# Topology of the World Tourism Web
## 2 Materials and Methods
### Analysis  
The tourism network is directed and defined as \(T=(V,A)\), with a set of vertices (nodes) \(V\) and a set of arches (links) \(a=\langle\stackrel{{\rightarrow}}{{u}},\stackrel{{ \rightarrow}}{{v}}\rangle\), which connect the source and target vertices. For the tourism network, the in-degree and out-degree differ for each vertex:  
\[n_{in}(v)\stackrel{{\mathrm{def}}}{{=}}\left\{w\;\in V(T)\Big{|}v \
eq w,\exists a=\langle\stackrel{{\rightarrow}}{{w}}, \stackrel{{\rightarrow}}{{v}}\rangle:\;a\;\in A(T)\right\} \tag{1}\]  
\[n_{out}(v)\stackrel{{\mathrm{def}}}{{=}}\left\{w\;\in V(T)\Big{|}v \
eq w,\exists a=\langle\stackrel{{\rightarrow}}{{v}}, \stackrel{{\rightarrow}}{{w}}\rangle:\;a\;\in A(T)\right\} \tag{2}\]  
and  
\[n(v)\stackrel{{\mathrm{def}}}{{=}}n_{inv}(v)\;\cup\;n_{out}(v), \tag{3}\]  
where \(n(v)\) stands for the set of neighboring vertices to the vertex \(v\)[47]. A weighted tourism network treats the number of arrivals as weights of the arcs, \(w(a)\), where the weighted in-degree and the weighted out-degree differ. The calculations are performed with and without using arrivals as weights. The statistics calculated for the network and the nodes are as follows:  
* Average in-degree is the average number of incoming links (the source countries from which tourists arrive to the target countries). In terms of the tourism network, the in-degree can be an indicator of how likely is that other countries will be drawn to the observed country (presumably) due to its touristic attractiveness. So, the average in-degree points out to the average number of countries drawn to one country in a tourism network. * Average out-degree is the average number of target countries to which tourists travel from the source countries. Given the data used, the average out-degree must match the average out-degree, but individual values of countries' in- and out-degrees do not have to be the same. Hence, standard deviations may be different, revealing different underlying distributions. * Average degree is the average number of countries to which a country is connected in the network. * Average weighted in-degree or inward link strength is the average number of tourist arrivals from source countries. It is the average of individual countries' weighted in-degree (the number of tourists drawn to a country). * Average weighted out-degree or outward link strength is the average number of tourist departures to target countries. * Average weighted degree is the average number of tourist arrivals and departures per country. The weighted degree illustrates a country's share in the tourism market, measured by the number of tourists. * Diameter [48] is a maximal distance in the network and shows how many connections it takes to traverse the network. It is a longest shortest path length. * Radius [48] is the minimum over distances of each country-node to the farthest country-node in the network (or a minimum over all eccentricity values), indicating how (dis)connected countries are. * Average path length [48] measures the average of the possible geodesic (shortest path) between all countries in the network and is an indication of network efficiency. * Density [41] is a share of links in the network in all possible connections in the network. Generally, values closer to 1 mean that it is a dense graph, while values closer to 0 reveal a sparse graph. * Average clustering coefficient [49; 50] is an average of individual clustering coefficients. Individual clustering coefficients measure the degree of connections between the countries to which a certain country is connected. If a clustering coefficient of a country is closer to 1, it means that almost all its connections are connected between themselves. This also indicates that the country is less important in the network. * Number of triangles [51] is a summation of open or closed triples in each country's network. * Number of paths (length 2) [51] is a summation of all paths of length 2 for each node.