367 lines
11 KiB
Python
367 lines
11 KiB
Python
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"""Group centrality measures."""
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from itertools import combinations
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import networkx as nx
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from networkx.utils.decorators import not_implemented_for
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__all__ = [
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"group_betweenness_centrality",
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"group_closeness_centrality",
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"group_degree_centrality",
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"group_in_degree_centrality",
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"group_out_degree_centrality",
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]
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def group_betweenness_centrality(G, C, normalized=True, weight=None):
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r"""Compute the group betweenness centrality for a group of nodes.
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Group betweenness centrality of a group of nodes $C$ is the sum of the
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fraction of all-pairs shortest paths that pass through any vertex in $C$
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.. math::
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c_B(C) =\sum_{s,t \in V-C; s<t} \frac{\sigma(s, t|C)}{\sigma(s, t)}
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where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
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shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of
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those paths passing through some node in group $C$. Note that
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$(s, t)$ are not members of the group ($V-C$ is the set of nodes
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in $V$ that are not in $C$).
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Parameters
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----------
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G : graph
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A NetworkX graph.
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C : list or set
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C is a group of nodes which belong to G, for which group betweenness
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centrality is to be calculated.
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normalized : bool, optional
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If True, group betweenness is normalized by `2/((|V|-|C|)(|V|-|C|-1))`
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for graphs and `1/((|V|-|C|)(|V|-|C|-1))` for directed graphs where `|V|`
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is the number of nodes in G and `|C|` is the number of nodes in C.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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Raises
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------
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NodeNotFound
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If node(s) in C are not present in G.
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Returns
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-------
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betweenness : float
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Group betweenness centrality of the group C.
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See Also
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--------
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betweenness_centrality
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Notes
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-----
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The measure is described in [1]_.
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The algorithm is an extension of the one proposed by Ulrik Brandes for
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betweenness centrality of nodes. Group betweenness is also mentioned in
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his paper [2]_ along with the algorithm. The importance of the measure is
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discussed in [3]_.
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The number of nodes in the group must be a maximum of n - 2 where `n`
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is the total number of nodes in the graph.
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For weighted graphs the edge weights must be greater than zero.
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Zero edge weights can produce an infinite number of equal length
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paths between pairs of nodes.
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References
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----------
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.. [1] M G Everett and S P Borgatti:
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The Centrality of Groups and Classes.
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Journal of Mathematical Sociology. 23(3): 181-201. 1999.
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http://www.analytictech.com/borgatti/group_centrality.htm
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.. [2] Ulrik Brandes:
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On Variants of Shortest-Path Betweenness
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Centrality and their Generic Computation.
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Social Networks 30(2):136-145, 2008.
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http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf
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.. [3] Sourav Medya et. al.:
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Group Centrality Maximization via Network Design.
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SIAM International Conference on Data Mining, SDM 2018, 126–134.
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https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf
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"""
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betweenness = 0 # initialize betweenness to 0
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V = set(G) # set of nodes in G
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C = set(C) # set of nodes in C (group)
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if len(C - V) != 0: # element(s) of C not in V
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raise nx.NodeNotFound(
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"The node(s) " + str(list(C - V)) + " are not " "in the graph."
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)
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V_C = V - C # set of nodes in V but not in C
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# accumulation
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for pair in combinations(V_C, 2): # (s, t) pairs of V_C
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try:
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paths = 0
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paths_through_C = 0
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for path in nx.all_shortest_paths(
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G, source=pair[0], target=pair[1], weight=weight
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):
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if set(path) & C:
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paths_through_C += 1
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paths += 1
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betweenness += paths_through_C / paths
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except nx.exception.NetworkXNoPath:
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betweenness += 0
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# rescaling
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v, c = len(G), len(C)
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if normalized:
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scale = 1 / ((v - c) * (v - c - 1))
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if not G.is_directed():
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scale *= 2
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else:
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scale = None
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if scale is not None:
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betweenness *= scale
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return betweenness
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def group_closeness_centrality(G, S, weight=None):
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r"""Compute the group closeness centrality for a group of nodes.
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Group closeness centrality of a group of nodes $S$ is a measure
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of how close the group is to the other nodes in the graph.
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.. math::
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c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}}
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d_{S, v} = min_{u \in S} (d_{u, v})
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where $V$ is the set of nodes, $d_{S, v}$ is the distance of
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the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes
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in $V$ that are not in $S$).
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Parameters
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----------
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G : graph
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A NetworkX graph.
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S : list or set
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S is a group of nodes which belong to G, for which group closeness
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centrality is to be calculated.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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Raises
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------
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NodeNotFound
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If node(s) in S are not present in G.
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Returns
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-------
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closeness : float
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Group closeness centrality of the group S.
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See Also
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--------
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closeness_centrality
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Notes
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-----
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The measure was introduced in [1]_.
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The formula implemented here is described in [2]_.
