1. How to Analyse Social Network? : Part 2 Social networks can be represented by complex networks.

2. Reviews Social network is a social structure made up of individuals (or organizations) called “nodes”, which are connected by one or more types of relationships, represented by “links”.  Friendship  Kinship  Common Interest  …. Graph-based structures are very complex. 2 Source: http://followingfactory.com/

3. Introduction Various nature and society systems can be described as complex networks  social systems, biological systems, and communication systems. 3 By presented as a graph, vertices (nodes) represent individuals or organizations and edges (links) represent interaction among them Source: http://www.fmsasg.com/SocialNetworkAnalysis

4. Introduction Why is network anatomy so important to characterize?  Because structure always affects function. For instance, the topology of social networks affects the spread of information. 4

5. Introduction Network Models  Regular Networks: chains, grids, lattices and fully- connected graphs  Random network model by Erdős and Rényi: ER model  Small-world phenomenon by Watts and Strogatz: WS model  Scale-free network model by Barabási and Albert: BA model Evolution mechanism of network structures are very interested among many researchers not only engineering but also physics communities. 5

6. Types of Network Models Regular Networks 1. Ring of ten nodes connected to their nearest neighbours. 2. Fully connected network of ten nodes 6

7. Types of Network Models Random Networks  placing n nodes on a plane,  joining pairs of them together at random until m links are used.  Nodes may be chosen more than once, or not at all. 7

8. Types of Network Models Random Networks  Erdös and Rényi studied how the expected topology of this random graph changes as a function of m.  When m is small, the graph is likely to be fragmented into many small clusters of nodes, called components.  As m increases, the components grow, at first by linking to isolated nodes and later by coalescing with other components. 8

9. Types of Network Models Random Networks A phase transition occurs at m = n/2, where many clusters crosslink spontaneously to form a single giant component. For m > n/2, this giant component contains on the order of n nodes (its size scales linearly with n), while its closest rival contains only about log n nodes. All nodes in the giant component are connected to each other by short paths: the maximum number of 'degrees of separation' between any two nodes grows slowly, like log n 9

10. Types of Network Models Random Networks  Gene networks  Ecosystems  Spread of infectious diseases  Computer viruses 10

11. Types of Network Models Small-World Networks  Watts and Strogatz studied a simple model that can be tuned through this middle ground: a regular lattice where the original links are replaced by random ones with some probability 0<p< 1. the slightest bit of rewiring transforms the network into a 'small world', with short paths between any two nodes, just as in the giant component of a random graph. 11

12. Types of Network Models Small-World Networks the network is much more highly clustered than a random graph, if A is linked to B and B is linked to C, there is a greatly increased probability that A will also be linked to C two properties — short paths and high clustering — for many natural and technological networks 12

13. Types of Network Models Small-World Networks  Starts with a ring of n nodes, each connected by undirected links to its nearest and next-nearest neighbours out to some range k.  Shortcut links are then added — rather than rewired — between randomly selected pairs of nodes, with probability p per link on the underlying lattice; thus there are typically nkp shortcuts in the graph How many steps are required to go from one node to another along the shortest route? 13

14. Types of Network Models Small-World Networks  how to actually find a short chain of acquaintances linking yourself to a random target person  search problems 14

15. Types of Network Models Scale-Free Networks  Some nodes are more highly connected than others are.  To quantify this effect, let p denote the fraction of nodes that have k links.  k is called the degree and p is the degree distribution.  connectivity probability distribution P(k) of a node connecting to k other nodes is a power-law degree distribution, where k is the degree of a node and γ is a scalar exponent 15

16. Types of Network Models Scale-Free Networks  The probability of attachment is proportional to the degree of the target node; thus richly connected nodes tend to get richer, leading to the formation of hubs and a skewed degree distribution with a heavy tail.  Red, k=33 links; blue, k=12; green, k=11. Here n=200 nodes, m=199 links.. 16

