(* ::Package:: *)

Needs["ErrorBarPlots`"]

(* ::Section:: *)

(*TODO*)

(* ::Text:: *)

(*- Triangle law exponent: gather more data*)

(**)

(*- Why does GCM-2 start with very low #triangles*)

(*  Do not only consider number of standard deviations but also relative number of triangles.*)

(*  Look at the following: for all triangles (v1, v2, v3) consider the degrees d1<d2<d3 and make a scatter plot of di vs dj. Make such a scatter plot for the initial GCM-2 graph and for a mixed graph and see how it changes.*)

(**)

(*- GCM success rates: for the degree sequences where it "always fails", look at the degree sequence. Does it have a low/high number of degree 1 nodes? Is the maximum degree very low/high?*)

(**)

(*- Does GCM start closer to uniform?*)

(*   (a) How close to uniform ? At least w.r.t. the measure of #triangles*)

(*   (b) How often does this procedure work/fail. Might still be faster to do switchings from Erdos-Gallai.*)

(*   (d) Time evolution for GCM on top of Erdos-Gallai time evolution.*)

(**)

(*- Count #moves that result in +-k triangles (one move could create many triangles at once!)*)

(**)

(*- For a graph snapshot: for all V shapes, compute the number of ways to make it into a triangle:*)

(*  Let u1,u2 be the endpoints of the V. For all neighbors v1 of u1 and v2 of u2, see of v1,v2 has an edge. Meaning, if we were to select randomly an u1 edge and an u2 edge, then whats the probability that it can be used to switch the V into a triangle.*)

(**)

(*- Improve runtime*)

(*   (a) Better direct triangle counting? (I doubt it)*)

(*   (b) Better triangle counting by only keeping track of CHANGES in #triangles*)

(* ::Subsection:: *)

(*Done*)

(* ::Text:: *)

(*- Do a single very long run: nothing weird seems to happen with the triangle counts. Tried 10 million steps.*)

(**)

(*- Compute  Sum over i<j<k  of  (1-Exp[- d_i d_j / (2E)]) * (1 - Exp[-d_j d_k / (2E)]) * (1 - Exp[-d_k d_i / (2E)]) .*)

(*  Computing the f(i,j) = (1-Exp[..]) terms is fine, but then computing Sum[ f(i,j) f(j,k) f(i,k) ) ] over n^3 entries is very slow.*)

(*  *)

(*- Improve runtime*)

(*   (a) Don't remove/add edges from the std::vector. Simply replace them. Done, is way faster for large n.*)

(*   (b) Do not choose the three permutations with 1/3 probability: choose the "staying" one with zero probability. Should still be a valid switch chain?*)

(*   *)

(*- Experimental mixing time as function of n. At (n,tau)=(1000,2.5) it seems to be between 10.000 and 20.000 steps.*)

(*   Done. Seems to be something like  (1/2)(32-26(tau-2))n  so we run it for that time without the factor (1/2).*)

minCount=Min[Map[Min[#[[2]]]&,selectedData]];

maxCount=Max[Map[Max[#[[2]]]&,selectedData]];

maxTime=Max[Map[Length[#[[2]]]&,selectedData]];

(* maxTime=30000; *)

skipPts=Max[1,Round[maxTime/500]]; (* Plotting every point is slow. Plot only once per `skipPts` timesteps *)

coarseData=Map[#[[2,1;;maxTime;;skipPts]]&,selectedData];

labels=Map["{n,tau} = "<>ToString[#[[1]]]&,selectedData];

ListPlot[coarseData,Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime},PlotLegends->labels]

(* Map[ListPlot[#,Joined->True,PlotRange\[Rule]{minCount,maxCount},DataRange\[Rule]{0,maxTime}]&,coarseData] *)

selectedData=gdata[[1,1]];

measureSkip=1;

minCount=Min[Map[Min[#[[2]]]&,selectedData]];

maxCount=Max[Map[Max[#[[2]]]&,selectedData]];

maxTime=Max[Map[Length[#[[2]]]&,selectedData]];

maxTime=30000;

skipPts=Max[1,Round[maxTime/100]]; (* Plotting every point is slow. Plot only once per `skipPts` timesteps *)

coarseData=Map[#[[2,1;;maxTime;;skipPts]]&,selectedData];

labels=Map["{n,tau} = "<>ToString[#[[1]]]&,selectedData];

plot1=ListPlot[coarseData[[1;;5]],Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime}]

plot2=ListPlot[coarseData[[6;;10]],Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime}]

plot3=ListPlot[coarseData[[11;;15]],Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime}]

plot4=ListPlot[coarseData[[16;;20]],Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime}]

(* For export *)

numPlots=20;

selectedData=gdata[[2,1]][[-numPlots;;-1]];

measureSkip=1;

minCount=Min[Map[Min[#[[2]]]&,selectedData]];

maxCount=Max[Map[Max[#[[2]]]&,selectedData]];

maxTime=Max[Map[Length[#[[2]]]&,selectedData]];

(* maxTime=30000; *)

skipPts=Max[1,Round[maxTime/5000]]; (* Plotting every point is slow. Plot only once per `skipPts` timesteps *)

coarseData=Map[#[[2,1;;maxTime;;skipPts]]&,selectedData];

labels=Map["{n,tau} = "<>ToString[#[[1]]]&,selectedData];

plotTimeEvol=ListPlot[coarseData,Joined->True,PlotRange->{0*minCount,maxCount},DataRange->{0,measureSkip*maxTime},Frame->True,FrameLabel->{"timesteps","number of triangles"},ImageSize->300]

