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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation .
Json Representation of a Sparse Matrix: Example and Test

Discussion
The example using a CppAD Json to represent the sparse matrix @[@ \partial_x f(x, p) = \left( \begin{array}{ccc} p_0 & 0 & 0 \\ 0 & p_1 & 0 \\ 0 & 0 & c_0 \end{array} \right) @]@ where @(@ c_0 @)@ is the constant 3.

Source Code 

# include <cppad/cppad.hpp>

bool sparse(void)
{   bool ok = true;
    using CppAD::vector;
    typedef vector<size_t> s_vector;
    typedef vector<double> d_vector;
    //
    // An AD graph example
    // node_1 : p[0]
    // node_2 : p[1]
    // node_3 : x[0]
    // node_4 : x[1]
    // node_5 : x[2]
    // node_6 : c[0]
    // node_7 : p[0] * x[0]
    // node_8 : p[1] * x[1]
    // node_9 : c[0] * x[2]
    // y[0]   = p[0] * x[0]
    // y[1]   = p[1] * x[1]
    // y[2]   = c[0] * x[0]
    // use single quote to avoid having to escape double quote
    std::string json =
        "{\n"
        "   'function_name'  : 'sparse example',\n"
        "   'op_define_vec'  : [ 1, [\n"
        "       { 'op_code':1, 'name':'mul', 'n_arg':2 } ]\n"
        "   ],\n"
        "   'n_dynamic_ind'  : 2,\n"
        "   'n_variable_ind' : 3,\n"
        "   'constant_vec'   : [ 1, [ 3.0 ] ],\n"
        "   'op_usage_vec'   : [ 3, [\n"
        "       [ 1, 1, 3 ] , \n"
        "       [ 1, 2, 4 ] , \n"
        "       [ 1, 6, 5 ] ] \n"
        "   ],\n"
        "   'dependent_vec'   : [ 3, [7, 8, 9] ] \n"
        "}\n";
    // Convert the single quote to double quote
    for(size_t i = 0; i < json.size(); ++i)
        if( json[i] == '\'' ) json[i] = '"';
    //
    CppAD::ADFun<double> fun;
    fun.from_json(json);
    ok &= fun.Domain() == 3;
    ok &= fun.Range()  == 3;
    ok &= fun.size_dyn_ind() == 2;
    //
    // Point at which we are going to evaluate the Jacobian of f(x, p).
    // The Jacobian is constant w.r.t. x, so the value of x does not matter.
    vector<double> p(2), x(3);
    p[0] = 1.0;
    p[1] = 2.0;
    for(size_t j = 0; j < x.size(); ++j)
        x[j] = 0.0;
    //
    // set the dynamic parameters
    fun.new_dynamic(p);
    //
    // compute the sparsity pattern for f_x(x, p)
    vector<bool> select_domain(3);
    vector<bool> select_range(3);
    for(size_t i = 0; i < 3; ++i)
    {   select_domain[i] = true;
        select_range[i]  = true;
    }
    bool transpose = false;
    CppAD::sparse_rc<s_vector> pattern;
    fun.subgraph_sparsity(select_domain, select_range, transpose, pattern);
    //
    // compute the entire Jacobian
    CppAD::sparse_rcv<s_vector, d_vector> subset(pattern);
    fun.subgraph_jac_rev(x, subset);
    //
    // information in the sparse Jacobian
    const s_vector& row( subset.row() );
    const s_vector& col( subset.col() );
    const d_vector& val( subset.val() );
    size_t   nnz       = subset.nnz();
    s_vector row_major = subset.row_major();
    //
    // check number of non-zero elements in sparse matrix
    ok      &= nnz == 3;
    //
    // check first element of matrix (in row major order)
    size_t k = row_major[0];
    ok      &= row[k] == 0;
    ok      &= col[k] == 0;
    ok      &= val[k] == p[0];
    //
    // check second element of matrix
    k        = row_major[1];
    ok      &= row[k] == 1;
    ok      &= col[k] == 1;
    ok      &= val[k] == p[1];
    //
    // check third element of matrix
    k        = row_major[2];
    ok      &= row[k] == 2;
    ok      &= col[k] == 2;
    ok      &= val[k] == 3.0;
    //
    return ok;
}
Input File: example/json/sparse.cpp