# Creating confidence interval plots for random graphs

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"];
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]]]]

• 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. Oct 1, 2015 at 6:08

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[
{{#[], #[[3, 1]]} & /@ dataPlot[[All, 1]],
{#[], #[[3, 1]] + 2*#[[3, 2]]} & /@ dataPlot[[All, 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

• Thanks Roel. I am not very familiar with Mathematica. Can you please tell me how to return two arguments from Cases? Oct 1, 2015 at 9:48
• 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]} ] Oct 2, 2015 at 18:34