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# Networks

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### Transcript

• 1. Introduction to Networks dsdht.wikispaces.com
• 2. Networks • a collection of individuals or entities, each called a vertex or node • a list of pairs of vertices that are neighbors, representing edges or links • vertices are Facebook users, edges represent Facebook friendships • Vertices of Drugs and Targets ,edges represent Drug target relations
• 3. Network Basic definitions N to denote the number of vertices in a network • Number of possible edges: N(N-1)/2 ~ N2/2 • The degree of a vertex is its number of neighbors • The distance between two vertices is the length of the shortest path connecting them. • If two vertices are in different components, their distance is undefined or infinite. • The diameter of a network is the average distance between pairs(It measures how near or far typical individuals are from each other)
• 4. Network Basic definitions Directed and Undirected Networks • If the pairs are unordered, then the graph is undirected: vertices = {A, B, C, D, E} edges = ({A, B}, {A, C}, {B, C}, {C, E}). • Otherwise it is directed: vertices = {A, B, C, D, E} edges = ((A, B),(A, C),(B, C),(C, E)). Example • I can follow you on Twitter without you following me • web page A may link to page B, but not vice-versa.
• 5. Representing Networks • adjacency matrix • edgelist 2, 3 2, 4 3, 2 3, 4 4, 5 5, 2 5, 1 • adjacency list 1: 2: 3 4 3: 2 4 4: 5 5: 1 2 1 2 3 45 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 A =
• 6. Degree Properties • Indegree how many directed edges (arcs) are incident on a node 2 • outdegree how many directed edges (arcs) originate at a node 2 • degree (in or out) number of edges incident on a node 3 • Degree distribution: A frequency count of the occurrence of each degree • In-degree distribution: [(2,3) (1,4) (0,1)] • Out-degree distribution: [(2,4) (1,3) (0,1)] • (undirected) distribution: [(3,3) (2,2) (1,3)]
• 7. Connected Components • Strongly connected components • Each node within the component can be reached from every other node in the component by following directed links Strongly connected components B C D E A G H F • Weakly connected components: every node can be reached from every other node by following links in either direction Weakly connected components A B C D E G H F • In undirected networks one talks simply about “connected components” B C D F H GA E A C D E F G H B
• 8. Giant Component In a network, a "component" is a group of nodes (people) that are all connected to each other, directly or indirectly. So if a network has a "giant component", that means almost every node is reachable from almost every other.
• 9. Centrality Measures • We often want to know important nodes in a network . • Centrality is one of the methods There are many but for this course we will look into : Degree - connectedness Closeness – ease of reaching other nodes Betweeness – role as an intermediary node Eigenvector - not what you know, but who you know Things to Remember • Centrality is a measure of an node, centralization is a measure of the network. • It matters whether you are considering a directed or an undirected network. • Most centrality measures work on binary/unweighted networks
• 10. Centrality Measures Node Degree Centrality : divide degree by the max. possible, i.e. (N-1) Degree Centralization: CD = CD (n* ) - CD (i)[ ]i=1 g å [(N -1)(N -2)] maximum value in the network CD = 0.167CD = 1.0
• 11. Centrality Measures Closeness Centrality • one still wants to be in the “middle” of network, not too far from the center. • Closeness is based on the length of the average shortest path between a node and all other nodes in the network CC ' (i) = (CC (i))/(N -1) Closeness Centrality: Normalized Closeness Centrality: Cc ' (A) = d(A, j) j=1 N å N -1 é ë ê ê ê ê ù û ú ú ú ú -1 = 1+ 2 +3+ 4 4 é ëê ù ûú -1 = 10 4 é ëê ù ûú -1 = 0.4 An actor who has very low closeness centrality takes many more steps to get to everyone.
• 12. Centrality Measures Betweeness Centrality • A node has a high betweenness centrality when they occupy a position in the geodesics connecting many pairs of other actors in the network • It is equal to the number of shortest paths from all vertices to all others that pass through that node. CB (i) = gjk (i)/gjk j<k å Where gjk = the number of shortest paths connecting jk gjk(i) = the number that actor i is on. Usually normalized by: CB ' (i) = CB (i )/[(n -1)(n -2)/2] number of pairs of vertices excluding the vertex itself
• 13. • non-normalized version: A lies between no two other vertices B lies between A and 3 other vertices: C, D, and E C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E) note that there are no alternate paths for these pairs to take, so C gets full credit A B C ED Betweeness Centrality
• 14. Betweeness Example why do C and D each have betweenness 1? They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: • ½+½ = 1
• 15. Eigen Vector Centrality Eigenvector centrality is calculated by assessing how well connected an individual is to the parts of the network with the greatest connectivity. Here yellow has more eigen vector centrality because it is connected to nodes with greatest connectivity Applications: High eigenvector centrality individuals are leaders of the network. They are often public figures with many connections to other high-profile individuals. Thus, they often play roles of key opinion leaders and shape public perception. A related example of this is Google’s page rank algorithm, which is closely related to eigenvector centrality calculated on websites based on links to them.
• 16. Resources Lada Adamic : https://www.coursera.org/instructor/~267 Network,crowds and Markets https://www.coursera.org/instructor/~267 http://www.cs.cornell.edu/home/kleinber/netwo rks-book/networks-book.pdf