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\title{Mathematics of Networks}
\date{14.03.2016}

\begin{document}
\titlepage

\begin{frame}
\frametitle{Definition of a network}
Our definition (Newman): A network (graph) is a collection of vertices (nodes) joined by edges (links).\\
\vspace{0.5in}
More precise definition (Bollob\`as): A graph $G$ is an ordered pair of disjoint sets $(V,E)$ such that $E$ (the edges) is a subset of the set $V^{(2)}$ of unordered pairs of $V$ (the vertices). 
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Edge list representation}
  \begin{columns}
    \column[c]{.40\textwidth}
% \emph{Edge list} representation:
\begin{itemize}
\item Vertices:\\
1,2,3,4,5,6\\
\item Edges:\\
(1,2), (1,3), (1,6), (2,3), (3,4), (4,5), (4,6)
\end{itemize}
Definition from graph theory:
\begin{itemize}
\item $n=6$ vertices: \emph{order} of the graph\\
\item $m=7$ edges: \emph{size} of the graph
\end{itemize}
    \column[c]{.60\textwidth}
\includegraphics[scale=0.15]{graph1a.png}
  \end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Adjacency matrix $\mathbf{A}$}
\begin{equation*}
A_{ij} = \begin{cases} 1 &\mbox{if there is an edge between vertices i and j} \\
0 & \mbox{otherwise}. \end{cases} 
\end{equation*}
\begin{columns}
    \column[c]{.40\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 1 & 1 & 0 & 0 & 1\\
1 & 0 & 1 & 0 & 0 & 0\\
1 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 1 & 1\\
0 & 0 & 0 & 1 & 0 & 0\\
1 & 0 & 0 & 1 & 0 & 0
\end{matrix}\right)
\end{equation*}
\begin{itemize}
\item Diagonal elements are zero.
\item Symmetric matrix.
\end{itemize}
    \column[c]{.60\textwidth}
\includegraphics[scale=0.15]{graph1a.png}
  \end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Multigraph}
There can be more than one edge between a pair of vertices.
  \begin{columns}
    \column[c]{.40\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 1 & 1 & 0 & 0 & 3\\
1 & 0 & 2 & 0 & 0 & 0\\
1 & 2 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 1 & 1\\
0 & 0 & 0 & 1 & 0 & 0\\
3 & 0 & 0 & 1 & 0 & 0
\end{matrix}\right)
\end{equation*}
    \column[c]{.60\textwidth}
\includegraphics[scale=0.15]{graph2a.png}
  \end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Self-edges}
There can be self-edges (also called self-loops).
  \begin{columns}
    \column[c]{.40\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 1 & 1 & 0 & 0 & 3\\
1 & 0 & 1 & 0 & 0 & 0\\
1 & 1 & 2 & 1 & 0 & 0\\
0 & 0 & 1 & 2 & 1 & 1\\
0 & 0 & 0 & 1 & 0 & 0\\
3 & 0 & 0 & 1 & 0 & 0
\end{matrix}\right)
\end{equation*}
    \column[c]{.60\textwidth}
\includegraphics[scale=0.15]{graph3a.png}
  \end{columns}
  \begin{itemize}
\item Diagonal elements can be non-zero:\\
Definition: $A_{ii}=2$ for one self-edge.
\end{itemize}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Weighted networks}
\emph{Weight} or \emph{strength} assigned to each edge.
  \begin{columns}
    \column[c]{.55\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 1.4 & 0.4 & 0 & 0 & 0.8\\
1.4 & 0 & 1.2 & 0 & 0 & 0\\
0.4 & 1.2 & 0 & 0.2 & 0 & 0\\
0 & 0 & 0.2 & 0 & 0.2 & 0\\
0 & 0 & 0 & 0.2 & 0 & 0\\
0.8 & 0 & 0 & 0.4 & 0 & 0
\end{matrix}\right)
\end{equation*}
    \column[c]{.45\textwidth}
\includegraphics[scale=0.13]{graph4a.png}
\end{columns}
Weights can be both positive or negative.
