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Diffstat (limited to '.venv/lib/python3.12/site-packages/networkx/algorithms/structuralholes.py')
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diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/structuralholes.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/structuralholes.py new file mode 100644 index 00000000..bae42d06 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/structuralholes.py @@ -0,0 +1,283 @@ +"""Functions for computing measures of structural holes.""" + +import networkx as nx + +__all__ = ["constraint", "local_constraint", "effective_size"] + + +@nx._dispatchable(edge_attrs="weight") +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 + + +@nx._dispatchable(edge_attrs="weight") +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 + + +@nx._dispatchable(edge_attrs="weight") +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 is 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()} + + See also + -------- + 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 + + +@nx._dispatchable(edge_attrs="weight") +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. + + See also + -------- + 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 + + +@nx._dispatchable(edge_attrs="weight") +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 + $\ell(u, v)$, 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``. + + See also + -------- + 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 |