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%%%%%%%%%%%%%%%%

\title[Networks With \textbf{R}]{Intro to Social Network Analysis\\}
\author[LJasny]{Lorien Jasny\inst{1}}
\institute[SESYNC]{\inst{1} University of Exeter\\
\texttt{L.Jasny@exeter.ac.uk}\\
				{\insertlogo}}
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}

\date{\emph{Online Workshop} \\
September 2026}
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\begin{document}
	\begin{frame}
	\maketitle
	\end{frame}


\section{Basic SNA Measures}

%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Descriptives}
\begin{columns}
\column{6cm} %first column
\begin{tikzpicture}
%\draw[help lines] (0,0) grid (10,6);
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\end{tikzpicture}
\column{6cm} %second column
\begin{itemize}
\uncover<2->{\item One isolate}
\uncover<3->{\item Two components}
\uncover<4->{\item Diameter is 5}
\uncover<5->{\item Medici is most popular}
\uncover<6->{\item Three triads}
\end{itemize}
\end{columns}
\end{frame}


%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Degree}
For each node, its degree is
\begin{itemize}
\uncover<2->{
\item the number of nodes adjacent to it }
\uncover<3->{
\item or, the number of lines incident with it}
\end{itemize}
\begin{columns}
\column{5cm}
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}
\uncover<8>{\draw [thick,blue] (.56,-.4) -- (-.35,-.1);
\draw [thick,blue] (.52,-.35) -- (.12,.25);
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\draw [thick,blue] (.52,-.38) -- (1.35,0);
\draw [thick,blue] (.52,-.35) -- (1.05,-1.08);
\draw [thick,blue] (.52,-.35) -- (.15,-1.15);
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\end{tikzpicture}
\column{5cm}
\begin{itemize}
\small{
\uncover<5->{
\item Pucci has degree 0}
\uncover<6->{
\item Lamberteschi has degree 1}
\uncover<7->{
\item Guadagni has degree 4}
\uncover<8->{
\item Medici has degree 6}
}
\end{itemize}
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Directed Degree}
In directed graphs,
\begin{itemize}
\uncover<2->{
\item \textit{Indegree} indicates the number of received ties}
\uncover<3->{
\item \textit{Outdegree} indicates the number of sent ties}
\end{itemize}
\begin{columns}
\column{4cm}
\begin{tikzpicture}
\node at (1,0) {A};
\node at (3,1) {B};
\node at (3,-1) {C};
\node at (5,0) {D};
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\column{5cm}
\begin{itemize}
\uncover<4->{\item A has 1 \color{blue}outdegree\color{black}}
\uncover<5->{\item B has 1 \color{blue}outdegree} \uncover<6->{\color{black}  and 3 \color{red}indegree \color{black}}
\uncover<7->{\item C has 2 \color{blue} outdegree} \uncover<8->{\color{black}  and 1 \color{red}indegree \color{black}}
\uncover<9->{\item D has 1 \color{blue}outdegree\color{black}}\uncover<10->{ and 1 \color{red}indegree\color{black}}
\end{itemize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Betweenness Centrality}
\begin{columns}
\column{3cm}
Proportion of shortest paths that the given node lies on
\pause
\column{7cm}
\begin{tikzpicture}
\node (russia) at (0,0) {\includegraphics[width=7cm,trim=0.4cm 1cm 0.5cm 0cm, clip=true]{../../images/russiaTrade.jpg}};
\uncover<3->{\draw [ultra thick, blue] (-.85,.5) circle [radius=0.3];}
\end{tikzpicture}
\tiny{Forrest Pitts (1978)"}
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Closeness Centrality -- Alternate Measure}
How far is each node from the other nodes in the graph\\
\begin{columns}
\column{4cm}
\begin{tikzpicture}
\node at (1,0) {A};
\node at (3,1) {B};
\node at (3,-1) {C};
\node at (5,0) {D};
\draw [->, ultra thick] (1.1,.1) -- (2.8,.8);
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\end{tikzpicture}
\column{5cm}
\begin{itemize}
\item $C^2(v)=\frac{\sum{\frac{1}{d(v,i)}}}{|V|-1}$
\item B's Distance to
\begin{itemize}
\item C = 1
\item D = 2
\item A = inf
\end{itemize}
\uncover<2->{\item B's Closeness Centrality = $\frac{\frac{1}{inf}+\frac{1}{1}+\frac{1}{2}}{3}=\frac{1}{2}$}
\end{itemize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Page Rank and Eigenvector Centrality}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Page Rank}
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\draw [black, ultra thick] (1.5,-.5) circle [radius=0.2];
\node at (0,-.4) {\small{Erica}};
\node at (0.5,1.4) {\small{Amy}};
\node at (2.5,1.4) {\small{Bryan}};
\node at (3,-.4) {\small{Carter}};
\node at (1.5,-.9) {\small{David}};
\draw [->,black,ultra thick] (0.7,1)--(2.3,1);
\draw [->,black,ultra thick] (0.65,.95)--(2.8,0.1);
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\node [below] at (1.5,-1) {1};
}
\end{tikzpicture}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Other Centrality Measures}
\begin{itemize}
\pause
\item Bonacich power centrality (1987)
\pause
\item Weighted measures (Opsahl, Agneessens and Skvoretz 2010)
\pause
\item Two-mode centrality measures (Everett and Borgatti 2005)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
\section{Graph Level Indices}
\begin{frame}
\frametitle{Graph Level Indices}
\end{frame}
%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Density}
\begin{itemize}
\uncover<2->{
\item Number of ties, expressed as a percentage of the number of possible ties}
\uncover<3->{
\item For directed graphs: \LARGE{$\frac{E}{N(N-1)}$}}
\uncover<4->{
\item \normalsize{For undirected graphs:} \LARGE{$\frac{\frac{E}{N(N-1)}}{2}$}}
\end{itemize}
\begin{tikzpicture}
%\draw[help lines,white] (0,0) grid (6,4);
\uncover<5->{
\draw [blue, ultra thick] (0,0) circle [radius=0.5];
\node [blue] at (0,0) {\textbf{1}};

