<commit_before>"""
Centrality measures.

"""
#    Copyright (C) 2004-2010 by 
#    <NAME> <<EMAIL>>
#    <NAME> <<EMAIL>>
#    <NAME> <<EMAIL>>
#    All rights reserved.
#    BSD license.
__author__ = """<NAME> (<EMAIL>)\<NAME> (<EMAIL>)\n<NAME> (<EMAIL>)"""

__all__ = ['betweenness_centrality',
           'betweenness_centrality_source',
           'load_centrality',
           'newman_betweenness_centrality',
           'brandes_betweenness_centrality',
           'edge_betweenness',
           'edge_load',
           'degree_centrality',
           'in_degree_centrality',
           'out_degree_centrality',
           'closeness_centrality',
           'eigenvector_centrality']


import heapq
from networkx.algorithms.traversal.path import predecessor, \
        single_source_shortest_path_length, \
        single_source_dijkstra_path_length, \
        dijkstra_predecessor_and_distance 
import networkx


def brandes_betweenness_centrality(G,normalized=True,weighted_edges=False):
    """Compute betweenness centrality for nodes.

    Betweenness centrality of a node is the fraction of all shortest 
    paths that pass through that node.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with betweeness centrality as the value.

        
    See Also
    --------
    load_centrality()

    Notes
    -----
    The algorithm is from Ulrik Brandes,
    A Faster Algorithm for Betweenness Centrality.
    Journal of Mathematical Sociology 25(2):163-177, 2001.
    http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf

"""
    betweenness=dict.fromkeys(G,0.0) # b[v]=0 for v in G
    for s in G:
        S=[]
        P={}
        for v in G:
            P[v]=[]
        sigma=dict.fromkeys(G,0)    # sigma[v]=0 for v in G
        D={}
        sigma[s]=1
        if not weighted_edges:  # use BFS
            D[s]=0
            Q=[s]
            while Q:   # use BFS to find shortest paths
                v=Q.pop(0)
                S.append(v)
                for w in G[v]:
#                for w in G.adj[v]: # speed hack, exposes internals
                    if w not in D:
                        Q.append(w)
                        D[w]=D[v]+1
                    if D[w]==D[v]+1:   # this is a shortest path, count paths
                        sigma[w]=sigma[w]+sigma[v]
                        P[w].append(v) # predecessors 
        else:  # use Dijkstra's algorithm for shortest paths,
               # modified from Eppstein
            push=heapq.heappush
            pop=heapq.heappop
            seen = {s:0} 
            Q=[]   # use Q as heap with (distance,node id) tuples
            push(Q,(0,s,s))
            while Q:   
                (dist,pred,v)=pop(Q)
                if v in D:
                    continue # already searched this node.
                sigma[v]=sigma[v]+sigma[pred] # count paths
                S.append(v)
                D[v] = dist
                for w,edgedata in G[v].iteritems():
                    vw_dist = D[v] + edgedata.get('weight',1)
                    if w not in D and (w not in seen or vw_dist < seen[w]):
                        seen[w] = vw_dist
                        push(Q,(vw_dist,v,w))
                        sigma[w]=0
                        P[w]=[v]
                    elif vw_dist==seen[w]:  # handle equal paths
                        sigma[w]=sigma[w]+sigma[v]
                        P[w].append(v)


        delta=dict.fromkeys(G,0) 
        while S:
            w=S.pop()
            for v in P[w]:
                delta[v]=delta[v]+\
                          (float(sigma[v])/float(sigma[w]))*(1.0+delta[w])
            if w != s:
                betweenness[w]=betweenness[w]+delta[w]
                    
    # normalize
    if normalized:
        order=len(betweenness)
        if order <=2:
            return betweenness # no normalization b=0 for all nodes
        scale=1.0/((order-1)*(order-2))
        for v in betweenness:
            betweenness[v] *= scale

    return betweenness            

betweenness_centrality=brandes_betweenness_centrality


def newman_betweenness_centrality(G,v=None,cutoff=None,
                           normalized=True,
                           weighted_edges=False):
    """Compute load centrality for nodes.

    The load centrality of a node is the fraction of all shortest 
    paths that pass through that node.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    cutoff : bool, optional
      If specified, only consider paths of length <= cutoff.

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with centrality as the value.

        
    See Also
    --------
    betweenness_centrality() 

    Notes
    -----
    Load centrality is slightly different than betweenness.
    For this load algorithm see the reference
    Scientific collaboration networks: II.
    Shortest paths, weighted networks, and centrality,
    <NAME>, Phys. Rev. E 64, 016132 (2001).

