In directed networks, reciprocal links have dramatic effects on dynamical processes, network growth, and higher-order structures such as motifs and communities. While the reciprocity of binary networks has been extensively studied, that of weighted networks is still poorly understood, implying an ever-increasing gap between the availability of weighted network data and our understanding of their dyadic properties. Here we introduce a general approach to the reciprocity of weighted networks, and define quantities and null models that consistently capture empirical reciprocity patterns at different structural levels. We show that, counter-intuitively, previous reciprocity measures based on the similarity of mutual weights are uninformative. By contrast, our measures allow to consistently classify different weighted networks according to their reciprocity, track the evolution of a network's reciprocity over time, identify patterns at the level of dyads and vertices, and distinguish the effects of flux (im)balances or other (a)symmetries from a true tendency towards (anti-)reciprocation.

An outstanding open problem is whether collective social phenomena occurring over short timescales can systematically reduce cultural heterogeneity in the long run, and whether offline and online human interactions contribute differently to the process. Theoretical models suggest that short-term collective behavior and long-term cultural diversity are mutually excluding, since they require very different levels of social influence. The latter jointly depends on two factors: the topology of the underlying social network and the overlap between individuals in multidimensional cultural space. However, while the empirical properties of social networks are intensively studied, little is known about the large-scale organization of real societies in cultural space, so that random input specifications are necessarily used in models. Here we use a large dataset to perform a high-dimensional analysis of the scientific beliefs of thousands of Europeans. We find that interopinion correlations determine a nontrivial ultrametric hierarchy of individuals in cultural space. When empirical data are used as inputs in models, ultrametricity has strong and counterintuitive effects. On short timescales, it facilitates a symmetry-breaking phase transition triggering coordinated social behavior. On long timescales, it suppresses cultural convergence by restricting it within disjoint groups. Moreover, ultrametricity implies that these results are surprisingly robust to modifications of the dynamical rules considered. Thus the empirical distribution of individuals in cultural space appears to systematically optimize the coexistence of short-term collective behavior and long-term cultural diversity, which can be realized simultaneously for the same moderate level of mutual influence in a diverse range of online and offline settings.

Real-world multilayer networks feature nontrivial dependencies among links of different layers. Here we argue that if links are directed, then 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 antialign as a result of what we call “multireciprocity,” i.e., the fact that links in one layer can be reciprocated by 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 us 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 and complexity metrics.

Objective The aim of this study was to investigate the performance of four frequency weightings for hand-transmitted vibration to predict the incidence of vibration-induced white finger (VWF). Methods In a longitudinal study of vibration-exposed forestry and stone workers (N=206), the incidence of VWF was related to measures of vibration exposure expressed in terms of 8-hour frequency-weighted energy-equivalent root-mean-square (rms) acceleration magnitude [ A (8)] and years of follow-up. To calculate A (8), the rms acceleration magnitudes of vibration were weighted by means of four frequency weightings: (i) W h (the frequency weighting specified in ISO 5349-1:2001); (ii) W h-bl (the band-limiting component of W h ); (iii) W hf (a frequency weighting based on finger vibration power absorption); and (iv) W hT (a frequency weighting based on a Japanese study of VWF prevalence). The relations of VWF to alternative measures of vibration exposure were assessed by the generalized estimating equations (GEE) method to account for the within-subject dependency of the observations over time. Results Data analysis with a GEE logistic model and a measure of statistical fit suggested that calculating A (8) by weighting the tool rms accelerations with W h-bl gave better predictions of the cumulative incidence of VWF than the other alternative measures of daily vibration exposure. Values of A (8) derived from the currently recommended ISO frequency weighting Wh produced poorer predictions of the incidence of VWF than those obtained with frequency weightings W hf or W hT . Conclusions This prospective cohort study suggests that measures of daily vibration exposure which give relatively more weight to intermediate and high frequency vibration are more appropriate for assessing the probability of VWF.

Spatially embedded networks are shaped by a combination of purely topological (space-independent) and space-dependent formation rules. While it is quite easy to artificially generate networks where the relative importance of these two factors can be varied arbitrarily, it is much more difficult to disentangle these two architectural effects in real networks. Here we propose a solution to this problem, by introducing global and local measures of spatial effects that, through a comparison with adequate null models, effectively filter out the spurious contribution of nonspatial constraints. Our filtering allows us to consistently compare different embedded networks or different historical snapshots of the same network. As a challenging application we analyze the World Trade Web, whose topology is known to depend on geographic distances but is also strongly determined by nonspatial constraints (degree sequence or gross domestic product). Remarkably, we are able to detect weak but significant spatial effects both locally and globally in the network, showing that our method succeeds in retrieving spatial information even when nonspatial factors dominate. We finally relate our results to the economic literature on gravity models and trade globalization.