AENC/switchchain Changeset - c9c22e41130d · Centrum Wiskunde & Informatica (CWI)

Changeset - c9c22e41130d

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0 10 1

Tom Bannink - 8 years ago 2017-06-11 13:29:41
tombannink@gmail.com

Move degree sequence generation to separate file

11 files changed with 153 insertions and 260 deletions:

cpp/graph_powerlaw.hpp

cpp/switchchain.cpp

cpp/switchchain.hpp

cpp/switchchain_dsp.cpp

cpp/switchchain_exponent.cpp

cpp/switchchain_initialtris.cpp

cpp/switchchain_mixingtime.cpp

cpp/switchchain_spectrum.cpp

cpp/switchchain_successrates.cpp

cpp/switchchain_timeevol.cpp

triangle_etmt_plots.m

0 comments (0 inline, 0 general)

cpp/graph_powerlaw.hpp

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@@ new file 100644 @@
 #pragma once
 #include "graph.hpp"
 #include "powerlaw.hpp"
 #include <algorithm>
 #include <iostream>
 template <typename RNG>
 void generatePowerlawGraph(int n, float tau, Graph& g, DegreeSequence& ds,
                            RNG& rng) {
     ds.resize(n);
     powerlaw_distribution degDist(tau, 1, n);
     // Generate a graph
     // might require multiple tries
     for (int i = 1;; ++i) {
         std::generate(ds.begin(), ds.end(),
                       [&degDist, &rng] { return degDist(rng); });
         // First make the sum even
         unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
         if (sum % 2) {
             continue;
             // Can we do this: ??
             ds.back()++;
+        }
         if (g.createFromDegreeSequence(ds))
             break;
         // When 10 tries have not worked, output a warning
         if (i % 10 == 0) {
             std::cerr << "Warning: could not create graph from "
                          "degree sequence. Trying again...\n";
+        }
+    }
+}

cpp/switchchain.cpp

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@@ @@ -2,8 +2,8 @@ @@
 #include "exports.hpp"
 #include "graph.hpp"
 #include "graph_gcm.hpp"
 #include "graph_powerlaw.hpp"
 #include "graph_spectrum.hpp"
 #include "powerlaw.hpp"
 #include <algorithm>
 #include <array>
 #include <fstream>
@@ @@ -65,37 +65,11 @@ int main(int argc, char* argv[]) { @@
     for (int numVertices = 500; numVertices <= 500; numVertices += 1000) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 5; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
 #if 0
                 //

cpp/switchchain.hpp

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 #pragma once
 #include "graph.hpp"
 #include <iostream>
 #include <random>