    """
    if v is not None:   # only one node
        betweenness=0.0
        for source in G: 
            ubetween=_node_betweenness(G,source,
                                       cutoff=cutoff,
                                       normalized=normalized,
                                       weighted_edges=weighted_edges)
            betweenness+=ubetween[v]
        return betweenness
    else:
        betweenness={}.fromkeys(G,0.0) 
        for source in betweenness: 
            ubetween=_node_betweenness(G,source,
                                       cutoff=cutoff,
                                       normalized=False,
                                       weighted_edges=weighted_edges)
            for vk in ubetween:
                betweenness[vk]+=ubetween[vk]
        if normalized:
            order=len(betweenness)
            if order <=2:
                return betweenness # no normalization b=0 for all nodes
            scale=1.0/((order-1)*(order-2))
            for v in betweenness:
                betweenness[v] *= scale
        return betweenness  # all nodes

def _node_betweenness(G,source,cutoff=False,normalized=True,weighted_edges=False):
    """Node betweenness helper:
    see betweenness_centrality for what you probably want.

    This actually computes "load" and not betweenness.
    See https://networkx.lanl.gov/ticket/103

    This calculates the load of each node for paths from a single source.
    (The fraction of number of shortests paths from source that go
    through each node.)

    To get the load for a node you need to do all-pairs shortest paths.

    If weighted_edges is True then use Dijkstra for finding shortest paths.
    In this case a cutoff is not implemented and so is ignored.

    """

    # get the predecessor and path length data
    if weighted_edges:
        (pred,length)=dijkstra_predecessor_and_distance(G,source) 
    else:
        (pred,length)=predecessor(G,source,cutoff=cutoff,return_seen=True) 

    # order the nodes by path length
    onodes = [ (l,vert) for (vert,l) in length.items() ]
    onodes.sort()
    onodes[:] = [vert for (l,vert) in onodes if l>0]
    
    # intialize betweenness
    between={}.fromkeys(length,1.0)

    while onodes:           
        v=onodes.pop()
        if v in pred:
            num_paths=len(pred[v])   # Discount betweenness if more than 
            for x in pred[v]:        # one shortest path.
                if x==source:   # stop if hit source because all remaining v  
                    break       #  also have pred[v]==[source]
                between[x]+=between[v]/float(num_paths)
    #  remove source
    for v in between:
        between[v]-=1
    # rescale to be between 0 and 1                
    if normalized:
        l=len(between)
        if l > 2:
            scale=1.0/float((l-1)*(l-2)) # 1/the number of possible paths
            for v in between:
                between[v] *= scale
    return between


load_centrality=newman_betweenness_centrality

def betweenness_centrality_source(G,normalized=True,
                                  weighted_edges=False,
                                  sources=None):
    """Compute betweenness centrality for a subgraph.

    Enchanced version of the method in centrality module that allows
    specifying a list of sources (subgraph).


    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    sources : node list 
      A list of nodes to consider as sources for shortest paths.
      

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with betweeness centrality as the value.


    Notes
    -----
    See Sec. 4 in 
    <NAME>,
    A Faster Algorithm for Betweenness Centrality.
    Journal of Mathematical Sociology 25(2):163-177, 2001.
    http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf

    This algorithm does not count the endpoints, i.e.
    a path from s to t does not contribute to the betweenness of s and t.
    """
    if sources == None:
        sources = G   # only used to iterate over nodes.

    betweenness=dict.fromkeys(G,0.0)
    for s in sources:
        S,P,D,sigma = _brandes_betweenness_helper(G,s,weighted_edges)

        delta=dict.fromkeys(G,0) # unnormalized betweenness
        while S:
            w=S.pop()
            for v in P[w]:
                delta[v] += (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
            if w == s:
                continue
            betweenness[w] = betweenness[w] + delta[w]
                   
    # normalize to size of entire graph
    if normalized and G.number_of_edges() > 1:
        order=len(betweenness)
        scale=1.0/((order-1)*(order-2))
        for v in betweenness:
            betweenness[v] *= scale

    return betweenness


def edge_betweenness(G,normalized=True,weighted_edges=False,sources=None):
    """Compute betweenness centrality for edges.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    sources : node list 
      A list of nodes to consider as sources for shortest paths.
      

    Returns
    -------
    nodes : dictionary
       Dictionary of edges with betweeness centrality as the value.
    """
    if sources is None:
        sources = G # only used to iterate over nodes

    betweenness=dict.fromkeys(G.edges(),0.0)
    if G.is_directed():
        for s in sources:
            S, P, D, sigma =_brandes_betweenness_helper(G,s,weighted_edges)
            delta=dict.fromkeys(G,0.0)
            while S:
                w=S.pop()
                for v in P[w]:
                    edgeFlow = (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
                    edge = (v,w)
                    delta[v]          += edgeFlow
                    betweenness[edge] += edgeFlow
    else:
        for s in sources:
            S, P, D, sigma =_brandes_betweenness_helper(G,s,weighted_edges)
            delta=dict.fromkeys(G,0.0)
            while S:
                w=S.pop()
                for v in P[w]:
                    edgeFlow = (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
                    edge = (v,w)
                    if edge not in betweenness:
                        edge = (w,v)
                    delta[v]          += edgeFlow
                    betweenness[edge] += edgeFlow

    size=len(betweenness)                
    if normalized and size > 1:
        # normalize to size of entire graph (beware of disconnected components)
        scale=1.0/((size-1)*(size-2))
        for edge in betweenness:
            betweenness[edge] *= scale

    return betweenness


def _brandes_betweenness_helper(G,root,weighted_edges):
    """
    Helper for betweenness centrality and edge betweenness centrality.

