Module Expr Str Import alias identifier ImportFrom alias Assign NameStore Call AttributeLoad Str attr ListLoad Str Str Assign NameStore ListLoad Str Str FunctionDef arguments args NameParam defaults body Expr Str Import alias Import alias Assign NameStore Call NameLoad NameLoad Assign NameStore Call AttributeLoad NameLoad attr NameLoad NameLoad Assign SubscriptStore NameLoad Index CompareNotEq NameLoad Num Num Assign TupleStore NameStore NameStore Call AttributeLoad AttributeLoad NameLoad attr attr NameLoad Assign NameStore Call AttributeLoad NameLoad attr NameLoad Assign NameStore Call NameLoad Call NameLoad NameLoad Call NameLoad Call NameLoad NameLoad Assign NameStore Dict For NameStore NameLoad body Assign SubscriptStore NameLoad Index NameLoad Dict For NameStore NameLoad body Assign NameStore Num Assign NameStore SubscriptLoad NameLoad Index NameLoad Assign NameStore SubscriptLoad NameLoad Index NameLoad For NameStore Call NameLoad Call NameLoad NameLoad body AugAssignAdd NameStore BinOpMult BinOpMult SubscriptLoad SubscriptLoad NameLoad ExtSlice Slice Index NameLoad Index TupleLoad NameLoad Num SubscriptLoad SubscriptLoad NameLoad ExtSlice Slice Index NameLoad Index TupleLoad NameLoad Num SubscriptLoad NameLoad Index NameLoad Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad Index NameLoad Call NameLoad NameLoad Return NameLoad decorator_list Call NameLoad Str Call NameLoad Str FunctionDef arguments args NameParam defaults body Expr Str Import alias Assign NameStore Call NameLoad NameLoad Assign NameStore Call AttributeLoad NameLoad attr NameLoad NameLoad Assign SubscriptStore NameLoad Index CompareNotEq NameLoad Num Num Assign NameStore Call AttributeLoad AttributeLoad NameLoad attr attr NameLoad Assign NameStore Call NameLoad Call NameLoad NameLoad Call NameLoad Call NameLoad NameLoad Assign NameStore Dict For NameStore NameLoad body Assign SubscriptStore NameLoad Index NameLoad Dict For NameStore NameLoad body Assign SubscriptStore SubscriptLoad NameLoad Index NameLoad Index NameLoad Call NameLoad SubscriptLoad NameLoad Index TupleLoad SubscriptLoad NameLoad Index NameLoad SubscriptLoad NameLoad Index NameLoad Return NameLoad decorator_list Call NameLoad Str Call NameLoad Str FunctionDef arguments args NameParam defaults body ImportFrom alias TryExcept body Import alias handlers ExceptHandler body Raise Call NameLoad Str TryExcept body Import alias handlers ExceptHandler body Raise Call NameLoad Str decorator_list
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Communicability.
 N/A networkx nx networkx.utils * N/A __author__ N/A N/A join N/A Aric Hagberg (hagberg@lanl.gov) Franck Kalala (franckkalala@yahoo.fr N/A __all__ N/A communicability communicability_exp communicability N/A N/A G N/A N/A N/A Return communicability between all pairs of nodes in G.

    The communicability between pairs of nodes in G is the sum of
    closed walks of different lengths starting at node u and ending at node v.

    Parameters
    ----------
    G: graph

    Returns
    -------
    comm: dictionary of dictionaries
        Dictionary of dictionaries keyed by nodes with communicability
        as the value.

    Raises
    ------
    NetworkXError
       If the graph is not undirected and simple.

    See Also
    --------
    communicability_exp:
       Communicability between all pairs of nodes in G  using spectral
       decomposition.
    communicability_betweenness_centrality:
       Communicability betweeness centrality for each node in G.