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Directed Networks (Digraphs)}
Edge is pointing from one vertex to another (\emph{directed edge}).
\begin{equation*}
A_{ij} = \begin{cases} 1 &\mbox{if there is an edge from $j$ to $i$} \\
0 & \mbox{otherwise}. \end{cases} 
\end{equation*}
  \begin{columns}
    \column[c]{.55\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
1 & 0 & 0 & 1 & 0 & 0
\end{matrix}\right)
\end{equation*}
    \column[c]{.45\textwidth}
\includegraphics[scale=0.11]{graph6a.png}
\end{columns}
In general the adjacency matrix of a directed network is asymetric.
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Degree}
\begin{itemize}
\item Degree $k_{i}$ of vertex $i$: Number of edges connected to $i$.
\item Average degree of the network: $\langle k \rangle$.
\end{itemize}
\vspace{0.25cm}
In terms of the adjacency matrix $\mathbf{A}$:
\begin{equation*}
k_{i} = \sum_{j=1}^{n}A_{ij}\quad , \quad
\langle k\rangle = \frac{1}{n}\sum_{i}k_{i} = \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}A_{ij}~.
\end{equation*}
\begin{columns}
\column[c]{.55\textwidth}
\begin{flalign*}
k_{5}&   =  1\\
k_{2}& = k_{6} = 2\\
k_{1}& = k_{3} = k_{4} =3\\
\langle k \rangle& = 2.33
\end{flalign*}
    \column[c]{.45\textwidth}
\includegraphics[scale=0.11]{graph1a.png}
\end{columns}
\end{frame}






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\begin{frame}
\frametitle{Examples}

\includegraphics[scale=0.35]{netscibook.png}

(from the free textbook "Network Science")
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Degree}
With $n$ the number of vertices in the graph, and $m$ the number of edges, it holds:
\begin{equation*}
2m = \sum_{i=1}^{n}k_{i}=
\sum_{i=1}^{n}\sum_{j=1}^{n}A_{ij}~.
\end{equation*}

For the average degree $\langle k \rangle$ of the graph this yields
\begin{equation*}
\langle k \rangle = \frac{1}{n}\sum_{i=1}^{n}k_{i}
=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}A_{ij}
=\frac{2m}{n}~.
\end{equation*}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Density / connectance}
Maximum possible number of edges in a simple graph with $n$ vertices:
\begin{equation*}
\frac{1}{2}n(n-1)~.
\end{equation*}
\emph{Density} or \emph{connectance} of a graph: Fraction of maximum possible number of edges which are present in a given graph:
\begin{equation*}
\rho = \frac{m}{\frac{1}{2}n(n-1)}=\frac{2m}{n(n-1)}=\frac{\langle k\rangle}{n-1}~.
\end{equation*}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Degree distribution}
Number of vertices with degree $k$ in a graph:$n_{k}$
\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph9.png}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph8.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Degree distribution}
Fraction of vertices in a graph that have degree $k$:
\begin{equation*}
p_{k}=\frac{n_{k}}{n}~.
\end{equation*}

\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph10.png}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph8.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Degree distribution}
\emph{Hubs}: well-connected vertices
\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph12.png}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph11.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Average degree from the degree distribution}
Degree distribution tells important information about a network, but doesn't contain the complete information.\\
\vspace{0.5cm}
The average degree of a graph can be easily calculated from the degree distribution:
\begin{equation*}
\langle k \rangle =
\frac{1}{n}\sum_{i=1}^{n}k_{i}=
\frac{1}{n}\sum_{k=0}^{k_{max}}n_{k}k=
\sum_{k=0}^{k_{max}}kp_{k}~.
\end{equation*}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Directed networks: in-degree, out-degree}
Number of vertices with $k$ ingoing / outgoing edges.
