Implement Katz centrality and related methods for graphs

[clienkaemper]

want to be able to compute Katz centrality, use Katz method for link prediction in graphs/networks (​https://en.wikipedia.org/wiki/Katz_centrality)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​99c2705

Adds functions for Katz centrality and related methods

[dcoudert]

Welcome to Sagemath

I have some comments to help you improving the code.

• unify nonedges_only and nonedgesonly. Actually, #24094 uses nonedgesonly. I don't know which is best, but should be the same.
• Returns the Katz matrix of G with parameter alpha. -> Return the Katz matrix of the graph with parameter alpha.
• only 1 empty line is needed after INPUT::
• `True` -> ``True``
• if true, eliminates common neighbors between adjacent vertices. Could you add that the value for each edge is set to 0 ?
• the Katz matrix of the graph self with parameter alpha -> the Katz matrix of the graph with parameter alpha
• G1 = Graph({1:[2,3], 2:[1,4], 3:[1,4], 4:[2,3]}) -> G1 = graphs.CycleGraph(4) ?
• since you don't have G2, you could rename G1 to G
• add a space between * and the text in *:meth:`~katz_centrality` and the line after
• add an empty line after TESTS::
• The examples you give in the TESTS could be in the examples section. In a TESTS block, we expect examples raising errors, with the empty graph, etc.
• the test alpha <= 0 should be made before any other computation
• if nonedges_only == True: -> if nonedges_only:

in method def katz_centrality(self, alpha):

• you could add an optional parameter u to return the centrality of a specified vertex. For that, add u=None and then the return statement becomes something like if u is None: return K else: return K[u]
• you must indent the text in 80 columns mode. Some code editors are really helpful to do that without effort
• You return a list. Wouldn't it be better to return a dictionary keyed by vertices ? How to know which value corresponds to a node ?

[dcoudert]

You can either rebase this ticket on last develop branch (and fix the merge conflicts), or create a new branch based on last beta.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​77310bb	Katz added
​2c0f41d	Katz added improved
​c7fff6c	fixed typos

[dcoudert]

Several issues (I'm skipping the coding style issues for the moment, but please have a look at ​http://doc.sagemath.org/html/en/developer/coding_basics.html and check how other methods are written).

In method katz_matrix:

• add some explanation of what is a Katz matrix
• please add parameter vertices similarly to method .adjacency_matrix(). This is important as in the future we will no longer ensure that the list of vertices is sorted (due to Python 3).

In method katz_centrality

• add some explanation of what is the Katz centrality, plus a link to relevant wikipedia page
• in the code, choose an order like verts = list(self), and pass it to katz_matrix. Here we don't need the list of vertices to be sorted, we just want to know the ordering, so we fix it.
• u not in self.vertices() -> u not in self
• You can simplify the construction of the dictionary

  -        for i in range(n):
  -            Kdict[vert[i]] = katz_values[0][i]
  +        Kdict = {u: katz_values[0][i] for i, u in enumerate(verts)}
  

• Do we really need to build dictionary Kdict when u is not None ?

Do not forget to check that all tests pass and that the documentation builds properly and displays nicely.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​66151a6

changes made in accordance to the review.

[gh-rajat1433]

Thanks for the prompt reply and suggesting the changes. Done all the improvements mentioned above and also made sure to pass all the tests. I have done the changes as per required by the documentation in generic_graph.py file. But could not figure out how to view those changes as they will actually be looking in documentation. And is there any test command for checking if format is correct for documentation.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​172082f

made additions for documentation.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​47d85e3

bug fix

[dcoudert]

I have done the changes as per required by the documentation in generic_graph.py file. But could not figure out how to view those changes as they will actually be looking in documentation. And is there any test command for checking if format is correct for documentation.

See ​http://doc.sagemath.org/html/en/developer/sage_manuals.html

You can rebuild the full documentation using

sage -b && sage --docbuild reference html

or rebuild only the documentation of the graph module

sage -b && sage --docbuild reference/graphs html

Then, you can look at the documentation with your web browser.

file://PATH_ON_YOUR_COMPUTER/sage/local/share/doc/sage/html/en/reference/graphs/index.html

For the coding style, please try to comply to ​https://www.python.org/dev/peps/pep-0008

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​67a2384

documentation build changes

[gh-rajat1433]

I have made changes so as to build the documentation properly and also passed all the tests and tried to follow the coding convention. Can you please review the code so that we can finalize it.

[dcoudert]

• You must ensure that all comments are formatted in 80 columns mode. Some code editors can be configured to ease this task

• minor stylistic issue

  -        - ``nonedgesonly`` -- Boolean (default: ``True``); if True, value for each
  -          edge present in the graph is set to zero.
  +        - ``nonedgesonly`` -- boolean (default: ``True``); if ``True``, value for each
  +          edge present in the graph is set to zero
  

• you can use for u in self or set(self) or if u in self, etc.

  -              set(vertices) != set(self.vertex_iterator())):
  +              set(vertices) != set(self)):
  

• pep8

  -        K =  (In - alpha * A.transpose()).inverse()-In
  +        K =  (In - alpha * A.transpose()).inverse() - In
  

-            Missing = onesmat - A- In
+            Missing = onesmat - A - In

• Please revise method katz_centrality: alignment, ensure that the code is correct, remove useless instructions, etc.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​05e8e6b	empty fix
​27579a2	fixed alignment and other parts as per review

[gh-rajat1433]

Thanks for the review. Its been really a learning experience. I have started following alignment and pep8 seriously and I have fixed the issues above.

