At least one embodiment of this disclosure includes a method of inferring attribute labels for a user in a social networking system based on the user's social connections and user-specified attribute labels in the social networking system. The method can include: establishing variational equations based on attribute labels of nodes in an ego network in a social graph of a social networking system; determining likelihood scores for at least a portion of the attribute labels of neighboring nodes from a focal user node in the ego network based on user-specified attribute labels from the social networking system; and calculating probability distributions of possible attribute labels for the focal user node of the ego network based on the variational equations and the likelihood scores.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A computer-implemented method, comprising: receiving one or more user-specified attribute labels of a user node and a set of probability distributions associated with possible attribute labels of nodes directly connected from the user node from a social graph in a social networking system; establishing multiple functions with variable likelihood scores of possible attribute labels of the user node and the set of the probability distributions as parameters; generating multiple equations representing explainability scores respectively corresponding to the nodes based on the multiple functions and the probability distributions associated with the nodes, each explainability score measuring a likelihood that a direct social connection between a directly connected node (“v”) and the user node is explained by at least a shared attribute label between the user node and the directly connected node; and ranking a first set of the possible attribute labels for the user node by maximizing the explainability scores utilizing the multiple equations respectively associated with the nodes, the explainability scores being implicit predictions of formation reasons of social connections between each of the nodes and the user node. 2. The computer-implemented method of claim 1, wherein ranking the first set includes calculating a probability score for each of the possible attribute labels in the first set; and wherein maximizing the explainability scores includes varying the variable likelihood scores of the user node to maximize the explainability scores, and each variable likelihood score corresponds to the probability score for each of the possible attribute labels. 3. The computer-implemented method of claim 1, wherein the multiple functions correspond to multinomial distributions for the user node, each multinomial distribution denoting probabilities over a type-specific subset of the possible attribute labels for a label type (“t”) of the user node, wherein the multinomial distributions are established to be consistent with the user-specified attribute labels of the user node. 4. The computer-implemented method of claim 3, wherein establishing each function corresponding to each multinomial distribution includes establishing a summation of products of futl and fvtl over the type-specific subset of the possible attribute labels of the label type t, futl representing a variable probability that the user node has an attribute label (“l”) of the label type t and fvtl representing a probability that the directly connected node v has the same attribute label l of the label type t. 5. The computer-implemented method of claim 4, wherein futl is set to be 1 when the user-specified attribute labels of the user node includes the attribute label l of the label type t and to be 0 when the user-specified attribute labels of the user node includes another attribute label of the label type t such that the other attribute label is mutual exclusive to the attribute label l. 6. The computer-implemented method of claim 4, wherein fvtl is set in accordance with the set of the probability distributions for the directly connected node v. 7. The computer-implemented method of claim 6, wherein fvtl is initially set to be a nominal value between 0 and 1 when a probability distribution for the label type t is absent from the set and no attribute label of the label type t is known in the social networking system for the directly connected node v. 8. The computer-implemented method of claim 4, wherein generating each equation corresponding to the explainability score of the directly connected node v is based on taking a softmax operation on the multiple functions corresponding to the multinomial distributions associated with the directly connected node v, the multiple functions being arguments to the softmax operation. 9. The computer-implemented method of claim 8, wherein the softmax operation is monotonically non-decreasing for each argument. 10. The computer-implemented method of claim 8, wherein the softmax operation returns a value close to a maximum value of the softmax operation whenever at least one of the arguments is close to the maximum value. 11. The computer-implemented method of claim 8, wherein the softmax operation is a function that is differentiable. 12. The computer-implemented method of claim 8, wherein the softmax operation is implemented as a sigmoid function that takes an input argument that is computed based a cumulative operation of the multinomial distributions. 13. The computer-implemented method of claim 8, wherein the softmax operation is implemented as softmax t ∈ T  ( r  ( u , v , t ) ) = σ  ( α   ∑ t ∈ T   r  ( u , v , t ) + c ) , the summation of r(u,v,t) representing the summation of the multiple functions, a and c being constant parameters. 14. The computer-implemented method of claim 1, wherein ranking the first set of the possible attribute labels includes identifying a cumulative maximum of the explainability scores. 15. The computer-implemented method of claim 1, further comprising associating a top ranked label in the first set of the possible attribute labels with the user node in the social networking system. 16. A computer-implemented method comprising: establishing variational equations based on attribute labels of nodes in an ego network in a social graph of a social networking system; determining likelihood scores for at least a portion of the attribute labels of neighboring nodes from a focal user node in the ego network based on user-specified attribute labels from the social networking system; computing probability distributions of possible attribute labels for the focal user node of the ego network based on the variational equations and the likelihood scores; and providing at least a portion of the computed probability distributions with an external process instance inferring labels for another ego network of the social networking system. 17. The computer-implemented method of claim 16, further comprising: receiving an update to a first attribute label of at least one of the neighboring nodes, wherein the update is a user-reported attribute label or a probability distribution update of possible attribute labels from the external process instance inferring labels for another ego network of the social networking system; and re-calculating the probability distributions based at least partly on the update. 18. The computer-implemented method of claim 16, further comprising: inferring an attribute label based on the probability distributions for the focal user node; and generating a user prompt based on the inferred attribute label for advertisement, friend suggestion, label suggestion, or affiliation suggestion. 19. The computer-readable storage medium of claim 16, wherein calculating the probability distribution includes inferring an explanation for a formation of a social connection in the ego network. 20. A computer system comprising: a component configured to establish variational equations based on attribute labels of nodes in a first ego network in a social graph of a social networking system; a component configured to set likelihood scores for at least a portion of the attribute labels of neighboring nodes from a focal user node in the ego network based on user-specified attribute labels from the social networking system; a component configured to calculate probability distributions of possible attribute labels for the focal user node of the ego network based on the variational equations and the likelihood scores; and a component configured to share at least a portion of the computed probability distributions with another process instance of a method inferring labels for a second ego network of the social networking system.