Encyclopedia of Social Network Analysis and Mining

Living Edition
| Editors: Reda Alhajj, Jon Rokne

Pajek and PajekXXL

  • Vladimir Batagelj
  • Andrej MrvarEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7163-9_310-1

Synonyms

Glossary

BOM (byte order mark)

A Unicode character used to signal the byte order of a text file or stream

CPM

Critical path method

GEDCOM (GEnealogical Data COMmunications)

A genealogical software interchange format

GUI

Graphical user interface

MDS

Multidimensional scaling

OR

Operations research

SN5

Network data set on social networks prepared for Viszards session at Sunbelt XXVIII, 2008

SNA

Social network analysis

STRAN

STRuctural ANalysis

SVG (Scalable Vector Graphics)

A WWW picture format

Unicode

The name of the international character set

VOS

Visualization of similarities

VR

Virtual reality

Tool’s ID Card

  • Tool name, title: Pajek and PajekXXL, program for analysis and visualization of large networks

  • Creation year: November 1996

  • Authors: Vladimir Batagelj and Andrej Mrvar

  • Range: general network problems with emphasis on large networks

  • Copyright: free for noncommercial use

  • Type: program

  • Scalability: Pajek, one billion vertices; PajekXXL, two billion vertices

  • Platforms: Windows, using emulators runs also on Linux and Mac

  • Programming language: Delphi pascal

  • Orientation: social network analysis, bibliometrics, analysis of genealogies, OR, bioinformatics

Introduction

In the first half of the 1990s, the multimedia computers connected through the Internet provided a new framework for data analysis. They enabled interactive visualization of data and large data sets started to move in the focus of research in data analysis. Pajek and PajekXXL are programs for analysis and visualization of large networks developed to provide tools for dealing with large networks.

Key Points

Most real-life large networks are sparse – the number of vertices (nodes) and lines (links) are of the same order. This property is also known as a Dunbar number (Hill and Dunbar 2002). The basic idea is that if each vertex has to spend for each link certain amount of “energy” to maintain the links to selected other vertices then, since it has a limited “energy” at its disposal, the number of links should be limited. In human networks, the Dunbar number is between 100 and 150.

From algorithms’ complexity theory (Cormen et al. 1990; Batagelj 2009a), it follows that only algorithms of subquadratic complexity are fast enough to be used for analysis of large networks with hundreds of thousands or millions of vertices. It turns out that for some problems for which the general algorithms are too slow, there exist the corresponding algorithms for sparse networks that are subquadratic. Development and implementation of subquadratic algorithms was one of the main goals in the development of Pajek.

From the Roman times, the standard approach to deal with something large is the “divide and conquer.” In network analysis, this means to support abstraction by (recursive) decomposition of a large network into several smaller subnetworks that can be treated further using more sophisticated methods and control over their interlinks. PajekXXL provides tools for identifying important parts of large networks and extract them for further analysis with Pajek or using other network analysis tools. Pajek also provides the user with some powerful visualization tools for automatic network drawing and for manual improvements of the obtained pictures.

Historical Background and Development

Pajek is a Slovenian word for spider. In Slovenian mathematics the graph theory is one of its strongest fields (Pisanski, Mohar, Klavžar, Marušič, Batagelj, and others). In the 1980s, VB (Vladimir Batagelj) had a series of research projects on graph theory algorithms (Batagelj 1987). A library graph for data structure graph was implemented in pascal. In 1990/1991, Anuška Ferligoj and VB were visiting Patrick Doreian at the University of Pittsburgh. They started to develop an optimization approach to blockmodeling (Batagelj et al. 1992a, b; Doreian et al. 2000, 2004) that was later generalized to types of links between clusters (Batagelj 1997) to two-mode networks (Doreian et al. 2004) and by Patrick Doreian and Andrej Mrvar also to signed networks (Doreian and Mrvar 1996, 2009, 2014, 2015, 2016; Mrvar and Doreian 2009; Brusco et al. 2011; Doreian et al. 2013). In Pittsburgh VB developed in pascal a package STRAN for blockmodeling, a very fast algorithm for computing Hummon-Doreian weights in acyclic networks (Hummon and Doreian 1989; Batagelj 2003) and a semiring-based algorithm for computing betweenness centralities (Batagelj 1994).

