Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale Network Theory: Computational Phenomena and Processes Social Network Analysis

Degree, Indegree, Outdegree Centrality Degree Centrality: Indegree Centrality: Outdegree Centrality:

Eigenvector Centrality=C E (i)= I’th entry of eigenvector e e = largest eigenvalue of adjacency matrix

Betweenness Centrality Normalized betweenness= = number of geodesic linking across i and j has pass through node k.

Closeness Centrality Scaling factor Adjustment

One-node network 1-Connection 0-No connection One-mode network: Actors are tied to one another considering one type of relationship; i.e Binary Adjacency matrix v1v2v3v4 v5 v v v v v501000

Two-node network Two-node network: Actors are tied to events. Incident network Bipartite graph e.g. Student attending classes

Affiliation network Actors are tied to ----Organization/Attributes; e.g. Affinity network, Homophily network Sociogram ≡ Org1……………………..…….Org n Attribute m1………………..Attribute mn {} Points individuals Lines relationship People Attributes

Centrality ij

Normalized Centrality Centrality: Normalized Centrality: A is more central than F A F D B CE ‘ 6-1 = = 80%,,,,

Directed network centrality: Prestigue of A=B=C=E=2 (Indegree) A F D C B ‘ 6-1 == 40% (Normalize centrality) E

Eigen vectors Vector X is a matrix with n rows and column, linear operator A, maps the vector X to matrix product AX A

Eigen Value

Second degree centrality Consider this 16 degree graph network: Node A Value.534 B.275 C.363 D.199 E F G H.102 I J.164 K L M N O.441 Eigen value centrality M N L K J C A B D G H I E F O

Betweenness Centrality Betweennes centrality measures the extent to which a vertex lies on paths between other vertices.

Normalized centrality: A D CE FB NodeNo of distinct path from the node Normalized (C’) E4.040% A3.535% D1.010% B0.55% C0.00% F0.00%

Closeness Centrality Closeness centrality is the mean distance from a vertex to other vertices. A D CE FB Node (i) A65/683% E75/772% D75/772% B85/863 % C95/955% F115/1146% f= farness c= closeness d= distance between i & j n= total number of nodes

Eigen vector

Page Rank Cetrality The numerical weight that it assigns to any given element E is referred to as the PageRank of E and denoted by PR(E). Page Rank Centrality:

Bonacich/Beta Centrality Both centrality and power were a function of the connections of the actors in one's neighborhood. The more connections the actors in your neighborhood have the more central you are. The fewer the connections the actors in your neighborhood, the more powerful you are. It is the weighted centrality

Density Density: It is the level of ties/connectedness in a network; It is a measure of a network’s distance from a complete graph. Complete graph: Every node is connected to every node in the network

L = number of links in network n = number of nodes in the network Ego Density

Structural Hole (Ron Burt) Let’s consider this, The gap between connected components is the hole Structural hole provides diversity of information for nodes that bridge them Without structural hole information becomes redundant and less available 12 Structural Hole gap

Brokering Brokering is bridging different group of individuals. 1.Coordinator (local brokers; Intragroup brokering) e.g. manger, mediating employees 2. Consultant (Intergroup brokering by an outsider e.g. middle man in business between buyers &seller, stock agent ) A B C B as Coordinator/ Broker Seller Consultant Buyer

Brokering 3. Representation (represents A when negotiating with C) e.g. hiring a mechanic to buy car for you 4. Gate Keeper(e.g a butler, chief of staff) Actor Producer Agent Actor Producer A B C

Dyadic Relation Dyads: Triads: when a triad consists of many ties, an open triad (triangle) is forbidden. A A A A B B B B C AB 0

Triad Relations (census)

Components Component is a group where all individuals are connected to one another by at least one path. Weak Component: A component ingoing direction of ties. Strong Component: A component with directional ties. Clique: A subgroup with mutual ties of three or more. who are directly connected to one another by mutual ties

Bonacich Centrality C BC = Degree Centrality High Degree + Low Betweenness : Ego Connection are redundant Low Closeness+ High Betweenness : Rare node but pivotal to many In triads, there is a structural force toward transitivity.

Reverse distance : Principle of strength of weak tie.(Granovetter, 1973): There is a social force that suggests transitivity. If A has ties to B and B to C, then there is tie from A to C. Bonacich

Integration: Reverse distance Network distance Degree to which a node’s inward ties integrate it into the network.

Radiality Degree to which a node’s outward ties connects the node with novel nodes.

Edge between-ness Number of shortest path from s to t that pass through edge e Number of shortest path from s to t This is important in diffusion studies like epidemics

Social Capital The network closure argument: Social Capital is created by a strongly interconnected network. The structural hole argument: Social Capital is created by a network of nodes who broker connections among disparate group.

Structural Equivalence= similarity of position in a network Euclidean Distance E.g., E A C B D have distance zero