1258 lines
62 KiB
TeX
1258 lines
62 KiB
TeX
% ! TeX program = lualatex
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\documentclass[a4paper,11pt]{article}
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% packages
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\usepackage{amsmath}
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\usepackage{minted}
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\newenvironment{code}{\captionsetup{type=listing}}{}
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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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\hrule
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\vfill
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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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\end{minipage}
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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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\medskip\hrule
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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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|
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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) \}
|
|
\end{align*}
|
|
\end{itemize}
|
|
|
|
% MISSING WEEK06-01 shit
|
|
|
|
\section{Centrality Measures}
|
|
Key nodes in a network can be identified through \textbf{centrality measures}: a way of assigning ``scores'' to nodes that represents their ``importance''.
|
|
However, what it means to be central depends on the context;
|
|
accordingly, in the context of network analysis, a variety of different centrality measures have been developed.
|
|
Measures of centrality include:
|
|
\begin{itemize}
|
|
\item \textbf{Degree centrality:} just the degree of the nodes, important in transport networks for example.
|
|
\item \textbf{Eigenvector centrality:} defined in terms of properties of the network's adjacency matrix.
|
|
\item \textbf{Closeness centrality:} defined in terms of a node's \textbf{distance} to other nodes in the network.
|
|
\item \textbf{Betweenness centrality:} defined in terms of \textbf{shortest paths}.
|
|
\end{itemize}
|
|
\subsection{Degree Centrality}
|
|
In a (simple) graph $G=(X,E)$ with $X=\{ 0, 1, \dots, n-1 \}$ and adjacency matrix $A=(a_{i,j})$, the \textbf{degree centrality} $c_i^D$ of node $i \in X$ is defined as:
|
|
\[
|
|
c_i^D = k_i = \sum_j a_{i,j}
|
|
\]
|
|
|
|
where $k_i$ is the degree of node $i$.
|
|
\\\\
|
|
In some cases, this measure can be misleading since it depends (among other things) on the order of the graph.
|
|
A better measure is the \textbf{normalised degree centrality}: the normalised degree centrality $C_I^D$ of node $i \in X$ is defined as:
|
|
\[
|
|
C_i^D = \frac{k_i}{n-1} = \frac{c_i^D}{n-1} \left( = \frac{\text{degree of centrality of node } i}{\text{number of potential neighbours of } i} \right)
|
|
\]
|
|
|
|
Note that in a directed graph, one distinguishes between the \textbf{in-degree} and the \textbf{out-degree}o f a node and defines the in-degree centrality and the out-degree centrality accordingly.
|
|
|
|
\subsection{Eigenvector Centrality}
|
|
Let $A$ be a square $n \times n$ matrix.
|
|
An $n$-dimensional vector, $v$, is called an \textbf{eigenvector} of $A$ if :
|
|
\[
|
|
Av = \lambda v
|
|
\]
|
|
|
|
for some scalar $\lambda$ which is called an \textbf{eigenvalue} of $A$.
|
|
\\\\
|
|
When $A$ is a real-valued matrix, one usually finds that $\lambda$ and $v$ are \textit{complex-valued}.
|
|
However, if $A$ is \textbf{symmetric}, then they are \textit{real-valued}.
|
|
$A$ may have up to $n$ eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$.
|
|
The \textbf{spectral radius} of $A$ is $\rho(A) := \text{max}(|\lambda_1|, \lambda_2|, \dots, |\lambda_n|)$.
|
|
If $v$ is an eigenvector associated with the eigenvalue $\lambda$, so too is any non-zero multiple of $v$.
|
|
\\\\
|
|
The basic idea of eigenvector centrality is that a node's ranking in a network should relate to the rankings of the nodes it is connected to.
|
|
More specifically, up to to some scalar $\lambda$, the centrality $c_i^E$ of node $i$ should be equal to the sum of the centralities $c_j^E$ of its neighbouring nodes $j$.