Andrej Mrvar (AM) was a computer science student at the University of Ljubljana. VB noticed him while he was presenting his home project on Saaty’s AH procedure. He became AM’s supervisor for diploma, master, and PhD theses. For his master thesis, AM implemented in pascal different graph visualization algorithms. Afterward he started to work on analysis of large networks, which was an emergent topic at that time, for his PhD.

To support AM’s work on his PhD, AM and VB in November 1996 decided to collect all their already developed graph analysis programs and to combine them into a single program – Pajek. They also decided to make it free for noncommercial use. The first version of Pajek was presented on January 29, 1997, on the Wednesday seminar at the Faculty of Mathematics and Physics, University of Ljubljana, and on XVII Sunbelt Conference in San Diego, USA, February 13–16, 1997 (Batagelj and Mrvar 1998). Among the first users of Pajek were Wouter de Nooy and Doug White. Since Pajek had no user-friendly documentation, VB and AM decided with Wouter to write a book Exploratory Social Network Analysis with Pajek. It was published in 2005 and the revised and expanded edition in 2011 by CUP (De Nooy et al. 2011). The book was also published in Japanese in 2009 and in Chinese in 2012. In collaboration with Doug White and later with Klaus Hamberger, different tools for analysis of genealogical data were added to Pajek (White et al. 1999; Batagelj and Mrvar 2008). For example, Pajek can read genealogies from GED files and supports (bipartite) p-graph representation of genealogies.

In the 1990s, VB and AM contributed several solutions to yearly graph drawing contests (see Fig. 1). They won many prizes but also included in Pajek some new tools to address the contests’ problems. In Batagelj et al. (1999), they argued that for dense networks, the matrix representation is more readable than the standard graphical representation. They also noticed that Seidman’s (1983) notion of cores is the only known formalization of dense parts of the network that can be efficiently determined. The chemists from MATH/CHEM/COMP conference were interested in identification of selected substructures in large organic molecules. For this purpose, the fragments (later reinvented and called motifs by physicists) searching procedure was added to Pajek in 1998.
Fig. 1

Snapshot of VR scene made with Pajek for graph drawing contest (1997)

VB was a PhD supervisor also to Matjaž Zaveršnik who studied methods for network decomposition. They developed algorithms for generalized cores (Batagelj and Zaveršnik 2011), islands (Zaveršnik and Batagelj 2004), and short cycle connectivity (Batagelj and Zaveršnik 2007). These methods were also included in Pajek.

Multiplication of networks is a very useful operation in network analysis, but it is dangerous when applied on large networks. The product of two large sparse networks need not to be sparse itself – it can “explode.” In many interesting cases (e.g., genealogical and scientometric networks (Batagelj and Mrvar 2008; Batagelj and Cerinšek 2013)), the sparseness of the product can be guaranteed. A fast network multiplication procedure for sparse networks was added to Pajek in 2005.

In summer of 2005, while VB was visiting NICTA in Sydney, Australia, he was trying to analyze the two-mode IMDB network that was one of the tasks for that year’s graph drawing contest. The two-mode hubs and authorities procedure did not give interesting results. There were no other appropriate tools in Pajek. So, VB asked AM to include in Pajek the counting of four rings and two-mode cores. The submitted solution (Ahmed et al. 2007) that was prepared in cooperation with NICTA’s group won the first prize.

In 2007 a fast hierarchical method for clustering with relational constraints (Ferligoj and Batagelj 1983) was developed based on the idea to consider only the values on the existing lines.

Around 2005 the 64-bit PC computers started to appear. Since they can have memories over 4G, they are very important for further development of Pajek. The problem was that Pajek had to wait for 64-bit Delphi pascal compiler, and for several years, the answer was “mañana.” Finally, in September 2011, the compiler became available.

Program Pajek was significantly reconstructed in the period 2011–2012: version Pajek 3 (32 and 64 bit) was shipped. Pajek 3 has a renewed and optimized menu structure. New community detection methods and layout algorithms for large networks were included. Additionally to “standard” Pajek, also version of PajekXXL with a compact data structure was finished in 2012. It is supposed to be used for huge networks (in which labeling of vertices is not important) that cannot be loaded into standard Pajek. PajekXXL can analyze networks with up to two billion vertices on everyday computers. Using PajekXXL we can identify and extract some smaller, interesting parts of a huge network that can be further analyzed (and visualized) with more sophisticated methods available in standard Pajek.

