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I have the following code where I create instances of random graphs and use their properties to compute the mean of a quantity labeled 'PoA'. In the last line, I use a range of values for the parameters 'nodes' and 'conn', and create a table for plotting the mean.

I now want to bring Standard deviation also into consideration. I would like the plots to have lines for mean and mean$\pm 2 \sigma$. Can someone help me to add those extra lines in the plot?

PoAGen[nodes_, conn_, n_: 1000] := 
 Module[{an = 100, al = 1, s, M, id, od, wd, x, poa, PoA}, 
  Cases[_?NumericQ]@
    Table[s = 
      DirectedGraph[RandomGraph[{nodes, IntegerPart[conn*nodes]}], 
       "Acyclic"];
     M = al*Transpose[AdjacencyMatrix[s]];
     id = an + al*VertexInDegree[s];
     od = al*VertexOutDegree[s];
     wd = -Log[1 - (od/id)];
 x = (DiagonalMatrix[id] - M).wd;
 poa = N[od - x, 4];
 PoA = Total@poa, {n}] // Mean]

dataPlot = Table[{nodes, conn, PoAGen[nodes, conn]}, {nodes, 20, 40, 2}, 
{conn,2.5, 5, 0.2}];
ListPlot[dataPlot[[All, 1, {1, 3}]], Frame -> True, Joined -> True, 
 FrameLabel -> {"|V|", "Inefficiency \[CapitalDelta]"}, 
 PlotLabel -> "|E|/|V|=: " ~~ ToString[dataPlot[[1, 1, 2]]]]
ListPlot[dataPlot[[2, All, {2, 3}]], Frame -> True, Joined -> True, 
 FrameLabel -> {"|E|/|V|", "Inefficiency \[CapitalDelta]"}, 
 PlotLabel -> "|V|: " ~~ ToString[dataPlot[[1, 1, 1]]]]
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  • $\begingroup$ Assign the result of Cases within the PoAGen function to a variable, then calculate its Mean and StandardDeviation and return those as a list. You will get a slightly more complex expression for dataPlot, but it should be relatively simple to extract the data you need and plot it, just like you extract the means. $\endgroup$
    – MarcoB
    Oct 1, 2015 at 6:08

1 Answer 1

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Follow the suggestion by MarcoB to make PoAGen return a list of the mean and standard deviation of the result of Cases. I always like to use Filling to show confidence intervals, for example:

ListPlot[
  {{#[[1]], #[[3, 1]]} & /@ dataPlot[[All, 1]],
   {#[[1]], #[[3, 1]] + 2*#[[3, 2]]} & /@ dataPlot[[All, 1]],
   {#[[1]], #[[3, 1]] - 2*#[[3, 2]]} & /@ dataPlot[[All, 1]]},
  Frame -> True, Joined -> True, 
  FrameLabel -> {"|V|", "Inefficiency \[CapitalDelta]"}, 
  PlotLabel -> "|E|/|V|=: " ~~ ToString[dataPlot[[1, 1, 2]]],
  PlotStyle -> {Black, Gray, Gray},
  Filling -> {1 -> {{2}, LightGray}, 1 -> {{3}, LightGray}}
]

NB. You should also reconsider your confidence intervals. The mean±2σ 'rule of thumb' is usually incorrect. For example, see https://en.wikipedia.org/wiki/Confidence_interval#Practical_example

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  • $\begingroup$ Thanks Roel. I am not very familiar with Mathematica. Can you please tell me how to return two arguments from Cases? $\endgroup$
    – Bravo
    Oct 1, 2015 at 9:48
  • $\begingroup$ You can do something like, PoAGen[nodes_, conn_, n_: 1000] := Module[{an = 100, al = 1, s, M, id, od, wd, x, poa, PoA, f}, f = Cases[_?NumericQ] (...) Total@poa, {n}]; {Mean[f],StandardDeviation[f]} ] $\endgroup$
    – Thanatomanic
    Oct 2, 2015 at 18:34

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