597 lines
19 KiB
Python
597 lines
19 KiB
Python
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"""
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Link prediction algorithms.
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"""
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from math import log
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = [
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"resource_allocation_index",
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"jaccard_coefficient",
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"adamic_adar_index",
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"preferential_attachment",
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"cn_soundarajan_hopcroft",
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"ra_index_soundarajan_hopcroft",
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"within_inter_cluster",
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"common_neighbor_centrality",
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]
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def _apply_prediction(G, func, ebunch=None):
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"""Applies the given function to each edge in the specified iterable
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of edges.
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`G` is an instance of :class:`networkx.Graph`.
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`func` is a function on two inputs, each of which is a node in the
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graph. The function can return anything, but it should return a
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value representing a prediction of the likelihood of a "link"
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joining the two nodes.
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`ebunch` is an iterable of pairs of nodes. If not specified, all
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non-edges in the graph `G` will be used.
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"""
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if ebunch is None:
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ebunch = nx.non_edges(G)
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return ((u, v, func(u, v)) for u, v in ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def resource_allocation_index(G, ebunch=None):
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r"""Compute the resource allocation index of all node pairs in ebunch.
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Resource allocation index of `u` and `v` is defined as
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.. math::
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\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{|\Gamma(w)|}
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where $\Gamma(u)$ denotes the set of neighbors of $u$.
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Parameters
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----------
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G : graph
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A NetworkX undirected graph.
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ebunch : iterable of node pairs, optional (default = None)
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Resource allocation index will be computed for each pair of
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nodes given in the iterable. The pairs must be given as
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2-tuples (u, v) where u and v are nodes in the graph. If ebunch
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is None then all non-existent edges in the graph will be used.
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Default value: None.
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Returns
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-------
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piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
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pair of nodes and p is their resource allocation index.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> preds = nx.resource_allocation_index(G, [(0, 1), (2, 3)])
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>>> for u, v, p in preds:
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... print(f"({u}, {v}) -> {p:.8f}")
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(0, 1) -> 0.75000000
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(2, 3) -> 0.75000000
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References
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----------
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.. [1] T. Zhou, L. Lu, Y.-C. Zhang.
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Predicting missing links via local information.
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Eur. Phys. J. B 71 (2009) 623.
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https://arxiv.org/pdf/0901.0553.pdf
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"""
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def predict(u, v):
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return sum(1 / G.degree(w) for w in nx.common_neighbors(G, u, v))
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return _apply_prediction(G, predict, ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def jaccard_coefficient(G, ebunch=None):
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r"""Compute the Jaccard coefficient of all node pairs in ebunch.
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Jaccard coefficient of nodes `u` and `v` is defined as
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.. math::
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\frac{|\Gamma(u) \cap \Gamma(v)|}{|\Gamma(u) \cup \Gamma(v)|}
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where $\Gamma(u)$ denotes the set of neighbors of $u$.
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Parameters
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----------
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G : graph
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A NetworkX undirected graph.
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ebunch : iterable of node pairs, optional (default = None)
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Jaccard coefficient will be computed for each pair of nodes
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given in the iterable. The pairs must be given as 2-tuples
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(u, v) where u and v are nodes in the graph. If ebunch is None
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then all non-existent edges in the graph will be used.
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Default value: None.
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Returns
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-------
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piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
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pair of nodes and p is their Jaccard coefficient.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> preds = nx.jaccard_coefficient(G, [(0, 1), (2, 3)])
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>>> for u, v, p in preds:
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... print(f"({u}, {v}) -> {p:.8f}")
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(0, 1) -> 0.60000000
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(2, 3) -> 0.60000000
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References
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----------
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.. [1] D. Liben-Nowell, J. Kleinberg.
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The Link Prediction Problem for Social Networks (2004).
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http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
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"""
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def predict(u, v):
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union_size = len(set(G[u]) | set(G[v]))
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if union_size == 0:
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return 0
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return len(list(nx.common_neighbors(G, u, v))) / union_size
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return _apply_prediction(G, predict, ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def adamic_adar_index(G, ebunch=None):
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r"""Compute the Adamic-Adar index of all node pairs in ebunch.
