Informetric methods seminar

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

2. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

3. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

4. 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.

5. Example *vertices 15 10 1 "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 1 11 12 15 2 12 14 15 3 14 4 11 15 5 12 13 6 13 7 11 15 8 11 12 14 9 11 12 13 14 15 10 11 12 15 • 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 See two_mode.net

6. 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-Modeto1-Mode>Columns”. • Network with or without loops can be generated: • “Net>Transform>2-Modeto1-Mode>IncludeLoops”. • 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-Modeto1-Mode>MultipleLine”.

7. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

8. 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.

9. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

10. Global and local views on network

11. 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.

12. Example • Import and export in 1994 among 80 countries are given. They is given in 1000$. (See Country_Imports.net) • Partition according to continents (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

13. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

14. 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.

15. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

16. 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”.

17. 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.

18. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

19. 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

20. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

21. 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

22. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

23. 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 subnetworkcan be extracted using “Operations>Extract from Network>Partition” and typing the lower and upper limit for the core number. • Example • See k_core.net

24. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

25. 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

26. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

27. 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”

28. Content • Two mode network • Basic information about a network • Global and local views on network • Degrees • Centrality and Centralization • Component • Biconnectedcomponents • Cores • Slice • Diameter • Clustering Coefficients

29. Clustering Coefficients • Local Clustering Coefficients • Let deg(v) denotes degree of vertex v, |E(G1(v))| number of lines among vertices in neighborhood of vertex v, MaxDegmaximum degree of vertex in a network. • cc1(v)=cc1’(v)= • Global Clustering Coefficients • cc= • Calculation of local Clustering Coefficients: • “Net>Vector>Clustering Coefficients>CC1”