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Lecture Notes

Page Class 1 and 2.
General Intro. Intro to Graph Theory: local structure of graphs, conn matrix.
Page Class 3: algorithm complexity
January 25. Going into the structure of the adjacency matrix we discovered we needed to recap linear algebra. In the introduction to MATLAB we saw we needed to understand loop control and recursion a lot better. Then we segued into complexity of algorithms using sorting as a classical example of divide and conquer.
Page Class 4: floodfill and connectedness; erdos-renyi
February 1. The floodfill algorithm and connectivity. The Erdos-Renyi model, giant component.
Page Class 5: erdos-renyi, floodfill and dijkstra
February 8. Finished the percolation transition and birth of the giant component. Discussed using floodfill to rearrange graph to sort by connected component. A whole enquiry ensued pursuing the connection between percolation, connected components, and epidemiology, discussing relevance of parameters measured and various models for spread of infectious diseases. Discussed the concept of geodesic and dijkstra algorithm.
Page Class 5 practice: floodfill, erdos-renyi, dijkstra, small-world
We did code walkthroughs of short matlab programs implementing floodfill, generating erdos-renyi, measuring the percolation transition, rearranging components. We tried our hand at dijkstra, explained that we get paths as well as distances.
Page Class 6: topological measures of real networks and network models
We reviewed the basics of the Erdos model, looking with more care at its definition, the expected values of the degree, degree distribution, giant component and geodesic path. We discussed what conclusions to extract when contrasting real measures with Erdos. We then introduced the concept of clustering and other potential motif density measures. Finally, we introduced the idea of small-world networks, and discussed in detail the rewiring process to get from regular networks to Erdos.
Page Lecture 7
We discussed the following subjects: - Eigenvector centrality, having seen previously betweeness centrality - Maslov randomization - Barabasi-Albert network growth model - Node duplication network growth model Finally, we discuss in some detail the following paper: "Formation of Regulatory Patterns During Signal Propagation in a Mammalian Cellular Network" Avi Ma'ayan, et al. Science 309, 1078 (2005); DOI: 10.1126/science.1108876
Page Class 8: finding motifs within a graph
Monday, March 8. Deep soul-searching was carried out to determine how to end the course and whether to start a second one. Then motif searching was debated, using as an example finding loops and feed-forward loops. Two ways of finding motifs were examined: one, a la floodfill, finds them by algorithm; the algebraic method examines the diagonal of the powers of the adjacency matrix. We discussed finding the feed-forward loop in the diagonal of M^2 M'.
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