Modular structure is ubiquitous in real-world complex networks, and its detection is important because it gives insights in the structure-functionality relationship. The standard approach is based on the optimization of a quality function, modularity, which is a relative quality measure for a partition of a network into modules. Recently some authors [1,2] have pointed out that the optimization of modularity has a fundamental drawback: the existence of a resolution limit beyond which no modular structure can be detected even though these modules might have own entity. The reason is that several topological descriptions of the network coexist at different scales, which is, in general, a fingerprint of complex systems. Here we propose a method that allows for multiple resolution screening of the modular structure. The method has been validated using synthetic networks, discovering the predefined structures at all scales. Its application to two real social networks allows to find the exact splits reported in the literature, as well as the substructure beyond the actual split.

Aims To assess tolerability and optimal time point for initiation of sacubitril/valsartan in patients stabilised after acute heart failure (AHF). Methods and results TRANSITION was a randomised, multicentre, open‐label study comparing two treatment initiation modalities of sacubitril/valsartan. Patients aged ≥ 18 years, hospitalised for AHF were stratified according to pre‐admission use of renin–angiotensin–aldosterone system inhibitors and randomised (n = 1002) after stabilisation to initiate sacubitril/valsartan either ≥ 12‐h pre‐discharge or between Days 1–14 post‐discharge. Starting dose (as per label) was 24/26 mg or 49/51 mg bid with up‐ or down‐titration based on tolerability. The primary endpoint was the proportion of patients attaining 97/103 mg bid target dose after 10 weeks. Median time of first dose of sacubitril/valsartan from the day of discharge was Day –1 and Day +1 in the pre‐discharge group and the post‐discharge group, respectively. Comparable proportions of patients in the pre‐ and post‐discharge initiation groups met the primary endpoint [45.4% vs. 50.7%; risk ratio (RR) 0.90; 95% confidence interval (CI) 0.79–1.02]. The proportion of patients who achieved and maintained for ≥ 2 weeks leading to Week 10, either 49/51 or 97/103 mg bid was 62.1% vs. 68.5% (RR 0.91; 95% CI 0.83–0.99); or any dose was 86.0% vs. 89.6% (RR 0.96; 95% CI 0.92–1.01). Discontinuation due to adverse events occurred in 7.3% vs. 4.9% of patients (RR 1.49; 95% CI 0.90–2.46). Conclusions Initiation of sacubitril/valsartan in a wide range of heart failure with reduced ejection fraction patients stabilised after an AHF event, either in hospital or shortly after discharge, is feasible with about half of the patients achieving target dose within 10 weeks. Clinical Trial Registration: http://ClinicalTrials.gov ID: NCT02661217

The ubiquity of modular structure in real-world complex networks is being the focus of attention in many trials to understand the interplay between network topology and functionality. The best approaches to the identification of modular structure are based on the optimization of a quality function known as modularity. However this optimization is a hard task provided that the computational complexity of the problem is in the NP-hard class. Here we propose an exact method for reducing the size of weighted (directed and undirected) complex networks while maintaining invariant its modularity. This size reduction allows the heuristic algorithms that optimize modularity for a better exploration of the modularity landscape. We compare the modularity obtained in several real complex-networks by using the Extremal Optimization algorithm, before and after the size reduction, showing the improvement obtained. We speculate that the proposed analytical size reduction could be extended to an exact coarse graining of the network in the scope of real-space renormalization.

No abstract

In this paper we apply a heuristic method based on artificial neural networks in order to trace out the efficient frontier associated to the portfolio selection problem. We consider a generalization of the standard Markowitz mean-variance model which includes cardinality and bounding constraints. These constraints ensure the investment in a given number of different assets and limit the amount of capital to be invested in each asset. We present some experimental results obtained with the neural network heuristic and we compare them to those obtained with three previous heuristic methods.Comment: 12 pages; submitted to "Computers & Operations Research