cpp/switchchain_dsp.cpp

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 #include "exports.hpp"
 #include "graph.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <fstream>
@@ @@ -9,7 +9,7 @@ @@
 #include <random>
 #include <vector>
-double getProperty(const DegreeSequence& ds) {
+double getDSTN(const DegreeSequence& ds) {
     std::vector<std::vector<double>> vals(ds.size());
     for (auto& v : vals) {
         v.resize(ds.size(), 0);
@@ @@ -38,56 +38,60 @@ double getProperty(const DegreeSequence& ds) { @@
     return result;
+}
 int main() {
     // Generate a random degree sequence
     std::mt19937 rng(std::random_device{}());
     // Goal:
     // Degrees follow a power-law distribution with some parameter tau
     // Expect:  #tri = const * n^{ something }
     // The goal is to find the 'something' by finding the number of triangles
     // for different values of n and tau
     float tauValues[] = {2.1f, 2.5f, 2.9f};
 int main(int argc, char* argv[]) {
     // Simulation parameters
     const int numVerticesMin = 100;
     const int numVerticesMax = 1000;
     const int numVerticesStep = 100;
     float tauValues[] = {2.1f, 2.2f, 2.3f, 2.4f, 2.5f, 2.6f, 2.7f, 2.8f, 2.9f};
     const int totalDegreeSamples = 2000;
     auto getMixingTime = [](int n, float tau) {
         return int(30.0f * (50.0f - 30.0f * (tau - 2.0f)) * n);
     };
     constexpr int measurements = 50;
     constexpr int measureSkip = 200; // Take a sample every ... steps
     // Output file
     std::ofstream outfile;
     if (argc >= 2)
         outfile.open(argv[1]);
     else
         outfile.open("graphdata_dsp.m");
     if (!outfile.is_open()) {
         std::cout << "ERROR: Could not open output file.\n";
         return 1;
+    }
     Graph g;
     // Output Mathematica-style comment to indicate file contents
     outfile << "(*\n";
     outfile << "n from " << numVerticesMin << " to " << numVerticesMax
             << " step " << numVerticesStep << std::endl;
     outfile << "tauValues: " << tauValues << std::endl;
     outfile << "degreeSamples: " << totalDegreeSamples << std::endl;
     outfile << "mixingTime: 30 * (50 - 30 (tau - 2)) n\n";
     outfile << "data:\n";
     outfile << "1: {n,tau}\n";
     outfile << "2: avgTriangles\n";
     outfile << "3: dstn\n";
     outfile << "*)" << std::endl;
     std::ofstream outfile("graphdata_dsp.m");
     outfile << '{';
     bool outputComma = false;
     for (int numVertices = 1000; numVertices <= 1000; numVertices += 1000) {
     std::mt19937 rng(std::random_device{}());
     Graph g;
     for (int numVertices = numVerticesMin; numVertices <= numVerticesMax;
          numVertices += numVerticesStep) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 2000; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
             for (int degreeSample = 0; degreeSample < totalDegreeSamples;
                  ++degreeSample) {
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 SwitchChain chain;
                 if (!chain.initialize(g)) {
@@ @@ -98,12 +102,8 @@ int main() { @@
                 std::cout << "Running n = " << numVertices << ", tau = " << tau
                           << ". \t" << std::flush;
                 int mixingTime = 32*(32.0f - 10.0f*(tau - 2.0f)) * numVertices; //40000;
                 constexpr int measurements = 50;
                 constexpr int measureSkip =
 ; // Take a sample every ... steps
                 int movesDone = 0;
                 int mixingTime = getMixingTime(numVertices,tau);
                 long long trianglesTotal = 0;
@@ @@ -117,24 +117,25 @@ int main() { @@
                             ++movesDone;
                     trianglesTotal += chain.g.countTriangles();
+                }
                 float avgTriangles =
                     float(trianglesTotal) / float(measurements);
                 std::cout << movesDone << '/' << mixingTime + measurements * measureSkip
                 std::cout << movesDone << '/'
                           << mixingTime + measurements * measureSkip
                           << " moves succeeded ("
                           << 100.0f * float(movesDone) /
                                  float(mixingTime + measurements * measureSkip)
                           << "%).";
                 std::cout << std::flush;
                 //std::cout << std::endl;
                 if (outputComma)
                     outfile << ',' << '\n';
                 outputComma = true;
                 float avgTriangles =
                     float(trianglesTotal) / float(measurements);
                 outfile << '{' << '{' << numVertices << ',' << tau << '}';
                 outfile << ',' << avgTriangles;
                 outfile << ',' << getProperty(ds) << '}' << std::flush;
                 outfile << ',' << getDSTN(ds);
                 outfile << '}' << std::flush;
                 std::cout << std::endl;
+            }

cpp/switchchain_exponent.cpp

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 #include "exports.hpp"
 #include "graph.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <fstream>
@@ @@ -28,37 +28,11 @@ int main() { @@
     for (int numVertices = 1000; numVertices <= 10000; numVertices += 1000) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 2000; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 SwitchChain chain;
                 if (!chain.initialize(g)) {

cpp/switchchain_initialtris.cpp

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 #include "exports.hpp"
 #include "graph.hpp"
 #include "graph_gcm.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <fstream>
@@ @@ -29,37 +29,11 @@ int main() { @@
     for (int numVertices = 200; numVertices <= 2000; numVertices += 400) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 200; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 std::cout << "Running n = " << numVertices << ", tau = " << tau
                           << "." << std::flush;

cpp/switchchain_mixingtime.cpp

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 #include "exports.hpp"
 #include "graph.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <array>
@@ @@ -43,37 +43,12 @@ int main(int argc, char* argv[]) { @@
     for (int numVertices = 100; numVertices <= 1000; numVertices += 100) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 200; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 // Multiple runs from the same degree sequence
                 for (int i = 0; i < 5; ++i) {