|
|
In terms of the adjacency matrix $A = (a_{i,j})$, this relationship is expressed as:
|
|
\[
|
|
\lambda c_i^E = \sum_j a_{i,j} c_j^E
|
|
\]
|
|
|
|
which, in turn, in matrix language is:
|
|
\[
|
|
\lambda c^E = Ac^E
|
|
\]
|
|
|
|
for the vector $c^E = (c_i^E)$ which then is an eigenvector of $A$.
|
|
So $c^E$ is an eigenvector of $A$ (but which one?).
|
|
|
|
\subsubsection{How to find $c^E$ and/or $\lambda$}
|
|
If the network is small, one could use the usual method (although it is almost never a good idea).
|
|
\begin{enumerate}
|
|
\item Find the \textit{characteristic polynomial} $p_A(x)$ of $A$ as \textit{determinant} of the matrix $xI -A$, where $I$ is the $n \times n$ identity matrix.
|
|
\item Find the roots $\lambda$ of $p_A(x)$ (i.e., scalars $\lambda$ such that $p_A(\lambda) = 0$).
|
|
\item Find a \textit{non-trivial solution} of the linear system $(\lambda I - A) c = 0$ (where $0$ stands for an all-$0$ column vector and $c = (c_1, \dots, c_n)$ is a column of \textit{unknowns}).
|
|
\end{enumerate}
|
|
|
|
For large networks, there is a much better algorithm, such as the \textbf{Power method}, which we will look at later.
|
|
|
|
\subsubsection{Perron-Frobenius Theory}
|
|
Presently, we'll lean that the adjacency matrix always has one eigenvalue which is greater than all the others.
|
|
\\\\
|
|
A matrix $A$ is called \textbf{reducible} if, for some simultaneous permutations of its rows and columns, it has the block form:
|
|
\[
|
|
A =
|
|
\begin{pmatrix}
|
|
A_{1,1} & A_{1,2} \\
|
|
o & A_{2,2}
|
|
\end{pmatrix}
|
|
\]
|
|
|
|
If $A$ is not reducible, we say that it is \textbf{irreducible}.
|
|
The adjacency matrix of a simple graph $G$ is \textbf{irreducible} if and only if $G$ is connected.
|
|
\\\\
|
|
A matrix $A=(a_{i,j})$ is \textbf{non-negative} is $a_{i,j \geq 0}$ for all $i$, $j$.
|
|
For simplicity, we usually write $A \geq 0$.
|
|
It is important to node that adjacency matrices are examples of non-negative matrices.
|
|
There are similar concepts of, say, positive matrices, negative matrices, etc.
|
|
Of particular importance are \textbf{positive vectors}: $v = (v_i)$ is positive for if $v_i > 0$ for all $i$.
|
|
We write $v \geq 0$.
|
|
\\\\
|
|
\textbf{Theorem:} suppose that $A$ is a square, non-negative, \textbf{irreducible} matrix.
|
|
Then:
|
|
\begin{itemize}
|
|
\item $A$ has a real eigenvalue $\lambda = \rho(A)$ and $\lambda > |\lambda'|$ for any other eigenvalue $\lambda'$ of $A$.
|
|
$\lambda$ is called the \textbf{Perron root} of $A$.
|
|
|
|
\item $\lambda$ is a simple root of the characteristic polynomial of $A$ (so has just one corresponding eigenvector).
|
|
|
|
\item There is an eigenvector, $v$, associated with $\lambda$ such that $v >0$.
|
|
\end{itemize}
|
|
|
|
For us, this means:
|
|
\begin{itemize}
|
|
\item The adjacency matrix of a connected graph has an eigenvalue that is positive and greater in magnitude than any other.
|
|
\item It has an eigenvector $v$ that is positive.
|
|
\item $v_i$ is the eigenvector centrality of the node $i$.
|
|
\end{itemize}
|
|
|
|
|
|
\subsection{Closeness Centrality}
|
|
A node $x$ in a network can be regarded as being central if it is \textbf{close} to (many) other nodes, as it can quickly interact with them.
|
|
Recalling that $d(i,j)$ is the distance between nodes $i$ and $j$ (i.e., the length of the shortest path between them).
