math1310_2020

math1310_2020, a guest lecture for MATH1310, "Graph Theory", Instructor: Jeffrey Wheeler, University of Pittsburgh, spring 2020, MWF 9:00-9:50, Room Thackeray 524.

The notes:

• math1310_2020.tex, the Latex document.
• math1310_2020.pdf, the PDF version.

Matlab:

• abc_check.m, abc_check() demonstrates that the transition matrix for the three field graph has three distinct eigenvalues of norm 1.
• abc_inc.m, A=abc_inc() returns the (sparse) incidence matrix for a directed graph on 3 nodes, the "three field pasture".
• abcd_inc.m, A=abcd_inc() returns the (sparse) incidence matrix for a directed graph on 4 nodes, the "four field pasture".
• five_inc.m, A=five_inc() returns the incidence matrix for a directed graph on 5 nodes.
• google_matrix.m, G=google_matrix(A,p) returns the Google matrix for a directed graph with incidence matrix A, and a jump probability p specified by the user. The value of p is usually 0.15.
• google_rank.m, r=google_rank(G) returns the page rankings based on the Google matrix G, iterated for 100 steps of the power method, with a jump probability of p=0.15.
• inc_to_google.m, G=inc_to_google(A) returns the Google matrix for a given incidence matrix A, using a jump probability of p=0.15.
• inc_to_transition.m, T=inc_to_transition(A) returns the transition matrix for a given incidence matrix A. If the i-th row of A is zero, then the i-th column of T has A(i,i)=1.
• moler_inc.m, A=moler_inc() returns the incidence matrix for a directed graph on 6 nodes, studied by Moler.
• plot_digraphs.m, a script that creates plots of the digraphs discussed in the presentation.
• power_method.m, r=power_method(A,its) uses its power method steps of the form r=A*r, starting from a random vector r.
• power_rank.m, r=power_rank(G) returns the page rankings based on the Google matrix G, iterated for 100 steps of the power method.
• r8vec_print.m, prints a vector.
• r8vec3_print.m, prints three vectors as a table.
• sauer_inc.m, A=sauer_inc() returns the incidence matrix for a directed graph on 15 nodes.
• surf_rank.m, r=surf_rank(A) returns the page rankings for a graph with incidence matrix A, based on the surfer algorithm, with a probability p=0.15 of jumping to a random page, and using 100,000 steps.
• transition_matrix.m, T=transition_matrix(A) returns the transition matrix for a given incidence matrix.

Images:

• abc.png, a plot of the 3-field pasture.
• abcd.png, a plot of the 4-field pasture.
• five.png, a plot of a 5-node directed graph.
• abcde1.png, an initial ranking for a 5 node network;
• abcde2.png, two updates to the ranking of the 5-node network;
• pagerank.png, a cartoon illustrating the pagerank process.
• sauer.png, a plot of Sauer's 15 node network.