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# random_kernel_graph

random_kernel_graph(n, kernel_integral, kernel_root=None, seed=None, *, create_using=None)

  Returns an random graph based on the specified kernel.

  The algorithm chooses each of the \([n(n-1)]/2\) possible edges with probability specified by a kernel \(\kappa(x,y)\) [1]. The kernel \(\kappa(x,y)\) must be a symmetric (in \(x,y\)), non-negative, bounded function.

  Parameters:
    n int

      The number of nodes

    kernel_integral function

      Function that returns the definite integral of the kernel \(\kappa(x,y)\), \(F(y,a,b) := \int_a^b \kappa(x,y)dx\)

    kernel_root: function (optional)

      Function that returns the root \(b\) of the equation \(F(y,a,b) = r\). If None, the root is found using `scipy.optimize.brentq()` (this requires SciPy).

    seed integer, random_state, or None (default)

      Indicator of random number generation state. See Randomness.

    create_using Graph constructor, optional (default=nx.Graph)

      Graph type to create. If graph instance, then cleared before populated. Multigraph and directed types are not supported and raise a `NetworkXError`.

  See also

  `gnp_random_graph`
  `expected_degree_graph`

  Notes

  The kernel is specified through its definite integral which must be provided as one of the arguments. If the integral and root of the kernel integral can be found in \(O(1)\) time then this algorithm runs in time \(O(n+m)\) where m is the expected number of edges [2].

  The nodes are set to integers from \(0\) to \(n-1\).

  References

  [1]

  Bollobás, Béla, Janson, S. and Riordan, O. “The phase transition in inhomogeneous random graphs”, Random Structures Algorithms, 31, 3–122, 2007.

  [2]

  Hagberg A, Lemons N (2015), “Fast Generation of Sparse Random Kernel Graphs”. PLoS ONE 10(9): e0135177, 2015. doi:10.1371/journal.pone.0135177

  Examples

  Generate an Erdős–Rényi random graph \(G(n,c/n)\), with kernel \(\kappa(x,y)=c\) where \(c\) is the mean expected degree.

      >>> def integral(u, w, z):
      ...     return c * (z - w)
      >>> def root(u, w, r):
      ...     return r / c + w
      >>> c = 1
      >>> graph = nx.random_kernel_graph(1000, integral, root)