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\uncover<6->{\node at (5,1) {\LARGE{$=\frac{\frac{2}{3(3-1)}}{2}=\frac{4}{6}$}};}

\end{tikzpicture}

\end{frame}



%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Mean Degree}
The mean degree, $\bar{d}$, of all nodes in the graph is
\vspace{.3cm}

\begin{center}\LARGE{
\uncover<2->{$\bar{d}(n)=\frac{\sum_{i=1}^{N}d(n_i)}{N}=\frac{2E}{N}$}\\
\vspace{.3cm}
\uncover<3->{$=\frac{1+2+1}{3}=\frac{4}{3}$}}
\end{center}
\begin{tikzpicture}
%\draw[help lines,white] (0,0) grid (6,4);
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\node [blue] at (0,0) {\textbf{1}};

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\node [blue] at (2,0) {\textbf{3}};

\draw [thick, blue] (.35,.35)--(.8,1.55);
\draw [thick, blue] (1.2,1.55)--(1.65,.35);



\end{tikzpicture}
\end{frame}

%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Size, Density, and Mean Degree}
\begin{columns}
\column{4cm}
If we hold the mean degree constant, but vary size, what happens to density?
\pause
\column{6cm}
\includegraphics[width=6cm]{../../images/meanDegree.jpg}
\end{columns}
\end{frame}


%%%%%%%%%%%%%%%%


\begin{frame}
\frametitle{Centralization}
\begin{itemize}
\pause
\item Extent to which centrality is concentrated on a single vertex (Freeman 1978)
\pause
\item Calculated as the sum of the differences between each node's centrality score and the maximum score
\pause
\item Most centralized structure is usually a star network
\begin{tikzpicture}
\draw[help lines,white] (0,0) grid (4,3);
\node (b1) at (-2,1) {\includegraphics[width=4cm]{../../images/b1.jpg}};
\node (b3) at (2,1) {\includegraphics[width=4cm]{../../images/b3.jpg}};
\end{tikzpicture}

