267 lines
8.2 KiB
Python
267 lines
8.2 KiB
Python
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import networkx as nx
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__all__ = ["degree_centrality", "betweenness_centrality", "closeness_centrality"]
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def degree_centrality(G, nodes):
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r"""Compute the degree centrality for nodes in a bipartite network.
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The degree centrality for a node `v` is the fraction of nodes
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connected to it.
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Parameters
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----------
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G : graph
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A bipartite network
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nodes : list or container
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Container with all nodes in one bipartite node set.
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Returns
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-------
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centrality : dictionary
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Dictionary keyed by node with bipartite degree centrality as the value.
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See Also
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--------
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betweenness_centrality
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closeness_centrality
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:func:`~networkx.algorithms.bipartite.basic.sets`
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:func:`~networkx.algorithms.bipartite.basic.is_bipartite`
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Notes
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-----
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The nodes input parameter must contain all nodes in one bipartite node set,
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but the dictionary returned contains all nodes from both bipartite node
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sets. See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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For unipartite networks, the degree centrality values are
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normalized by dividing by the maximum possible degree (which is
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`n-1` where `n` is the number of nodes in G).
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In the bipartite case, the maximum possible degree of a node in a
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bipartite node set is the number of nodes in the opposite node set
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[1]_. The degree centrality for a node `v` in the bipartite
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sets `U` with `n` nodes and `V` with `m` nodes is
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.. math::
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d_{v} = \frac{deg(v)}{m}, \mbox{for} v \in U ,
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d_{v} = \frac{deg(v)}{n}, \mbox{for} v \in V ,
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where `deg(v)` is the degree of node `v`.
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References
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----------
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.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
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Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
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of Social Network Analysis. Sage Publications.
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https://dx.doi.org/10.4135/9781446294413.n28
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"""
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top = set(nodes)
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bottom = set(G) - top
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s = 1.0 / len(bottom)
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centrality = {n: d * s for n, d in G.degree(top)}
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s = 1.0 / len(top)
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centrality.update({n: d * s for n, d in G.degree(bottom)})
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return centrality
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def betweenness_centrality(G, nodes):
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r"""Compute betweenness centrality for nodes in a bipartite network.
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Betweenness centrality of a node `v` is the sum of the
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fraction of all-pairs shortest paths that pass through `v`.
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Values of betweenness are normalized by the maximum possible
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value which for bipartite graphs is limited by the relative size
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of the two node sets [1]_.
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Let `n` be the number of nodes in the node set `U` and
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`m` be the number of nodes in the node set `V`, then
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nodes in `U` are normalized by dividing by
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.. math::
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\frac{1}{2} [m^2 (s + 1)^2 + m (s + 1)(2t - s - 1) - t (2s - t + 3)] ,
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where
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.. math::
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s = (n - 1) \div m , t = (n - 1) \mod m ,
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and nodes in `V` are normalized by dividing by
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.. math::
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\frac{1}{2} [n^2 (p + 1)^2 + n (p + 1)(2r - p - 1) - r (2p - r + 3)] ,
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where,
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.. math::
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p = (m - 1) \div n , r = (m - 1) \mod n .
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Parameters
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----------
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G : graph
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A bipartite graph
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nodes : list or container
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Container with all nodes in one bipartite node set.
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Returns
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-------
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betweenness : dictionary
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Dictionary keyed by node with bipartite betweenness centrality
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as the value.
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See Also
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--------
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degree_centrality
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closeness_centrality
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:func:`~networkx.algorithms.bipartite.basic.sets`
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:func:`~networkx.algorithms.bipartite.basic.is_bipartite`
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Notes
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-----
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The nodes input parameter must contain all nodes in one bipartite node set,
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but the dictionary returned contains all nodes from both node sets.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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References
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----------
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.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
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Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
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of Social Network Analysis. Sage Publications.
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https://dx.doi.org/10.4135/9781446294413.n28
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"""
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top = set(nodes)
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bottom = set(G) - top
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n = len(top)
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m = len(bottom)
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s, t = divmod(n - 1, m)
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bet_max_top = (
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((m**2) * ((s + 1) ** 2))
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+ (m * (s + 1) * (2 * t - s - 1))
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- (t * ((2 * s) - t + 3))
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) / 2.0
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p, r = divmod(m - 1, n)
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bet_max_bot = (
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((n**2) * ((p + 1) ** 2))
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+ (n * (p + 1) * (2 * r - p - 1))
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- (r * ((2 * p) - r + 3))
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) / 2.0
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betweenness = nx.betweenness_centrality(G, normalized=False, weight=None)
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for node in top:
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betweenness[node] /= bet_max_top
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for node in bottom:
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betweenness[node] /= bet_max_bot
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return betweenness
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def closeness_centrality(G, nodes, normalized=True):
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r"""Compute the closeness centrality for nodes in a bipartite network.
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The closeness of a node is the distance to all other nodes in the
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graph or in the case that the graph is not connected to all other nodes
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in the connected component containing that node.
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Parameters
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----------
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G : graph
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A bipartite network
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nodes : list or container
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Container with all nodes in one bipartite node set.
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normalized : bool, optional
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If True (default) normalize by connected component size.
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Returns
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-------
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closeness : dictionary
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Dictionary keyed by node with bipartite closeness centrality
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as the value.
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See Also
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--------
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betweenness_centrality
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degree_centrality
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:func:`~networkx.algorithms.bipartite.basic.sets`
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:func:`~networkx.algorithms.bipartite.basic.is_bipartite`
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Notes
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-----
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The nodes input parameter must contain all nodes in one bipartite node set,
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but the dictionary returned contains all nodes from both node sets.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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Closeness centrality is normalized by the minimum distance possible.
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In the bipartite case the minimum distance for a node in one bipartite
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node set is 1 from all nodes in the other node set and 2 from all
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other nodes in its own set [1]_. Thus the closeness centrality
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for node `v` in the two bipartite sets `U` with
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`n` nodes and `V` with `m` nodes is
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.. math::
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c_{v} = \frac{m + 2(n - 1)}{d}, \mbox{for} v \in U,
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c_{v} = \frac{n + 2(m - 1)}{d}, \mbox{for} v \in V,
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where `d` is the sum of the distances from `v` to all
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other nodes.
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Higher values of closeness indicate higher centrality.
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As in the unipartite case, setting normalized=True causes the
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values to normalized further to n-1 / size(G)-1 where n is the
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number of nodes in the connected part of graph containing the
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node. If the graph is not completely connected, this algorithm
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computes the closeness centrality for each connected part
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separately.
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References
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----------
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.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
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Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
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of Social Network Analysis. Sage Publications.
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https://dx.doi.org/10.4135/9781446294413.n28
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"""
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closeness = {}
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path_length = nx.single_source_shortest_path_length
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top = set(nodes)
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bottom = set(G) - top
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n = len(top)
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m = len(bottom)
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for node in top:
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sp = dict(path_length(G, node))
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totsp = sum(sp.values())
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if totsp > 0.0 and len(G) > 1:
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closeness[node] = (m + 2 * (n - 1)) / totsp
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if normalized:
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s = (len(sp) - 1) / (len(G) - 1)
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closeness[node] *= s
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else:
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closeness[node] = 0.0
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for node in bottom:
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sp = dict(path_length(G, node))
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totsp = sum(sp.values())
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if totsp > 0.0 and len(G) > 1:
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closeness[node] = (n + 2 * (m - 1)) / totsp
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if normalized:
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s = (len(sp) - 1) / (len(G) - 1)
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closeness[node] *= s
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else:
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closeness[node] = 0.0
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return closeness
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