Node Metadata Can Produce Predictability Crossovers in Network Inference Problems

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Node Metadata Can Produce Predictability Crossovers in Network Inference Problems

Oscar Fajardo-Fontiveros, Roger Guimerà, and Marta Sales-Pardo

Phys. Rev. X 12, 011010 – Published 14 January 2022

Abstract

Network inference is the process of learning the properties of complex networks from data. Besides using information about known links in the network, node attributes and other forms of network metadata can help solve network inference problems. Indeed, several approaches have been proposed to introduce metadata into probabilistic network models and to use them to make better inferences. However, we know little about the effect of such metadata in the inference process. Here, we investigate this issue. We find that, rather than affecting inference gradually, adding metadata causes a crossover in the inference process and in our ability to make accurate predictions, from a situation in which metadata do not play any role to a situation in which metadata completely dominate the inference process. When network data and metadata are partly correlated, metadata optimally contributes to the inference process at the crossover between data-dominated and metadata-dominated regimes.

Received 8 April 2021
Accepted 16 November 2021

DOI:https://doi.org/10.1103/PhysRevX.12.011010

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Network phase transitions Network structure

Interconnected & interdependent networks Multilayer & multiplex networks Real world networks Social networks Stochastic networks

Block models Data analysis

NetworksStatistical Physics & Thermodynamics

Authors & Affiliations

Oscar Fajardo-Fontiveros ^1,‡, Roger Guimerà ^2,1,*, and Marta Sales-Pardo ^1,†

¹Department of Chemical Engineering, Universitat Rovira i Virgili, 43007 Tarragona, Catalonia
²ICREA, 08010 Barcelona, Catalonia

^*Corresponding author. roger.guimera@urv.cat
^†Corresponding author. marta.sales@urv.cat
^‡oscar.fajardo@urv.cat

Popular Summary

Predicting whether two drugs have a harmful interaction or whether someone is going to like a certain movie are examples of network inference problems. In these problems, the goal is to predict new interactions (between drugs or between people and movies) based on some previously observed interactions. Having additional information about the network nodes or their metadata (for example, the mechanism of action of the drugs or the age of the individuals) helps to make better predictions, though it is not clear why or how. Here, we explore how that improvement happens.

We study a very general network inference problem and show that node metadata do not affect the inference problem gradually. Rather, even when the importance assigned to the metadata increases smoothly, the inference process crosses over from a data-dominated regime to a metadata-dominated regime. These crossovers show some similarities to transitions driven by temperature, where one finds energy- and entropy-dominated regimes. Importantly, optimal inference is often encountered exactly at this crossover.

Our study opens the door to better understanding the role of metadata in network inference problems and, more broadly, establishes further connections between general inference problems and physical concepts such as phase transitions.

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Vol. 12, Iss. 1 — January - March 2022

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Figure 1
Multipartite mixed-membership stochastic block model with labeled links. In panel (a), we cast the recommendation problem (in which one aims to predict how users will rate certain items) into a network inference problem. Here, users rate movies with three possible ratings (green, orange, or red). Additionally, we have excluding attributes for users (two excluding genders and three excluding age groups, represented by different shades of the same color) and nonexcluding attributes for movies (two movie genres; the connection to these attributes is binary, yes or no, but, in general, it does not need to be). Similar to ratings, we represent these attributes as bipartite networks. Although we frame our description of the model in terms of recommendations or link prediction in a bipartite network, the problem of link prediction in regular unipartite networks is just a particular case in which user nodes and item nodes are the same. In panel (b), each bipartite network in the multipartite network is modeled using a mixed-membership stochastic block model (see text). The individual block models are coupled by the user and item membership vectors ( $θ$ and $η$ , respectively), shown in panel (c) along with all other model parameters and their dimensions (see text).
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Figure 2
Predictive performance and effect of metadata on synthetic ratings. We create synthetic ratings from 200 users on 200 items, with different levels of correlation $c$ between ratings and node attributes (see text). We then use fivefold cross-validation to calculate the performance of the expectation-maximization equations at predicting unobserved ratings. We use accuracy $a$ (that is, the fraction of correctly predicted ratings) as our performance metric; we take as a reference the predictive accuracy $a_{0} = 0.468$ of the algorithm when all attributes are ignored ( $λ_{user} = λ_{item} = 0$ ) and measure relative accuracy $α$ for a given pair $(λ_{user}, λ_{item})$ as the log-ratio $α (λ_{user}, λ_{item}) = \log [a (λ_{user}, λ_{item}) / a_{0}]$ . The value $α (λ_{user}, λ_{item}) = 0$ (dashed line) thus indicates no change with respect to the reference $a_{0} = 0.468$ , and $α (λ_{user}, λ_{item}) > 0$ [respectively, $α (λ_{user}, λ_{item}) < 0$ ] indicates predictions that are more (less) accurate than those obtained by ignoring node attributes. The maximum possible relative performance ( $a_{\max} = 0.580$ ; dotted line) is obtained when each rating is assigned the exact probability that was used to generate it. For each value of the correlation [(a,b) full correlation, $c = 1$ ; (c,d) $c = 0.75$ ; (e,f) $c = 0.50$ ; (g,h) no correlation, $c = 0$ ], we show the variation of $α (λ_{user}, λ_{item})$ with $λ_{item}$ for different values of $λ_{user}$ (left), and the whole dependence of $α (λ_{user}, λ_{item})$ on both $λ_{user}$ and $λ_{item}$ (right).
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Figure 3
Crossover between data-dominated and metadata-dominated inference regimes. For the synthetic data in Fig. 2, we plot the log-posterior $π (θ, η, ζ, p, q, \hat{q} | R^{O}, A^{O})$ as a function of the hyperparameter $λ = λ_{item} = λ_{user}$ for three models: the model that maximizes the data likelihood $L^{R}$ , the model that maximizes the metadata likelihood $L^{A}$ , and the model that maximizes the posterior when two previous cases cross (that is, have equal posteriors). The position of the crossing coincides with the crossover and the maxima observed in Fig. 2.
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Figure 4
Predictive performance and effect of metadata on the MovieLens data set. As in Fig. 2, we take as a reference the predictive accuracy $a_{0}$ of the algorithm when all attributes are ignored ( $λ_{user} = λ_{item} = 0$ ), and we measure relative accuracy $α$ for a given pair $(λ_{user}, λ_{item})$ as the log-ratio $α (λ_{user}, λ_{item}) = \log [a (λ_{user}, λ_{item}) / a_{0}]$ . Accuracy is the fraction of correctly predicted ratings in cross-validation experiments, and $a_{0} = 0.448$ . We consider three different attributes for user nodes: (a,b) age; (c,d) gender; and (e,f) age and gender combined as a single attribute. We plot the whole range of $λ_{user}$ (left) and zoom into the intermediate (shaded) region of $λ_{user}$ in which predictions are significantly more accurate than the reference (right).
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Figure 5
Predictive performance and effect of metadata on the U.S. Congress data set. As in Fig. 2, we take as a reference the predictive accuracy $a_{0}$ of the algorithm when all attributes are ignored ( $λ_{user} = 0$ ) and measure relative accuracy $α$ for a given $λ_{user}$ as the log-ratio $α (λ_{user}) = \log [a (λ_{user}) / a_{0}]$ . Accuracy is the fraction of correctly predicted ratings in cross-validation experiments, and $a_{0} = 0.677$ . We consider three different attributes for user nodes: party, state, and party and state simultaneously.
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