\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%


\begin{frame}
\frametitle{Dyad Census}
\begin{tikzpicture}
\uncover<2->{
%\draw[help lines,black] (0,0) grid (10,6);
\draw [blue, ultra thick] (1,5) circle [radius=0.3];
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\draw [<->, ultra thick,blue] (1.5,5) -- (2.5,5);
\node at (6.1,5) {\LARGE{Mutual (M)}};
}

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%\draw[help lines,black] (0,0) grid (10,6);
\draw [blue, ultra thick] (1,3) circle [radius=0.3];
\draw [blue, ultra thick] (3,3) circle [radius=0.3];
\draw [<-, ultra thick,blue] (1.5,3) -- (2.5,3);
\node at (6.5,3) {\LARGE{Assymetric (A)}};
}

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%\draw[help lines,black] (0,0) grid (10,6);
\draw [blue, ultra thick] (1,1) circle [radius=0.3];
\draw [blue, ultra thick] (3,1) circle [radius=0.3];
\node at (5.6,1) {\LARGE{Null (N)}};
}
\end{tikzpicture}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Reciprocity}
\begin{itemize}
\pause
\item Dyadic: the proportion of dyads that are symmetric \LARGE{$\frac{M+N}{M+A+N}$}
\pause
\item \normalsize{Dyadic non-null: the proportion of non-null dyads that are reciprocal} \LARGE{$\frac{M}{M+A}$}
\pause
\item \normalsize{Edgewise:} \LARGE{$\frac{2*M}{2*M+A}$}
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Triad Census}
\begin{tikzpicture}
\node (tc) at (0,0) {\includegraphics[width=9.5cm]{../../images/triadcensus.jpg}};
\end{tikzpicture}
\scriptsize{Holland and Leinhardt 1974}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Triad Census}
\begin{tikzpicture}
\draw (2,0) rectangle (8,4);
\node at (4,4.4) {16 different triad types};
\node [rotate=90] at (1.5,2) {One row per network};
\uncover<2->{
\node at (4,3.6) {i,j cell is the number};
\node at (3.4,3.2) {of triad type j}; 
\node at (3.2,2.8) {in network i};
}
\end{tikzpicture}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Triad Census}
\begin{tikzpicture}
\node (tc) at (0,0) {\includegraphics[width=10cm]{../../images/faust.jpg}};
\uncover<2>{
\draw [blue, ultra thick] (-2.5,1.5) circle [radius=1];
\node at (1,1.5) {Dominance Relationships};
}
\uncover<3->{
\draw [red, ultra thick] (.5,-1.5) circle [radius=1];
\node at (1.7,0.2) {Friendship, Assistance};
}
\end{tikzpicture}
\scriptsize{Faust 2007}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Transitivity}
\begin{columns}
\column{4cm}
\begin{tikzpicture}
%\draw[help lines,white] (0,0) grid (6,4);
\uncover<2->{
\draw [blue, ultra thick] (0,0) circle [radius=0.5];
\node [blue] at (0,0) {\textbf{1}};

\draw [blue, ultra thick] (1,2) circle [radius=0.5];
\node [blue] at (1,2) {\textbf{2}};

\draw [blue, ultra thick] (2,0) circle [radius=0.5];
\node [blue] at (2,0) {\textbf{3}};
}
\uncover<3->{
\draw [->,ultra thick, blue] (.2,.45)--(.8,1.5);}
\uncover<4->{
\draw [->,ultra thick, blue] (1.2,1.5)--(1.8,.45);}
\uncover<5->{
\draw [->,ultra thick, blue] (.5,0)--(1.5,0);}

\end{tikzpicture}
\uncover<6->{
\column{6cm}
\begin{itemize}
\item Usually calculated as the fraction of completed two-paths}
\uncover<7->{
\item Related to Grannovetter's `forbidden triad' (1983)}
\uncover<8->{
\item Can be directed or undirected}
\end{itemize}
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example}
\includegraphics[width=10cm]{../../images/janetNDino.jpg}\\
\scriptsize{Janet Box-Steffensmeier and Dino Christenson\\
``The evolution and formation of amicus curiae networks"\\ \textit{Social Networks} 2012}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example}
\includegraphics<1>[width=10cm]{../../images/echoChamber1.png}
\includegraphics<2>[width=10cm]{../../images/echoChamber2.png}
\includegraphics<3>[width=10cm]{../../images/inhofe.jpg}
\scriptsize{Jasny et al 2015}

