From ec187be71a24e95c9c5265982c400fd0944eb581 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Tue, 6 Oct 2020 17:00:12 +0200 Subject: [PATCH] mp2: done 1-7.3 and MATLAB's part of 7.4 --- .../degcentrality.m | 14 +++++ mp2/Project.2.Maggioni.Claudio/ex3.m | 13 +--- mp2/Project.2.Maggioni.Claudio/ex6.m | 59 ++++++++++++++++++ mp2/project.2.Maggioni.Claudio.pdf | Bin 1354320 -> 1362614 bytes mp2/project.2.Maggioni.Claudio.tex | 56 +++++++++++++++++ 5 files changed, 130 insertions(+), 12 deletions(-) create mode 100644 mp2/Project.2.Maggioni.Claudio/degcentrality.m create mode 100644 mp2/Project.2.Maggioni.Claudio/ex6.m diff --git a/mp2/Project.2.Maggioni.Claudio/degcentrality.m b/mp2/Project.2.Maggioni.Claudio/degcentrality.m new file mode 100644 index 0000000..4c4a31e --- /dev/null +++ b/mp2/Project.2.Maggioni.Claudio/degcentrality.m @@ -0,0 +1,14 @@ +function degcentrality(names, A) + counts = full(sum(A, 2)); + ranks = sortrows([counts, (1:size(counts,1))'], 'descend'); + + for i = 1:size(ranks, 1) + fprintf("%14s %2d: ", names(ranks(i, 2)), ranks(i, 1)-1); + for j = 1:size(A, 2) + if ranks(i, 2) ~= j && A(ranks(i, 2), j) > 0 + fprintf("%s, ", names(j)); + end + end + fprintf("\n"); + end +end \ No newline at end of file diff --git a/mp2/Project.2.Maggioni.Claudio/ex3.m b/mp2/Project.2.Maggioni.Claudio/ex3.m index 23ff354..4b93cf4 100644 --- a/mp2/Project.2.Maggioni.Claudio/ex3.m +++ b/mp2/Project.2.Maggioni.Claudio/ex3.m @@ -2,15 +2,4 @@ clear; clc; load('householder/housegraph.mat') names = split(strtrim(convertCharsToStrings(name'))); -counts = full(sum(A, 2)); -ranks = sortrows([counts, (1:size(counts,1))'], 'descend'); - -for i = 1:size(ranks, 1) - fprintf("%14s %2d: ", names(ranks(i, 2)), ranks(i, 1)-1); - for j = 1:size(A, 2) - if ranks(i, 2) ~= j && A(ranks(i, 2), j) > 0 - fprintf("%s, ", names(j)); - end - end - fprintf("\n"); -end +degcentrality(names, A); \ No newline at end of file diff --git a/mp2/Project.2.Maggioni.Claudio/ex6.m b/mp2/Project.2.Maggioni.Claudio/ex6.m new file mode 100644 index 0000000..a932c87 --- /dev/null +++ b/mp2/Project.2.Maggioni.Claudio/ex6.m @@ -0,0 +1,59 @@ +clc; +clear; +fileID = fopen('karate.adj','r'); +row = split(strtrim(fgetl(fileID))); +n = size(row,1); +frewind(fileID); + +ii = []; +jj = []; +vv = []; + +for i = 1:n + row = split(strtrim(fgetl(fileID))); + for j = 1:n + num = str2double(row(j)); + if num ~= 0 + ii(end+1) = i; + jj(end+1) = j; + vv(end+1) = num; + end + end +end +fclose(fileID); + +A = sparse(ii,jj,vv,n,n); +names = string(1:n); + +disp("Exercise 6.1:"); +degcentrality(names,A); + +disp("Exercise 6.2:"); +pagerank(names,A); + +disp("Exercise 6.4:"); +x = randperm(n); +gs = 450; + +%group1 = [1 2 3 4 5 6 7 8 11 12 13 14 17 18 20 22]; +%group2 = [9 10 15 16 19 21 23:34]; + +deg = sum(A); +L = full(diag(deg) - A); +[V, D] = eig(L); +fprintf("lambda_2: %d\n", D(2,2)); +plot(sort(V(:,2)), '.-'); + +[ignore, p] = sort(V(:,2)); +figure; +subplot(1,2,1) +spy(A); +subplot(1,2,2) +spy(A(p,p)); +fprintf("Group 1: "); +display(sort(p(1:16))'); +fprintf("Group 2: "); +display(sort(p(17:end))'); + + + diff --git a/mp2/project.2.Maggioni.Claudio.pdf b/mp2/project.2.Maggioni.Claudio.pdf index 78329172c808b60c17ec7d14a750d4b0df4ca5be..4f4750d14568f68d5debec6b600b423ce61d1795 100644 GIT binary patch delta 121805 zcmV(|K+(U@)Ka#UQ-Fj4gaU*Egam{Iga(8Mgb0KQgbIWUgbaiYv<^q412H!