cpp/switchchain_spectrum.cpp

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@@ @@ -3,7 +3,7 @@ @@
 #include "graph.hpp"
 #include "graph_gcm.hpp"
 #include "graph_spectrum.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include <algorithm>
 #include <array>
 #include <fstream>
@@ @@ -42,37 +42,11 @@ int main(int argc, char* argv[]) { @@
     for (int numVertices = 500; numVertices <= 500; numVertices += 1000) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 5; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 SwitchChain chain;
                 if (!chain.initialize(g)) {

cpp/switchchain_successrates.cpp

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Show inline comments

 #include "exports.hpp"
 #include "graph.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <array>
@@ @@ -41,37 +41,11 @@ int main(int argc, char* argv[]) { @@
     for (int numVertices = 1000; numVertices <= 1000; numVertices += 1000) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 2000; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 SwitchChain chain;
                 if (!chain.initialize(g)) {

cpp/switchchain_timeevol.cpp

➞

Show inline comments

 #include "exports.hpp"
 #include "graph.hpp"
-#include "powerlaw.hpp"
+#include "graph_powerlaw.hpp"
 #include "switchchain.hpp"
 #include <algorithm>
 #include <array>
@@ @@ -41,37 +41,11 @@ int main(int argc, char* argv[]) { @@
     for (int numVertices = 1000; numVertices <= 1000; numVertices += 1000) {
         for (float tau : tauValues) {
             DegreeSequence ds(numVertices);
             powerlaw_distribution degDist(tau, 1, numVertices);
             //std::poisson_distribution<> degDist(12);
             // For a single n,tau take samples over several instances of
             // the degree distribution.
             // 500 samples seems to give reasonable results
             for (int degreeSample = 0; degreeSample < 5; ++degreeSample) {
                 // Generate a graph
                 // might require multiple tries
                 for (int i = 1; ; ++i) {
                     std::generate(ds.begin(), ds.end(),
                                   [&degDist, &rng] { return degDist(rng); });
                     // First make the sum even
                     unsigned int sum = std::accumulate(ds.begin(), ds.end(), 0);
                     if (sum % 2) {
                         continue;
                         // Can we do this: ??
                         ds.back()++;
+                    }
                     if (g.createFromDegreeSequence(ds))
                         break;
                     // When 10 tries have not worked, output a warning
                     if (i % 10 == 0) {
                         std::cerr << "Warning: could not create graph from "
                                      "degree sequence. Trying again...\n";
+                    }
+                }
                 DegreeSequence ds;
                 generatePowerlawGraph(numVertices, tau, g, ds, rng);
                 // Multiple runs from the same degree sequence
                 for (int i = 0; i < 5; ++i) {