\end{frame}
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Code Time}
\begin{itemize}
\item Section 3-4.7
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Forbidden Triad or\\ Structural Hole?}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm,trim={1cm 0 1cm 0},clip]{../../images/forbiddenTriad.png}\\
\column{7cm}
\pause
\begin{itemize}
\item Granovetter, Mark S. 1973.\\  ``The Strength of Weak Ties"
\end{itemize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Forbidden Triad or\\ Structural Hole?}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm,trim={6cm 0 6cm 0},clip]{../../images/weakTies.png}\\
\column{7cm}
\begin{itemize}
\item Granovetter, Mark S. 1973.\\  ``The Strength of Weak Ties"
\end{itemize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Forbidden Triad or\\ Structural Hole?}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm,trim={0 0 0 0},clip]{../../images/structuralHoles.png}\\
\column{7cm}
\begin{itemize}
\item Burt, Ronald S. 2004.\\  ``Structural Holes: The Social Structure of Competition"
\end{itemize}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Extensions}
Attributes!
\begin{itemize}
\pause
\item Properties of nodes, edges, or even networks
\pause
\item Pretty much anything you can measure could be an attribute
\pause
\item Extension based on node attributes: Brokerage
\pause
\item Extension based on edge attributes: Structural Balance
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Brokerage}
\begin{itemize}
\item Brokerage is a process ``by which intermediary actors facilitate transactions between other actors lacking access to or trust in one another" (Marsden 1982)
\pause
\item Brokers play a crucial role in knitting together diverse groups of people, organizations, parties
\pause
\item Brokers can gain a lot -- early access to information, prestige
\pause
\item But can also be distrusted by everyone
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Brokerage: Formal Concept}
\begin{columns}
\column{4cm} %first column
\uncover<1->{In a network $N$ with edges $E$, }\uncover<2->{\\node \color{red}$j$ \color{black}  brokers }\uncover<3->{\\nodes \color{blue}$i$ \color{black} and \color{green}$k$ \color{black} }\uncover<4->{\\if $e_{ij}\in E$ }\uncover<5->{\\and $e_{jk}\in E$ }\uncover<6->{\\but $e_{ik} \color{orange} \notin \color{black} E$}


\column{4cm} %second column
\begin{tikzpicture}
\uncover<1->{
\draw [black, ultra thick] (0,3) circle [radius=0.5];
\draw [black, ultra thick] (3,3) circle [radius=0.5];
\draw [black, ultra thick] (1.5,1) circle [radius=0.5];
}
\uncover<2->{
\node [red] at (3,3) {\textbf{\textit{j}}};
}
\uncover<3->{
\node [blue] at (0,3) {\textbf{\textit{i}}};
\node [green] at (1.5,1) {\textbf{\textit{k}}};
}
\uncover<4->{
\draw [->,ultra thick] (0.5,3) -- (2.5,3);
}
\uncover<5->{
\draw [->, ultra thick] (2.65, 2.65) -- (1.8,1.45);
}
\uncover<6->{
\draw [<->, ultra thick, dotted, orange] (0.2,2.5) -- (1.2, 1.45);
}
\end{tikzpicture}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Brokerage: Formal Concept}
Gould and Fernandez (1989, 1994)
\pause
\begin{itemize}
\item{formalized the concept}
\pause
\item{added a vertex attribute component}
\pause
\item{compared empirical brokerage counts to counts from random graphs conditioned on the number of edges}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Brokerage}
\begin{tikzpicture}[scale=.7]
\draw [fill=blue, thick] (1,5) circle [radius=0.3];
\draw [fill=blue, thick] (1,3) circle [radius=0.3];
\draw [fill=blue, thick] (1,1) circle [radius=0.3];
\draw [->, ultra thick] (1,4.7)--(1,3.3);
\draw [->, ultra thick] (1,2.7)--(1,1.3);
\node at (1,0) {\tiny{Coordinator}};

