4
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My goal is to find such properties of a sparse matrix as the maximum/average number of non-zero elements per row.

The brute-force way of doing this is via converting the sparse array into a regular one:

MaxSpar[matr_] := Module[{curr, ms = 0},
   Do[
    curr = Length[Cases[matr[[k]], 0]];
    If[curr > ms, ms = curr];
    , {k, 1, Length[matr]}
    ];
   Return[ms];
   ];

MaxSpar[Normal[SomeSparseMatrix]]

How can we do the same without using Normal[]?

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6
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To obtain the number of nonzero entry of the row with fewest zeros:

Max[Length /@ SomeSparseMatrix["AdjacencyLists"]]

There are other useful strings. "Methods" shows which are availble:

SomeSparseMatrix["Methods"]

{"AdjacencyLists", "Background", "ColumnIndices", "Density", "MatrixColumns", "MethodInformation", "Methods", "NonzeroPositions", "NonzeroValues", "PatternArray", "PatternValues", "Properties", "RowPointers"}

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3
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maxNonZero = Max[Length /@ #["MatrixColumns"]] &;
aveNonZero = Mean[Length /@ #["MatrixColumns"] ] &
SeedRandom[1]
sa = SparseArray[RandomInteger[3, {7, 10}]];
sa // MatrixForm // TeXForm

$\left( \begin{array}{cccccccccc} 3 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 2 & 0 & 1 & 2 & 0 & 0 \\ 3 & 3 & 3 & 1 & 1 & 0 & 0 & 1 & 3 & 0 \\ 2 & 0 & 1 & 1 & 3 & 3 & 3 & 2 & 3 & 2 \\ 0 & 1 & 3 & 3 & 0 & 1 & 0 & 1 & 0 & 3 \\ 0 & 2 & 3 & 0 & 2 & 2 & 0 & 1 & 3 & 2 \\ 1 & 2 & 0 & 0 & 0 & 2 & 1 & 2 & 1 & 0 \\ \end{array} \right)$

maxNonZero[sa]

9

N @ aveNonZero[sa]

6.285714285714

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1
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m = 100000;
n = 2000000;
A = SparseArray[
   RandomInteger[{1, m}, {n, 2}] -> RandomReal[{-1, 1}, n],
   {m, m}, 0.
   ];

Maximum number of nonempty elements per row:

a = Max[Unitize[A].ConstantArray[1, Dimensions[A][[2]]]]; // RepeatedTiming // First
b = Max[Length /@ A["AdjacencyLists"]]; // RepeatedTiming // First

0.122

0.053

A faster way (that works only for rows) is

c = Max[Differences[A["RowPointers"]]]; // RepeatedTiming // First
a == b == c

0.000642

True

Analogously, the mean of the numbers of nonempty elements per row can be obtain as follows:

Mean[N[Differences[A["RowPointers"]]]]
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