|
|
Then, we can use $\frac{1}{d(i,j)}$ as a measure of ``closeness'';
|
|
in a simple, \textit{connected} graph $G=(X,E)$ of order $n$, the \textbf{closeness centrality}, $c^C_i$ of node $i$ is defined as:
|
|
\[
|
|
c_i^C = \frac{1}{\sum_{j \in X} d(i,j)} = \frac{1}{s(i)}
|
|
\]
|
|
|
|
where $s(i)$ is the \textbf{distance sum} for node $i$.
|
|
As is usually the case, there is a \textbf{normalised} version of this measure;
|
|
the \textbf{normalised closeness centrality} is defined as:
|
|
\[
|
|
C_i^C = (n-1)c_i^C = \frac{n-1}{\sum_{j \in X} d(i,j)} = \frac{n-1}{s(i)}
|
|
\]
|
|
|
|
Note that $0 \leq C_i^C \leq 1$.
|
|
\\\\
|
|
The \textbf{distance matrix} of a graph, $G$, of order $n$ is the $n \times n$ matrix $D=(d_{i,j})$ such that:
|
|
\[
|
|
d_{i,j} = d(i,j)
|
|
\]
|
|
|
|
We'll return to how to compute $D$ later, but for now we note:
|
|
\begin{itemize}
|
|
\item $s(i)$ is the sum of row $i$ of $D$.
|
|
\item If $s$ is the vector of distance sums, then $s = De$ where $e = (1,1, \dots, 1)^T$.
|
|
\end{itemize}
|
|
|
|
\subsection{Betweenness Centrality}
|
|
In a simple, connected graph $G$, the \textbf{betweenness centrality} $c_i^B$ of node $i$ is defined as:
|
|
\[
|
|
c_i^B = \sum_j \sum_k \frac{n_i(j,k)}{n(j,k)}, \quad j \neq k \neq 1
|
|
\]
|
|
|
|
where $n(j,k)$ denotes the \textit{number} of shortest paths from node $j$ to node $k$, where $n_i(j,k)$ denotes the number of those shortest paths \textit{passing through} node $i$.
|
|
\\\\
|
|
In a simple, connected graph $G$, the \textbf{normalised betweenness centrality} $c_i^B$ of node $i$ is defined as:
|
|
\[
|
|
C_i^B = \frac{c_i^B}{(n-1)(n-2)}
|
|
\]
|
|
|
|
\section{Random Graphs}
|
|
A \textbf{random graph} is mathematical model of a family of networks, where certain parameters (like the number of nodes \& edges) have fixed values, but other aspects (like the actual edges) are randomly assigned.
|
|
Although a random graph is not a specific object, many of its properties can be described precisely in the form of expected values or probability distributions.
|
|
|
|
\subsection{Random Samples}
|
|
Suppose our network $G = (X,E)$ has $|X| = n$ nodes.
|
|
Then, we know that the greatest number of edges it can have is:
|
|
\begin{align*}
|
|
\binom{n}{2} = \frac{n!}{(n-2)! 2!} = \frac{n(n-1)}{2}
|
|
\end{align*}
|
|
|
|
Our goal is to randomly select edges on the vertex set $X$, that is, pick random elements from the set $\binom{X}{2}$ of pairs of nodes.
|
|
So, we need a procedure for selecting $m$ from $N$ objects randomly, in such a way that each of the $\binom{N}{m}$ subsets of the $N$ objects is an equally likely outcome.
|
|
We first discuss sampling $m$ values in the range $\{0,1, \dots, N-1 \}$.
|
|
\begin{enumerate}
|
|
\item Suppose that we choose a natural number $N$ and a real number $p \in [0,1]$.
|
|
\item Then, iterate over each element of the set $\{0,1 \dots, N-1\}$.
|
|
\item For each, we pick a random number $x \in [0,1]$.
|
|
\item If $x < p$, we keep that number.
|
|
Otherwise, remove it from the set.
|
|
\end{enumerate}
|
|
|
|
When we are done, how many elements do we expect in the set if $p = \frac{m}{N}$ for some chosen $m$?
|
|
And what is the likelihood of there being, say, $K$ elements in the set?