\pause

\draw [fill=blue, thick] (4,5) circle [radius=0.3];
\draw [fill=blue, thick] (4,3) circle [radius=0.3];
\draw [fill=red, thick] (4,1) circle [radius=0.3];
\draw [->, ultra thick] (4,4.7)--(4,3.3);
\draw [->, ultra thick] (4,2.7)--(4,1.3);
\node at (4,0) {\tiny{Representative}};

\pause

\draw [fill=red, thick] (7,5) circle [radius=0.3];
\draw [fill=blue, thick] (7,3) circle [radius=0.3];
\draw [fill=blue, thick] (7,1) circle [radius=0.3];
\draw [->, ultra thick] (7,4.7)--(7,3.3);
\draw [->, ultra thick] (7,2.7)--(7,1.3);
\node at (7,0) {\tiny{Gatekeeper}};

\pause

\draw [fill=blue, thick] (10,5) circle [radius=0.3];
\draw [fill=red, thick] (10,3) circle [radius=0.3];
\draw [fill=blue, thick] (10,1) circle [radius=0.3];
\draw [->, ultra thick] (10,4.7)--(10,3.3);
\draw [->, ultra thick] (10,2.7)--(10,1.3);
\node at (10,0) {\tiny{Itinerant}};

\pause

\draw [fill=blue, thick] (13,5) circle [radius=0.3];
\draw [fill=red, thick] (13,3) circle [radius=0.3];
\draw [fill=green, thick] (13,1) circle [radius=0.3];
\draw [->, ultra thick] (13,4.7)--(13,3.3);
\draw [->, ultra thick] (13,2.7)--(13,1.3);
\node at (13,0) {\tiny{Liaison}};