17. Types of Network Models Scale-Free Networks  Resistant to random failures because a few hubs dominate their topology 17

18. Types of Network Models Most large networks have been demonstrated that they have scale-free features according to the BA network properties.  There are two issues of realistic networks that are not related in both ER and WS network properties.  The first issue is a network grows. Both network models start with a fixed number of nodes (size of network) without modifying it.  It means the size of network is constant. 18

19. Types of Network Models Most real networks are growing continuously; new nodes are added in the system in anytime  World-Wide-Web network grows by increasing the new documents. The second issue is a connectivity probability.  Two nodes are connected together with randomly selection in the random network. Most real networks illustrate a preferential connection.  New documents in the World-Wide-Web network will link to popular documents with already high connectivity. 19

20. Types of Network Models The BA properties can support these issues of realistic networks:  The network expands continuously following a power law distribution.  The new nodes are added and connected with existing nodes in the network.  The new nodes are connected with the existing one based on a preferential attachment; Higher connectivity probability to a node that has a large number of connections. 20

21. Types of Network Models The network of co-authorship relationships in SEG's journal Geophysics is scale- free SEG's journal Geophysics 21 Source: http://www.agilegeoscience.com/journal/tag/networks

22. Graph Representation of Networks  Simple Graphs DEF: A simple graph G = (V,E ) consists of a non- empty set V of vertices (or nodes) and a set E (possibly empty) of edges where each edge is a subset of V with cardinality 2 (an unordered pair). 22 How to analyse social networks?

23. Graph Representation of Networks  Multigraphs allow multiple edges, but still no self-loops  Pseudographs If self-loops are allowed. 23 How to analyse social networks?

24. L2324 Undirected Graphs Terminology Vertices are adjacent if they are the endpoints of the same edge. Q: Which vertices are adjacent to 1? How about adjacent to 2, 3, and 4? 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6

25. L2325 Undirected Graphs Terminology A:1 is adjacent to 2 and 3 2 is adjacent to 1 and 3 3 is adjacent to 1 and 2 4 is not adjacent to any vertex 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6

26. L2326 Undirected Graphs Terminology A vertex is incident with an edge (and the edge is incident with the vertex) if it is the endpoint of the edge. Q: Which edges are incident to 1? How about incident to 2, 3, and 4? 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6

27. L2327 Undirected Graphs Terminology A:e 1, e 2, e 3, e 6 are incident with 1 2 is incident with e 1, e 2, e 4, e 5, e 6 3 is incident with e 3, e 4, e 5 4 is not incident with any edge 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6

28. L2328 Digraphs Last time introduced digraphs as a way of representing relations: Q: What type of pair should each edge be (multiple edges not allowed)? 1 2 3 4

29. L2329 Digraphs A: Each edge is directed so an ordered pair (or tuple) rather than unordered pair. Thus the set of edges E is just the represented relation on V. 1 2 3 4 (1,2) (1,1) (2,2) (2,4) (1,3) (2,3) (3,4) (3,3) (4,4)

30. L2330 Digraphs DEF: A directed graph (or digraph) G = (V,E ) consists of a non-empty set V of vertices (or nodes) and a set E of edges with E  V  V. The edge (a,b) is also denoted by a  b and a is called the source of the edge while b is called the target of the edge.

31. Degree:  The degree of a vertex counts the number of edges that Oriented Degree when Edges Directed:  The in-degree of a vertex (deg - ) counts the number of edges that stick in to the vertex.  The out-degree (deg + ) counts the number sticking out. 31 Network Analysis

32. Handshaking Theorem THM: In an undirected graph In a directed graph 32 Network Analysis

33. For a directed graph G = (V,E ) define matrix A G by: Rows, Columns –one for each vertex in V Value at i th row and j th column is  1 if i th vertex connects to j th vertex (i  j )  0 otherwise For a directed multigraph G = (V,E ) define the matrix A G by: Rows, Columns –one for each vertex in V Value at i th row and j th column is  The number of edges with source the i th vertex and target the j th vertex 33 Adjacency Matrix

