Informetric methods seminar Tutorial 2: Using Pajek for network properties Qi Yu

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Two-mode network  One-mode network  each vertex can be related to each other vertex.  Two-mode network  vertices are divided into two sets and vertices can only be related to vertices in the other set.

Example  Suppose we have data as below:  P1: Au1, Au2, Au5  P2: Au2, Au4, Au5  P3: Au4  P4: Au1, Au5  P5: Au2, Au3  P6: Au3  P7: Au1, Au5  P8: Au1, Au2, Au4  P9: Au1, Au2, Au3, Au4, Au5  P10: Au1, Au2, Au5 *vertices "P1" 2 "P2" 3 "P3" 4 "P4" 5 "P5" 6 "P6" 7 "P7" 8 "P8" 9 "P9" 10 "P10" 11 "Au1" 12 "Au2" 13 "Au3" 14 "Au5" 15 "Au5" *edgeslist See two_mode.net

Transforming to valued networks  The network is transformed into an ordinary network, where the vertices are elements from the first subset, using  “Net>Transform>2-Mode to 1-Mode>Rows”.  If we want to get a network with elements from the second subset we use  “Net>Transform>2-Mode to 1-Mode>Columns”.  Network with or without loops can be generated:  “Net>Transform>2-Mode to 1-Mode>Include Loops”.  We store values of loops into vector using “ Net>Vector>Get Loops” and use later this vector to determine  We can generate network with multiple lines – for each common event a line between corresponding two women:  “Net>Transform>2-Mode to 1-Mode>Multiple Line”.

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Basic information about a network  Basic information can be obtained by “Info>Network>General” which is available in the main window of the program. We get  number of vertices  number of arcs, number of directed loops  number of edges, number of undirected loops  density of lines  Additionally we must answer the question:  Input 1 or 2 numbers: +/highest, -/lowest where we enter the number of lines with the highest/lowest value or interval of values that we want to output.  If we enter 10, 10 lines with the highest value will be displayed. If we enter -10, 10 lines with the lowest value will be displayed. If we enter 3 10, lines with the highest values from rank 3 to 10 will be displayed.

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Global and local views on network

 Local view is obtained by extracting sub-network induced by selected cluster of vertices.  Global view is obtained by shrinking vertices in the same cluster to new (compound) vertex. In this way relations among clusters of vertices are shown.  Combination of local and global view is contextual view: Relations among clusters of vertices and selected vertices are shown.

Example  Import and export in 1994 among 80 countries are given. They is given in 1000$. (See Country_Imports.net)See Country_Imports.net  Partition according to continents (see Country_Continent.clu)see Country_Continent.clu  1 – Africa, 2 – Asia, 3 – Europe, 4 – N. America, 5 – Oceania, 6 – S. America.  Operations>Extract from Network>Partition  Operations>Shrink Network>Partition

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Degrees  In Pajek degrees are computed using “Net>Partitions>Degree” and selecting Input, Output or All.  Vertices with the highest degree can be displayed using Info>Partition.  For smaller networks the result can be displayed by:  double clicking the partition window, or  selecting “File>Partition>Edit” or  selecting the corresponding icon.  Normalized degree can be found in the vector window.

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Centrality and Centralization  Degree centrality is available in “Net>Partitions>Degree”;  Closeness centrality can be found in “Net>Vector>Centrality>Closeness”.  Betweenness centrality can be found in “Net>Vector>Centrality>Betweenness”.  When computing centrality according to degree and closeness we must additionally select Input, Output or All.  In the report window the network centralization index is given.  List of the selected number of most central vertices can be obtained by “Info>vector”.

Centralization  Centralization expresses the extent to which a network has a center.  A network is more centralized if the vertices vary more with respect to their centrality. More variation in the centrality scores of vertices yields a more centralized the network.  Take degree centralization for example:  Degree centralization of a network is the variation in the degrees of vertices divided by the maximum degree variation which is possible in a network of the same size. (Variation is the summed (absolute) differences between the centrality scores of the vertices and the maximum centrality score among them) Star- and line-networks.

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Component  Strongly connected components are computed using “Net>Components>Strong”  Weakly connected components using “Net>Components>Weak”.  Result is represented by a partition  vertices that belong to the same component have the same number in the partition.  Example  See component.net

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Biconnected components  A cut-vertex is a vertex whose deletion increases the number of components in the network.  A bi-component is a component of minimum size 3 that does not contain a cut-vertex.  To compute bicomponents use: “Net>Components>Bi-components”  Biconnected components are stored to hierarchy.  Example  See bicomponent.net

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Cores  A subset of vertices is called a k-core if every vertex from the subset is connected to at least k vertices from the same subset.  Cores can be computed using “Net>Partitions>Core” and selecting Input, Output or All core.  Result is a partition: for every vertex its core number is given.  In most cases we are interested in the highest core(s) only. The corresponding subnetwork can be extracted using “Operations>Extract from Network>Partition” and typing the lower and upper limit for the core number.  Example  See k_core.net

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Slice  An m-slice is a maximal subnetwork containing the lines with a multiplicity equal to or greater than m and the vertices incident with these lines.  M-slice can be calculated in Pejek by following:  1. select “Use max instead of sum” in option “Net>Partitions>Valued Core”;  2. select “Net>Partitions>Valued Core>First Threshold and Step”  Result is a partition with class numbers corresponding to the highest m-slice each vertex belongs to.  Example  See metformin_nonloop.net

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Diameter  Diameter means the length of the longest shortest path in network.  Calcultation: “net>Paths between 2 vertices>diameter”  Full search is performed, so the operation may be slow for very large networks (number of vertices larger than 2000)  Result is – the length of the longest shortest path in network and corresponding two vertices  Then we can find all the longest shortest path between above two vertices from “net>Paths between 2 vertices> All Shortest”

Content  Two mode network  Basic information about a network  Global and local views on network  Degrees  Centrality and Centralization  Component  Biconnected components  Cores  Slice  Diameter  Clustering Coefficients

Clustering Coefficients