    Runs single-source shortest path from root node.

    weighted_edges:: consider edge weights 

    Finds::

    S=[] list of nodes reached during traversal
    P={} predecessors, keyed by child node
    D={} distances
    sigma={} indexed by node, is the number of paths to root
    going through the node
    """
    S=[]
    P={}
    for v in G:
        P[v]=[]
    sigma=dict.fromkeys(G,0.0)
    D={}
    sigma[root]=1

    if not weighted_edges:  # use BFS
        D[root]=0
        Q=[root]
        while Q:   # use BFS to find shortest paths
            v=Q.pop(0)
            S.append(v)
            for w in G[v]: #  for w in G.adj[v]: # speed hack, exposes internals
                if w not in D:
                    Q.append(w)
                    D[w]=D[v]+1
                if D[w]==D[v]+1:   # this is a shortest path, count paths
                    sigma[w]=sigma[w]+sigma[v]
                    P[w].append(v) # predecessors
    else:  # use Dijkstra's algorithm for shortest paths,
           # modified from Eppstein
        push=heapq.heappush
        pop=heapq.heappop
        seen = {root:0}
        Q=[]   # use Q as heap with (distance,node id) tuples
        push(Q,(0,root,root))
        while Q:   
            (dist,pred,v)=pop(Q)
            if v in D:
                continue # already searched this node.
            sigma[v]=sigma[v]+sigma[pred] # count paths
            S.append(v)
            D[v] = dist
            for w,edgedata in G[v].iteritems(): 
                vw_dist = D[v] + edgedata.get('weight',1)
                if w not in D and (w not in seen or vw_dist < seen[w]):
                    seen[w] = vw_dist
                    sigma[w] = 0
                    push(Q,(vw_dist,v,w))
                    P[w]=[v]
                elif vw_dist==seen[w]:  # handle equal paths
                    sigma[w]=sigma[w]+sigma[v]
                    P[w].append(v)
    return S, P, D, sigma


def edge_load(G,nodes=None,cutoff=False):
    """Compute edge load.

    WARNING:

    This module is for demonstration and testing purposes.

    """
    betweenness={} 
    if not nodes:         # find betweenness for every node  in graph
        nodes=G.nodes()   # that probably is what you want...
    for source in nodes:
        ubetween=_edge_betweenness(G,source,nodes,cutoff=cutoff)
        for v in ubetween.keys():
            b=betweenness.setdefault(v,0)  # get or set default
            betweenness[v]=ubetween[v]+b    # cumulative total
    return betweenness

def _edge_betweenness(G,source,nodes,cutoff=False):
    """
    Edge betweenness helper.
    """
    between={}
    # get the predecessor data
    #(pred,length)=_fast_predecessor(G,source,cutoff=cutoff) 
    (pred,length)=predecessor(G,source,cutoff=cutoff,return_seen=True) 
    # order the nodes by path length
    onodes = [ nn for dd,nn in sorted( (dist,n) for n,dist in length.iteritems() )]
    # intialize betweenness, doesn't account for any edge weights
    for u,v in G.edges(nodes):
        between[(u,v)]=1.0
        between[(v,u)]=1.0

    while onodes:           # work through all paths
        v=onodes.pop()
        if v in pred:
            num_paths=len(pred[v])   # Discount betweenness if more than 
            for w in pred[v]:        # one shortest path.
                if w in pred:
                    num_paths=len(pred[w])  # Discount betweenness, mult path  
                    for x in pred[w]: 
                        between[(w,x)]+=between[(v,w)]/num_paths
                        between[(x,w)]+=between[(w,v)]/num_paths
    return between


def _degree_centrality(degree,degree_iter,G,v=None):
    """Internal function to consolidate *-degree centrality functions."""
    if v is not None:
        d = G.__getattribute__(degree)
        return d(v)/(G.order()-1.0)
    centrality={}
    s=1.0/(G.order()-1.0)
    d_iter = G.__getattribute__(degree_iter)
    for n,deg in d_iter():
        centrality[n]=deg*s
    return centrality
 

def degree_centrality(G,v=None):
    """Compute the degree centrality for nodes.

    The degree centrality for a node v is the fraction of nodes it
    is connected to.

    Parameters
    ----------
    G : graph
      A networkx graph 

    v : node, optional
      Return only the value for node v.
      
    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with degree centrality as the value.

    See Also
    --------
    betweenness_centrality(), load_centrality(), eigenvector_centrality() 

    Notes
    -----
    The degree centrality is normalized to the maximum possible degree
    in the graph G.  That is, G.degree(v)/(G.order()-1).

    """
    return _degree_centrality('degree', 'degree_iter', G, v)


def in_degree_centrality(G,v=None):
    """Compute the in-degree centrality for nodes.

    The in-degree centrality for a node v is the fraction of nodes its 
    incoming edges are connected to.

    Parameters
    ----------
    G : graph
        A NetworkX graph

    v : node, optional
        Return only the value for node `v'.

    Returns
    -------
    nodes : dictionary
        Dictionary of nodes with in-degree centrality as values.