|
|
Since we are creating random samples, where the size of each is a random number, $k$, we expect that $E[k] = Np = m$; this is a \textbf{binomial distribution}:
|
|
\begin{itemize}
|
|
\item The probability of a specific subset of size $K$ to be chosen is $p^k(1-p)^{N-k}$.
|
|
\item There are $\binom{N}{k}$ subsets of size $k$, so the probability $P(k)$ of the sample to have size $k$ is $P(k) = \binom{N}{k}p^k (1-p)^{N-k}$.
|
|
\end{itemize}
|
|
|
|
We use the following facts:
|
|
\begin{itemize}
|
|
\item $j\binom{N}{j}p^i = Np \binom{N-1}{j-1}p^{i-1}$.
|
|
\item $(1-p)^{N-j} = (1-p)^{(N-1) - (j-1)}$.
|
|
\item $(p + (1 -p))^r = 1$ for all $r$.
|
|
\end{itemize}
|
|
|
|
The expected value is:
|
|
|
|
\begin{align*}
|
|
E[k] =& \sum^N_{j=1}jP(j) \quad &\text{ weighted average of } j \\
|
|
=& \sum^N_{j=0} j \binom{N}{j} p^j (1-p)^{N-j} \quad \text{ formula for } P(j) \\
|
|
= & Np \sum^{N-1}_{l=0} \binom{N-1}{l} p^l (1-p)^{(N-1) - l} = Np \quad \text{ substituting } l=k-1
|
|
\end{align*}
|
|
|
|
\subsection{Erd\"os-Rényi Models}
|
|
\subsubsection{Model A: $G_{ER}(n,m)$ --- Uniformly Selected Edges}
|
|
Let $n \geq 1$, let $N = \binom{n}{2}$ and let $0 \leq m \leq N$.
|
|
The model $G_{ER}(n,m)$ consists of the ensemble of graphs $G$ on the $n$ nodes $X = \{0,1, \dots, n-1\}$, and $M$ randomly selected edges, chosen uniformly from the $N = \binom{n}{2}$ possible edges.
|
|
Equivalently, one can choose uniformly at random one network in the set $G(n,m)$ of \textit{all} networks on a given set of $n$ nodes with exactly $m$ edges.
|
|
\\\\
|
|
Equivalently, one can choose uniformly at random one network in the \textbf{set} $G(n,m)$ of \textit{all} networks on a given set of $n$ nodes with \textit{exactly} $m$ edges.
|
|
One could think of $G(n,m)$ as a probability distribution $P: G(n,m) \rightarrow \mathbb{R}$ that assigns to each network $G \in G(n,m)$ the same probability
|
|
\[
|
|
P(G) = \binom{N}{m}^-1
|
|
\]
|
|
where $N = \binom{n}{2}$.
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/gnm.png}
|
|
\caption{ Some networks drawn from $G_{ER}(20,15)$ }
|
|
\end{figure}
|
|
|
|
|
|
\subsubsection{Model B: $G_{ER}(n,p)$ --- Randomly Selected Edges}
|
|
Let $n \geq 1$, let $N = \binom{n}{2}$ and let $0 \leq p \leq 1$.
|
|
The model $G_{ER}(n,p)$ consists of the ensemble of graphs $G$ on the $n$ nodes $X=\{0,1, \dots, n-1\}$ with each of the possible $N=\binom{n}{2}$ edges chosen with probability $p$.
|
|
\\\\
|
|
The probability $P(G)$ of a particular graph $G=(X,E)$ with $X=\{0,1, \dots, n-1\}$ and $m = |E|$ edges in the $G_{ER}(n,p)$ model is
|
|
\[
|
|
P(G) = p^m(1-p)^{N-m}
|
|
\]
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{./images/gnm2005.png}
|
|
\caption{ Some networks drawn from $G_{ER}(20,0.5)$ }
|
|
\end{figure}
|
|
|
|
Of the two models, $G_{ER}(n,p)$ is the more studied.
|
|
There are many similarities, but they do differ.
|
|
For example:
|
|
\begin{itemize}
|
|
\item $G_{ER}(n,m)$ will have $m$ edges with probability 1.
