
|
|
We now have the code as a list of length $n-2$.
|
|
\end{enumerate}
|
|
|
|
A tree can be re-constructed from its Pr\"ufer code as the degree of a node $x$ is $1$ plus the number of entries $x$ in the Pr\"ufer code of $T$.
|
|
A tree can be computed from a Pr\"ufer code $a$ (where the list $a$ is a list of length $n-2$ with all entries numbered $0$ to $n-1$) as follows:
|
|
\begin{enumerate}
|
|
\item Set $G$ to be a graph with node list $[0,1,2, \dots, n-1]$ and no edges yet.
|
|
\item Compute the list of node degrees $d$ from the code.
|
|
\item For $k=0,1,\dots, n-2$:
|
|
\begin{enumerate}[label=\arabic*.]
|
|
\item Set $y = a[k]$.
|
|
\item Set $x$to be the node with the smallest degree in $d$.
|
|
\item Add the edge $(x,y)$ to $G$.
|
|
\item Set $d[x]=d[x]-1$and $d[y]=d[y]-1$ (that is, decrease the degrees of both $x$ and $y$ by one).
|
|
\end{enumerate}
|
|
\item Finally, connect the remaining two nodes of degrees 1 by an edge.
|
|
\end{enumerate}
|
|
|
|
Since we know now that there is a bijection between labelled trees and Pr\"ufer codes, we can prove Cayley's theorem easily:
|
|
\begin{enumerate}
|
|
\item A tree with $n$ nodes has a Pr\"ufer code of length $n-2$.
|
|
\item There are $n$ choices for each entry in the code.
|
|
\item So, there are $n^{n-2}$ possible codes for a tree with $n$ nodes.
|
|
\item So, there are $n^{n-2}$ possible trees with $n$ nodes.
|
|
\end{enumerate}
|
|
|
|
\subsection{Graph \& Tree Traversal}
|
|
Often, one has to search through a network to check properties of nodes such as to find the node with the largest degree.
|
|
For large unstructured networks, this can be challenging;
|
|
fortunately, there are simple \& efficient algorithms to achieve this:
|
|
\begin{itemize}
|
|
\item DFS.
|
|
\item BFS.
|
|
\end{itemize}
|
|
|
|
\subsubsection{Depth-First Search}
|
|
\textbf{Depth-first search (DFS)} works by starting at a root node and travelling as far along one of its branches as it can, then returning to the last unexplored branch.
|
|
The main data structure needed to implement DFS is a \textbf{stack}, also known as a Last-In-First-Out (LIFO) queue.
|
|
Given a rooted tree $T$ with root $x$, to visit all nodes in the tree:
|
|
\begin{enumerate}
|
|
\item Start with an empty stack $S$.
|
|
\item Push $x$ onto $S$.
|
|
\item While $S \neq \emptyset$:
|
|
\begin{enumerate}[label=\arabic*.]
|
|
\item Pop node $y$ from the stack.
|
|
\item Visit $y$.
|
|
\item Push $y$'s children onto the stack.
|
|
\end{enumerate}
|
|
\end{enumerate}
|
|
|
|
\subsubsection{Breadth-First Search}
|
|
\textbf{Breadth-first search (BFS)} works by starting at a root node and exploring all the neighbouring nodes (on the same level) first.
|
|
Next, it searches their neighbours (level 2), etc.
|
|
The main data structure needed to implement BFS is a \textbf{queue}, also known as a First-In-First-Out (FIFO) queue.
|
|
Given a rooted tree $T$ with root $x$, to visit all nodes in the tree:
|
|
\begin{itemize}
|
|
\item Start with an empty queue $Q$.
|
|
\item Push $x$ onto $Q$.
|
|
\item While $Q \neq \emptyset$:
|
|
\begin{enumerate}[label=\arabic*.]
|
|
\item Pop node $y$ from $Q$.
|
|
\item Visit node $y$.
|
|
\item Push $y$'s children onto $Q$.
|
|
\end{enumerate}
|
|
\end{itemize}
|
|
|
|
Many questions on networks regarding distance \& connectivity can be answered by a versatile strategy involving a subgraph which is a tree and then searching that; such a tree is called \textbf{spanning tree} of the underlying graph.
|
|
|
|
\subsubsection{Graph Diameter}
|
|
A natural problem arising in many practical applications is the following: given a pair of nodes $x,y$, find one or all the paths from $x$ to $y$ with the fewest number of edges possible.
|
|
This is a somewhat complex measure on a network (compared to, say, statistics on node degrees) and we will therefore need a more complex procedure, that is, an algorithm, in order to solve such problems systematically.
|
|
\\\\
|
|
\textbf{Definition:} let $G=(X,E)$ be a simple graph and let $x,y \in X$.
|
|
Let $P(x,y)$ be the set of all paths from $x$ to $y$.
|
|
Then:
|
|
\begin{itemize}
|
|
\item The \textbf{distance} $d(x,y)$ from $x$ to $y$ is
|
|
\begin{align*}
|
|
d(x,y) = \text{min}\{ l(p) : p \in P(x,y) \},
|
|
\end{align*}
|
|
the shortest possible length of a path from $x$ to $y$, and a \textbf{shortest path} from $x$ to $y$ is a path $p \in P(x,y)$ of length $l(p) = d(x,y)$.
|
|
|
|
\item The \textbf{diameter} $\text{diam}(G)$ of the network $G$ is the length of the longest shortest path between any two nodes:
|
|
\begin{align*}
|
|
\text{diam}(G) = \text{max}\{ l(p) : p \in P(x,y) \}
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\end{align*}
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\end{itemize}
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\end{document}
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