    See Also
    --------
    degree_centrality(), out_degree_centrality()

    """
    return _degree_centrality('in_degree', 'in_degree_iter', G, v)


def out_degree_centrality(G,v=None):
    """Compute the out-degree centrality for nodes.

    The out-degree centrality for a node v is the fraction of nodes its 
    outgoing edges are connected to.

    Parameters
    ----------
    G : graph
        A NetworkX graph

    v : node, optional
        Return only the value for node `v'.

    Returns
    -------
    nodes : dictionary
        Dictionary of nodes with out-degree centrality as values.

    See Also
    --------
    degree_centrality(), in_degree_centrality()

    """
    return _degree_centrality('out_degree', 'out_degree_iter', G, v)


def closeness_centrality(G,v=None,weighted_edges=False):
    """Compute closeness centrality for nodes.

    Closeness centrality at a node is 1/average distance to all 
    other nodes.

    Parameters
    ----------
    G : graph
      A networkx graph 

    v : node, optional
      Return only the value for node v.

    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.
      
    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with closeness centrality as the value.

    See Also
    --------
    betweenness_centrality(), load_centrality(), eigenvector_centrality(),
    degree_centrality()

    Notes
    -----
    The closeness centrality is normalized to to n-1 / size(G)-1 where 
    n is the number of nodes in the connected part of graph containing
    the node.  If the graph is not completely connected, this
    algorithm computes the closeness centrality for each connected
    part separately.  

    """
    if weighted_edges:
        path_length=single_source_dijkstra_path_length
    else:
        path_length=single_source_shortest_path_length

    if v is None:
        closeness_centrality={}
        for n in G:
            sp=path_length(G,n)
            totsp=sum(sp.values())
            if totsp > 0.0 and len(G) > 1:
                # normalize to number of nodes-1 in connected part
                s=(len(sp)-1.0) / ( len(G) - 1 )
                closeness_centrality[n]= s / (totsp/(len(sp)-1.0))
            else:                                                                
                closeness_centrality[n]=0.0           
        return closeness_centrality
    else: # only compute for v
        sp=path_length(G,v)
        totsp=sum(sp.values())
        if totsp > 0.0 and len(G) > 1: 
            # normalize to number of nodes-1 in connected part
            return ( (len(sp)-1.0)/(len(G) - 1) )/ ( totsp / (len(sp) - 1.0) )
        else:
            return 0.0

def eigenvector_centrality(G,max_iter=100,tol=1.0e-6,nstart=None):
    """Compute the eigenvector centrality for the graph G.

    Uses the power method to find the eigenvector for the 
    largest eigenvalue of the adjacency matrix of G.

    Parameters
    ----------
    G : graph
      A networkx graph 

    max_iter : interger, optional
      Maximum number of iterations in power method.

    tol : float, optional
      Error tolerance used to check convergence in power method iteration.

    nstart : dictionary, optional
      Starting value of eigenvector iteration for each node. 

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with eigenvector centrality as the value.

    Examples
    --------
    >>> G=nx.path_graph(4)
    >>> centrality=nx.eigenvector_centrality(G)
    >>> print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
    ['0 0.19', '1 0.31', '2 0.31', '3 0.19']

    Notes
    ------
    The eigenvector calculation is done by the power iteration method
    and has no guarantee of convergence.  The iteration will stop
    after max_iter iterations or an error tolerance of
    number_of_nodes(G)*tol has been reached.

    For directed graphs this is "right" eigevector centrality.  For
    "left" eigenvector centrality, first reverse the graph with
    G.reverse().

    See Also
    --------
    pagerank()

    """
    if type(G) == networkx.MultiGraph or type(G) == networkx.MultiDiGraph:
        raise Exception("eigenvector_centrality() not defined for graphs with multiedges.")

#    if not G.weighted:
#        raise Exception("eigenvector_centrality(): input graph must be weighted")

    if nstart is None:
        # choose starting vector with entries of 1/len(G) 
        x=dict([(n,1.0/len(G)) for n in G])
    else:
        x=nstart
    # normalize starting vector
    s=1.0/sum(x.values())
    for k in x: x[k]*=s
    nnodes=G.number_of_nodes()
    # make up to max_iter iterations        
    for i in range(max_iter):
        xlast=x
        x=dict.fromkeys(xlast.keys(),0)
        # do the multiplication y=Ax
        for n in x:
            for nbr in G[n]:
                x[n]+=xlast[nbr]*G[n][nbr].get('weight',1)
        # normalize vector
        s=1.0/sum(x.values())
        for n in x: x[n]*=s
        # check convergence            
        err=sum([abs(x[n]-xlast[n]) for n in x])
        if err < nnodes*tol:
            return x

    raise networkx.NetworkXError("eigenvector_centrality(): power iteration failed to converge in %d iterations."%(i+1))

<commit_msg>Catch case of zero eigenvector in eigenvector_centrality.<commit_after>"""
Centrality measures.

