Source code for networkx.algorithms.link_prediction

"""
Link prediction algorithms.
"""


from math import log

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = [
    "resource_allocation_index",
    "jaccard_coefficient",
    "adamic_adar_index",
    "preferential_attachment",
    "cn_soundarajan_hopcroft",
    "ra_index_soundarajan_hopcroft",
    "within_inter_cluster",
    "common_neighbor_centrality",
]


def _apply_prediction(G, func, ebunch=None):
    """Applies the given function to each edge in the specified iterable
    of edges.

    `G` is an instance of :class:`networkx.Graph`.

    `func` is a function on two inputs, each of which is a node in the
    graph. The function can return anything, but it should return a
    value representing a prediction of the likelihood of a "link"
    joining the two nodes.

    `ebunch` is an iterable of pairs of nodes. If not specified, all
    non-edges in the graph `G` will be used.

    """
    if ebunch is None:
        ebunch = nx.non_edges(G)
    return ((u, v, func(u, v)) for u, v in ebunch)


[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def resource_allocation_index(G, ebunch=None): r"""Compute the resource allocation index of all node pairs in ebunch. Resource allocation index of `u` and `v` is defined as .. math:: \sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{|\Gamma(w)|} where $\Gamma(u)$ denotes the set of neighbors of $u$. Parameters ---------- G : graph A NetworkX undirected graph. ebunch : iterable of node pairs, optional (default = None) Resource allocation index 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. 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 resource allocation index. Examples -------- >>> G = nx.complete_graph(5) >>> preds = nx.resource_allocation_index(G, [(0, 1), (2, 3)]) >>> for u, v, p in preds: ... print(f"({u}, {v}) -> {p:.8f}") (0, 1) -> 0.75000000 (2, 3) -> 0.75000000 References ---------- .. [1] T. Zhou, L. Lu, Y.-C. Zhang. Predicting missing links via local information. Eur. Phys. J. B 71 (2009) 623. https://arxiv.org/pdf/0901.0553.pdf """ def predict(u, v): return sum(1 / G.degree(w) for w in nx.common_neighbors(G, u, v)) return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def jaccard_coefficient(G, ebunch=None): r"""Compute the Jaccard coefficient of all node pairs in ebunch. Jaccard coefficient of nodes `u` and `v` is defined as .. math:: \frac{|\Gamma(u) \cap \Gamma(v)|}{|\Gamma(u) \cup \Gamma(v)|} where $\Gamma(u)$ denotes the set of neighbors of $u$. Parameters ---------- G : graph A NetworkX undirected graph. ebunch : iterable of node pairs, optional (default = None) Jaccard coefficient 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. 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 Jaccard coefficient. Examples -------- >>> G = nx.complete_graph(5) >>> preds = nx.jaccard_coefficient(G, [(0, 1), (2, 3)]) >>> for u, v, p in preds: ... print(f"({u}, {v}) -> {p:.8f}") (0, 1) -> 0.60000000 (2, 3) -> 0.60000000 References ---------- .. [1] D. Liben-Nowell, J. Kleinberg. The Link Prediction Problem for Social Networks (2004). http://www.cs.cornell.edu/home/kleinber/link-pred.pdf """ def predict(u, v): union_size = len(set(G[u]) | set(G[v])) if union_size == 0: return 0 return len(list(nx.common_neighbors(G, u, v))) / union_size return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def adamic_adar_index(G, ebunch=None): r"""Compute the Adamic-Adar index of all node pairs in ebunch. Adamic-Adar index of `u` and `v` is defined as .. math:: \sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{\log |\Gamma(w)|} where $\Gamma(u)$ denotes the set of neighbors of $u$. This index leads to zero-division for nodes only connected via self-loops. It is intended to be used when no self-loops are present. Parameters ---------- G : graph NetworkX undirected graph. ebunch : iterable of node pairs, optional (default = None) Adamic-Adar index 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. 