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Adamic-Adar index of `u` and `v` is defined as
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.. math::
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\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{\log |\Gamma(w)|}
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where $\Gamma(u)$ denotes the set of neighbors of $u$.
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This index leads to zero-division for nodes only connected via self-loops.
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It is intended to be used when no self-loops are present.
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Parameters
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----------
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G : graph
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NetworkX undirected graph.
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ebunch : iterable of node pairs, optional (default = None)
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Adamic-Adar index will be computed for each pair of nodes given
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in the iterable. The pairs must be given as 2-tuples (u, v)
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where u and v are nodes in the graph. If ebunch is None then all
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non-existent edges in the graph will be used.
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Default value: None.
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Returns
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-------
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piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
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pair of nodes and p is their Adamic-Adar index.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> preds = nx.adamic_adar_index(G, [(0, 1), (2, 3)])
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>>> for u, v, p in preds:
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... print(f"({u}, {v}) -> {p:.8f}")
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(0, 1) -> 2.16404256
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(2, 3) -> 2.16404256
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References
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----------
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.. [1] D. Liben-Nowell, J. Kleinberg.
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The Link Prediction Problem for Social Networks (2004).
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http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
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"""
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def predict(u, v):
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return sum(1 / log(G.degree(w)) for w in nx.common_neighbors(G, u, v))
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return _apply_prediction(G, predict, ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def common_neighbor_centrality(G, ebunch=None, alpha=0.8):
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r"""Return the CCPA score for each pair of nodes.
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Compute the Common Neighbor and Centrality based Parameterized Algorithm(CCPA)
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score of all node pairs in ebunch.
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CCPA score of `u` and `v` is defined as
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.. math::
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\alpha \cdot (|\Gamma (u){\cap }^{}\Gamma (v)|)+(1-\alpha )\cdot \frac{N}{{d}_{uv}}
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where $\Gamma(u)$ denotes the set of neighbors of $u$, $\Gamma(v)$ denotes the
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set of neighbors of $v$, $\alpha$ is parameter varies between [0,1], $N$ denotes
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total number of nodes in the Graph and ${d}_{uv}$ denotes shortest distance
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between $u$ and $v$.
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This algorithm is based on two vital properties of nodes, namely the number
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of common neighbors and their centrality. Common neighbor refers to the common
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nodes between two nodes. Centrality refers to the prestige that a node enjoys
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in a network.
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.. seealso::
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:func:`common_neighbors`
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Parameters
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----------
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G : graph
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NetworkX undirected graph.
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ebunch : iterable of node pairs, optional (default = None)
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Preferential attachment score will be computed for each pair of
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nodes given in the iterable. The pairs must be given as
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2-tuples (u, v) where u and v are nodes in the graph. If ebunch
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is None then all non-existent edges in the graph will be used.
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Default value: None.
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alpha : Parameter defined for participation of Common Neighbor
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and Centrality Algorithm share. Values for alpha should
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normally be between 0 and 1. Default value set to 0.8
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because author found better performance at 0.8 for all the
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dataset.
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Default value: 0.8
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Returns
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-------
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piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
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pair of nodes and p is their Common Neighbor and Centrality based
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Parameterized Algorithm(CCPA) score.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> preds = nx.common_neighbor_centrality(G, [(0, 1), (2, 3)])
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>>> for u, v, p in preds:
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... print(f"({u}, {v}) -> {p}")
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(0, 1) -> 3.4000000000000004
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(2, 3) -> 3.4000000000000004
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References
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----------
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.. [1] Ahmad, I., Akhtar, M.U., Noor, S. et al.
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Missing Link Prediction using Common Neighbor and Centrality based Parameterized Algorithm.
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Sci Rep 10, 364 (2020).
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https://doi.org/10.1038/s41598-019-57304-y
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"""
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# When alpha == 1, the CCPA score simplifies to the number of common neighbors.