"""
#    Copyright (C) 2004-2010 by 
#    <NAME> <<EMAIL>>
#    <NAME> <<EMAIL>>
#    <NAME> <<EMAIL>>
#    All rights reserved.
#    BSD license.
__author__ = """<NAME> (<EMAIL>)\<NAME> (<EMAIL>)\n<NAME> (<EMAIL>)"""

__all__ = ['betweenness_centrality',
           'betweenness_centrality_source',
           'load_centrality',
           'newman_betweenness_centrality',
           'brandes_betweenness_centrality',
           'edge_betweenness',
           'edge_load',
           'degree_centrality',
           'in_degree_centrality',
           'out_degree_centrality',
           'closeness_centrality',
           'eigenvector_centrality']


import heapq
from networkx.algorithms.traversal.path import predecessor, \
        single_source_shortest_path_length, \
        single_source_dijkstra_path_length, \
        dijkstra_predecessor_and_distance 
import networkx


def brandes_betweenness_centrality(G,normalized=True,weighted_edges=False):
    """Compute betweenness centrality for nodes.

    Betweenness centrality of a node is the fraction of all shortest 
    paths that pass through that node.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with betweeness centrality as the value.

        
    See Also
    --------
    load_centrality()

    Notes
    -----
    The algorithm is from <NAME>,
    A Faster Algorithm for Betweenness Centrality.
    Journal of Mathematical Sociology 25(2):163-177, 2001.
    http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf

"""
    betweenness=dict.fromkeys(G,0.0) # b[v]=0 for v in G
    for s in G:
        S=[]
        P={}
        for v in G:
            P[v]=[]
        sigma=dict.fromkeys(G,0)    # sigma[v]=0 for v in G
        D={}
        sigma[s]=1
        if not weighted_edges:  # use BFS
            D[s]=0
            Q=[s]
            while Q:   # use BFS to find shortest paths
                v=Q.pop(0)
                S.append(v)
                for w in G[v]:
#                for w in G.adj[v]: # speed hack, exposes internals
                    if w not in D:
                        Q.append(w)
                        D[w]=D[v]+1
                    if D[w]==D[v]+1:   # this is a shortest path, count paths
                        sigma[w]=sigma[w]+sigma[v]
                        P[w].append(v) # predecessors 
        else:  # use Dijkstra's algorithm for shortest paths,
               # modified from Eppstein
            push=heapq.heappush
            pop=heapq.heappop
            seen = {s:0} 
            Q=[]   # use Q as heap with (distance,node id) tuples
            push(Q,(0,s,s))
            while Q:   
                (dist,pred,v)=pop(Q)
                if v in D:
                    continue # already searched this node.
                sigma[v]=sigma[v]+sigma[pred] # count paths
                S.append(v)
                D[v] = dist
                for w,edgedata in G[v].iteritems():
                    vw_dist = D[v] + edgedata.get('weight',1)
                    if w not in D and (w not in seen or vw_dist < seen[w]):
                        seen[w] = vw_dist
                        push(Q,(vw_dist,v,w))
                        sigma[w]=0
                        P[w]=[v]
                    elif vw_dist==seen[w]:  # handle equal paths
                        sigma[w]=sigma[w]+sigma[v]
                        P[w].append(v)


        delta=dict.fromkeys(G,0) 
        while S:
            w=S.pop()
            for v in P[w]:
                delta[v]=delta[v]+\
                          (float(sigma[v])/float(sigma[w]))*(1.0+delta[w])
            if w != s:
                betweenness[w]=betweenness[w]+delta[w]
                    
    # normalize
    if normalized:
        order=len(betweenness)
        if order <=2:
            return betweenness # no normalization b=0 for all nodes
        scale=1.0/((order-1)*(order-2))
        for v in betweenness:
            betweenness[v] *= scale

    return betweenness            

betweenness_centrality=brandes_betweenness_centrality


def newman_betweenness_centrality(G,v=None,cutoff=None,
                           normalized=True,
                           weighted_edges=False):
    """Compute load centrality for nodes.

    The load centrality of a node is the fraction of all shortest 
    paths that pass through that node.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    cutoff : bool, optional
      If specified, only consider paths of length <= cutoff.

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with centrality as the value.

        
    See Also
    --------
    betweenness_centrality() 

    Notes
    -----
    Load centrality is slightly different than betweenness.
    For this load algorithm see the reference
    Scientific collaboration networks: II.
    Shortest paths, weighted networks, and centrality,
    M. <NAME>, Phys. Rev. E 64, 016132 (2001).

    """
    if v is not None:   # only one node
        betweenness=0.0
        for source in G: 
            ubetween=_node_betweenness(G,source,
                                       cutoff=cutoff,
                                       normalized=normalized,
                                       weighted_edges=weighted_edges)
            betweenness+=ubetween[v]
        return betweenness
    else:
        betweenness={}.fromkeys(G,0.0) 
        for source in betweenness: 
            ubetween=_node_betweenness(G,source,
                                       cutoff=cutoff,
                                       normalized=False,
                                       weighted_edges=weighted_edges)
            for vk in ubetween:
                betweenness[vk]+=ubetween[vk]
        if normalized:
            order=len(betweenness)
            if order <=2:
                return betweenness # no normalization b=0 for all nodes
            scale=1.0/((order-1)*(order-2))
            for v in betweenness:
                betweenness[v] *= scale
        return betweenness  # all nodes

def _node_betweenness(G,source,cutoff=False,normalized=True,weighted_edges=False):
    """Node betweenness helper:
    see betweenness_centrality for what you probably want.

