mp2: done 1-7.3 and MATLAB's part of 7.4

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

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));
+degcentrality(names, A);
-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

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))');
+
+
+

mp2/project.2.Maggioni.Claudio.tex

@@ -366,6 +366,62 @@ The PageRank values for all authors were computing by using the scripts
 
 \section{Zachary's karate club: social network of friendships 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.