(* ::Package:: *)

Needs["ErrorBarPlots`"]

(* ::Section:: *)

(*TODO*)

(* ::Text:: *)

(*- Use different starting point for switch chain that is closer to uniform:*)

(*   Do configuration model, starting with the vertex with highest degree and keeping track of a "forbidden list" meaning dont pair something that is not allowed*)

(*   (a) How close is this 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.*)

(**)

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

(* ::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.*)

(*  *)

(*  *)

(* ::Section:: *)

(*Visualize graphs*)

gsraw=Import[NotebookDirectory[]<>"graphdata.m"];

ListPlot[gsraw[[2]],Joined->True,PlotRange->All,AxesLabel->{"Step","Triangles"}]

gs=Map[Graph[#,GraphLayout->"CircularEmbedding"]&,gsraw[[1]]];

gs2=Map[Graph[#,GraphLayout->Automatic]&,gsraw[[1]]];

Grid[Partition[gs,10],Frame->All]

(* ::Section:: *)

(*Plot triangle counts*)

(* ::Subsection:: *)

(*Data import and data merge*)

gsraw=Import[NotebookDirectory[]<>"data/graphdata_merged.m"];

gsraw=SortBy[gsraw,#[[1,1]]&]; (* Sort by n *)

gdata=GatherBy[gsraw,{#[[1,2]]&,#[[1,1]]&}];

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

newData=Import[NotebookDirectory[]<>"data/graphdata_3.m"];

mergedData=Import[NotebookDirectory[]<>"data/graphdata_merged.m"];

Export[NotebookDirectory[]<>"data/graphdata_merged_new.m",Join[mergedData,newData]]

(* ::Subsection:: *)

(*Plot empirical distribution of maximum degree*)

maxDegrees=Map[{#[[1]],Max[#[[3]]]}&,gsraw];

maxDegrees=GatherBy[maxDegrees,{#[[1,2]]&,#[[1,1]]&}];

(* maxDegrees[[ tau index, n index, run index,  ntau or dmax ]] *)

Histogram[maxDegrees[[1,-1,All,2]],PlotRange->{{0,2000},{0,100}},AxesLabel->{"d_max","frequency"}]

Histogram[maxDegrees[[2,-1,All,2]],PlotRange->{{0,2000},{0,100}},AxesLabel->{"d_max","frequency"}]

Histogram[maxDegrees[[3,-1,All,2]],PlotRange->{{0,2000},{0,100}},AxesLabel->{"d_max","frequency"}]

(* ::Subsection:: *)

(*Plot #trianges vs some degree-sequence-property*)

getProperty[ds1_]:=Module[{ds,n=Length[ds1],tmp=ConstantArray[0,{Length[ds1],Length[ds1]}]},

ds=N[ds1/Sqrt[N[Total[ds1]]]]; (* scale degrees by 1/Sqrt[total] *)

(* The next table contains unneeded entries, but its faster to have a square table for the sum *)

tmp=Table[1.-Exp[-ds[[i]]ds[[j]]],{i,1,n},{j,1,n}];

Sum[tmp[[i,j]]*tmp[[j,k]]*tmp[[i,k]],{i,3,n},{j,2,i-1},{k,1,j-1}] (* somehow i>j>k is about 60x faster than doing i<j<k !!! *)

(* This sparser table is slower

tmp=Table[1.-Exp[-ds[[i]]ds[[j]]],{i,1,n-1},{j,i+1,n}];

(* tmp[[a,b]] is now with  ds[[a]]*ds[[a+b]] *)

Sum[tmp[[i,j-i]]*tmp[[j,k-j]]*tmp[[i,k-i]],{i,1,n-2},{j,i+1,n-1},{k,j+1,n}]

*)

];

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

avgAndProp=ParallelMap[{getProperty[#[[3]]],Mean[#[[2,1;;-1]]]}&,gdata[[2,4,1;;100]]];

Show[ListPlot[avgAndProp,AxesOrigin->{0,0},AxesLabel->{"degree-sequence-property","<#triangles>"},AspectRatio->1],Plot[x,{x,1,1000}]]

(* ::Subsection:: *)

(*Plot triangle count over "time" in Markov chain instances*)

numPlots=20;

selectedData=gsraw[[-numPlots;;-1]];

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

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

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

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

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

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

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

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

(* ::Subsection:: *)