chore: import upstream snapshot with attribution
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wehub-resource-sync
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"""Utils for computing graph label informativeness"""
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from . import to_bidirected
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try:
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import torch
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except ImportError:
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HAS_TORCH = False
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else:
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HAS_TORCH = True
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__all__ = ["edge_label_informativeness", "node_label_informativeness"]
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def check_pytorch():
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"""Check if PyTorch is the backend."""
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if HAS_TORCH is False:
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raise ModuleNotFoundError(
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"This function requires PyTorch to be the backend."
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)
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def edge_label_informativeness(graph, y, eps=1e-8):
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r"""Label informativeness (:math:`\mathrm{LI}`) is a characteristic of
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labeled graphs proposed in the `Characterizing Graph Datasets for Node
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Classification: Homophily-Heterophily Dichotomy and Beyond
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<https://arxiv.org/abs/2209.06177>`__
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Label informativeness shows how much information about a node's label we
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get from knowing its neighbor's label. Formally, assume that we sample an
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edge :math:`(\xi,\eta) \in E`. The class labels of nodes :math:`\xi` and
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:math:`\eta` are then random variables :math:`y_\xi` and :math:`y_\eta`.
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We want to measure the amount of knowledge the label :math:`y_\eta` gives
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for predicting :math:`y_\xi`. The entropy :math:`H(y_\xi)` measures the
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`hardness' of predicting the label of :math:`\xi` without knowing
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:math:`y_\eta`. Given :math:`y_\eta`, this value is reduced to the
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conditional entropy :math:`H(y_\xi|y_\eta)`. In other words, :math:`y_\eta`
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reveals :math:`I(y_\xi,y_\eta) = H(y_\xi) - H(y_\xi|y_\eta)` information
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about the label. To make the obtained quantity comparable across different
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datasets, label informativeness is defined as the normalized mutual
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information of :math:`y_{\xi}` and :math:`y_{\eta}`:
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.. math::
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\mathrm{LI} = \frac{I(y_\xi,y_\eta)}{H(y_\xi)}
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Depending on the distribution used for sampling an edge
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:math:`(\xi, \eta)`, several variants of label informativeness can be
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obtained. Two of them are particularly intuitive: in edge label
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informativeness (:math:`\mathrm{LI}_{edge}`), edges are sampled uniformly
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at random, and in node label informativeness (:math:`\mathrm{LI}_{node}`),
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first a node is sampled uniformly at random and then an edge incident to it
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is sampled uniformly at random. These two versions of label informativeness
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differ in how they weight high/low-degree nodes. In edge label
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informativeness, averaging is over the edges, thus high-degree nodes are
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given more weight. In node label informativeness, averaging is over the
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nodes, so all nodes are weighted equally.
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This function computes edge label informativeness.
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Parameters
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----------
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graph : DGLGraph
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The graph.
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y : torch.Tensor
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The node labels, which is a tensor of shape (|V|).
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eps : float, optional
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A small constant for numerical stability. (default: 1e-8)
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Returns
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-------
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float
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The edge label informativeness value.
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Examples
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--------
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>>> import dgl
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>>> import torch
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>>> graph = dgl.graph(([0, 1, 2, 2, 3, 4], [1, 2, 0, 3, 4, 5]))
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>>> y = torch.tensor([0, 0, 0, 0, 1, 1])
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>>> dgl.edge_label_informativeness(graph, y)
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0.25177597999572754
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"""
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check_pytorch()
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graph = to_bidirected(graph.cpu()).to(y.device)
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degrees = graph.in_degrees().float()
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num_classes = y.max() + 1
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class_degree_weighted_probs = torch.zeros(num_classes).to(y.device)
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class_degree_weighted_probs.index_add_(dim=0, index=y, source=degrees)
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class_degree_weighted_probs /= class_degree_weighted_probs.sum()
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edge_probs = torch.zeros(num_classes, num_classes).to(y.device)
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labels_u = y[graph.edges()[0].long()]
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labels_v = y[graph.edges()[1].long()]
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edge_probs.index_put_(
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indices=(labels_u, labels_v),
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values=torch.ones(graph.num_edges()).to(y.device),
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accumulate=True,
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)
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edge_probs /= edge_probs.sum()
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edge_probs += eps
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numerator = (edge_probs * torch.log(edge_probs)).sum()
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denominator = (
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class_degree_weighted_probs * torch.log(class_degree_weighted_probs)
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).sum()
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li_edge = 2 - numerator / denominator
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return li_edge.item()
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def node_label_informativeness(graph, y, eps=1e-8):
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r"""Label informativeness (:math:`\mathrm{LI}`) is a characteristic of
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labeled graphs proposed in the `Characterizing Graph Datasets for Node
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Classification: Homophily-Heterophily Dichotomy and Beyond
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<https://arxiv.org/abs/2209.06177>`__
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Label informativeness shows how much information about a node's label we
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get from knowing its neighbor's label. Formally, assume that we sample an
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edge :math:`(\xi,\eta) \in E`. The class labels of nodes :math:`\xi` and
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:math:`\eta` are then random variables :math:`y_\xi` and :math:`y_\eta`.