|
|
\item A graph in $G_{ER}(n,p)$ will have $m$ edges with probability $\binom{N}{m}p^m(p-1)^{N-m}$.
|
|
\end{itemize}
|
|
|
|
\subsubsection{Properties}
|
|
We'd like to investigate (theoretically \& computationally) the properties of such graphs.
|
|
For example:
|
|
\begin{itemize}
|
|
\item When might it be a tree?
|
|
\item Does it contain a tree, or other cycles? If so, how many?
|
|
\item When does it contain a small complete graph?
|
|
\item When does it contain a \textbf{large component}, larger than all other components?
|
|
\item When does the network form a single \textbf{connected component}?
|
|
\item How do these properties depend on $n$ and $m$ (or $p$)?
|
|
\end{itemize}
|
|
|
|
Denote by $G_n$ the set of \textit{all} graphs the $n$ nodes $X=\{0, \dots, n-1\}$.
|
|
Set $N=\binom{n}{2}$ the maximal number of edges of a graph $G \in \textsl{G}$.
|
|
Regard the ER models A \& B as \textbf{probability distributions} $P : \mathcal{G}_n \rightarrow \mathbb{R}$
|
|
\\\\
|
|
Denote $m(G)$ as the number of edges of a graph $G$.
|
|
As we have seen, the probability of a specific graph $G_{ER}$ to be sampled from the model $G(n,m)$ is:
|
|
\begin{align*}
|
|
P(G) =
|
|
\begin{cases}
|
|
\binom{N}{m}^{-1} & \text{if } m(G)= m, \\
|
|
0 & \text{otherwise}
|
|
\end{cases}
|
|
\end{align*}
|
|
|
|
And the probability of a specific graph $G$ to be sampled from the model $G(n,p)$ is
|
|
\begin{align*}
|
|
P(G) = n^m(1-n)&{N-m}
|
|
\end{align*}
|
|
|
|
\subsubsection{Expected Size \& Average Degree}
|
|
Let's use the following notation:
|
|
\begin{itemize}
|
|
\item $\bar{a}$ is the expected value of property $a$ (that is, as the graphs vary across the ensemble produced by the model).
|
|
\item $<a>$ is the average of property $a$ over all the nodes of a graph.
|
|
\end{itemize}
|
|
|
|
In $G(n,m)$ the expected \textbf{size} is
|
|
\begin{align*}
|
|
\bar{m} = m
|
|
\end{align*}
|
|
as every graph $G$ in $G(n,m)$ has exactly $m$ edges.
|
|
The expected \textbf{average degree} is
|
|
\begin{align*}
|
|
\langle k \rangle = \frac{2m}{n}
|
|
\end{align*}
|
|
|
|
as every graph has average degree $\frac{2m}{n}$.
|
|
Other properties of $G(n,m)$ are less straightforward, and it is easier to work with the $G(n,p)$.
|
|
\\\\
|
|
In $G(n,m)$, the \textbf{expected size} (i.e., expected number of edges) is
|
|
\begin{align*}
|
|
\bar{m} = pN
|
|
\end{align*}
|
|
|
|
Also, variance is $\sigma^2_m = Np(1-p)$.
|
|
\\\\
|
|
The expected \textbf{average degree} is
|
|
\begin{align*}
|
|
\langle k\rangle = p(n-1)
|
|
\end{align*}
|
|
with standard deviation $\sigma_k = \sqrt{p(1-p) (n-1)}$.
|
|
|
|
|
|
\subsubsection{Degree Distribution}
|
|
The \textbf{degree distribution} $p: \mathbb{N}_0 \to \mathbb{R}, k \mapsto p_k$ of a graph $G$ is defined as
|
|
\begin{align*}
|
|
p_k = \frac{n_k}{n}
|
|
\end{align*}
|
|
where, for $k \geq 0$, $n_k$ is the number of nodes of degree $k$ in $G$.