34. Complete Graphs – K n  A simple graph is complete if every pair of distinct vertices share an edge. Cycles Graphs – C n  The cycle graph C n is a circular graph. Wheels Graphs- W n  The wheel graph W n is just a cycle graph with an extra vertex in the middle Bipartite Graphs  A simple graph is bipartite if V can be partitioned into V = V 1  V 2 so that any two adjacent vertices are in different parts of the partition. No two vertices of the same party are adjacent. 34 Other Types of Graphs

35. There are various measures of the centrality of a vertex within a graph that determine the relative importance of a vertex within the graph  how important a person is within a social network who is the most well-known author in the citation network 35 Centrality Measures

36. Degree centrality  Degree centrality is defined as the number of links incident upon a node (i.e., the number of ties that a node has).  Degree is often interpreted in terms of the immediate risk of node for catching whatever is flowing through the network such as a virus, or some information.  If the network is directed (meaning that ties have direction), then we usually define two separate measures of degree centrality, namely indegree and outdegree. 36 Centrality Measures

37. Degree centrality  Indegree is a count of the number of ties directed to the node.  Outdegree is the number of ties that the node directs to others. For positive relations such as friendship or advice, we normally interpret indegree as a form of popularity, and outdegree as gregariousness. 37 Centrality Measures

38. Degree centrality  An entity with high degree centrality: Is generally an active player in the network. Is often a connector or hub in the network. Is not necessarily the most connected entity in the network (an entity may have a large number of relationships, the majority of which point to low-level entities). May be in an advantaged position in the network. May have alternative avenues to satisfy organizational needs, and consequently may be less dependent on other individuals. Can often be identified as third parties or deal makers. 38 Centrality Measures

39. Degree centrality  An entity with high degree centrality:  Alice has the highest degree centrality, which means that she is quite active in the network. However, she is not necessarily the most powerful person because she is only directly connected within one degree to people in her clique—she has to go through Rafael to get to other cliques. 39 Centrality Measures Source: http://www.fmsasg.com/SocialNetworkAnalysis/

40. Degree centrality 40 Centrality Measures

41. Betweenness Centrality  Betweenness is a centrality measure of a vertex within a graph.  Vertices that occur on many shortest paths between other vertices have higher betweenness than those that do not. 41 Centrality Measures

42. Betweenness Centrality  An entity with a high betweenness centrality generally: Holds a favored or powerful position in the network. Represents a single point of failure—take the single betweenness spanner out of a network and you sever ties between cliques. Has a greater amount of influence over what happens in a network. 42 Centrality Measures

43. Betweenness Centrality  An entity with a high betweenness centrality generally:  Rafael has the highest betweenness because he is between Alice and Aldo, who are between other entities. Alice and Aldo have a slightly lower betweenness because they are essentially only between their own cliques. Therefore, although Alice has a higher degree centrality, Rafael has more importance in the network in certain respects. 43 Centrality Measures Source: http://www.fmsasg.com/SocialNetworkAnalysis/

44. Betweenness centrality 44 Centrality Measures

45. Closeness Centrality Closeness is one of the basic concepts in a topological space.  We say two sets are close if they are arbitrarily near to each other.  The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances. 45 Centrality Measures

46. Closeness Centrality  Closeness is a centrality measure of a vertex within a graph. Vertices that are 'shallow' to other vertices (that is, those that tend to have short geodesic distances to other vertices with in the graph) have higher closeness.  Closeness is preferred in network analysis to mean shortest-path length, as it gives higher values to more central vertices, and so is usually positively associated with other measures such as degree.  Closeness centrality measures how quickly an entity can access more entities in a network 46 Centrality Measures

47. Closeness Centrality  An entity with a high closeness centrality generally: Has quick access to other entities in a network. Has a short path to other entities. Is close to other entities. Has high visibility as to what is happening in the network. 47 Centrality Measures