    Notes
    -----
    This algorithm uses a spectral decomposition of the adjacency matrix.
    Let G=(V,E) be a simple undirected graph.  Using the connection between
    the powers  of the adjacency matrix and the number of walks in the graph,
    the communicability  between nodes `u` and `v` based on the graph spectrum
    is [1]_

    .. math::
        C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},

    where `\phi_{j}(u)` is the `u\rm{th}` element of the `j\rm{th}` orthonormal
    eigenvector of the adjacency matrix associated with the eigenvalue
    `\lambda_{j}`.

    References
    ----------
    .. [1] Ernesto Estrada, Naomichi Hatano,
       "Communicability in complex networks",
       Phys. Rev. E 77, 036111 (2008).
       http://arxiv.org/abs/0707.0756

    Examples
    --------
    >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
    >>> c = nx.communicability(G)
     N/A numpy N/A scipy.linalg N/A nodelist N/A list G N/A A N/A N/A nx to_numpy_matrix G nodelist N/A N/A A N/A N/A A 0.0 1 N/A N/A w vec N/A N/A N/A numpy linalg eigh A N/A expw N/A N/A numpy exp w N/A mapping N/A dict N/A zip nodelist N/A range N/A len nodelist N/A c N/A N/A u G N/A N/A N/A c N/A u N/A N/A v G N/A N/A s 0 N/A p N/A mapping N/A u N/A q N/A mapping N/A v N/A j N/A range N/A len nodelist N/A N/A s N/A N/A N/A N/A vec N/A N/A N/A j N/A N/A p 0 N/A N/A vec N/A N/A N/A j N/A N/A q 0 N/A expw N/A j N/A N/A N/A c N/A u N/A v N/A float s N/A c N/A N/A not_implemented_for directed N/A not_implemented_for multigraph communicability_exp N/A N/A G N/A N/A N/A Return communicability between all pairs of nodes in G.

    Communicability between pair of node (u,v) of node in G is the sum of
    closed walks of different lengths starting at node u and ending at node v.

    Parameters
    ----------
    G: graph

    Returns
    -------
    comm: dictionary of dictionaries
        Dictionary of dictionaries keyed by nodes with communicability
        as the value.

    Raises
    ------
    NetworkXError
        If the graph is not undirected and simple.

    See Also
    --------
    communicability:
       Communicability between pairs of nodes in G.
    communicability_betweenness_centrality:
       Communicability betweeness centrality for each node in G.

    Notes
    -----
    This algorithm uses matrix exponentiation of the adjacency matrix.

    Let G=(V,E) be a simple undirected graph.  Using the connection between
    the powers  of the adjacency matrix and the number of walks in the graph,
    the communicability between nodes u and v is [1]_,

    .. math::
        C(u,v) = (e^A)_{uv},

    where `A` is the adjacency matrix of G.

    References
    ----------
    .. [1] Ernesto Estrada, Naomichi Hatano,
       "Communicability in complex networks",
       Phys. Rev. E 77, 036111 (2008).
       http://arxiv.org/abs/0707.0756

    Examples
    --------
    >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
    >>> c = nx.communicability_exp(G)
     N/A scipy.linalg N/A nodelist N/A list G N/A A N/A N/A nx to_numpy_matrix G nodelist N/A N/A A N/A N/A A 0.0 1 N/A expA N/A N/A N/A scipy linalg expm A N/A mapping N/A dict N/A zip nodelist N/A range N/A len nodelist N/A c N/A N/A u G N/A N/A N/A c N/A u N/A N/A v G N/A N/A N/A N/A c N/A u N/A v N/A float N/A expA N/A N/A N/A mapping N/A u N/A mapping N/A v N/A c N/A N/A not_implemented_for directed N/A not_implemented_for multigraph setup_module N/A N/A module N/A N/A nose SkipTest N/A N/A N/A numpy N/A N/A N/A N/A N/A SkipTest NumPy not available N/A N/A N/A scipy N/A N/A N/A N/A N/A SkipTest SciPy not available N/A