\vspace{0.5cm}
\begin{columns}
\column[c]{.6\textwidth}
\begin{equation*}
k_{i}^{in}=\sum_{j=1}^{n}A_{ij}\quad , \quad  k_{i}^{out}=\sum_{j=1}^{n}A_{ji}
\end{equation*}
\includegraphics[scale=0.085]{graph14.png}
\includegraphics[scale=0.085]{graph15.png}
    \column[c]{.4\textwidth}
\includegraphics[scale=0.12]{graph13.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks}
Often a system can be represented as a network consisting of two kinds of vertices, with edges only between vertices of different types (group membership).
\vspace{0.5cm}
Examples:
\begin{itemize}
\item Film actors: Actors, group: Cast of a film
\item Coauthorship: Authors, group: Authors of an article
\item Rail connections: Train stations, group: Route
\item Brazilian soccer players: Players, group: Clubs
\item Blinkist: Users, group: Readers of a book
\end{itemize}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks: Adjacency matrix}
\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
\begin{equation*}
\mathbf{A}=\left(\begin{matrix}
0 & 0 & 0 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1\\
1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0
\end{matrix}\right)
\end{equation*}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.15]{graph16.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks: Incidence matrix $\mathbf{B}$}
Vertices of type $1$:$i=1,2,\ldots,n_{1}$ (often groups)\\
Vertices of type $2$:$j=1,2,\ldots,n_{2}$ (often people)\\
\begin{equation*}
B_{ij} = \begin{cases} 1 &\mbox{if there is an edge between vertices i and j} \\
0 & \mbox{otherwise}. \end{cases} 
\end{equation*}
\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
\begin{equation*}
\mathbf{B}=\left(\begin{matrix}
1 & 1 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 1 & 1 & 1
\end{matrix}\right)
\end{equation*}
\begin{itemize}
\item $n_{1}\times n_{2}$ matrix
\item In general asymetric
\end{itemize}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.12]{graph17.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks: One-mode projections}
Projection to a (weighted) network only with vertices of the second type:
\begin{equation*}
P_{ij} = \sum_{k=1}^{n_{1}}B_{ki}B_{kj}~.
\end{equation*}
That is $\mathbf{P}=\mathbf{B}^{T}\mathbf{B}$. Note:
\begin{equation*}
P_{ii}=\sum_{k=1}^{n_{1}}B_{ki}B_{ki}=\sum_{k=1}^{n_{1}}B_{ki}~.
\end{equation*}
 Adjacency matrix ($n_{2}\times n_{2}$):
\begin{equation*}
A_{ij} = \begin{cases} P_{ij} &\mbox{if $i\neq j$} \\
0 & \mbox{if $i=j$}. \end{cases} 
\end{equation*}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks: One-mode projections}
Two one-mode projections based on
\begin{equation*}
\mathbf{P} = \mathbf{B}^{T}\mathbf{B}\quad , \quad
\mathbf{P}' = \mathbf{B}\mathbf{B}^{T}~.
\end{equation*}
\begin{columns}
\column[c]{.45\textwidth}
\includegraphics[scale=0.09]{graph17.png}
    \column[c]{.55\textwidth}
\includegraphics[scale=0.08]{graph24.png}
\includegraphics[scale=0.08]{graph25.png}\\
Note: Union of "cliques".
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Bipartite networks - Example:Blinkist}
One-mode projections to the networks of books, with an edge between two vertices if there more than $1000$ / $1500$ users have read both books.\\
\vspace{0.5cm}
\begin{center}
\includegraphics[scale=0.28]{graph_1000.pdf}
\includegraphics[scale=0.28]{graph_1500.pdf}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Accessing edge labels with the incidence matrix}
For a given network one can consider the edges as one type of vertices of a corresponding bipartite network, with the orginal vertices representing the second type.\\
\vspace{0.5cm}
Useful for directed networks, where heads and tails of directed edges are represented in the incidence matrix by $-1$ and $1$, respectively.
