Entropy of labeled versus unlabeled networks.

The structure of a network is an unlabeled graph, yet graphs in most models of complex networks are labeled by meaningless random integers. Is the associated labeling noise always negligible, or can it overpower the network-structural signal? To address this question, we introduce and consider the sparse unlabeled versions of popular network models and compare their entropy against the original labeled versions. We show that labeled and unlabeled Erdos-Rényi graphs are entropically equivalent, even though their degree distributions are very different. The labeled and unlabeled versions of the configuration model may have different prefactors in their leading entropy terms, although this remains conjectural. Our main results are upper and lower bounds for the entropy of labeled and unlabeled one-dimensional random geometric graphs. We show that their unlabeled entropy is negligible in comparison with the labeled entropy. This means that in sparse networks the entropy of meaningless labeling may dominate the entropy of the network structure. The main implication of this result is that the common practice of using exchangeable models to reason about real-world networks with distinguishable nodes may introduce uncontrolled aberrations into conclusions made about these networks, suggesting a need for a thorough reexamination of the statistical foundations and key results of network science.

Parameterless detection of liquid-liquid interfaces with sub-micron resolution in single-molecule localization microscopy.

HYPOTHESIS: Knowing the exact location of soft interfaces, such as between water and oil, is essential to the study of nanoscale wetting phenomena. Recently, iPAINT was used to visualize soft interfaces in situ with minimal invasiveness, but computing the exact location of the interface remains challenging. We propose a new method to determine the interface with high accuracy. By modelling the localizations as points generated by two homogeneous Poisson processes, the exact location of the interface can be determined using a maximum likelihood estimator (MLE). EXPERIMENTS: An MLE was constructed to estimate the location of the interface based on the discontinuity in localization density at the interface. To test the MLE, we collected experimental data through iPAINT experiments of oil-water interfaces and generated simulated data using the Monte Carlo method. FINDINGS: Simulations show that the interface given by the MLE rapidly converges to the true interface location. The error of the MLE drops below the experimental localization precision. Furthermore, we show that the MLE remains accurate even if the field-of-view is reduced or when one or more particles are on the interface within the field-of-view. This work provides a key step towards the in situ, sub-micron characterization of (nanoparticle-laden) interfaces with minimal invasiveness.

Phase transitions for scaling of structural correlations in directed networks.

Analysis of degree-degree dependencies in complex networks, and their impact on processes on networks requires null models, i.e., models that generate uncorrelated scale-free networks. Most models to date, however, show structural negative dependencies, caused by finite size effects. We analyze the behavior of these structural negative degree-degree dependencies, using rank based correlation measures, in the directed erased configuration model. We obtain expressions for the scaling as a function of the exponents of the distributions. Moreover, we show that this scaling undergoes a phase transition, where one region exhibits scaling related to the natural cutoff of the network while another region has scaling similar to the structural cutoff for uncorrelated networks. By establishing the speed of convergence of these structural dependencies we are able to assess statistical significance of degree-degree dependencies on finite complex networks when compared to networks generated by the directed erased configuration model.