\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gould and Fernandez' Findings}
\pause
\begin{itemize}
\item{the benefits of brokerage are mediated both by the type of organization (the node sets) and the type of brokerage chain}
\pause
\item{non-governmental organizations were found to have more influence when they held any type of brokerage position}
\pause
\item{governmental organizations gained influence only when they held ``outsider" brokerage roles in itinerant and liaison chains}
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Structural Balance}
\includegraphics[width=9.5cm,trim={0 0 0 0},clip]{../../images/balance.png}\\
\scriptsize{Cartwright and Harary 1956}
\end{frame}
%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Subgroups}
\begin{columns}
\column{5cm}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width=5cm,trim=1.5cm 1.8cm 2.5cm 0cm, clip=true]{../../images/componentsEG.png}};
\end{tikzpicture}
\pause
\column{6cm}
\begin{itemize}
\uncover<2->{\item Component: A maximal connected subgraph}
\uncover<3->{\item A subgraph is \textit{maximal} with respect to some property if it has the property, but loses it with the addition of more nodes or edges}
\end{itemize}
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cliques}
\begin{columns}
\column{5cm}
\begin{tikzpicture}
%\draw[help lines] (0,0) grid (5,6);
\node at (0,0) {\includegraphics[width=5cm,trim=1.5cm 1.8cm 2.5cm 0cm, clip=true]{../../images/componentsEG.png}};
\uncover<5>{
\draw [blue, fill=blue, ultra thick] (0.47,0.05) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (-0.27,0.77) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (0.468,1.475) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (1.23,0.76) circle [radius=0.1];
}
\uncover<6>{
\draw [blue, fill=blue, ultra thick] (0.67,-.58) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (0.67,-1.48) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (2.1,-.58) circle [radius=0.1];
}
\uncover<7>{
\draw [blue, fill=blue, ultra thick] (2.09,-1.48) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (0.67,-1.48) circle [radius=0.1];
\draw [blue, fill=blue, ultra thick] (2.1,-.58) circle [radius=0.1];
}
\end{tikzpicture}
\pause
\column{6cm}
\begin{itemize}
\uncover<2->{\item A maximally complete subgraph of 3 or more}
\uncover<3->{\item All nodes are adjacent to all others in the subgraph}
\uncover<4->{\item No nodes can be added that have that property}
\end{itemize}
\end{columns}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Limitations of Cliques}
\begin{itemize}
\item A very strict definition
\item Often networks contain many small overlapping cliques
\item Cliques have no internal structure
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Relaxations of Cliques}
Subgroups based on reachability or diameter
\begin{itemize}
\pause
\item $n$-clique: a maximal subgraph in which the \textbf{largest geodesic distance} between any two nodes is no greater than $n$
\pause
\item $n$-clan: an $n$-clique in which the geodesic distance between all nodes in the subgraph is no greater than $n$ \textbf{for paths within the subgraph}
\pause
\item $n$-club: a maximal subgraph of diameter $n$
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Relaxations of Cliques}
Subgroups based on nodal degree
\begin{itemize}
\pause
\item $k$-plex: a subgrah of $g$ nodes in whcich each node is adjacent to no fewer than $g-k$ ndoes in the subgraph
\pause
\item $k$-core: a subgraph in which each node is adjacent to at least $k$ other nodes in the subgroup
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{What to do with these subgroup definitions?}
\begin{itemize}
\pause
\item characterize the overall structure of the network
\pause
\item analyze the pattern of co-membership in cliques, etc.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{K-Core Example}
\begin{center}
\includegraphics<1>[width=8cm]{../../images/sandraCore4.png}
\includegraphics<2>[width=8cm]{../../images/sandraCore1.png}
\includegraphics<3>[width=8cm]{../../images/sandraCore2.png}
\includegraphics<4>[width=6cm]{../../images/sandraCore3.png}
\scriptsize{Barberá et al 2015}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\pause
\begin{itemize}
\item Attribute Based
\pause
\end{itemize}
\includegraphics[width=5cm]{../../images/boysNGirls.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\end{itemize}
\includegraphics[width=5cm]{../../images/boysNGirlsBlock.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\item Relation Based
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\item Relation Based
\end{itemize}
\includegraphics[width=5cm]{../../images/block1.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\item Relation Based
\end{itemize}
\includegraphics[width=5cm]{../../images/block2.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\item Relation Based
\end{itemize}
\includegraphics[width=5cm]{../../images/block3.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Block Models}
\begin{itemize}
\item Attribute Based
\item Relation Based
\end{itemize}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm]{../../images/block4.png}
\pause
\column{5cm}
\includegraphics[width=5cm]{../../images/block5.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Core-Periphery Models}
\includegraphics[width=5cm]{../../images/core1.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Core-Periphery Models}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm]{../../images/core2.png}
\pause
\column{5cm}
