AENC/switchchain Changeset - 9c99596eadb5 · Centrum Wiskunde & Informatica (CWI)

@@ -9,21 +9,20 @@ Needs["ErrorBarPlots`"]

(* ::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.*)

(*  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.*)

(* TODO: Investigate #triangles not only in number of standard deviations but also percentage above/below average.*)

(**)

(*- 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.*)

(**)

@@ -62,42 +61,35 @@ Needs["ErrorBarPlots`"]

(*The success rates, conditioned on the degree sequence being graphical, is almost always higher using GCM2. For certain degree sequences the success rate of GCM2 can be 0.9 higher than that of GCM1. (i.e. amost always works vs almost always fails).*)

(*For tau > ~2.3 the success rate of GCM2 seems to be higher than 80% for most sequences.*)

(*For tau < ~2.3 the success rate of GCM2 can drop to less than 10% for some sequences but for many sequences it is still larger than 80%.*)

(**)

(*Success rate of GCM seems to be correlated with mixing time from Erdos-Gallai: higher success rate implies lower mixing time.  *)

(**)

(*The initial #triangles in GCM2 is somewhere between 0 and 5 standard deviations below the average #triangles, whereas the #triangles in Erdos-Gallai can be as high as 100 standard deviations above the average.*)

(* ::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]

(*Initial #triangles in both GCM1 and GCM2 is always below the average #triangles whereas Erdos-Gallai is usually many times higher than average.*)

(* ::Section:: *)

(*Data import*)

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

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

(* 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. *) *)

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

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

(* 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

*)

nlabels=Map["n = "<>ToString[#]&,gdata[[1,All,1,1,1]]];

taulabels=Map["tau = "<>ToString[#]&,gdata[[All,1,1,1,2]]];

(* ::Subsection:: *)

(*Merge data*)

@@ -119,14 +111,14 @@ Export[NotebookDirectory[]<>"data/graphdata_merged_new.m",Join[mergedData,newDat

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"}]

Histogram[maxDegrees[[4,-1,All,2]],PlotRange->{{0,2000},{0,100}},AxesLabel->{"d_max","frequency"}]

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

(* ::Subsection:: *)

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

@@ -152,13 +144,13 @@ Show[ListPlot[avgAndProp,AxesOrigin->{0,0},AxesLabel->{"degree-sequence-property

(* ::Subsection:: *)

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

numPlots=20;

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

selectedData=gdata[[4,-1]][[-numPlots;;-1]];

measureSkip=1;

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

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

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

skipPts=Max[1,Round[maxTime/200]]; (* Plotting every point is slow. Plot only once per `skipPts` timesteps *)

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

@@ -213,32 +205,78 @@ TableForm[Transpose[histograms],TableHeadings->{nlabels,taulabels}]

(* ::Section:: *)

(*Greedy configuration model*)

(* ::Subsection:: *)

(*#triangles(GCM) distribution vs #triangles(SwitchChain)*)

(*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];

getStats[run_]:=Module[{avg,stddev},

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]],stddev/avg,(run[[2,1]])/avg,Map[N[#/avg]&,run[[4]]]}

    {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 *)

singleStats=Map[getSingleStats,gdata,{2}];

(* Yellow: GCM1 (take new highest everytime *)

(* Blue: GCM2 (finish highest first, more similar to EG) *)

histogramsSingle=Map[Histogram[{#[[4]],#[[5]]},PlotRange->{{0,5},Automatic},ImageSize->300,PlotLabel->"ErdosGallai="<>ToString[NumberForm[#[[3]],3]]<>"\[Cross]average. stddev="<>ToString[NumberForm[#[[2]],3]]<>"\[Cross]average"]&,singleStats,{2}];

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

 (* Consider all runs *)

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

getAverage[run_]:=Module[{avg,stddev},

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

    Mean[run[[4]]]/avg,(* GCM1 counts *)

    Mean[run[[5]]]/avg, (* GCM2 counts *)

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

stats=Map[getStats,gdata,{3}];

getTotalStats[runs_]:=Transpose[Map[getAverage,runs]];

totalStats=Map[getTotalStats,gdata,{2}];

(* Yellow: GCM1 (take new highest everytime *)

(* Blue: GCM2 (finish\[AliasDelimiter] highest first, more similar to EG) *)

histogramsTotal=Map[Histogram[#,{0.1},PlotRange->{{0,5},Automatic},ImageSize->300]&,totalStats,{2}];

histograms=Map[Histogram[{#[[1,4]]},PlotRange->{{0,5},Automatic},PlotLabel->"ErdosGallai="<>ToString[NumberForm[#[[1,3]],3]]<>"\[Cross]average. stddev="<>ToString[NumberForm[#[[1,2]],3]]<>"\[Cross]average"]&,stats,{2}];

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

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

(* ::Subsection:: *)

(*Greedy CM success rates*)

(*GCM1 vs GCM2 success rates*)

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

successrates=Map[{Length[#[[4]]],Length[#[[5]]]}&,gdata,{3}];

successrates=Map[Transpose,successrates,{2}];

successratesDelta=Map[Length[#[[5]]]-Length[#[[4]]]&,gdata,{3}];