### diff env/lib/python3.9/site-packages/networkx/algorithms/structuralholes.py @ 0:4f3585e2f14bdraftdefaulttip

author shellac Mon, 22 Mar 2021 18:12:50 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/env/lib/python3.9/site-packages/networkx/algorithms/structuralholes.py	Mon Mar 22 18:12:50 2021 +0000
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+"""Functions for computing measures of structural holes."""
+
+import networkx as nx
+
+__all__ = ["constraint", "local_constraint", "effective_size"]
+
+
+def mutual_weight(G, u, v, weight=None):
+    """Returns the sum of the weights of the edge from u to v and
+    the edge from v to u in G.
+
+    weight is the edge data key that represents the edge weight. If
+    the specified key is None or is not in the edge data for an edge,
+    that edge is assumed to have weight 1.
+
+    Pre-conditions: u and v must both be in G.
+
+    """
+    try:
+        a_uv = G[u][v].get(weight, 1)
+    except KeyError:
+        a_uv = 0
+    try:
+        a_vu = G[v][u].get(weight, 1)
+    except KeyError:
+        a_vu = 0
+    return a_uv + a_vu
+
+
+def normalized_mutual_weight(G, u, v, norm=sum, weight=None):
+    """Returns normalized mutual weight of the edges from u to v
+    with respect to the mutual weights of the neighbors of u in G.
+
+    norm specifies how the normalization factor is computed. It must
+    be a function that takes a single argument and returns a number.
+    The argument will be an iterable of mutual weights
+    of pairs (u, w), where w ranges over each (in- and
+    out-)neighbor of u. Commons values for normalization are
+    sum and max.
+
+    weight can be None or a string, if None, all edge weights
+    are considered equal. Otherwise holds the name of the edge
+    attribute used as weight.
+
+    """
+    scale = norm(mutual_weight(G, u, w, weight) for w in set(nx.all_neighbors(G, u)))
+    return 0 if scale == 0 else mutual_weight(G, u, v, weight) / scale
+
+
+def effective_size(G, nodes=None, weight=None):
+    r"""Returns the effective size of all nodes in the graph G.
+
+    The *effective size* of a node's ego network is based on the concept
+    of redundancy. A person's ego network has redundancy to the extent
+    that her contacts are connected to each other as well. The
+    nonredundant part of a person's relationships it's the effective
+    size of her ego network [1]_.  Formally, the effective size of a
+    node $u$, denoted $e(u)$, is defined by
+
+    .. math::
+
+       e(u) = \sum_{v \in N(u) \setminus \{u\}}
+       \left(1 - \sum_{w \in N(v)} p_{uw} m_{vw}\right)
+
+    where $N(u)$ is the set of neighbors of $u$ and $p_{uw}$ is the
+    normalized mutual weight of the (directed or undirected) edges
+    joining $u$ and $v$, for each vertex $u$ and $v$ [1]_. And $m_{vw}$
+    is the mutual weight of $v$ and $w$ divided by $v$ highest mutual
+    weight with any of its neighbors. The *mutual weight* of $u$ and $v$
+    is the sum of the weights of edges joining them (edge weights are
+    assumed to be one if the graph is unweighted).
+
+    For the case of unweighted and undirected graphs, Borgatti proposed
+    a simplified formula to compute effective size [2]_
+
+    .. math::
+
+       e(u) = n - \frac{2t}{n}
+
+    where t is the number of ties in the ego network (not including
+    ties to ego) and n is the number of nodes (excluding ego).
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        The graph containing v. Directed graphs are treated like
+        undirected graphs when computing neighbors of v.
+
+    nodes : container, optional
+        Container of nodes in the graph G to compute the effective size.
+        If None, the effective size of every node is computed.
+
+    weight : None or string, optional
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+
+    Returns
+    -------
+    dict
+        Dictionary with nodes as keys and the effective size of the node as values.
+
+    Notes
+    -----
+    Burt also defined the related concept of *efficiency* of a node's ego
+    network, which is its effective size divided by the degree of that
+    node [1]_. So you can easily compute efficiency:
+
+    >>> G = nx.DiGraph()
+    >>> G.add_edges_from([(0, 1), (0, 2), (1, 0), (2, 1)])
+    >>> esize = nx.effective_size(G)
+    >>> efficiency = {n: v / G.degree(n) for n, v in esize.items()}
+
+    --------
+    constraint
+
+    References
+    ----------
+    .. [1] Burt, Ronald S.
+           *Structural Holes: The Social Structure of Competition.*
+           Cambridge: Harvard University Press, 1995.
+
+    .. [2] Borgatti, S.
