Solving for multi-class using orthogonal coding matrices

A common method of generalizing binary to multi-class classification is the error correcting code (ECC). ECCs may be optimized in a number of ways, for instance by making them orthogonal. Here we test two types of orthogonal ECCs on seven different datasets using three types of binary classifier and compare them with three other multi-class methods: 1 vs. 1, one-versus-the-rest and random ECCs. The first type of orthogonal ECC, in which the codes contain no zeros, admits a fast and simple method of solving for the probabilities. Orthogonal ECCs are always more accurate than random ECCs as predicted by recent literature. Improvments in uncertainty coefficient (U.C.) range between 0.4--17.5% (0.004--0.139, absolute), while improvements in Brier score between 0.7--10.7%. Unfortunately, orthogonal ECCs are rarely more accurate than 1 vs. 1. Disparities are worst when the methods are paired with logistic regression, with orthogonal ECCs never beating 1 vs. 1. When the methods are paired with SVM, the losses are less significant, peaking at 1.5%, relative, 0.011 absolute in uncertainty coefficient and 6.5% in Brier scores. Orthogonal ECCs are always the fastest of the five multi-class methods when paired with linear classifiers. When paired with a piecewise linear classifier, whose classification speed does not depend on the number of training samples, classifications using orthogonal ECCs were always more accurate than the the remaining three methods and also faster than 1 vs. 1. Losses against 1 vs. 1 here were higher, peaking at 1.9% (0.017, absolute), in U.C. and 39% in Brier score. Gains in speed ranged between 1.1% and over 100%. Whether the speed increase is worth the penalty in accuracy will depend on the application

Multiplexity and multireciprocity in directed multiplexes

Real-world multi-layer networks feature nontrivial dependencies among links of different layers. Here we argue that, if links are directed, dependencies are twofold. Besides the ordinary tendency of links of different layers to align as the result of `multiplexity', there is also a tendency to anti-align as the result of what we call `multireciprocity', i.e. the fact that links in one layer can be reciprocated by \emph{opposite} links in a different layer. Multireciprocity generalizes the scalar definition of single-layer reciprocity to that of a square matrix involving all pairs of layers. We introduce multiplexity and multireciprocity matrices for both binary and weighted multiplexes and validate their statistical significance against maximum-entropy null models that filter out the effects of node heterogeneity. We then perform a detailed empirical analysis of the World Trade Multiplex (WTM), representing the import-export relationships between world countries in different commodities. We show that the WTM exhibits strong multiplexity and multireciprocity, an effect which is however largely encoded into the degree or strength sequences of individual layers. The residual effects are still significant and allow to classify pairs of commodities according to their tendency to be traded together in the same direction and/or in opposite ones. We also find that the multireciprocity of the WTM is significantly lower than the usual reciprocity measured on the aggregate network. Moreover, layers with low (high) internal reciprocity are embedded within sets of layers with comparably low (high) mutual multireciprocity. This suggests that, in the WTM, reciprocity is inherent to groups of related commodities rather than to individual commodities. We discuss the implications for international trade research focusing on product taxonomies, the product space, and fitness/complexity metrics.Comment: 20 pages, 8 figure

Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints

We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Pareto-optimal allocation satisfies envy-freeness up to one good, and that it is NP-hard to decide whether such an allocation exists, even for binary valuations. We also show that, for a path, it is NP-hard to find a Pareto-optimal allocation that satisfies maximin share, but show that a moving-knife algorithm can find such an allocation when agents have binary valuations that have a non-nested interval structure.Comment: 21 pages, full version of paper at AAAI-201

Null Models of Economic Networks: The Case of the World Trade Web

In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the World Trade Web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models.Comment: 39 pages, 46 figures, 2 table