|
|
This definition can be extended to ensembles of graphs with $n$ nodes (like the random graphs $G(n,m)$ and $G(n,p)$) by setting
|
|
\begin{align*}
|
|
p_k \frac{\bar{n}_k}{n}
|
|
\end{align*}
|
|
|
|
where $\bar{n}_k$ denotes the expected value of the random graph $n_k$ over the ensemble of graphs.
|
|
\\\\
|
|
The degree distribution in a random graph $G(n,p)$ is a \textbf{binomial distribution}:
|
|
\begin{align*}
|
|
p_k = \binom{n-1}{k}p^k (1-p)^{n-1-k} = \text{bin}(n-1,p,k)
|
|
\end{align*}
|
|
|
|
That is, in the $G(n,p)$ model, the probability that a nodes has degree $k$ is $p_k$.
|
|
Also, the \textbf{average degree} of a randomly chosen node is
|
|
\begin{align*}
|
|
\langle k \rangle = \sum^{n-1}_{k=0} kp_k = p(n-1)
|
|
\end{align*}
|
|
|
|
(with standard deviation $\sigma_k = \sqrt{p(1-p)(n-1))}$.
|
|
\\\\
|
|
In general, it is not so easy to compute
|
|
\[
|
|
\binom(n-1)(k) p^k (1-p)^{n-1-k}
|
|
\]
|
|
|
|
However, in the limit $n \to \infty$ with $\langle k \rangle k = p(n-1)$ kept constant, the binomial distribution $\text{bin}(n-1,p,k)$ is well-approximated by the \textbf{Poisson distribution}:
|
|
\[
|
|
p_k = e^{-\lambda} \frac{\lambda^k}{k!} = \text{Pois}(\lambda, k)
|
|
\]
|
|
|
|
where $\lambda = p(n-1)$.
|
|
|
|
\section{Giant Components \& Small Worlds}
|
|
Recall that a network may be made up of several \textbf{connected components}, and any connected network has a single connected component.
|
|
It is common in large networks to observe a \textbf{giant component}: a connected component which has a large proportion of the network's nodes.
|
|
This is particularly the case with graphs in $G_{ER}(n,p)$ with large enough $p$.
|
|
More formally, a connected component of a graph $G$ is called a \textbf{giant component} if its number of nodes increases with the order $n$ of $G$ as some positive power of $n$.
|
|
Suppose that $p(n) = cn^{-1}$ for some positive constant $c$;
|
|
then, the average degree $\langle k \rangle = pn = c$ remains fixed as $n \to \infty$.
|
|
For graphs $G_{ER}(n,p)$:
|
|
\begin{itemize}
|
|
\item If $c < 1$, the graph contains many small components with orders bounded by $O(\ln(n))$.
|
|
\item If $c=1$ the graph has large components of order $S = O(n^\frac{2}{3})$.
|
|
\item If $c > 1$, there is a unique \textbf{giant component} of order $S = O(n)$.
|
|
\end{itemize}
|
|
|
|
\subsection{Small World Network}
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Many real-world networks are \textbf{small world networks}, wherein most pairs of nodes are only a few steps away from each other, and where nodes to form \textit{cliques}, i.e., subgraphs in which all nodes are connected to each other.
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Three network attributes that measure these small-world effects are:
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\begin{itemize}
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\item \textbf{Characteristic path length}, $L$: the average length of all shortest paths in the network.
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\item \textbf{Transitivity}, $T$: the proportion of \textit{triads} that form triangles.
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\item \textbf{Clustering coefficient}, $C$: the average node clustering coefficient.
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\end{itemize}
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A network is called a \textbf{small world network} if it has:
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\begin{itemize}
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\item A small \textbf{average shortest path length} $L$ (scaling with $\log(n)$, where $n$ is the number of nodes) and
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\item A high \textbf{clustering coefficient} $C$.
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\end{itemize}
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It turns out that ER random networks do have a small average shortest path length, but not a high clustering coefficient.
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This observation justifies the need for a different model of random networks, if they are to be used to model the clustering behaviour of real-world networks.
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\subsubsection{Distance}
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We have seen how BFS can determine the length of a shortest path from a given node $x$ to any node $y$ in a \textit{connected network}.
