Selected Topics in Data Networking Explore Social Networks: Center and Periphery

Introduction A concepts of Centrality and Centralization  Most social networks contain people or organizations that are central. Because of their position, they have better access to information and better opportunities to spread information. This is known as the ego-centered approach to centrality. 2

Introduction Phenomena of or affecting individual entities across different settings (networks)  The ego-centered approach Individual people, organizations Different patterns of interaction within defined groups (networks)  The socio-centered approach 3

Introduction Centrality to refer to positions of individual vertices within the network => ego-centered approach Centralization to characterize an entire network => socio-centered approach  A network is highly centralized if there is a clear boundary between the center and the periphery. In a highly centralized network, information spreads easily but the center is indispensable for the transmission of information. 4

Introduction Centrality and centralization: information may easily reach people who are central in a communication network.  People are central if information may easily reach them. The simplest indicator of centrality is the number of its neighbors, which is his or her degree in a simple undirected network 5

Degree Centralization Degree centralization  The variation in the degree of vertices divided by the maximum variation in degree which is possible given the number of vertices in the network.  Degree centralization ranges from zero (no variation) to 1 (maximum variation) 6

Degree Centralization (possible) Maximum degree = (n-1) (possible) Minimum degree = n-(n-1) (possible) Maximum degree variation = (n-1)*[(possible) maximum degree – (possible) minimum degree]  v5 has degree 4, which is the maximum degree  Four vertices have minimum degree, which is 1  The degree variation amounts to 12: (vertices v1 to v4 contribute) 4 ×(4 − 1)=12 and (vertex v5 contributes) 1 × (4 − 4)=0.  Degree centralization = maximum degree variation/ (possible) maximum degree variation =12/12 =1 7

Degree Centralization 8

 v1 and v2 have a degree of 1 and the other vertices have a degree of 2.  2 is the maximum degree in this network the degree variation equals 2 × (2 − 1)=2 (for vertices v1, v2) and 3 × (2 − 2)=0 (for vertices v3 to v5), which is 2. the degree centralization = maximum degree variation/ (possible) maximum degree variation = 2/12 =  If we add a line between v1 and v2, degree centralization becomes minimal (0.00) because all vertices have equal degree, so variation in degree is zero and degree centralization is zero The degree centralization = maximum degree variation/ (possible) maximum degree variation = 0/12 = 0 9

Degree Centralization In a network with multiple lines or loops, the degree of a vertex is not equal to the number of its neighbors.  The star-network does not necessarily have maximum variation and we may obtain centralization scores over

Distance in the Networks In a communication network, information will reach a person more easily if it does not have to “travel a long way.”  This brings us to the concept of distance in networks the number of steps or intermediaries needed for someone to reach another person in the network.  The shorter the distance between vertices, the easier it is to exchange information. 11

Distance in the Networks  Paths as a sequence of lines in which no vertex in between the first and last vertices occurs more than once. Via a path, we can reach another person in the network We can inform our neighbor, who passes the information on to his neighbor, who in turn passes it on, until the information finally reaches its destination.  A person is reachable from another person if there is a path from the latter to the former. Two persons are mutually reachable if they are connected by a path in an undirected network, but that two paths (one in each direction) are needed in a directed network. 12

Distance in the Networks The distance between two vertices is simply the number of lines or steps in the shortest path that connects the vertices.  A shortest path is also called a geodesic. In a directed network, the geodesic from one person to another is different from the geodesic in the reverse direction, so the distances may be different. 13

Closeness Centralization 14

Closeness Centralization The closeness centrality of a vertex is based on the total distance between one vertex and all other vertices, where larger distances yield lower closeness centrality scores. The closer a vertex is to all other vertices, the easier information may reach it, the higher its centrality. 15

Closeness Centralization Degree and closeness centrality are based on the reachability of a person within a network A person is more central if he or she is more important as an intermediary in the communication network.  How crucial is a person to the transmission of information through a network? 16

Betweeness Centralization The concept of betweenness  The centrality of a person depends on the extent to which he or she is needed as a link in the chains of contacts that facilitate the spread of information within the network. 17

Betweeness Centralization 18

Selected Topics in Data Networking Explore Social Networks: Broker and Bridge

Introduction A person with many friends and acquaintances has better chances of getting help or information. People in crucial positions in the information network may also spread or retain information strategically because they have control over the diffusion of information. 20

Bridge 21

Bridge  Two brokerage roles involve mediation between members of one group. The mediator is also a member of the group: coordinator role Two members of a group use a mediator from outside, an itinerant broker  Three brokerage roles describe mediation between members of different groups. 22

Bridge  The mediator acts as a representative of his group  The mediator is a gatekeeper, who regulates the flow of information or goods to his or her group.  The liaison is a person who mediates between members of different groups but who does not belong to these groups himself or herself.  The five types of brokerage roles have been conceived for directed networks, namely transaction networks. the direction of relations is only needed to distinguish between the representative and the gatekeeper. The other brokerage roles are also apparent in undirected relations, 23

Bridge 24

Bridge  Bob plays the coordinator role. Ike and Mike, Hal, John, and Lanny  Bob bridges many structural holes between his group of English- speaking young employees and the Hispanic workers or the older employees. Bob is a representative and for information flowing toward members of his group, he is a gatekeeper.  He may mediate between Alejandro and Norm, that is, between the Hispanics and the older workers. In this role, he is a liaison. The only brokerage role that Bob cannot play given the ties in the network, is the role of an itinerant broker because he has no ties with two or more members of any group other than his own. 25

Practice Download files: lj.si/pub/networks/data/esna/strike.htmhttp://vlado.fmf.uni- lj.si/pub/networks/data/esna/strike.htm 26 Network Centralizations - Degree Centralization - Closeness Centralization - Betweenness Centralization

Practice 27 Bridge Cut Node

Practice  Load Network Connection  Load Clustering File  Operations>Transform>Remove Lines>Between Clusters  Select both Network Connection and Clustering File  Operations>Brokerage Roles  Info>Partition 28

Practice 29

Practice Remove Lines between Clusters 30

Practice 31 Cluster: 3 Cluster: 2 Cluster: 1

Practice Operations>Brokerage Roles  Coordinator  Representative  Gatekeeper  Liaison  Itinerant 32

Practice Alejandro (10), Bob (9), Norm (14), and Ozzie (15) are the only employees who also have other types of brokerage roles because they are the only ones who are connected to members of different groups. 33

Practice Info> Partition 34

Practice Info> Partition 35

Practice Info> Partition 36

Practice Info> Partition 37

Practice Info> Partition 38

References Wouter de Nooy, Andrej Mrvar, and Vladimir Batagelj, Exploratory Social Network Analysis with Pajek, Cambridge 39