\begin{columns}
\column[c]{.5\textwidth}
\begin{equation*}
\mathbf{B}=\left(\begin{matrix}
1 & -1 & 0 & 0\\
0 & 1 & -1 & 0\\
0 & 1 & 0 & -1\\
0 & 0 & 1 & -1
\end{matrix}\right)
\end{equation*}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.12]{graph19.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Remark: Bipartite networks and cluster synchronization}
Sometimes it is interesting to look for a (mostly) bipartite "colouring" of a network.\\
\vspace{0.5cm}
Example: Cluster synchronization of coupled map networks:
\begin{equation*}
x_{i}(t+1) = (1-\epsilon)f[x_{i}(t)]+\frac{\epsilon}{n}\sum_{j}f[x_{j}(t)]~,
\end{equation*}
with $f$ a chaotic map, for instance
\begin{equation*}
f(x) = 2x^2-1~.
\end{equation*}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Remark: Bipartite networks and cluster synchronization}
\includegraphics[scale=0.20]{graph20.png}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Trees}
A \emph{tree} is a connected, undirected network that contains\\ no closed loops.\\
\vspace{0.1cm}
\begin{center}
\includegraphics[scale=0.15]{graph21.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Trees}
\begin{columns}
\column[c]{.5\textwidth}
\begin{itemize}
\item A collection of trees is called a \emph{forest}.
\item Trees play an import role for random graph models.
\item In a tree, there is exactly one path between any pair of vertices.
\item A tree of $n$ vertices always has exactly $n-1$ edges.
\item Any connected network with~$n$ vertices and $n-1$ edges is a tree.
\end{itemize}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.12]{graph22.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Planar networks}
A \emph{planar network} is a network that can be drawn on a plane without having any edges cross.\\
\vspace{0.5cm}
\begin{columns}
\column[c]{.5\textwidth}
Examples:
\begin{itemize}
\item Trees
\item Road networks (approximately)
\item Power grids (approximately)
\item Shared borders between countries, etc.
\end{itemize}
    \column[c]{.5\textwidth}
\includegraphics[scale=0.12]{graph23.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Planar networks - Four-color theorem}
"In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color." [Wikipedia]
\begin{center}
\includegraphics[scale=0.35]{4colors.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Power Grid expansion optimisation}
Expansion condition: planar graph
\begin{center}
\includegraphics[scale=0.4]{grid_germany.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Paths}
\begin{itemize}
\item Route through the network, from vertex to vertex along the edges
\item Defined for both directed and undirected networks
\item Special case: self-avoiding paths
\item \emph{Length} of a path: number of edges along the path ("hops")
\item Number of paths of length $r$ between vertices $i$ and $j$:
\begin{equation*}
N_{ij}^{(r)}=\left[\mathbf{A}^{r}\right]_{ij}
\end{equation*}
\item Total number $L_{r}$ of loops of length $r$ anywhere in the network:
\begin{equation*}
L_{r}=\sum_{i=1}^{n}\left[\mathbf{A}^{r}\right]_{ii}=\text{Tr}\mathbf{A}^{r}~.
\end{equation*}
\end{itemize}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Geodesic / shortest paths}
\begin{columns}
\column[c]{.6\textwidth}
\begin{itemize}
\item A path between two vertices such that no shorter path exists
\item Geodesic distance between vertices $i$ and $j$ is the smallest value of $r$ such that 
$\left[\mathbf{A}^{r}\right]_{ij}>0$.