There exist several algorithms for finding communities but not all of them are suitable for analysis of very large networks. We decided to include in Pajek two community detection methods: Louvain method and VOS clustering. Louvain method is one of the most often used (and also among the fastest) methods. In its basic form, it was defined by Blondel et al. (2008) as a multilevel coarsening community detection method. In Pajek some improvements and extensions of this method suggested by Rotta and Noack (2011) were implemented. The second community detection method included in Pajek is VOS clustering (Waltman et al. 2010) that can be easily modified to be used also for visualization of networks – VOS mapping technique is included in Pajek as well. It is especially well suited for visualization of denser networks.

In the period 2014–2016, Pajek and PajekXXL were further optimized to be better compatible with the latest compilers and libraries. Pajek and PajekXXL version 4 was introduced. Some new options for analyzing acyclic networks were implemented, e.g., the main path analysis was generalized as suggested by Liu and Lu (2012) (key-Route searches). Several improvements were done on the visualization part. Some of them include:
  • Possibility to put the visualization window into a Fisheye mode was added (hovering the mouse pointer over the visualization window magnifies area around the mouse pointer).

  • Showing labels of vertices on demand only – as tooltips/hints – was included (label appears only when mouse pointer touches the vertex).

  • Assigning Unicode symbols to partition clusters and marking vertices with these symbols was added (see Pajek: using symbols in additional reading).

  • Three additional shapes were added in EPS/SVG export (“house,” “man,” and “woman”). Some visualizations with additional vertex shapes are available in Mrvar and Batagelj (2016). See also Pajek: shapes of vertices in additional reading.

  • Drawing vertices and lines transparently in SVG was enabled in Pajek version 4 as well.

In 2016 first steps in development of Pajek3XL were done. It is planned that Pajek3XL will be able to analyze networks where the number of vertices is larger than 232 (in the first version, the limit is set to 10 billion).

For additional details on Pajek’s background, see Pajek history (2016) wiki page, and on Pajek’s evolution, see the history file at Pajek’s site.

Pajek

In Pajek analysis and visualization are performed using six data types:
  • Network (graph)

  • Partition (nominal or ordinal properties of vertices)

  • Vector (numerical properties of vertices)

  • Cluster (subset of vertices)

  • Permutation (reordering of vertices, ordinal properties)

  • Hierarchy (general tree structure on vertices)

The power of Pajek is based on several transformations that support different transitions among these data structures. Also the menu structure of the main Pajek’s window is based on them. Pajek’s main window uses a “calculator” paradigm with a list accumulator for each data type. The operations are performed on the currently active (selected) data and are also returning the results through accumulators. The procedures are available through the main window menus. The menu options are organized with respect to the type(s) of input data. Scalars are treated as vectors of length 1.

Formats

Pajek has its own input data format (see the left side of Fig. 2). To describe a network, we first list the set of vertices followed by a list of directed lines or arcs and/or list of undirected lines or edges. The list of vertices starts with a keyword *vertices n where n is the total number of vertices in a network. Each vertex is described in its own line by its number (in the range 1 … n) followed by its label. Vertex descriptions can be omitted. The list of arcs starts with a keyword *arcs. Each arc is described with a triple u υ w saying that there is an arc with the initial vertex u, the terminal vertex υ, and the weight w. The weight w can be omitted – w gets the default value w = 1. Similarly in the list of edges, the triple u υ w tells that there is an edge linking vertices u and υ with the weight w. In the same network, we can mix arcs and edges and there can be parallel lines linking the same pair of vertices. Pajek’s format supports (BOM marked) Unicode. An input line starting with % is a comment.
Fig. 2

Pajek input formats and picture of network

Pajek supports also a compact description of line lists in the form of lists of vertex neighbors (*arcslist and *edgeslist).

To enable easy input of older network data in a form of a matrix, the list(s) of lines in a network description can be replaced by a matrix – see the network description on the right side of Fig. 2.

Besides standard networks, Pajek supports also some extended types of networks: two-mode networks, bipartite (valued) graphs, networks between two disjoint sets of vertices; multirelational networks, different relations on the same set of vertices; and temporal networks, dynamic graphs, networks changing over time.