    This actually computes "load" and not betweenness.
    See https://networkx.lanl.gov/ticket/103

    This calculates the load of each node for paths from a single source.
    (The fraction of number of shortests paths from source that go
    through each node.)

    To get the load for a node you need to do all-pairs shortest paths.

    If weighted_edges is True then use Dijkstra for finding shortest paths.
    In this case a cutoff is not implemented and so is ignored.

    """

    # get the predecessor and path length data
    if weighted_edges:
        (pred,length)=dijkstra_predecessor_and_distance(G,source) 
    else:
        (pred,length)=predecessor(G,source,cutoff=cutoff,return_seen=True) 

    # order the nodes by path length
    onodes = [ (l,vert) for (vert,l) in length.items() ]
    onodes.sort()
    onodes[:] = [vert for (l,vert) in onodes if l>0]
    
    # intialize betweenness
    between={}.fromkeys(length,1.0)

    while onodes:           
        v=onodes.pop()
        if v in pred:
            num_paths=len(pred[v])   # Discount betweenness if more than 
            for x in pred[v]:        # one shortest path.
                if x==source:   # stop if hit source because all remaining v  
                    break       #  also have pred[v]==[source]
                between[x]+=between[v]/float(num_paths)
    #  remove source
    for v in between:
        between[v]-=1
    # rescale to be between 0 and 1                
    if normalized:
        l=len(between)
        if l > 2:
            scale=1.0/float((l-1)*(l-2)) # 1/the number of possible paths
            for v in between:
                between[v] *= scale
    return between


load_centrality=newman_betweenness_centrality

def betweenness_centrality_source(G,normalized=True,
                                  weighted_edges=False,
                                  sources=None):
    """Compute betweenness centrality for a subgraph.

    Enchanced version of the method in centrality module that allows
    specifying a list of sources (subgraph).


    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    sources : node list 
      A list of nodes to consider as sources for shortest paths.
      

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with betweeness centrality as the value.


    Notes
    -----
    See Sec. 4 in 
    <NAME>,
    A Faster Algorithm for Betweenness Centrality.
    Journal of Mathematical Sociology 25(2):163-177, 2001.
    http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf

    This algorithm does not count the endpoints, i.e.
    a path from s to t does not contribute to the betweenness of s and t.
    """
    if sources == None:
        sources = G   # only used to iterate over nodes.

    betweenness=dict.fromkeys(G,0.0)
    for s in sources:
        S,P,D,sigma = _brandes_betweenness_helper(G,s,weighted_edges)

        delta=dict.fromkeys(G,0) # unnormalized betweenness
        while S:
            w=S.pop()
            for v in P[w]:
                delta[v] += (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
            if w == s:
                continue
            betweenness[w] = betweenness[w] + delta[w]
                   
    # normalize to size of entire graph
    if normalized and G.number_of_edges() > 1:
        order=len(betweenness)
        scale=1.0/((order-1)*(order-2))
        for v in betweenness:
            betweenness[v] *= scale

    return betweenness


def edge_betweenness(G,normalized=True,weighted_edges=False,sources=None):
    """Compute betweenness centrality for edges.

    Parameters
    ----------
    G : graph
      A networkx graph 

    normalized : bool, optional
      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
      n is the number of nodes in G.
       
    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.

    sources : node list 
      A list of nodes to consider as sources for shortest paths.
      

    Returns
    -------
    nodes : dictionary
       Dictionary of edges with betweeness centrality as the value.
    """
    if sources is None:
        sources = G # only used to iterate over nodes

    betweenness=dict.fromkeys(G.edges(),0.0)
    if G.is_directed():
        for s in sources:
            S, P, D, sigma =_brandes_betweenness_helper(G,s,weighted_edges)
            delta=dict.fromkeys(G,0.0)
            while S:
                w=S.pop()
                for v in P[w]:
                    edgeFlow = (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
                    edge = (v,w)
                    delta[v]          += edgeFlow
                    betweenness[edge] += edgeFlow
    else:
        for s in sources:
            S, P, D, sigma =_brandes_betweenness_helper(G,s,weighted_edges)
            delta=dict.fromkeys(G,0.0)
            while S:
                w=S.pop()
                for v in P[w]:
                    edgeFlow = (1.0+delta[w])*sigma[v]/sigma[w] # 1.0 converts all to float
                    edge = (v,w)
                    if edge not in betweenness:
                        edge = (w,v)
                    delta[v]          += edgeFlow
                    betweenness[edge] += edgeFlow

    size=len(betweenness)                
    if normalized and size > 1:
        # normalize to size of entire graph (beware of disconnected components)
        scale=1.0/((size-1)*(size-2))
        for edge in betweenness:
            betweenness[edge] *= scale

    return betweenness


def _brandes_betweenness_helper(G,root,weighted_edges):
    """
    Helper for betweenness centrality and edge betweenness centrality.