\item Self-avoiding
\item In general not unique
\item \emph{Diameter} of a network: Length of the longest geodesic path between any pair of vertices
\end{itemize}
\column[c]{.4\textwidth}
\includegraphics[scale=0.12]{graph26.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Shortest paths -- some examples}
Oracle of Bacon: https://oracleofbacon.org/\\
\begin{itemize}
\item Network of movie actors (joint appearance in a movie, based on IMDB)
\item Geodesic distance to Kevin Bacon
\end{itemize}
\begin{center}
\includegraphics[scale=0.2]{bacon.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Shortest paths -- some examples}
Erd\"os number: Consult http://wwwp.oakland.edu/enp/
\begin{itemize}
\item Coauthorship network
\item Geodesic distance to Paul Erd\"os
\end{itemize}
\begin{center}
\includegraphics[scale=0.2]{erdos.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Shortest paths -- "Six degrees of separation"}
\begin{itemize}
\item Classic experiment by Stanley Milgram (also known for "obedience to authority")
\item Average path lengths in social networks
\end{itemize}
\begin{center}
\includegraphics[scale=0.35]{milgram.png}
\end{center}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Shortest paths and breadth-first search}
\begin{itemize}
\item Single run of the algorithm: Finds shortest (geodesic) distance from a source vertex $s$ to \emph{every other} vertex in the same component of the network
\item In a second step the algorithm also finds shortest paths by construction the so-called \emph{shortest path tree}
\end{itemize}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Acyclic directed network}
\begin{columns}
\column[c]{.6\textwidth}
\begin{itemize}
\item Directed network without closed loops of edges (DAG)
\item Examples: power flow in an electricity grid, citation network of papers
\item Topological ordering: For every directed edge $i\to j$, vertex $i$ comes before $j$ in the ordering:\\(1,2,3,4,6,9,10,11,12,8,7,5,13)
\item With a topological ordering, the adjacency matrix of an acyclic directed network is \emph{strictly triangular}
\end{itemize}
\column[c]{.4\textwidth}
\includegraphics[scale=0.12]{graph13.png}
\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Components of networks}
\begin{columns}
\column[c]{.6\textwidth}
\begin{itemize}
\item Subgroups of vertices with no connections between the respective groups
\item \emph{Disconnected} network
\item Subgroups: \emph{components}
\item Adjacency matrix: Block-diagonal form
\end{itemize}
\column[c]{.4\textwidth}
\includegraphics[scale=0.12]{graph22.png}\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Components in directed networks}
\begin{columns}
\column[c]{.5\textwidth}
\begin{itemize}
\item \emph{Weakly connected components}: connected in the sense of an undirected network
\item \emph{Strongly connected components}: directed path in both directions between every pair in the subset
\end{itemize}
\column[c]{.5\textwidth}
\includegraphics[scale=0.14]{graph27.png}\end{columns}
\end{frame}
%-----%-----%-----%-----%-----%-----%-----%-----%
\begin{frame}
\frametitle{Components in directed networks}
\begin{columns}
\column[c]{.55\textwidth}
\begin{itemize}
\item \emph{Out-component} of a vertex $i$:\\ set of vertices which are reachable via directed paths starting form $i$, including the vertex $i$ itself
\item \emph{In-component} of a vertex $i$:\\ set of vertices from which there is a directed path to $i$, including the vertex $i$ itself
\item One often considers the out- or in-component of a strongly connected component
\end{itemize}
\column[c]{.45\textwidth}
\includegraphics[scale=0.13]{graph13.png}\end{columns}
\end{frame}
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\begin{frame}
\frametitle{Network of Global Corporate Control}
Ownership network of transnational corporations (TNCs)\\
Vitali et al., PLOS One, 6 (2011)\\
\vspace{0.5cm}
\begin{itemize}
\item Ownership matrix $\mathbf{W}$:\\
$W_{ij}$ is the percentage of ownership that the owner (shareholder) $i$ holds in firm $j$
\item If $W_{ij}>0$ and $W_{jl}>0$, then vertex $i$ has an \emph{indirect} ownership of firm $l$
\item Data: Orbis 2007 database
\item Resulting network: $600508$ vertices (economics actors), containing $43060$ TNCs, $1006987$ edges (ownership ties)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Network of Global Corporate Control}
Vitali et al., PLOS One, 6 (2011)\\
\begin{center}
\includegraphics[scale=0.5]{corporate_control.png}
\end{center}
\end{frame}
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\end{document}