For example, in the description of a two-mode network (see Fig. 3), the list of vertices starts with *vertices n k where k is the number of vertices in the first set. In the list of vertices, we first list all vertices from the first set, followed by all vertices in the second set. Each line has to have one end vertex in the first set and the other vertex in the second set.
Fig. 3

Two-mode network and its description

Types of networks can be combined – for example, we can have a two-mode, multirelational temporal network.

Pajek’s input format allows also detailed specification of elements’ (vertices and lines) graphical attributes. This enables the user to produce network layouts as is the Petri net presented on the left side of Fig. 4. For details see Pajek’s manual.
Fig. 4

Petri net picture using Pajek’s file graphic attributes and an enhanced picture with Unicode labels

To describe a network, we have to provide (beside its graph structure) also data about properties of its vertices. All three types of property files have the same structure:

*vertices n
p 1
p n

The ith vertex has value p i with the following meaning:
  • Vectors (VEC) – numeric data about vertices, p i ∈ ℝ: the property has value p i on vertex i.

  • Partitions or clusterings (CLU) – nominal or ordinal data about vertices, p i ∈ ℕ: vertex i belongs to the cluster p i .

  • Permutations (PER) – ordering of vertices, p i ∈ ℕ: vertex i is at the p i -th position.

All types of data files can be combined into a single file – Pajek’s project file (PAJ).

Pajek can read data also in some other formats: UCINET DL files, genealogies in GEDCOM format, and molecular data in formats BS, MAC, and MOL.

Methods

This is a list of the main network analysis methods implemented in Pajek:
  • Basic operations on Pajek’s structures (extraction, shrinking, combinations, conversions, arithmetics, etc.)

  • Transforming temporal and multirelational networks into collections of networks (time slices, single-relation networks)

  • Connectivities: weak, strong, biconnectivity, and periodic, condensation (Tarjan 1983; Harary et al. 1965)

  • Shortest paths, k-neighbors, and flow

  • Measures of vertex’s importance: degrees, closeness, betweenness, and (corrected) clustering coefficient

  • Kleinberg’s hubs and authorities (also for two-mode networks) (Kleinberg 1998)

  • McCabe software metrics (McCabe 2003)

  • Structural holes (Burt 1992)

  • Brokerage

  • Vertex and line cuts

  • Vertex and line islands (Zaveršnik and Batagelj 2004)

  • (Generalized) cores (Batagelj and Zaveršnik 2011), two-mode cores (Ahmed et al. 2007), triadic spectrum; three-ring and four-ring weights (Batagelj and Mrvar 2001; Ahmed et al. 2007)

  • Pathfinder skeleton (Schvaneveldt et al. 1988.

  • Fragment (motif) searching

  • Clustering of small networks

  • (Generalized) blockmodeling of small networks (Doreian et al. 2000, 2004)

  • Hierarchical clustering with relational constraints (Ferligoj and Batagelj 1983)

  • Community detection methods (Blondel et al. 2008; Waltman et al. 2010)

  • Methods for partitioning signed networks (Doreian and Mrvar 1996)

  • Basic operations on acyclic networks (depth, topological ordering, CPM)

  • Kinship analysis (White et al. 1999; Batagelj and Mrvar 2008)

  • Hummon-Doreian weights in acyclic networks (Hummon and Doreian 1989; Batagelj 2003)

  • Computing probabilistic flow in acyclic networks

  • Generalized main path analysis of acyclic networks (so called key-Route searches as defined by Liu and Lu 2012)

  • Multiplication of networks (Batagelj and Mrvar 2008; Batagelj and Cerinšek 2013)

  • Normalizations of weighted networks (Batagelj and Mrvar 2003)

  • Basic support for Petri nets (Peterson 1981)

  • Generation of different types of random networks (small world, scale-free, etc.) (Batagelj and Brandes 2005; Pennock et al. 2002)

  • Computing different network indices (e.g., modularity, assortativity, relinking index, E-I index, etc.)

The “granularity” of Pajek’s methods is high – usually we need to perform a sequence of operations to achieve the intended result.

Visualization

Pajek’s layout is basically in a 3D unit cube x, y, z ∈ [0, 1]. In 2D layouts, the third dimension is not considered.