[dcoudert]

Please check carefully your code, that comments are in 80 columns mode, if possible that you follow the PEP8 coding style, that the comments are sufficient and helpful to the user, and that your code is best possible.

for instance, you have apparently not double check this part:

-        if u :
-            idx = verts.index(u)
-            return katz_values[0][idx]
-
-        Kdict = {}
-        Kdict = {u: katz_values[0][i] for i, u in enumerate(verts)}
-
-        if u is None:
-            return Kdict
-        else :
-            return Kdict[u]
+        if u:
+            return katz_values[0][verts.index(u)]
+        return {u: katz_values[0][i] for i, u in enumerate(verts)}

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f85cf1b

improved code

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​686b15e	improved code
​42019ac	improved code

[dcoudert]

• except in the names of the method, use Katz and not katz everywhere.
• remove trailing white spaces
• you should rephrase the description of method katz_matrix to first give the definition of this centrality measure, and then explain what's the returned matrix.
• Avoid having too many empty lines like this. 1 is often enough

  +
  +
  +
  

• remove the . at the end of error messages
• if alpha <= 0: -> if alpha <= 0:

• -        We compute katz_centrality for the undirected 4-cycle again (note that 
  +        We compute the Katz centrality of a 4-cycle (note that 
  

• -        Return the katz centrality of the vertex u of the graph.
  +        Return the Katz centrality of vertex `u`.
  

• there is a α. It should be ok, but it might be safer to use \alpha.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​cb9e45b

fixed the coding style

[gh-rajat1433]

Thanks for the review , I have fixed the coding style bugs.

[dcoudert]

• to use latex in the documentation, you must write `\alpha`, and not only \alpha. Using r""" and then `...` indicates to use latex.

• in katz_centrality, when you ask for the centrality of a vertex u, you first compute katz_values = (M*matrix(QQ, n, 1, lambda i, j: 1)).transpose(). However, if I understand well the definition of the Katz matrix, it suffices to make the some of the values of a particular row (or column), right ? If so, you could avoid the matrix multiplication when asking for a single value.

• also, a . and a to remove

  -          reciprocal of the spectral radius of the graph. (the maximum
  -          absolute eigenvalue of the adjacency matrix )
  +          reciprocal of the spectral radius of the graph (the maximum
  +          absolute eigenvalue of the adjacency matrix)
  

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3d9569b

Improved Code

[gh-rajat1433]

I have used sum function now and eliminated the multiplication step. I guess it give better complexity now in terms of space and time specially for a single vertex case.

[dcoudert]

This is much better now. We certainly same some computation time (but see point below). For the space complexity, it remains in O(n^2) as we compute the Katz matrix in all cases.

• The test if u and u not in self: must be done before computing verts = list(self) and more importantly the computation of the Katz matrix.

I have now more algorithmic questions (more interesting than coding style :P):

• What is the expected answer when the graph is not connected ? May be we can save computation time if the answer is always 0 for vertices in different components ?

• Same question for strongly connected digraphs? In the example, vertices 1, 2, 3, 7, 9 have no in-neighbors and vertex 4 has no out-neighbor. When you will get an answer, you should document the example explaining why some values are 0.

[gh-rajat1433]

What is the expected answer when the graph is not connected ? May be we can save computation time if the answer is always 0 for vertices in different components ?

• When graph is not connected answer for a particular vertex is only dependent on the vertices in its connected component only and not in the other component. So yes we really can do optimize here. Will work on that.

Same question for strongly connected digraphs? In the example, vertices 1, 2, 3, 7, 9 have no in-neighbors and vertex 4 has no out-neighbor. When you will get an answer, you should document the example explaining why some values are 0.

• for strongly connected Digraph the answer will be non zero for every node. Also for any node with in-neighbors 0 the answer will be 0 as its the very essence of katz centrality to measure influence of an actor(node here) in a social network.

Will document the example with reasoning.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c796415

optimized

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e2f562c

optimized

[dcoudert]

• You can get the connected component containing vertex u directly using C = G.connected_component_containing_vertex(u, sort=False).

Then you must compute the Katz matrix of the subgraph, i.e., self.subgraph(C).

if u:
    if self.is_connected():
        G = self
    else:
        G = self.subgraph(self.connected_component_containing_vertex(u, sort=False))
        verts = list(G)
    M = G.katz_matrix(alpha, nonedgesonly=False, vertices=verts)
    return sum(M[verts.index(u)])

• When you compute the Katz centrality of all vertices, you want the union of the Katz centralities of each subgraph. You can use self.connected_components_subgraphs()

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​13c2821

optimized the code

[gh-rajat1433]

I have optimized the code as per the review.

[dcoudert]

Please double check your code and test it ! it contains obvious errors.

You can write directly for g in self.connected_components_subgraphs()

Add a doctest with a non connected graph, e.g., graphs.PathGraph(3) + graphs.PathGraph(4).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f47a3dd

fixed the errors

[gh-rajat1433]

I am really sorry about the previous errors. Now its fixed and well tested.

[dcoudert]

Could you change the example to sage: (graphs.PathGraph(3) + graphs.PathGraph(4)).katz_centrality(1/20). This is to show that the values for the vertices in the P3 and in the P4 are the same than in previous examples.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1a87f58

minor addition

[gh-rajat1433]

Could you change the example to sage: (graphs.PathGraph?(3) + graphs.PathGraph?(4)).katz_centrality(1/20). This is to show that the values for the vertices in the P3 and in the P4 are the same than in previous examples.

Done!

[dcoudert]

Further speed up improvements can certainly be obtained, but I don't know how.

For me this patch is good to go.