    Runs single-source shortest path from root node.

    weighted_edges:: consider edge weights 

    Finds::

    S=[] list of nodes reached during traversal
    P={} predecessors, keyed by child node
    D={} distances
    sigma={} indexed by node, is the number of paths to root
    going through the node
    """
    S=[]
    P={}
    for v in G:
        P[v]=[]
    sigma=dict.fromkeys(G,0.0)
    D={}
    sigma[root]=1

    if not weighted_edges:  # use BFS
        D[root]=0
        Q=[root]
        while Q:   # use BFS to find shortest paths
            v=Q.pop(0)
            S.append(v)
            for w in G[v]: #  for w in G.adj[v]: # speed hack, exposes internals
                if w not in D:
                    Q.append(w)
                    D[w]=D[v]+1
                if D[w]==D[v]+1:   # this is a shortest path, count paths
                    sigma[w]=sigma[w]+sigma[v]
                    P[w].append(v) # predecessors
    else:  # use Dijkstra's algorithm for shortest paths,
           # modified from Eppstein
        push=heapq.heappush
        pop=heapq.heappop
        seen = {root:0}
        Q=[]   # use Q as heap with (distance,node id) tuples
        push(Q,(0,root,root))
        while Q:   
            (dist,pred,v)=pop(Q)
            if v in D:
                continue # already searched this node.
            sigma[v]=sigma[v]+sigma[pred] # count paths
            S.append(v)
            D[v] = dist
            for w,edgedata in G[v].iteritems(): 
                vw_dist = D[v] + edgedata.get('weight',1)
                if w not in D and (w not in seen or vw_dist < seen[w]):
                    seen[w] = vw_dist
                    sigma[w] = 0
                    push(Q,(vw_dist,v,w))
                    P[w]=[v]
                elif vw_dist==seen[w]:  # handle equal paths
                    sigma[w]=sigma[w]+sigma[v]
                    P[w].append(v)
    return S, P, D, sigma


def edge_load(G,nodes=None,cutoff=False):
    """Compute edge load.

    WARNING:

    This module is for demonstration and testing purposes.

    """
    betweenness={} 
    if not nodes:         # find betweenness for every node  in graph
        nodes=G.nodes()   # that probably is what you want...
    for source in nodes:
        ubetween=_edge_betweenness(G,source,nodes,cutoff=cutoff)
        for v in ubetween.keys():
            b=betweenness.setdefault(v,0)  # get or set default
            betweenness[v]=ubetween[v]+b    # cumulative total
    return betweenness

def _edge_betweenness(G,source,nodes,cutoff=False):
    """
    Edge betweenness helper.
    """
    between={}
    # get the predecessor data
    #(pred,length)=_fast_predecessor(G,source,cutoff=cutoff) 
    (pred,length)=predecessor(G,source,cutoff=cutoff,return_seen=True) 
    # order the nodes by path length
    onodes = [ nn for dd,nn in sorted( (dist,n) for n,dist in length.iteritems() )]
    # intialize betweenness, doesn't account for any edge weights
    for u,v in G.edges(nodes):
        between[(u,v)]=1.0
        between[(v,u)]=1.0

    while onodes:           # work through all paths
        v=onodes.pop()
        if v in pred:
            num_paths=len(pred[v])   # Discount betweenness if more than 
            for w in pred[v]:        # one shortest path.
                if w in pred:
                    num_paths=len(pred[w])  # Discount betweenness, mult path  
                    for x in pred[w]: 
                        between[(w,x)]+=between[(v,w)]/num_paths
                        between[(x,w)]+=between[(w,v)]/num_paths
    return between


def _degree_centrality(degree,degree_iter,G,v=None):
    """Internal function to consolidate *-degree centrality functions."""
    if v is not None:
        d = G.__getattribute__(degree)
        return d(v)/(G.order()-1.0)
    centrality={}
    s=1.0/(G.order()-1.0)
    d_iter = G.__getattribute__(degree_iter)
    for n,deg in d_iter():
        centrality[n]=deg*s
    return centrality
 

def degree_centrality(G,v=None):
    """Compute the degree centrality for nodes.

    The degree centrality for a node v is the fraction of nodes it
    is connected to.

    Parameters
    ----------
    G : graph
      A networkx graph 

    v : node, optional
      Return only the value for node v.
      
    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with degree centrality as the value.

    See Also
    --------
    betweenness_centrality(), load_centrality(), eigenvector_centrality() 

    Notes
    -----
    The degree centrality is normalized to the maximum possible degree
    in the graph G.  That is, G.degree(v)/(G.order()-1).

    """
    return _degree_centrality('degree', 'degree_iter', G, v)


def in_degree_centrality(G,v=None):
    """Compute the in-degree centrality for nodes.

    The in-degree centrality for a node v is the fraction of nodes its 
    incoming edges are connected to.

    Parameters
    ----------
    G : graph
        A NetworkX graph

    v : node, optional
        Return only the value for node `v'.