Pajek provides a collection of general graph drawing algorithms such as Kamada-Kawai, Fruchterman-Reingold, VOS, and MDS.

In Pajek vectors can be displayed as size (width and height) of vertex (figure), as its coordinate, and partitions as color or shape of the figure, Unicode symbol, or as a vertex label (content, font size, and color).

The weights on lines can be displayed as value, thickness, or gray level. Nominal line values can be assigned as label, color, or line pattern (see Pajek manual, Sect. 4.3).

The layouts can be exported in different formats: bitmap, JPEG, EPS, SVG, X3D, MOL, and Kinemages. Pictures of network in EPS or SVG can be imported in vector graphics programs such as Inkscape, Adobe Illustrator, or CorelDRAW. They can be used to enhance the pictures (see the right side of Fig. 4). Nice examples of this approach are visualizations produced by FAS Research, Vienna. Darko Brvar developed a program SVGanim for dynamic visualization of temporal networks.

The important parts of network identified by Pajek’s methods are often dense and the traditional dots and lines layout is unreadable. In such cases, a better solution is the matrix display for the “right” ordering of vertices (usually determined with clustering or blockmodeling). In Fig. 5 the network of collaboration between countries in EU projects on simulation is presented.
Fig. 5

Collaboration between countries in EU projects on simulation

A natural display of an acyclic network is by layers – all arcs are pointing in the same direction (up or down). In Fig. 6 the main island in the citation network on “social networks” is presented – it is an acyclic network.
Fig. 6

Main island in SN5

Interoperability

In the development of Pajek, we often encountered the need to include in Pajek different statistical procedures. We decided not to do this, but to provide a connection to statistical programs such as R, SPSS, or Excel. Other statistical programs can import results obtained by Pajek via export to tab-delimited files. The menu option Tools allows the user to connect Pajek with other programs – for example, viewers. This is also the way in which PajekXXL calls Pajek after the large network is decomposed to smaller (manageable) parts which can be analyzed and visualized with more sophisticated methods which are available in Pajek (but not in PajekXXL).

Many network analysis programs support (at least partially) Pajek’s input format. There exist also programs to transform specific types of data into Pajek’s network data – for example, Text2Pajek and WoS2Pajek.

Special Options

In Pajek frequently used sequences of operations can be defined and saved as macros. This allows also the adaptations of Pajek to groups of users from different areas (social networks, chemistry, genealogy, computer science, mathematics, etc.) for specific tasks. Pajek supports also repetitive operations on series (collections) of networks and/or other data objects.

Documentation

All the capabilities of Pajek are concisely described in its manual that is available at Pajek’s download page. A friendly introduction into SNA and corresponding Pajek’s commands is provided in the book ESNAExploratory Social Network Analysis with Pajek (De Nooy et al. 2011) – that was published also in Japanese and Chinese.

Basic info about Pajek and introduction to Pajek are available in different languages on the main Pajek’s page (e.g., Greek, Polish, Spanish, Portuguese, German, Persian, French, Japanese, Chinese, Italian, English, Slovenian). Several video lectures on using Pajek can be found as well.

Users can also join Pajek’s mailing list and discuss their problems with other users (http://list.fmf.uni-lj.si/cgi-bin/mailman/listinfo/pajek).

Key Applications

  • Pajek was used in many different fields. In September 2017, the book ESNA had in Google Scholar over 3000 citations, and other papers about Pajek had over 3200 citations. As examples we present some applications in analysis of genealogies, in bibliometrics, and in analysis of hyperlink graphs.

Genealogies

  • Pajek was successfully applied to analysis of large genealogies. When reading GEDCOM files, Pajek can produce three types of networks: Ore graphs, p-graphs, and bipartite p-graphs. Each representation has some advantages for special uses.

  • Many interesting subnetworks can be found in genealogies, e.g., searching for the shortest genealogical paths between people, longest patrilineages, different statistics on marriages, and births. Batagelj and Mrvar (2008) identified 16 fragments that represent all possible relinking marriages (blood marriages or not) on at most six vertices in p-graphs. Using general fragment searching in Pajek, we obtain frequency distributions of these fragment counts which can be used to compare different genealogies. Relinking index as a measure of relinking by marriages among persons belonging to the same families was defined in the paper as well.

