Add note on how to estimate appropriate values for alpha

Fixes #2307.

networkx/algorithms/centrality/katz.py

@@ -32,9 +32,9 @@ def katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1.0e-6,
 
         x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
 
-    where `A` is the adjacency matrix of graph G with eigenvalues `\lambda`.
+    where `A` is the adjacency matrix of graph G with eigenvalues $\lambda$.
 
-    The parameter `\beta` controls the initial centrality and
+    The parameter $\beta$ controls the initial centrality and
 
     .. math::
 
@@ -46,11 +46,11 @@ def katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1.0e-6,
     to the node under consideration through these immediate neighbors.
 
     Extra weight can be provided to immediate neighbors through the
-    parameter :math:`\beta`.  Connections made with distant neighbors
-    are, however, penalized by an attenuation factor `\alpha` which
+    parameter $\beta$.  Connections made with distant neighbors
+    are, however, penalized by an attenuation factor $\alpha$ which
     should be strictly less than the inverse largest eigenvalue of the
     adjacency matrix in order for the Katz centrality to be computed
-    correctly. More information is provided in [1]_ .
+    correctly. More information is provided in [1]_.
 
     Parameters
     ----------
@@ -100,10 +100,10 @@ def katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1.0e-6,
     --------
     >>> import math
     >>> G = nx.path_graph(4)
-    >>> phi = (1+math.sqrt(5))/2.0 # largest eigenvalue of adj matrix
-    >>> centrality = nx.katz_centrality(G,1/phi-0.01)
+    >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
+    >>> centrality = nx.katz_centrality(G, 1/phi - 0.01)
     >>> for n, c in sorted(centrality.items()):
-    ...    print("%d %0.2f"%(n, c))
+    ...    print("%d %0.2f" % (n, c))
     0 0.37
     1 0.60
     2 0.60
@@ -122,18 +122,20 @@ def katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1.0e-6,
     Katz centrality was introduced by [2]_.
 
     This algorithm it uses the power method to find the eigenvector
-    corresponding to the largest eigenvalue of the adjacency matrix of G.
-    The constant alpha should be strictly less than the inverse of largest
+    corresponding to the largest eigenvalue of the adjacency matrix of ``G``.
+    The parameter ``alpha`` should be strictly less than the inverse of largest
     eigenvalue of the adjacency matrix for the algorithm to converge.
-    The iteration will stop after max_iter iterations or an error tolerance of
-    number_of_nodes(G)*tol has been reached.
+    You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
+    eigenvalue of the adjacency matrix.
+    The iteration will stop after ``max_iter`` iterations or an error tolerance of
+    ``number_of_nodes(G) * tol`` has been reached.
 
-    When `\alpha = 1/\lambda_{max}` and `\beta=0`, Katz centrality is the same
+    When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same
     as eigenvector centrality.
 
     For directed graphs this finds "left" eigenvectors which corresponds
     to the in-edges in the graph. For out-edges Katz centrality
-    first reverse the graph with G.reverse().
+    first reverse the graph with ``G.reverse()``.
 
     References
     ----------
@@ -206,26 +208,25 @@ def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True,
 
         x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
 
-    where `A` is the adjacency matrix of graph G with eigenvalues `\lambda`.
+    where `A` is the adjacency matrix of graph G with eigenvalues $\lambda$.
 
-    The parameter `\beta` controls the initial centrality and
+    The parameter $\beta$ controls the initial centrality and
 
     .. math::
 
         \alpha < \frac{1}{\lambda_{max}}.
 
-
     Katz centrality computes the relative influence of a node within a
     network by measuring the number of the immediate neighbors (first
     degree nodes) and also all other nodes in the network that connect
     to the node under consideration through these immediate neighbors.
 
     Extra weight can be provided to immediate neighbors through the
-    parameter :math:`\beta`.  Connections made with distant neighbors
-    are, however, penalized by an attenuation factor `\alpha` which
+    parameter $\beta$.  Connections made with distant neighbors
+    are, however, penalized by an attenuation factor $\alpha$ which
     should be strictly less than the inverse largest eigenvalue of the
     adjacency matrix in order for the Katz centrality to be computed
-    correctly. More information is provided in [1]_ .
+    correctly. More information is provided in [1]_.
 
     Parameters
     ----------
@@ -261,10 +262,10 @@ def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True,
     --------
     >>> import math
     >>> G = nx.path_graph(4)
-    >>> phi = (1 + math.sqrt(5))/2.0  # largest eigenvalue of adj matrix
+    >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
     >>> centrality = nx.katz_centrality_numpy(G, 1/phi)
     >>> for n, c in sorted(centrality.items()):
-    ...    print("%d %0.2f"%(n, c))
+    ...    print("%d %0.2f" % (n, c))
     0 0.37
     1 0.60
     2 0.60
@@ -283,14 +284,17 @@ def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True,
     Katz centrality was introduced by [2]_.
 
     This algorithm uses a direct linear solver to solve the above equation.
-    The constant alpha should be strictly less than the inverse of largest
-    eigenvalue of the adjacency matrix for there to be a solution.  When
-    `\alpha = 1/\lambda_{max}` and `\beta=0`, Katz centrality is the same as
-    eigenvector centrality.
+    The parameter ``alpha`` should be strictly less than the inverse of largest
+    eigenvalue of the adjacency matrix for there to be a solution.
+    You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
+    eigenvalue of the adjacency matrix.
+
+    When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same
+    as eigenvector centrality.
 
     For directed graphs this finds "left" eigenvectors which corresponds
     to the in-edges in the graph. For out-edges Katz centrality
-    first reverse the graph with G.reverse().
+    first reverse the graph with ``G.reverse()``.
 
     References
     ----------