\includegraphics[width=5cm]{../../images/core3.png}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Even more blockmodeling}
\includegraphics[width=10cm]{../../images/batagelj1.png}
\small{Doreian, Batagelj, and Ferligoj \textit{Generalized Blockmodeling}, 2005}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Even more blockmodeling}
\begin{columns}
\column{5cm}
\includegraphics[width=5cm]{../../images/batagelj2.png}
\column{5cm}
\includegraphics[width=5cm]{../../images/batagelj3.png}
\end{columns}
\vspace{5pt}
\small{Doreian, Batagelj, and Ferligoj \textit{Generalized Blockmodeling}, 2005}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Structural Equivalence}
\pause
\includegraphics[width=9.5cm]{../../images/se1.png}
\scriptsize{Sailer 1978}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Structural Equivalence}
\includegraphics[width=9.5cm]{../../images/se2.png}
\scriptsize{Sailer 1978}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Automorphic Equivalence}
\includegraphics[width=9.5cm]{../../images/se_automorphic.png}
\scriptsize{Sailer 1978}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Regular Equivalence}
\includegraphics[width=9.5cm]{../../images/se_regular.png}
\scriptsize{Sailer 1978}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Structural Equivalence Example}
\includegraphics[width=10cm]{../../images/carter_se.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Regular Equivalence Example}
\includegraphics[width=10cm]{../../images/carter_re.png}
\scriptsize{Everett and Borgatti 1994}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Community Structure}
\begin{itemize}
\pause
\item Modularity index measures the `goodness' of an assignment of nodes to subgroups by comparing the number of ties within groups as compared to the number of expected ties
\pause
$$Q=\frac{1}{4m}\sum_{ij}(A_{ij}-\frac{k_ik_j}{2m})s_is_j$$
\pause
\item where $m$ is the number of nodes, $A$ is the adjacency matrix, $k$ is the node's degree, and $s$ is an indicator of which group the node is assigned 
\pause
\item find the assigments of $s$ that maximizes $Q$
\pause
\item M.E.J. Newman ``Modularity and community structure in networks" \textit{Proc Natl Acad Sci} 2006
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{References}
\scriptsize{
\begin{itemize}
\item Pitts, Forrest R. ``The medieval river trade network of Russia revisited." Social networks 1.3 (1978): 285-292.
\item Bonacich, Phillip. ``Power and centrality: A family of measures." American journal of sociology 92.5 (1987): 1170-1182.
\item Opsahl, Tore, Filip Agneessens, and John Skvoretz. ``Node centrality in weighted networks: Generalizing degree and shortest paths." Social networks 32.3 (2010): 245-251.
\item Everett, Martin G., and Stephen P. Borgatti. ``Extending centrality." Models and methods in social network analysis 35.1 (2005): 57-76.
\item Freeman, Linton C. ``Centrality in social networks conceptual clarification." Social networks 1.3 (1978): 215-239.
\item Holland, Paul W., and Samuel Leinhardt. ``The statistical analysis of local structure in social networks." (1974).
\item Faust, Katherine. ``Very local structure in social networks." Sociological methodology 37.1 (2007): 209-256.
\item Granovetter, Mark. ``The strength of weak ties: A network theory revisited." Sociological theory (1983): 201-233.
\item Box-Steffensmeier, Janet M., and Dino P. Christenson. ``The evolution and formation of amicus curiae networks." Social Networks 36 (2014): 82-96.
\end{itemize}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{References}
\scriptsize{
\begin{itemize}
\item Jasny, Lorien, Joseph Waggle, and Dana R. Fisher. ``An empirical examination of echo chambers in US climate policy networks." Nature Climate Change 5.8 (2015): 782-786.
\item Granovetter, Mark S. ``The strength of weak ties." American journal of sociology 78.6 (1973): 1360-1380.
\item Burt, Ron S. ``Structural Holes: The Social Structure of Competition." Harvard University Press (1992).
\item Marsden, P. Brokerage Behavior in Restricted Exchange Networks. In: Social Structure and Network
Analysis, P. V. Marsden and N. Lin, Eds., 201-18, Beverly Hills: Sage (1982).
\item Gould, Roger V., and Roberto M. Fernandez. ``Structures of mediation: A formal approach to brokerage in transaction networks." Sociological methodology (1989): 89-126.
\item Fernandez, Roberto M., and Roger V. Gould. ``A dilemma of state power: Brokerage and influence in the national health policy domain." American journal of Sociology 99.6 (1994): 1455-1491.
\item Cartwright, Dorwin, and Frank Harary. ``Structural balance: a generalization of Heider's theory." Psychological review 63.5 (1956): 277.

\end{itemize}}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{References}
\scriptsize{
\begin{itemize}
\item Barberá, Pablo, et al. ``The critical periphery in the growth of social protests." PloS one 10.11 (2015): e0143611.

\item Doreian, Patrick, Vladimir Batagelj, and Anuska Ferligoj. ``Generalized blockmodeling." No. 25. Cambridge university press, 2005.
\item Sailer, Lee Douglas. ``Structural equivalence: Meaning and definition, computation and application." Social networks 1.1 (1978): 73-90.
\item Everett, Martin G., and Stephen P. Borgatti. ``Regular equivalence: General theory." Journal of mathematical sociology 19.1 (1994): 29-52.
\item Newman, Mark EJ. ``Modularity and community structure in networks." Proceedings of the national academy of sciences 103.23 (2006): 8577-8582.
\end{itemize}}
\end{frame}
\end{document}