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An application to all nodes $x$ yields the shortest distances between all pairs of nodes.
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Recall that the \textbf{distance matrix} of a connected graph $G = (X,E)$ is $\mathcal{D} = (d_{i,j})$ where entry $d_{i,j}$ is the length of the shortest path from node $i \in X$ to node $j \in X$.
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(Note that $d_{i,i} = 0$ for all $i$).
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There are a number of graph (and node) attributes that can be defined in terms of this matrix:
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\begin{itemize}
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\item The \textbf{eccentricity} $e_i$ of a node $i \in X$ is the maximum distance between $i$ and any other vertex in $G$, so $e_i = \text{max}_j(d_{i,j})$.
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\item The \textbf{graph radius} $R$ is the minimum eccentricity, $R = \text{min}_i(e_i)$.
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\item The \textbf{graph diameter} $D$ is the maximum eccentricity: $D = \text{max}_i(e_i) = - \text{max}_{i,j} (d_{i,j})$.
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\end{itemize}
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Note that one shouldn't think that the ``diameter is twice the radius'', but rather diameter is the distance between the points furthest from each other and radius is the distance from the ``centre'' to the furthest point from it.
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It can be helpful to think about $P_n$.
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\subsubsection{Characteristic Path Length}
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The \textbf{characteristic path length} (i.e., the average shortest path length) $L$ of a graph $G$ is the average distance between pairs of nodes:
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\[
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L = \frac{1}{n(n-1)} \sum_{i \neq j} d_{i,j}
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\]
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For graphs drawn from $G_{ER}(n,m)$ and $G_{ER}(n,p)$, $L = \frac{\ln(n)}{\ln( \langle k \rangle)}$, where $\langle k \rangle$ is the average degree of the network.
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\subsubsection{Clustering}
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In contrast to random graphs, real-world networks also contain \textbf{many triangles}: it is not uncommon that a friend of one of my friends is also my friend.
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This \textbf{degree of transitivity} can be measured in several different ways.
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For the first, we need two concepts:
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\begin{itemize}
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\item The number of \textbf{triangles} in $G$, denoted $n_\Delta$, is the number of subgraphs of $G$ that are isomorphic to $C_3$.
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\item The number of \textbf{triads} in $G$, denoted $n_\land$, is the number of pairs of edges with a shared node.
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\end{itemize}
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There is an easy way to count the number of \textbf{triads} in a network:
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if node $i$ has degree $k_i = \text{deg}(i)$, then it is involved in $\binom{k_i}{2}$ triads,
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so the total number of triads is $n_\land = \sum_i \binom{k_i}{2}$.
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\\\\
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The \textbf{transitivity} $T$ of a graph $G = (X,E)$ is the proportion of \textbf{transitive} triads, i.e., triads which are subgraphs of \textbf{triangles}.
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This proportion can be computed as follows:
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\[
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T = 3 \frac{n_\Delta}{n_\land}
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\]
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where $n_\Delta$ is the number of triangles in $G$ and $n_\land$ is the number of triads.
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\subsubsection{Small World Behaviour}
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A network $G = (X,E)$ is said to exhibit \textbf{small world behaviour} if its characteristic path length $L$ grows proportionally to the logarithm of the number of nodes of $G$:
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\[
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L \sim \ln(n)
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\]
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In this sense, the ensembles $G(n,m)$ \& $G(n,p)$ of random graphs do exhibit small world behaviour (as $n \to \infty$).
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\subsubsection{Transitivity}
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The \textbf{transitivity} $T$ of a graph $G=(X,E)$ is the proportion of \textbf{transitive} triads, i.e., triads which are subgraphs of \textbf{triangles}.
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This proportion can be computed as:
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\[
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T = \frac{3n_\Delta}{n_\land}
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\]
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where $n_\Delta$ is the number of triangles in $G$ and $n_\land$ is the number of triads.
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\\\\
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The transitivity of a graph in $G_{ER}(n,p)$ is easy to estimate:
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for every triad, the ``third'' edge is present with probability $p$, so:
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\[
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T = p
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\]
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Or, compute $\frac{3n_\Delta}{n_\land}$ using the explicit formulas from the previous lecture:
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$n_\Delta = \binom{n}{3} p^3$ and $n_\land = 3 \binom{n}{3}p^2$.