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if alpha == 1:
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def predict(u, v):
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if u == v:
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raise nx.NetworkXAlgorithmError("Self links are not supported")
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return sum(1 for _ in nx.common_neighbors(G, u, v))
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else:
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spl = dict(nx.shortest_path_length(G))
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inf = float("inf")
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def predict(u, v):
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if u == v:
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raise nx.NetworkXAlgorithmError("Self links are not supported")
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path_len = spl[u].get(v, inf)
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return alpha * sum(1 for _ in nx.common_neighbors(G, u, v)) + (
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1 - alpha
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) * (G.number_of_nodes() / path_len)
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return _apply_prediction(G, predict, ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def preferential_attachment(G, ebunch=None):
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r"""Compute the preferential attachment score of all node pairs in ebunch.
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Preferential attachment score of `u` and `v` is defined as
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.. math::
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|\Gamma(u)| |\Gamma(v)|
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where $\Gamma(u)$ denotes the set of neighbors of $u$.
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Parameters
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----------
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G : graph
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NetworkX undirected graph.
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ebunch : iterable of node pairs, optional (default = None)
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Preferential attachment score will be computed for each pair of
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nodes given in the iterable. The pairs must be given as
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2-tuples (u, v) where u and v are nodes in the graph. If ebunch
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is None then all non-existent edges in the graph will be used.
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Default value: None.
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Returns
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-------
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piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
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pair of nodes and p is their preferential attachment score.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> preds = nx.preferential_attachment(G, [(0, 1), (2, 3)])
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>>> for u, v, p in preds:
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... print(f"({u}, {v}) -> {p}")
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(0, 1) -> 16
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(2, 3) -> 16
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References
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----------
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.. [1] D. Liben-Nowell, J. Kleinberg.
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The Link Prediction Problem for Social Networks (2004).
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http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
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"""
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def predict(u, v):
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return G.degree(u) * G.degree(v)
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return _apply_prediction(G, predict, ebunch)
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def cn_soundarajan_hopcroft(G, ebunch=None, community="community"):
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r"""Count the number of common neighbors of all node pairs in ebunch
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using community information.
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For two nodes $u$ and $v$, this function computes the number of
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common neighbors and bonus one for each common neighbor belonging to
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the same community as $u$ and $v$. Mathematically,
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.. math::
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|\Gamma(u) \cap \Gamma(v)| + \sum_{w \in \Gamma(u) \cap \Gamma(v)} f(w)
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where $f(w)$ equals 1 if $w$ belongs to the same community as $u$
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and $v$ or 0 otherwise and $\Gamma(u)$ denotes the set of
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neighbors of $u$.
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Parameters
|
||
|
|
----------
|
||
|
|
G : graph
|
||
|
|
A NetworkX undirected graph.
|
||
|
|
|
||
|
|
ebunch : iterable of node pairs, optional (default = None)
|
||
|
|
The score will be computed for each pair of nodes given in the
|
||
|
|
iterable. The pairs must be given as 2-tuples (u, v) where u
|
||
|
|
and v are nodes in the graph. If ebunch is None then all
|
||
|
|
non-existent edges in the graph will be used.
|
||
|
|
Default value: None.
|
||
|
|
|
||
|
|
community : string, optional (default = 'community')
|
||
|
|
Nodes attribute name containing the community information.
|
||
|
|
G[u][community] identifies which community u belongs to. Each
|
||
|
|
node belongs to at most one community. Default value: 'community'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
piter : iterator
|
||
|
|
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
|
||
|
|
pair of nodes and p is their score.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> G = nx.path_graph(3)
|
||
|
|
>>> G.nodes[0]["community"] = 0
|
||
|
|
>>> G.nodes[1]["community"] = 0
|
||
|
|
>>> G.nodes[2]["community"] = 0
|
||
|
|
>>> preds = nx.cn_soundarajan_hopcroft(G, [(0, 2)])
|
||
|
|
>>> for u, v, p in preds:
|
||
|
|
... print(f"({u}, {v}) -> {p}")
|
||
|
|
(0, 2) -> 2
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Sucheta Soundarajan and John Hopcroft.
|
||
|
|
Using community information to improve the precision of link
|
||
|
|
prediction methods.
|
||
|
|
In Proceedings of the 21st international conference companion on
|
||
|
|
World Wide Web (WWW '12 Companion). ACM, New York, NY, USA, 607-608.