  • Pajek generates three relations when reading genealogies as Ore graphs: …is a mother of…, …is a father of…, and …is a spouse of…. Using these three relations and network multiplication in Pajek, we can obtain other kinship relations (…is an aunt of…, …is a grandfather of…, …is a niece of…, etc.) as derived networks.

  • VB and AM collaborate with other researchers on analysis of genealogies: Klaus Hamberger (EHESS) is developing a specialized program PUCK for searching for matrimonial circuits in kinship networks (Hamberger 2016). The program is compatible with Pajek. Pajek is intensively used also by Douglas White (UCI). On his web site, we can find several references for using Pajek in anthropological research (White 2016).

Bibliographic Data

  • Pajek is a powerful tool for analyzing all kinds of bibliographical data (Batagelj and Mrvar 2000; Batagelj and Cerinšek 2013). Loet Leydesdorff produced several specialized programs for creation and analysis of bibliographical data that can be well combined with Pajek (Leydesdorff 2016).

  • Pajek is also well connected to program VOSviewer (VOSviewer 2017). VOSviewer is primarily intended to be used for analyzing bibliometric networks. As explained in previous sections, there are several algorithms in Pajek that produce partitions and vectors. Pajek can export networks, partitions, and vectors directly to VOSviewer and use its additional visualization techniques.

Hyperlink Graph

  • PajekXXL was successfully applied to analysis of huge hyperlink networks. It was used in Web Data Commons – Hyperlink Graphs project (network containing 43 million vertices and 623 million arcs). For details see http://webdatacommons.org/hyperlinkgraph/index.html.

Future Directions

Besides new algorithmic development, there are many things still missing in Pajek and waiting for implementation. For example:
  • Further increasing the upper limit of vertices that PajekXXL can handle, developing Pajek3XL

  • GUI control of elements’ graphical attributes

  • Internationalization of GUI and messages

  • Simple programming language for macros (with variables and control structure)

  • Support for vertex idents and namespaces

  • More interactive visualization

  • Support for additional input/output formats (GraphML, JSON, HTML 5, D3.js, etc.)

  • Computations with weights based on semirings (Batagelj 1994)

  • Genealogies of data

  • Layout styles

  • Support for analysis of temporal networks (Batagelj and Praprotnik 2016)

Cross-References

Notes

Acknowledgments

The work was supported in part by the ARRS, Slovenia, grant P1-0294, as well as by grant N1-0011 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation.

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Recommended Reading

  1. Batagelj V (2009a) Complex networks, visualization of. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, New York/London, pp 1253–1268CrossRefGoogle Scholar
  2. Batagelj V (2009b) Social network analysis, large-scale. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, New York/London, pp 8245–8265CrossRefGoogle Scholar
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  4. Batagelj V, Mrvar A (2003) Pajek – analysis and visualization of large networks. In: Juenger M, Mutzel P (eds) Graph drawing software. Mathematics and visualization. Springer, Berlin, pp 77–103Google Scholar
  5. Batagelj V, Doreian P, Ferligoj A, Kejžar N (2014) Understanding large temporal networks and spatial networks: exploration, pattern searching, visualization and network evolution. Wiley, Hoboken, New JerseyGoogle Scholar
  6. De Nooy W, Mrvar A, Batagelj V (2011) Exploratory social network analysis with Pajek, revised and expanded 2nd edn. Structural analysis in the social sciences. Cambridge University Press, New York, Sept 2011. Translation in Japanese 2008, Denkyo Press, Tokyo. Translation in Chinese 2012, Beijing World Publishing CorporationGoogle Scholar
  7. Doreian P, Batagelj V, Ferligoj A (2004) Generalized blockmodeling. Structural analysis in the social sciences. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  8. Mrvar A, Batagelj V (2016) Analysis and visualization of large networks with program package Pajek. Complex Adapt Syst Model 4:6. SpringerOpenCrossRefzbMATHGoogle Scholar
  9. Wasserman S, Faust K (1994) Social network analysis: methods and applications. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Faculty of Social SciencesUniversity of LjubljanaLjubljanaSlovenia

Section editors and affiliations

  • Vladimir Batagelj
    • 1
    • 2
  1. 1.Department of Theoretical Computer ScienceInstitute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  2. 2.University of Primorska, Andrej Marušič InstituteKoperSlovenia