(* Map[ListPlot[#,Joined->True,PlotRange\[Rule]{minCount,maxCount},DataRange\[Rule]{0,maxTime}]&,coarseData] *)

Export[NotebookDirectory[]<>"plots/timeevol.pdf",plotTimeEvol]

movingAvg=Map[MovingAverage[#[[2]],2000][[1;;-1;;skipPts]]-Mean[#[[2,-20000;;-1]]]&,selectedData[[1;;-1;;5]]];

plotMovingAvg=ListPlot[movingAvg,Joined->True,PlotRange->All,DataRange->{0,measureSkip*maxTime},Frame->True,FrameLabel->{"timesteps","number of triangles"}]

(* ::Subsection:: *)

(*Plot success rate over "time"*)

numPlots=10;

selectedData=gdata[[1,-1]][[-numPlots;;-1]];

measureSkip=100;

maxTime=Max[Map[Length[#[[4]]]&,selectedData]];

maxTime=10000;

coarseData=Map[#[[4,1;;maxTime/measureSkip]]&,selectedData];

labels=Map["{n,tau} = "<>ToString[#[[1]]]&,selectedData];

ListPlot[coarseData,Joined->True,PlotRange->{0,100},DataRange->{0,maxTime},PlotLegends->labels]

(* Map[ListPlot[#,Joined->True,PlotRange\[Rule]{minCount,maxCount},DataRange\[Rule]{0,maxTime}]&,coarseData] *)

(* ::Subsection:: *)

(*Correlation of avgsuccess rate vs other things*)

compare1=Map[{Mean[#[[4]]],Mean[#[[2]]]}&,gdata,{3}];

(* { GCM1 rate, GCM2 rate, mixing time from ErdosGallai } *)

scatterPlots=Map[ListPlot[#,PlotRange->{{0,100},All},PlotStyle->PointSize[Large]]&,compare1,{2}];

TableForm[scatterPlots,TableHeadings->{taulabels,nlabels}]

(* ::Subsection:: *)

(*Compute 'mixing time'*)

(* Compute average of last part and check when the value drops below that for the first time *)

getMixingTime[values_]:=Module[{avg},

    (* average over the last 20 percent *)

    avg=Mean[values[[-Round[Length[values]/5];;-1]]];

    FirstPosition[values,_?(#<=avg&)][[1]]

]

(* gdata[[ tau index, n index, run index , {ntau, #tris, ds} ]] *)

measureSkip=1;

mixingTimes=Map[{#[[1,1]],(1/#[[1,1]])measureSkip * getMixingTime[#[[2]]]}&,gdata,{3}];

mixingTimesBars=Map[

    {{#[[1,1]],Mean[#[[All,2]]]},ErrorBar[StandardDeviation[#[[All,2]]](*/Sqrt[Length[#]]*)]}&

,mixingTimes,{2}];

ErrorListPlot[mixingTimesBars,Joined->True,PlotMarkers->Automatic,AxesLabel->{"n","~~mixing time divided by n"},PlotLegends->taulabels]

(* For n fixed, look at function of tau *)

measureSkip=1;

mixingTimesBars=Map[

    {{#[[1,1]],Mean[#[[All,2]]]},ErrorBar[StandardDeviation[#[[All,2]]]]}&

Show[

,Plot[(32-20(tau-2)),{tau,2,3}]]

(* ::Subsection:: *)

(*Plot #triangles distribution for specific (n,tau)*)

plotRangeByTau[tau_]:=Piecewise[{{{0,30000},tau<2.3},{{0,4000},2.3<tau<2.7},{{0,800},tau>2.7}},Automatic]

histograms=Map[Histogram[#[[All,2]],PlotRange->{plotRangeByTau[#[[1,1,2]]],Automatic}]&,averagesGrouped,{2}];

(* TableForm[histograms,TableHeadings->{taulabels,nlabels}] *)

TableForm[Transpose[histograms],TableHeadings->{nlabels,taulabels}]

(* ::Section:: *)

(*Greedy configuration model*)

(* ::Subsection:: *)

(*Distribution of initial #triangles for GCM1,GCM2,EG compared to average #triangles.*)

(* Data format: *)

(* gdata[[ tau index, n index, run index , datatype index ]] *)

(* datatype index:

1: {n,tau}

2: #triangles time sequence

3: degree sequence

4: GCM1 starting triangle counts

5: GCM2 starting triangle counts

6: GCM1 time sequence

7: GCM2 time sequence

*)

 (* Stats for a single run at every (n,tau) *)

timeWindow=Round[Length[gdata[[1,1,1,2]]]/10];

skipPts=Max[1,Round[timeWindow/100]];

getSingleStats[runs_]:=Module[{run,avg,stddev},

    run=runs[[1]]; (* Select some run *)

    avg=N[Mean[run[[2,-timeWindow;;-1]]]];

    stddev=N[StandardDeviation[run[[2,timeWindow;;-1]]]];

    {run[[1]], (* {n,tau} *)

    stddev/avg,

    (run[[2,1]])/avg, (* EG starting point *)

    Map[N[#/avg]&,run[[4]]],  (* GCM1 counts *)

    Map[N[#/avg]&,run[[5]]],  (* GCM2 counts *)