|
||
|
|
http://doi.acm.org/10.1145/2187980.2188150
|
||
|
|
"""
|
||
|
|
|
||
|
|
def predict(u, v):
|
||
|
|
Cu = _community(G, u, community)
|
||
|
|
Cv = _community(G, v, community)
|
||
|
|
cnbors = list(nx.common_neighbors(G, u, v))
|
||
|
|
neighbors = (
|
||
|
|
sum(_community(G, w, community) == Cu for w in cnbors) if Cu == Cv else 0
|
||
|
|
)
|
||
|
|
return len(cnbors) + neighbors
|
||
|
|
|
||
|
|
return _apply_prediction(G, predict, ebunch)
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
@not_implemented_for("multigraph")
|
||
|
|
def ra_index_soundarajan_hopcroft(G, ebunch=None, community="community"):
|
||
|
|
r"""Compute the resource allocation index of all node pairs in
|
||
|
|
ebunch using community information.
|
||
|
|
|
||
|
|
For two nodes $u$ and $v$, this function computes the resource
|
||
|
|
allocation index considering only common neighbors belonging to the
|
||
|
|
same community as $u$ and $v$. Mathematically,
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{f(w)}{|\Gamma(w)|}
|
||
|
|
|
||
|
|
where $f(w)$ equals 1 if $w$ belongs to the same community as $u$
|
||
|
|
and $v$ or 0 otherwise and $\Gamma(u)$ denotes the set of
|
||
|
|
neighbors of $u$.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : graph
|
||
|
|
A NetworkX undirected graph.
|
||
|
|
|
||
|
|
ebunch : iterable of node pairs, optional (default = None)
|
||
|
|
The score will be computed for each pair of nodes given in the
|
||
|
|
iterable. The pairs must be given as 2-tuples (u, v) where u
|
||
|
|
and v are nodes in the graph. If ebunch is None then all
|
||
|
|
non-existent edges in the graph will be used.
|
||
|
|
Default value: None.
|
||
|
|
|
||
|
|
community : string, optional (default = 'community')
|
||
|
|
Nodes attribute name containing the community information.
|
||
|
|
G[u][community] identifies which community u belongs to. Each
|
||
|
|
node belongs to at most one community. Default value: 'community'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
piter : iterator
|
||
|
|
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
|
||
|
|
pair of nodes and p is their score.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> G = nx.Graph()
|
||
|
|
>>> G.add_edges_from([(0, 1), (0, 2), (1, 3), (2, 3)])
|
||
|
|
>>> G.nodes[0]["community"] = 0
|
||
|
|
>>> G.nodes[1]["community"] = 0
|
||
|
|
>>> G.nodes[2]["community"] = 1
|
||
|
|
>>> G.nodes[3]["community"] = 0
|
||
|
|
>>> preds = nx.ra_index_soundarajan_hopcroft(G, [(0, 3)])
|
||
|
|
>>> for u, v, p in preds:
|
||
|
|
... print(f"({u}, {v}) -> {p:.8f}")
|
||
|
|
(0, 3) -> 0.50000000
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Sucheta Soundarajan and John Hopcroft.
|
||
|
|
Using community information to improve the precision of link
|
||
|
|
prediction methods.
|
||
|
|
In Proceedings of the 21st international conference companion on
|
||
|
|
World Wide Web (WWW '12 Companion). ACM, New York, NY, USA, 607-608.
|
||
|
|
http://doi.acm.org/10.1145/2187980.2188150
|
||
|
|
"""
|
||
|
|
|
||
|
|
def predict(u, v):
|
||
|
|
Cu = _community(G, u, community)
|
||
|
|
Cv = _community(G, v, community)
|
||
|
|
if Cu != Cv:
|
||
|
|
return 0
|
||
|
|
cnbors = nx.common_neighbors(G, u, v)
|
||
|
|
return sum(1 / G.degree(w) for w in cnbors if _community(G, w, community) == Cu)
|
||
|
|
|
||
|
|
return _apply_prediction(G, predict, ebunch)
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
@not_implemented_for("multigraph")
|
||
|
|
def within_inter_cluster(G, ebunch=None, delta=0.001, community="community"):
|
||
|
|
"""Compute the ratio of within- and inter-cluster common neighbors
|
||
|
|
of all node pairs in ebunch.