>m%*G6 zDSzEqOOM+&5WeSE_*NNMt9KfrsyLL(ggxE$w8KBn}{qKvb-DT*LQ|P%JM_I z3$%x#CsV_jZ-ztio1xicHDQz2FPQ&*b@$@ipHea5EY4ZZCwI$9CStDmB-JeDnV#G& zCLbb^PiI_f7O|Ky4dX~mXM!c^@Ru_wqklgOYZlW_cfSDqj3=?sIYCJFT!=sF)inT~ zJs04WslQqWG1GA(vojpVdmuysurZ+Di*9WTRN6T>&SR14Q@E3;2q3!v`pvv*+S(jr zO3z{v&t6;8t_($U1t&R&;Uh4=nz}ZJ(Rk8}&0-@!KbyL>rOS4IsNh6#ccOHb<9~^o zg}g}_b2w}iDkh&Z5&cz^>tZT-)E>UsO=l9oKjSCJSkxAsAz4}N=eN_D%2~8)K*XZb z78T9arbFciG;5sMXPRi1^t)`WsTaGo-Fgpm_~8+naWXTe#+fXViZnK6Gk0!GyQ7&8 zn$gtO$s1dDyH6(zA0!}~WO1G(3V+uR2U$qS1`jLexOS#>^nNPxsI_j5TELRs4;;Iy zKw(*$zv|esx{ue zGO(5L5q}>lgiz^bD|T9$1=|$8W)B6 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between 34 members [50 points]} +\subsection{Write a Matlab code that ranks the five nodes with the largest +degree centrality? What are their degrees?} + +Results found here can be computed using the file \texttt{ex6.m}. + +Please find the top 5 nodes by degree centrality, with their degree and their +neighbours listed below: + +\begin{verbatim} +Node Degree: Neighbours... + 34 16: 9, 10, 14, 15, 16, 19, 20, 21, 23, 24, 27, 28, 29, 30, 31, 32, 33, + 1 15: 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 18, 20, 22, 32, + 33 11: 3, 9, 15, 16, 19, 21, 23, 24, 30, 31, 32, 34, + 3 9: 1, 2, 4, 8, 9, 10, 14, 28, 29, 33, + 2 8: 1, 3, 4, 8, 14, 18, 20, 22, 31, +\end{verbatim} + +\subsection{Rank the five nodes with the largest eigenvector centrality. What are +their (properly normalized) eigenvector centralities?} + +Results found here can be computed using the file \texttt{ex6.m}. + +Please find the top 5 nodes by eigenvector centrality (page-rank column) +listed below: + +\begin{verbatim} + page-rank in out author + 34 0.1009 17 17 34 + 1 0.0970 16 16 1 + 33 0.0717 12 12 33 + 3 0.0571 10 10 3 + 2 0.0529 9 9 2 +\end{verbatim} + +\subsection{Are the rankings in (a) and (b) identical? Give a brief verbal +explanation of the similarities and differences.} + +The rankings found are identical, even though if we normalize the degree +centrality to the greatest eigenvector centrality we find slighly different +values ($[0.1009, 0.0946, 0.0694, 0.0568, 0.0505]$) w.r.t the actual eigenvector +centrality. + +The identical rankings may be explained by the fact that by computing the +eigenvector centrality we are effectively applying PageRank to a symmetrical +matrix, i.e. to a graph with bidirectional links. Since the links are +bidirectional, we effectively make all the nodes in the graph of the same +``importance'' to the eyes of PageRank, thus avoiding a case where a node has +high PageRank thank to connections with few, but very ``important'' nodes. +Therefore PageRank is simply reduced to a priotarization of nodes with many +edges, i.e. the degree centrality ranking. + +\subsection{Use spectral graph partitioning to find a near-optimal split of the +network into two groups of 16 and 18 nodes, respectively. List the nodes in the +two groups. How does spectral bisection compare to the real split observed by +Zachary?} + \begin{thebibliography}{99} \bibitem{karate} The social network of a karate club at a US university, M.~E.~J. Newman and M. Girvan, Phys. Rev. E 69,026113 (2004) pp. 219-229.