Mathematics For A Connected Age: Networks and Their Many Uses

1. Mathematics For A Connected Age: Networks and Their Many Uses Jonathan Choate Groton School jchoate@groton.org www.zebragraph.com

5. http://www.boston.com/bostonglobe/ideas/articles/2010/12/26/visualizing_friendspace/?page=1http://www.boston.com/bostonglobe/ideas/articles/2010/12/26/visualizing_friendspace/?page=1

11. http://www.orgnet.com/tnet.html

12. Cell Phone Network Basics

14. http://www.google.com/patents?vid=4144411

16. Typical Cell Cluster

17. http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/pr4/article1.html#Cellshttp://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/pr4/article1.html#Cells

18. In order to serve the most customers the average cellsize is roughly 10 square miles. Each cell can service approximately 70-80 users at once because - Each cell is alloted 832 frequencies or channels - 42 channels are used for control issues- 790 are available for voice and data transmission.- Cell phones are duplex devices and need 2 frequencies per user unlike walkie talkies.

19. - Each cell is surrounded by 6 other cells so in order to avoid interference issues there has to be seven separate sets of frequencies. - 395/7 is roughly 56 so for each cell there are 56 sets of frequencies so each cell can handle 56 users at once. - In a 7 cell cluster, 392 people can be handled.

20. Problem 1. What is the closest any two cells using the same set of frequencies can be? What are some of the possible configurations for clusters?

21. Hexagon Geometry R Cell Area =

22. 2H 1H Ru 120 degrees

23. Let H = distance between two centers of adjacent hexagons. Let Ru = Distance between two cells with same set of frequences. Using the law of cosines, you get

24. C Rc B D Rc Rc Rc Rc E Ru A Let Rc be the cluster radius. Rc = AB =AD=DE and AE = Ru, <ADE=120

25. The area of the cluster can be calculated in two ways. Let C be the number of cells in the cluster Therefore, C = 7

26. This shows that the possible cluster configurations contain i2 + j2 +ij cells where i and j are the displacements used to get to the nearest cell that can have the same set of frequencies.

27. I = 1, j =1 I = 2, j = 0 I = 2 , j =1 I = 3 , j = 0

28. i=3, j = 2 9 + 4 + 6 = 19

29. Some Useful Network Concepts A graph is a collection V of vertices and E of edges. A network is one type of graph.

30. Adjacency Matrices

31. Neighborhoods The set of all vertices adjacent to a given vertex is called the neighborhood of that vertex. The neighborhood of vertex 2 is vertices 1, 3, and 4.

32. Degree of a Vertex If the graph is un-directed the number of edges meeting at a vertex is known as the degree of the vertex. The degree of vertex 3 is 4.

33. Paths and Diameter A path is a set of edges joining adjacent vertices. The length of a path is the number of edges in the path. The diameter of a graph is defined to be the maximum of all the shortest paths between pairs of vertices.

34. Connected Graphs If in a given graph there is a path joining every pair of vertices the graph is said to be connected.

35. Sub-Graphs A subset W of a graph {V,E} is called a sub-graph. In the sample network, the vertices 1,2,3,4 and the adjacent edges form a sub-graph

36. Complete Graphs If in a graph or a sub-graph every pair of vertices are adjacent then the graph is said to be complete. The subgraph described in the previous slide is complete but the whole network is not complete.

37. Graph Density

38. Airline Networks

45. FaceBook Basics

46. A Simple Network

47. Cliques • A clique is a subgraph with a density of 1. In a social network this means that everyone know s everyone else. In the previous example, friends 4, 7, 8 and 10 form a clique since there are 4 people and the maximum number of friendships is 6 and all 6 exist.

48. Clusters • A cluster is a sub-graph that is almost a clique. The density of the sub-graph is almost 1.

50. How FaceBook Finds Friends for You