48. Closeness Centrality Rafael has the highest closeness centrality because he can reach more entities through shorter paths. As such, Rafael's placement allows him to connect to entities in his own clique, and to entities that span cliques. 48 Centrality Measures Source: http://www.fmsasg.com/SocialNetworkAnalysis/

49. Hub and Authority (for directed graph) If an entity has a high number of relationships pointing to it, it has a high authority value, and generally:  Is a knowledge or organizational authority within a domain.  Acts as definitive source of information. Hubs are entities that point to a relatively large number of authorities. They are essentially the mutually reinforcing analogues to authorities. Authorities point to high hubs. Hubs point to high authorities. You cannot have one without the other. 49 Centrality Measures Source: http://www.fmsasg.com/SocialNetworkAnalysis/

50. Eigenvector Centrality  Eigenvector centrality is a measure of the importance of a node in a network. It assigns relative scores to all nodes in the network based on the principle that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.  Google's PageRank is a variant of the Eigenvector centrality measure. 50 Centrality Measures

51. Eigenvector Centrality 51 Centrality Measures

52. Eigenvector Centrality 52 Centrality Measures

53. 53 Centrality Measures

54. 54 RFID Datenvolumen Centrality Measures oPageRank oOnly Structure Consideration oKnowledge of Global Network Structure oBroken Link Problems

55. KONECT: the Koblenz Network Collection  contains 168 network datasets (for instance) Animal networks are networks of contacts between animals. Authorship networks are unweighted bipartite networks consisting of links between authors and their works. Citation networks consist of documents that reference each other. Coauthorship networks are unipartite network connecting authors who have written works together. Communication networks contain edges that represent individual messages between persons.  consists of Matlab code to generate statistics and plots about them 55 Social Network Analysis Software Source: konect.uni-koblenz.de/networks

56. “Pajek”: Large Network Analysis Software

57. 57 Introduction to Slovenian Spider: PajekSlovenian Spider: Pajek  http://vlado.fmf.uni-lj.si/pub/networks/pajek/  Free software  Windows 32 bit Pajek 2.05 “Whom would you choose as a friend ?”

58. 58 Introduction Its applications:  Communication networks: links among pages or servers on Internet, usage of phone calls  Transportation networks  Flow graphs of programs  Bibliographies, citation networks

59. 59 Data Structures Six data structures:  Network(*.net) – main object (vertices and lines - arcs, edges)  Partition(*.clu) – nominal property of vertices (gender);  Vector(*.vec) – numerical property of vertices;  permutation (*.per) – reordering of vertices;  cluster (*.cls) – subset of vertices (e.g. a cluster from partition);  hierarchy (*.hie) – hierarchically ordered clusters and vertices.

60. 60 Introduction Pajek 2.05

61. 61 Network Definitions Graph Theory  Graphs represent the structure of networks Directed and undirected graphs  Lists of vertices arcs and edges, where each arch and edge has a value. To view the network data files: NotePad, EditPlus

62. 62 Network Data File 62 Open Network Data File (*.net) Number of Vertices

63. 63 Transform

64. 64 Report Information

65. 65 Visualization Energy – Idea: the network is represented like a physical system, and we are searching for the state with minimal energy.  Two algorithms are included: Layout/Energy/Kamada-Kawai – slower Layout/Energy/Fruchterman-Reingold – faster, drawing in a plane or space (2D or 3D), and selecting the repulsion factor

66. 66 Network Creation 66

67. 67 Partitions File name: *.clu

68. 68 Degree

69. Social Network Analysis: Theory and Applications Graphs (ppt), Zeph Grunschlag, 2001-2002. KONECT:  http://konect.uni-koblenz.de/networks Pajek:  http://pajek.imfm.si/doku.php?id=download http://pajek.imfm.si/doku.php?id=download http://www.fmsasg.com/SocialNetworkAnalysis/ 69 References