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 Adamic-Adar index. Examples -------- >>> G = nx.complete_graph(5) >>> preds = nx.adamic_adar_index(G, [(0, 1), (2, 3)]) >>> for u, v, p in preds: ... print(f"({u}, {v}) -> {p:.8f}") (0, 1) -> 2.16404256 (2, 3) -> 2.16404256 References ---------- .. [1] D. Liben-Nowell, J. Kleinberg. The Link Prediction Problem for Social Networks (2004). http://www.cs.cornell.edu/home/kleinber/link-pred.pdf """ def predict(u, v): return sum(1 / log(G.degree(w)) for w in nx.common_neighbors(G, u, v)) return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def common_neighbor_centrality(G, ebunch=None, alpha=0.8): r"""Return the CCPA score for each pair of nodes. Compute the Common Neighbor and Centrality based Parameterized Algorithm(CCPA) score of all node pairs in ebunch. CCPA score of `u` and `v` is defined as .. math:: \alpha \cdot (|\Gamma (u){\cap }^{}\Gamma (v)|)+(1-\alpha )\cdot \frac{N}{{d}_{uv}} where $\Gamma(u)$ denotes the set of neighbors of $u$, $\Gamma(v)$ denotes the set of neighbors of $v$, $\alpha$ is parameter varies between [0,1], $N$ denotes total number of nodes in the Graph and ${d}_{uv}$ denotes shortest distance between $u$ and $v$. This algorithm is based on two vital properties of nodes, namely the number of common neighbors and their centrality. Common neighbor refers to the common nodes between two nodes. Centrality refers to the prestige that a node enjoys in a network. .. seealso:: :func:`common_neighbors` Parameters ---------- G : graph NetworkX undirected graph. ebunch : iterable of node pairs, optional (default = None) Preferential attachment 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. alpha : Parameter defined for participation of Common Neighbor and Centrality Algorithm share. Values for alpha should normally be between 0 and 1. Default value set to 0.8 because author found better performance at 0.8 for all the dataset. Default value: 0.8 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 Common Neighbor and Centrality based Parameterized Algorithm(CCPA) score. Examples -------- >>> G = nx.complete_graph(5) >>> preds = nx.common_neighbor_centrality(G, [(0, 1), (2, 3)]) >>> for u, v, p in preds: ... print(f"({u}, {v}) -> {p}") (0, 1) -> 3.4000000000000004 (2, 3) -> 3.4000000000000004 References ---------- .. [1] Ahmad, I., Akhtar, M.U., Noor, S. et al. Missing Link Prediction using Common Neighbor and Centrality based Parameterized Algorithm. Sci Rep 10, 364 (2020). https://doi.org/10.1038/s41598-019-57304-y """ spl = dict(nx.shortest_path_length(G)) inf = float("inf") def predict(u, v): if u == v: raise nx.NetworkXAlgorithmError("Self links are not supported") path_len = spl[u].get(v, inf) return alpha * sum(1 for _ in nx.common_neighbors(G, u, v)) + (1 - alpha) * ( G.number_of_nodes() / path_len ) return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def preferential_attachment(G, ebunch=None): r"""Compute the preferential attachment score of all node pairs in ebunch. Preferential attachment score of `u` and `v` is defined as .. math:: |\Gamma(u)| |\Gamma(v)| where $\Gamma(u)$ denotes the set of neighbors of $u$. Parameters ---------- G : graph NetworkX undirected graph. ebunch : iterable of node pairs, optional (default = None) Preferential attachment 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. 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 preferential attachment score. Examples -------- >>> G = nx.complete_graph(5) >>> preds = nx.preferential_attachment(G, [(0, 1), (2, 3)]) >>> for u, v, p in preds: ... print(f"({u}, {v}) -> {p}") (0, 1) -> 16 (2, 3) -> 16 References ---------- .. [1] D. Liben-Nowell, J. Kleinberg. The Link Prediction Problem for Social Networks (2004). http://www.cs.cornell.edu/home/kleinber/link-pred.pdf """ def predict(u, v): return G.degree(u) * G.degree(v) return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def cn_soundarajan_hopcroft(G, ebunch=None, community="community"): r"""Count the number of common neighbors of all node pairs in ebunch using community information. For two nodes $u$ and $v$, this function computes the number of common neighbors and bonus one for each common neighbor belonging to the same community as $u$ and $v$. Mathematically, .. math:: |\Gamma(u) \cap \Gamma(v)| + \sum_{w \in \Gamma(u) \cap \Gamma(v)} f(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.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)
[docs]@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)
[docs]@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