triangle_etmt_plots.m

➞

Show inline comments

@@ @@ -3,7 +3,7 @@ @@
 Needs["ErrorBarPlots`"]
-gsraw=Import[NotebookDirectory[]<>"data/graphdata_etmt_partial.m"];
+gsraw=Import[NotebookDirectory[]<>"data/graphdata_etmt.m"];
 (* gsraw=SortBy[gsraw,{#[[1,1]]&,#[[1,2]]&}]; (* Sort by n and then by tau. The {} forces a *stable* sort because otherwise Mathematica sorts also on triangle count and other things. *) *)
@@ @@ -14,17 +14,34 @@ gdata=GatherBy[gsraw,{#[[1,2]]&,#[[1,1]]&}]; @@
 : {n,tau}
 : etmt
 *)
 tauvalues=gdata[[All,1,1,1,2]];
 nlabels=Map["n = "<>ToString[#]&,gdata[[1,All,1,1,1]]];
 taulabels=Map["\[Tau] = "<>ToString[#]&,gdata[[All,1,1,1,2]]];
 etmtMean=Map[Mean[N[#[[All,2]]]]&,gdata,{2}];
 etmtSD=Map[StandardDeviation[N[#[[All,2]]]]&,gdata,{2}];
 etmtQuantile=Map[Quantile[#[[All,2]],99/100]&,gdata,{2}];
 histograms=Map[Histogram[#[[All,2]]]&,gdata,{2}];
 histogramsWithLine=Map[Histogram[#[[All,2]],Epilog->Line[{{Mean[N[#[[All,2]]]]+StandardDeviation[N[#[[All,2]]]],0},{Mean[N[#[[All,2]]]]+StandardDeviation[N[#[[All,2]]]],500}}]]&,gdata,{2}];
 histogramsWithLines=MapIndexed[
 Histogram[#[[All,2]],PlotRange->All,
 Epilog->Line[{
 {{etmtMean[[#2/.List->Sequence]],0},
 {etmtMean[[#2/.List->Sequence]],500}},
 {{etmtMean[[#2/.List->Sequence]]+etmtSD[[#2/.List->Sequence]],0},
 {etmtMean[[#2/.List->Sequence]]+etmtSD[[#2/.List->Sequence]],500}},
 {{etmtQuantile[[#2/.List->Sequence]],0},
 {etmtQuantile[[#2/.List->Sequence]],500}}
 }]
+]
 &,gdata,{2}];
-TableForm[histograms,TableHeadings->{taulabels,nlabels}]
+TableForm[histogramsWithLines,TableHeadings->{taulabels,nlabels}]
 gdataSwitched=Transpose[gdata];
 mixingTimesBars=Map[{{#[[1,1,1]],Mean[#[[All,2]]]},ErrorBar[StandardDeviation[#[[All,2]]]]}&,gdata,{2}];
 mixingTimesQuantiles=Map[{#[[1,1,1]],Quantile[#[[All,2]],99/100]}&,gdata,{2}];
 ErrorListPlot[mixingTimesBars[[{1,2,3,5,8}]],Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","ETMT"},PlotLegends->taulabels]
 plot1=ErrorListPlot[mixingTimesBars[[{1,2,3,5,8}]],Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","ETMT"},PlotLegends->taulabels];
 plot2=ListPlot[mixingTimesQuantiles[[{1,2,3,5,8}]],Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","ETMT"},PlotLegends->taulabels];
 Show[plot2,plot1]
 mixingTimesDivN=Map[{#[[1,1]],#[[1,2]]/(#[[1,1]])}&,mixingTimesBars,{2}];
 mixingTimesDivNlogN=Map[{#[[1,1]],#[[1,2]]/(#[[1,1]]*Log[#[[1,1]]])}&,mixingTimesBars,{2}];
 etmtQuantileDivN=Map[{#[[1]],#[[2]]/(#[[1]])}&,mixingTimesQuantiles,{2}];
 etmtQuantileDivNlogN=Map[{#[[1]],#[[2]]/(#[[1]]*Log[#[[1]]])}&,mixingTimesQuantiles,{2}];
 plotN=ListPlot[mixingTimesDivN,Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","\[LeftAngleBracket]ETMT\[RightAngleBracket]/n"},PlotLegends->taulabels,ImageSize->300]
 plotNlogN=ListPlot[mixingTimesDivNlogN,Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","\[LeftAngleBracket]ETMT\[RightAngleBracket]/(n log n)"},PlotLegends->taulabels,ImageSize->300]
 plotQuantileN=ListPlot[etmtQuantileDivN,Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","q(ETMT,99%)/n"},PlotLegends->taulabels,ImageSize->300]
 plotQuantileNlogN=ListPlot[etmtQuantileDivNlogN,Joined->True,PlotMarkers->Automatic,Frame->True,FrameLabel->{"n","q(ETMT,99%)/(n log n)"},PlotLegends->taulabels,ImageSize->300]
 Export[NotebookDirectory[]<>"plots/ETMTdivN.pdf",plotN]
 etmtQuantileDivNmax=Map[Max[#[[All,2]]]&,etmtQuantileDivN];
 etmtQuantileDivNmax=Transpose[{tauvalues,etmtQuantileDivNmax}];
 mixingTimesDivNmax=Map[Max[#[[All,2]]]&,mixingTimesDivN];
 mixingTimesDivNmax=Transpose[{tauvalues,mixingTimesDivNmax}];
 Show[
 Plot[{(50-30(tau-2)),32-26(tau-2)},{tau,2,3},AxesOrigin->{2,0}],
 ListPlot[{etmtQuantileDivNmax,mixingTimesDivNmax},Joined->True,PlotMarkers->Automatic,PlotRange->{{2,3},All}]
+]

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