+           "Structural Holes: Unpacking Burt's Redundancy Measures"
+           CONNECTIONS 20(1):35-38.
+           http://www.analytictech.com/connections/v20(1)/holes.htm
+
+    """
+
+    def redundancy(G, u, v, weight=None):
+        nmw = normalized_mutual_weight
+        r = sum(
+            nmw(G, u, w, weight=weight) * nmw(G, v, w, norm=max, weight=weight)
+            for w in set(nx.all_neighbors(G, u))
+        )
+        return 1 - r
+
+    effective_size = {}
+    if nodes is None:
+        nodes = G
+    # Use Borgatti's simplified formula for unweighted and undirected graphs
+    if not G.is_directed() and weight is None:
+        for v in nodes:
+            # Effective size is not defined for isolated nodes
+            if len(G[v]) == 0:
+                effective_size[v] = float("nan")
+                continue
+            E = nx.ego_graph(G, v, center=False, undirected=True)
+            effective_size[v] = len(E) - (2 * E.size()) / len(E)
+    else:
+        for v in nodes:
+            # Effective size is not defined for isolated nodes
+            if len(G[v]) == 0:
+                effective_size[v] = float("nan")
+                continue
+            effective_size[v] = sum(
+                redundancy(G, v, u, weight) for u in set(nx.all_neighbors(G, v))
+            )
+    return effective_size
+
+
+def constraint(G, nodes=None, weight=None):
+    r"""Returns the constraint on all nodes in the graph G.
+
+    The *constraint* is a measure of the extent to which a node *v* is
+    invested in those nodes that are themselves invested in the
+    neighbors of *v*. Formally, the *constraint on v*, denoted c(v),
+    is defined by
+
+    .. math::
+
+       c(v) = \sum_{w \in N(v) \setminus \{v\}} \ell(v, w)
+
+    where N(v) is the subset of the neighbors of v that are either
+    predecessors or successors of v and \ell(v, w) is the local
+    constraint on v with respect to w [1]_. For the definition of local
+    constraint, see :func:local_constraint.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        The graph containing v. This can be either directed or undirected.
+
+    nodes : container, optional
+        Container of nodes in the graph G to compute the constraint. If
+        None, the constraint of every node is computed.
+
+    weight : None or string, optional
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+
+    Returns
+    -------
+    dict
+        Dictionary with nodes as keys and the constraint on the node as values.
+
+    --------
+    local_constraint
+
+    References
+    ----------
+    .. [1] Burt, Ronald S.
+           "Structural holes and good ideas".
+           American Journal of Sociology (110): 349–399.
+
+    """
+    if nodes is None:
+        nodes = G
+    constraint = {}
+    for v in nodes:
+        # Constraint is not defined for isolated nodes
+        if len(G[v]) == 0:
+            constraint[v] = float("nan")
+            continue
+        constraint[v] = sum(
+            local_constraint(G, v, n, weight) for n in set(nx.all_neighbors(G, v))
+        )
+    return constraint
+
+
+def local_constraint(G, u, v, weight=None):
+    r"""Returns the local constraint on the node u with respect to
+    the node v in the graph G.
+
+    Formally, the *local constraint on u with respect to v*, denoted
+    $\ell(v)$, is defined by
+
+    .. math::
+
+       \ell(u, v) = \left(p_{uv} + \sum_{w \in N(v)} p_{uw} p{wv}\right)^2,
+
+    where $N(v)$ is the set of neighbors of $v$ and $p_{uv}$ is the
+    normalized mutual weight of the (directed or undirected) edges
+    joining $u$ and $v$, for each vertex $u$ and $v$ [1]_. The *mutual
+    weight* of $u$ and $v$ is the sum of the weights of edges joining
+    them (edge weights are assumed to be one if the graph is
+    unweighted).
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        The graph containing u and v. This can be either
+        directed or undirected.
+
+    u : node
+        A node in the graph G.
+
+    v : node
+        A node in the graph G.
+
+    weight : None or string, optional
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+
+    Returns
+    -------
+    float
+        The constraint of the node v in the graph G.
+
+    --------
+    constraint
+
+    References
+    ----------
+    .. [1] Burt, Ronald S.
+           "Structural holes and good ideas".
+           American Journal of Sociology (110): 349–399.
+
+    """
+    nmw = normalized_mutual_weight
+    direct = nmw(G, u, v, weight=weight)
+    indirect = sum(
+        nmw(G, u, w, weight=weight) * nmw(G, w, v, weight=weight)
+        for w in set(nx.all_neighbors(G, u))
+    )
+    return (direct + indirect) ** 2