Numeric Input Relations for Relational Learning with Applications to Community Structure Analysis

Most work in the area of statistical relational learning (SRL) is focussed on discrete data, even though a few approaches for hybrid SRL models have been proposed that combine numerical and discrete variables. In this paper we distinguish numerical random variables for which a probability distribution is defined by the model from numerical input variables that are only used for conditioning the distribution of discrete response variables. We show how numerical input relations can very easily be used in the Relational Bayesian Network framework, and that existing inference and learning methods need only minor adjustments to be applied in this generalized setting. The resulting framework provides natural relational extensions of classical probabilistic models for categorical data. We demonstrate the usefulness of RBN models with numeric input relations by several examples. In particular, we use the augmented RBN framework to define probabilistic models for multi-relational (social) networks in which the probability of a link between two nodes depends on numeric latent feature vectors associated with the nodes. A generic learning procedure can be used to obtain a maximum-likelihood fit of model parameters and latent feature values for a variety of models that can be expressed in the high-level RBN representation. Specifically, we propose a model that allows us to interpret learned latent feature values as community centrality degrees by which we can identify nodes that are central for one community, that are hubs between communities, or that are isolated nodes. In a multi-relational setting, the model also provides a characterization of how different relations are associated with each community

Evaluating Overfit and Underfit in Models of Network Community Structure

A common data mining task on networks is community detection, which seeks an unsupervised decomposition of a network into structural groups based on statistical regularities in the network's connectivity. Although many methods exist, the No Free Lunch theorem for community detection implies that each makes some kind of tradeoff, and no algorithm can be optimal on all inputs. Thus, different algorithms will over or underfit on different inputs, finding more, fewer, or just different communities than is optimal, and evaluation methods that use a metadata partition as a ground truth will produce misleading conclusions about general accuracy. Here, we present a broad evaluation of over and underfitting in community detection, comparing the behavior of 16 state-of-the-art community detection algorithms on a novel and structurally diverse corpus of 406 real-world networks. We find that (i) algorithms vary widely both in the number of communities they find and in their corresponding composition, given the same input, (ii) algorithms can be clustered into distinct high-level groups based on similarities of their outputs on real-world networks, and (iii) these differences induce wide variation in accuracy on link prediction and link description tasks. We introduce a new diagnostic for evaluating overfitting and underfitting in practice, and use it to roughly divide community detection methods into general and specialized learning algorithms. Across methods and inputs, Bayesian techniques based on the stochastic block model and a minimum description length approach to regularization represent the best general learning approach, but can be outperformed under specific circumstances. These results introduce both a theoretically principled approach to evaluate over and underfitting in models of network community structure and a realistic benchmark by which new methods may be evaluated and compared.Comment: 22 pages, 13 figures, 3 table

Structured probabilistic inference

AbstractProbabilistic inference is among the main topics with reasoning in uncertainty in AI. For this purpose, Bayesian Networks (BNs) is one of the most successful and efficient Probabilistic Graphical Model (PGM) so far. Since the mid-90s, a growing number of BNs extensions have been proposed. Object-oriented, entity-relationship and first-order logic are the main representation paradigms used to extend BNs. While entity-relationship and first-order models have been successfully used for machine learning in defining lifted probabilistic inference, object-oriented models have been mostly underused. Structured inference, which exploits the structural knowledge encoded in an object-oriented PGM, is a surprisingly unstudied technique. In this paper we propose a full object-oriented framework for PRM and propose two extensions of the state-of-the-art structured inference algorithm: SPI which removes the major flaws of existing algorithms and SPISBB which largely enhances SPI by using d-separation