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We want to measure the amount of knowledge the label :math:`y_\eta` gives
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for predicting :math:`y_\xi`. The entropy :math:`H(y_\xi)` measures the
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`hardness' of predicting the label of :math:`\xi` without knowing
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:math:`y_\eta`. Given :math:`y_\eta`, this value is reduced to the
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conditional entropy :math:`H(y_\xi|y_\eta)`. In other words, :math:`y_\eta`
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reveals :math:`I(y_\xi,y_\eta) = H(y_\xi) - H(y_\xi|y_\eta)` information
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about the label. To make the obtained quantity comparable across different
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datasets, label informativeness is defined as the normalized mutual
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information of :math:`y_{\xi}` and :math:`y_{\eta}`:
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.. math::
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\mathrm{LI} = \frac{I(y_\xi,y_\eta)}{H(y_\xi)}
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Depending on the distribution used for sampling an edge
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:math:`(\xi, \eta)`, several variants of label informativeness can be
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obtained. Two of them are particularly intuitive: in edge label
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informativeness (:math:`\mathrm{LI}_{edge}`), edges are sampled uniformly
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at random, and in node label informativeness (:math:`\mathrm{LI}_{node}`),
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first a node is sampled uniformly at random and then an edge incident to it
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is sampled uniformly at random. These two versions of label informativeness
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differ in how they weight high/low-degree nodes. In edge label
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informativeness, averaging is over the edges, thus high-degree nodes are
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given more weight. In node label informativeness, averaging is over the
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nodes, so all nodes are weighted equally.
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This function computes node label informativeness.
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Parameters
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----------
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graph : DGLGraph
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The graph.
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y : torch.Tensor
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The node labels, which is a tensor of shape (|V|).
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eps : float, optional
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A small constant for numerical stability. (default: 1e-8)
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Returns
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-------
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float
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The node label informativeness value.
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Examples
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--------
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>>> import dgl
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>>> import torch
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>>> graph = dgl.graph(([0, 1, 2, 2, 3, 4], [1, 2, 0, 3, 4, 5]))
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>>> y = torch.tensor([0, 0, 0, 0, 1, 1])
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>>> dgl.node_label_informativeness(graph, y)
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0.3381872773170471
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"""
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check_pytorch()
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graph = to_bidirected(graph.cpu()).to(y.device)
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degrees = graph.in_degrees().float()
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num_classes = y.max() + 1
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class_probs = torch.zeros(num_classes).to(y.device)
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class_probs.index_add_(
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dim=0, index=y, source=torch.ones(graph.num_nodes()).to(y.device)
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)
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class_probs /= class_probs.sum()
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class_degree_weighted_probs = torch.zeros(num_classes).to(y.device)
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class_degree_weighted_probs.index_add_(dim=0, index=y, source=degrees)
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class_degree_weighted_probs /= class_degree_weighted_probs.sum()
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num_nonzero_degree_nodes = (degrees > 0).sum()
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edge_probs = torch.zeros(num_classes, num_classes).to(y.device)
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labels_u = y[graph.edges()[0].long()]
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labels_v = y[graph.edges()[1].long()]
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degrees_u = degrees[graph.edges()[0].long()]
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edge_probs.index_put_(
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indices=(labels_u, labels_v),
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values=1 / (num_nonzero_degree_nodes * degrees_u),
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accumulate=True,
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)
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edge_probs += eps
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log = torch.log(
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edge_probs
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/ (class_probs[:, None] * class_degree_weighted_probs[None, :])
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)
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numerator = (edge_probs * log).sum()
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denominator = (class_probs * torch.log(class_probs)).sum()
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li_node = -numerator / denominator
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return li_node.item()
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