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Higher values of closeness indicate greater centrality.
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It is assumed that 1 / 0 is 0 (required in the case of directed graphs,
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or when a shortest path length is 0).
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The number of nodes in the group must be a maximum of n - 1 where `n`
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is the total number of nodes in the graph.
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For directed graphs, the incoming distance is utilized here. To use the
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outward distance, act on `G.reverse()`.
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For weighted graphs the edge weights must be greater than zero.
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Zero edge weights can produce an infinite number of equal length
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paths between pairs of nodes.
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References
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----------
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.. [1] M G Everett and S P Borgatti:
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The Centrality of Groups and Classes.
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Journal of Mathematical Sociology. 23(3): 181-201. 1999.
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http://www.analytictech.com/borgatti/group_centrality.htm
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.. [2] J. Zhao et. al.:
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Measuring and Maximizing Group Closeness Centrality over
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Disk Resident Graphs.
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WWWConference Proceedings, 2014. 689-694.
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http://wwwconference.org/proceedings/www2014/companion/p689.pdf
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"""
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if G.is_directed():
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G = G.reverse() # reverse view
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closeness = 0 # initialize to 0
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V = set(G) # set of nodes in G
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S = set(S) # set of nodes in group S
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V_S = V - S # set of nodes in V but not S
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shortest_path_lengths = nx.multi_source_dijkstra_path_length(G, S, weight=weight)
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# accumulation
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for v in V_S:
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try:
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closeness += shortest_path_lengths[v]
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except KeyError: # no path exists
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closeness += 0
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try:
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closeness = len(V_S) / closeness
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except ZeroDivisionError: # 1 / 0 assumed as 0
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closeness = 0
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return closeness
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def group_degree_centrality(G, S):
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"""Compute the group degree centrality for a group of nodes.
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Group degree centrality of a group of nodes $S$ is the fraction
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of non-group members connected to group members.
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Parameters
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----------
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G : graph
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A NetworkX graph.
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S : list or set
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S is a group of nodes which belong to G, for which group degree
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centrality is to be calculated.
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Raises
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------
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NetworkXError
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If node(s) in S are not in G.
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Returns
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-------
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centrality : float
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Group degree centrality of the group S.
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See Also
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--------
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degree_centrality
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group_in_degree_centrality
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group_out_degree_centrality
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Notes
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-----
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The measure was introduced in [1]_.
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The number of nodes in the group must be a maximum of n - 1 where `n`
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is the total number of nodes in the graph.
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References
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----------
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.. [1] M G Everett and S P Borgatti:
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The Centrality of Groups and Classes.
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Journal of Mathematical Sociology. 23(3): 181-201. 1999.
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http://www.analytictech.com/borgatti/group_centrality.htm
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"""
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centrality = len(set().union(*list(set(G.neighbors(i)) for i in S)) - set(S))
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centrality /= len(G.nodes()) - len(S)
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return centrality
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@not_implemented_for("undirected")
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def group_in_degree_centrality(G, S):
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"""Compute the group in-degree centrality for a group of nodes.
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Group in-degree centrality of a group of nodes $S$ is the fraction
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of non-group members connected to group members by incoming edges.
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Parameters
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----------
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G : graph
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A NetworkX graph.
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S : list or set
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S is a group of nodes which belong to G, for which group in-degree
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centrality is to be calculated.
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Returns
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-------
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centrality : float
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Group in-degree centrality of the group S.
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Raises
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------
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NetworkXNotImplemented
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If G is undirected.
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NodeNotFound
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If node(s) in S are not in G.
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See Also
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--------
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degree_centrality
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group_degree_centrality
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group_out_degree_centrality
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Notes
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-----
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The number of nodes in the group must be a maximum of n - 1 where `n`
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is the total number of nodes in the graph.
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`G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
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so for group in-degree centrality, the reverse graph is used.
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"""
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return group_degree_centrality(G.reverse(), S)
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@not_implemented_for("undirected")
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def group_out_degree_centrality(G, S):
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"""Compute the group out-degree centrality for a group of nodes.
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Group out-degree centrality of a group of nodes $S$ is the fraction
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of non-group members connected to group members by outgoing edges.
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Parameters
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----------
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G : graph
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A NetworkX graph.
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S : list or set
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S is a group of nodes which belong to G, for which group in-degree
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centrality is to be calculated.
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Returns
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-------
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centrality : float
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Group out-degree centrality of the group S.
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Raises
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------
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NetworkXNotImplemented
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If G is undirected.
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NodeNotFound
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If node(s) in S are not in G.
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See Also
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--------
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degree_centrality
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group_degree_centrality
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group_in_degree_centrality
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Notes
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-----
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The number of nodes in the group must be a maximum of n - 1 where `n`
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is the total number of nodes in the graph.
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`G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
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so for group out-degree centrality, the graph itself is used.
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"""
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return group_degree_centrality(G, S)
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