    Returns
    -------
    nodes : dictionary
        Dictionary of nodes with in-degree centrality as values.

    See Also
    --------
    degree_centrality(), out_degree_centrality()

    """
    return _degree_centrality('in_degree', 'in_degree_iter', G, v)


def out_degree_centrality(G,v=None):
    """Compute the out-degree centrality for nodes.

    The out-degree centrality for a node v is the fraction of nodes its 
    outgoing edges are connected to.

    Parameters
    ----------
    G : graph
        A NetworkX graph

    v : node, optional
        Return only the value for node `v'.

    Returns
    -------
    nodes : dictionary
        Dictionary of nodes with out-degree centrality as values.

    See Also
    --------
    degree_centrality(), in_degree_centrality()

    """
    return _degree_centrality('out_degree', 'out_degree_iter', G, v)


def closeness_centrality(G,v=None,weighted_edges=False):
    """Compute closeness centrality for nodes.

    Closeness centrality at a node is 1/average distance to all 
    other nodes.

    Parameters
    ----------
    G : graph
      A networkx graph 

    v : node, optional
      Return only the value for node v.

    weighted_edges : bool, optional
      Consider the edge weights in determining the shortest paths.
      If False, all edge weights are considered equal.
      
    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with closeness centrality as the value.

    See Also
    --------
    betweenness_centrality(), load_centrality(), eigenvector_centrality(),
    degree_centrality()

    Notes
    -----
    The closeness centrality is normalized to to n-1 / size(G)-1 where 
    n is the number of nodes in the connected part of graph containing
    the node.  If the graph is not completely connected, this
    algorithm computes the closeness centrality for each connected
    part separately.  

    """
    if weighted_edges:
        path_length=single_source_dijkstra_path_length
    else:
        path_length=single_source_shortest_path_length

    if v is None:
        closeness_centrality={}
        for n in G:
            sp=path_length(G,n)
            totsp=sum(sp.values())
            if totsp > 0.0 and len(G) > 1:
                # normalize to number of nodes-1 in connected part
                s=(len(sp)-1.0) / ( len(G) - 1 )
                closeness_centrality[n]= s / (totsp/(len(sp)-1.0))
            else:                                                                
                closeness_centrality[n]=0.0           
        return closeness_centrality
    else: # only compute for v
        sp=path_length(G,v)
        totsp=sum(sp.values())
        if totsp > 0.0 and len(G) > 1: 
            # normalize to number of nodes-1 in connected part
            return ( (len(sp)-1.0)/(len(G) - 1) )/ ( totsp / (len(sp) - 1.0) )
        else:
            return 0.0

def eigenvector_centrality(G,max_iter=100,tol=1.0e-6,nstart=None):
    """Compute the eigenvector centrality for the graph G.

    Uses the power method to find the eigenvector for the 
    largest eigenvalue of the adjacency matrix of G.

    Parameters
    ----------
    G : graph
      A networkx graph 

    max_iter : interger, optional
      Maximum number of iterations in power method.

    tol : float, optional
      Error tolerance used to check convergence in power method iteration.

    nstart : dictionary, optional
      Starting value of eigenvector iteration for each node. 

    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with eigenvector centrality as the value.

    Examples
    --------
    >>> G=nx.path_graph(4)
    >>> centrality=nx.eigenvector_centrality(G)
    >>> print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
    ['0 0.19', '1 0.31', '2 0.31', '3 0.19']

    Notes
    ------
    The eigenvector calculation is done by the power iteration method
    and has no guarantee of convergence.  The iteration will stop
    after max_iter iterations or an error tolerance of
    number_of_nodes(G)*tol has been reached.

    For directed graphs this is "right" eigevector centrality.  For
    "left" eigenvector centrality, first reverse the graph with
    G.reverse().

    See Also
    --------
    pagerank()

    """
    if type(G) == networkx.MultiGraph or type(G) == networkx.MultiDiGraph:
        raise Exception("eigenvector_centrality() not defined for graphs with multiedges.")

#    if not G.weighted:
#        raise Exception("eigenvector_centrality(): input graph must be weighted")

    if nstart is None:
        # choose starting vector with entries of 1/len(G) 
        x=dict([(n,1.0/len(G)) for n in G])
    else:
        x=nstart
    # normalize starting vector
    s=1.0/sum(x.values())
    for k in x: x[k]*=s
    nnodes=G.number_of_nodes()
    # make up to max_iter iterations        
    for i in range(max_iter):
        xlast=x
        x=dict.fromkeys(xlast.keys(),0)
        # do the multiplication y=Ax
        for n in x:
            for nbr in G[n]:
                x[n]+=xlast[nbr]*G[n][nbr].get('weight',1)
        # normalize vector
        try:
            s=1.0/sum(x.values())
        except ZeroDivisionError:
            s=1.0
        for n in x: x[n]*=s
        # check convergence            
        err=sum([abs(x[n]-xlast[n]) for n in x])
        if err < nnodes*tol:
            return x

    raise networkx.NetworkXError("eigenvector_centrality(): power iteration failed to converge in %d iterations."%(i+1))


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