|
||
|
|
|
||
|
|
For two nodes `u` and `v`, if a common neighbor `w` belongs to the
|
||
|
|
same community as them, `w` is considered as within-cluster common
|
||
|
|
neighbor of `u` and `v`. Otherwise, it is considered as
|
||
|
|
inter-cluster common neighbor of `u` and `v`. The ratio between the
|
||
|
|
size of the set of within- and inter-cluster common neighbors is
|
||
|
|
defined as the WIC measure. [1]_
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : graph
|
||
|
|
A NetworkX undirected graph.
|
||
|
|
|
||
|
|
ebunch : iterable of node pairs, optional (default = None)
|
||
|
|
The WIC measure will be computed for each pair of nodes given in
|
||
|
|
the iterable. The pairs must be given as 2-tuples (u, v) where
|
||
|
|
u and v are nodes in the graph. If ebunch is None then all
|
||
|
|
non-existent edges in the graph will be used.
|
||
|
|
Default value: None.
|
||
|
|
|
||
|
|
delta : float, optional (default = 0.001)
|
||
|
|
Value to prevent division by zero in case there is no
|
||
|
|
inter-cluster common neighbor between two nodes. See [1]_ for
|
||
|
|
details. Default value: 0.001.
|
||
|
|
|
||
|
|
community : string, optional (default = 'community')
|
||
|
|
Nodes attribute name containing the community information.
|
||
|
|
G[u][community] identifies which community u belongs to. Each
|
||
|
|
node belongs to at most one community. Default value: 'community'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
piter : iterator
|
||
|
|
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
|
||
|
|
pair of nodes and p is their WIC measure.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> G = nx.Graph()
|
||
|
|
>>> G.add_edges_from([(0, 1), (0, 2), (0, 3), (1, 4), (2, 4), (3, 4)])
|
||
|
|
>>> G.nodes[0]["community"] = 0
|
||
|
|
>>> G.nodes[1]["community"] = 1
|
||
|
|
>>> G.nodes[2]["community"] = 0
|
||
|
|
>>> G.nodes[3]["community"] = 0
|
||
|
|
>>> G.nodes[4]["community"] = 0
|
||
|
|
>>> preds = nx.within_inter_cluster(G, [(0, 4)])
|
||
|
|
>>> for u, v, p in preds:
|
||
|
|
... print(f"({u}, {v}) -> {p:.8f}")
|
||
|
|
(0, 4) -> 1.99800200
|
||
|
|
>>> preds = nx.within_inter_cluster(G, [(0, 4)], delta=0.5)
|
||
|
|
>>> for u, v, p in preds:
|
||
|
|
... print(f"({u}, {v}) -> {p:.8f}")
|
||
|
|
(0, 4) -> 1.33333333
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Jorge Carlos Valverde-Rebaza and Alneu de Andrade Lopes.
|
||
|
|
Link prediction in complex networks based on cluster information.
|
||
|
|
In Proceedings of the 21st Brazilian conference on Advances in
|
||
|
|
Artificial Intelligence (SBIA'12)
|
||
|
|
https://doi.org/10.1007/978-3-642-34459-6_10
|
||
|
|
"""
|
||
|
|
if delta <= 0:
|
||
|
|
raise nx.NetworkXAlgorithmError("Delta must be greater than zero")
|
||
|
|
|
||
|
|
def predict(u, v):
|
||
|
|
Cu = _community(G, u, community)
|
||
|
|
Cv = _community(G, v, community)
|
||
|
|
if Cu != Cv:
|
||
|
|
return 0
|
||
|
|
cnbors = set(nx.common_neighbors(G, u, v))
|
||
|
|
within = {w for w in cnbors if _community(G, w, community) == Cu}
|
||
|
|
inter = cnbors - within
|
||
|
|
return len(within) / (len(inter) + delta)
|
||
|
|
|
||
|
|
return _apply_prediction(G, predict, ebunch)
|
||
|
|
|
||
|
|
|
||
|
|
def _community(G, u, community):
|
||
|
|
"""Get the community of the given node."""
|
||
|
|
node_u = G.nodes[u]
|
||
|
|
try:
|
||
|
|
return node_u[community]
|
||
|
|
except KeyError as err:
|
||
|
|
raise nx.NetworkXAlgorithmError("No community information") from err
|