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\subsection{Clustering}
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The concept of \textbf{clustering} measures the transitivity of a node, or of an entire graph in a different way.
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To define it, we need the concept of an \textbf{induced subgraph}.
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\subsubsection{Induced Subgraph}
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Given $G = (X,E)$ and $Y \subset X$, the \textbf{induced subgraph} of $G$ on $Y$ is the graph $H = \left( Y, E \cap \binom{Y}{2} \right)$.
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That is:
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\begin{itemize}
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\item $H$ is a subgraph of $G$ with node set $Y$.
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\item $H$ has all possible edges in $G$ for which both nodes are in $Y$.
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\end{itemize}
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\subsubsection{Clustering Coefficient}
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For a node $i \in X$ of a graph $G = (X,E)$, denote by $G_i$ the subgraph induced on the neighbours of $i$ in $G$, and by $m(G_i)$ its number of edges;
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the \textbf{node clustering coefficient} $c_i$ of node $i$ is defined as:
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\[
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c_i =
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\begin{cases}
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\binom{k_i}{2}^{-1} m(G_i) & k_i \geq 2 \\
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0 & \text{otherwise}
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\end{cases}
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\]
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That is, the node clustering coefficient measures the proportion of existing edges in its \textbf{social graph} among the possible edges.
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\\\\
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The \textbf{graph clustering coefficient} $C$ of $G$ is the average node clustering coefficient:
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\[
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C = \langle c \rangle c = \frac{1}{n} \sum^n_{i=1} c_i
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\]
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By definition, $0 \leq c_i \leq 1$ for all nodes $i \in X$, and $0 \leq C \leq 1$.
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\\\\\
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The \textbf{node clustering coefficient} of any node $i$ in a $G_{ER}(n,p)$ \textbf{random graph} is $c_i = p$, i.e., in any selection of potential edges, by construction a proportion $p$ of them is present in the random graph;
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this is true in particular for the $\binom{k}{2}$ potential edges between the $k$ neighbours of a node of degree $k$.
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The \textbf{graph clustering coefficient} of a $G_{ER}(n,p)$ \textbf{random graph} is:
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\[
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C = p
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\]
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Note that when $p(n) = \langle k \rangle n^{-1}$ for a fixed expected average degree $\langle k \rangle$, then $C = \frac{\langle k \rangle}{n} \to 0$ for $n \to \infty$;
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that is, in large $G_{ER}$ random graphs, the number of triangles is negligible.
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In real-world networks, one often observers that $\frac{C}{\langle k \rangle}$ does not depend on $n$ (as $n \to \infty$).
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\subsubsection{Clustering versus Transitivity}
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For a node $i \in X$, denote by $n^\land_i = \binom{k_i}{2}$ the number of triads containing $i$ as their central node, and by $n_i^\Delta$ the actual number of triangles containing $i$;
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then, the node clustering coefficient is:
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\begin{align*}
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c_i = \frac{n_i^\Delta}{n_i^\land} \quad \text{ or,} \\
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n_i^\Delta = n_i^\land c_i
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\end{align*}
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Moreover, $3n_\Delta = \sum_i n_i^\Delta$ and $n_\land = \sum_i n_i^\land$;
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it follows that $T = \frac{3n_\Delta}{n_\land} = \frac{1}{n_\land} \sum_i n_i^\land c_i$, in contrast to $C = \frac{1}{n} \sum_i c_i$.
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That is, $C$ is the (plain) \textbf{average} of the node clustering coefficients, whereas $T$ is a \textbf{weighted average} of node clustering coefficients, giving higher weight to high-degree nodes.
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\\\\
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The fact that ER random networks tend to have low transitivity \& clustering shows the need for a new kind of (random) network construction that is better at modelling real-world networks.
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One idea is to start with some \textbf{regular network} that naturally has \textit{high clustering}, and then to randomly distort its edges to introduce some \textbf{short paths}.
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\end{document}
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