Learning the mechanisms of network growth.

We propose a novel model-selection method for dynamic networks. Our approach involves training a classifier on a large body of synthetic network data. The data is generated by simulating nine state-of-the-art random graph models for dynamic networks, with parameter range chosen to ensure exponential growth of the network size in time. We design a conceptually novel type of dynamic features that count new links received by a group of vertices in a particular time interval. The proposed features are easy to compute, analytically tractable, and interpretable. Our approach achieves a near-perfect classification of synthetic networks, exceeding the state-of-the-art by a large margin. Applying our classification method to real-world citation networks gives credibility to the claims in the literature that models with preferential attachment, fitness and aging fit real-world citation networks best, although sometimes, the predicted model does not involve vertex fitness.

Can structure predict function at individual level in the human connectome?

Several studies predicting Functional Connectivity (FC) from Structural Connectivity (SC) at individual level have been published in recent years, each promising increased performance and utility. We investigated three of these studies, analyzing whether the results truly represent a meaningful individual-level mapping from SC to FC. Using data from the Human Connectome Project shared accross the three studies, we constructed a predictor by averaging FC of training data and analyzed its performance in the same way. In each case, we found that group average FC is an equivalent or better predictor of individual FC than the predictive models in terms of raw prediction performance. Furthermore, we showed that additional analyses performed by the authors of the three studies, in which they attempt to show that their predicted FC has value beyond raw prediction performance, could also be reproduced using the group average FC predictor. This makes it unclear whether any of the three methods represent a meaningful individual-level predictive model. We conclude that either the methods are not appropriate for the data, that the sample size is too small, or that the data does not contain sufficient information to learn a mapping from SC to FC. We advise future individual-level studies to explicitly report results in comparison to the performance of the group average, and carefully demonstrate that their predictions contain meaningful individual-level information. Finally, we believe that investigating alternatives for the construction of SC and FC may improve the chances of developing a meaningful individual-level mapping from SC to FC.

Working memory performance in glioma patients is associated with functional connectivity between the right dorsolateral prefrontal cortex and default mode network.

In healthy subjects, activity in the default mode network (DMN) and the frontoparietal network (FPN) has consistently been associated with working memory (WM). In particular, the dorsolateral prefrontal cortex (DLPFC) is important for WM. The functional-anatomical basis of WM impairment in glioma patients is, however, still poorly understood. We investigated whether WM performance of glioma patients is reflected in resting-state functional connectivity (FC) between the DMN and FPN, additionally focusing on the DLPFC. Resting-state functional MRI data were acquired from 45 glioma patients prior to surgery. WM performance was derived from a pre-operative N-back task. Scans were parcellated into ROIs using both the Gordon and Yeo atlas. FC was calculated as the average Pearson correlation between functional time series. The FC between right DLPFC and DMN was inversely related to WM performance for both the Gordon and Yeo atlas (p = .010). No association was found for FC between left DLPFC and DMN, nor between the whole FPN and DMN. The results are robust and not dependent on atlas choice or tumor location, as they hold for both the Gordon and Yeo atlases, and independently of location variables. Our findings show that WM performance of glioma patients can be quantified in terms of interactions between regions and large-scale networks that can be measured with resting-state fMRI. These group-based results are a necessary step toward development of biomarkers for clinical management of glioma patients, and provide additional evidence that global disruption of the DMN relates to cognitive impairment in glioma patients.

The role of inter-regional mobility in forecasting SARS-CoV-2 transmission.

In this paper, we present a method to forecast the spread of SARS-CoV-2 across regions with a focus on the role of mobility. Mobility has previously been shown to play a significant role in the spread of the virus, particularly between regions. Here, we investigate under which epidemiological circumstances incorporating mobility into transmission models yields improvements in the accuracy of forecasting, where we take the situation in The Netherlands during and after the first wave of transmission in 2020 as a case study. We assess the quality of forecasting on the detailed level of municipalities, instead of on a nationwide level. To model transmissions, we use a simple mobility-enhanced SEIR compartmental model with subpopulations corresponding to the Dutch municipalities. We use commuter information to quantify mobility, and develop a method based on maximum likelihood estimation to determine the other relevant parameters. We show that taking inter-regional mobility into account generally leads to an improvement in forecast quality. However, at times when policies are in place that aim to reduce contacts or travel, this improvement is very small. By contrast, the improvement becomes larger when municipalities have a relatively large amount of incoming mobility compared with the number of inhabitants.

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.

Annealed inhomogeneities in random ferromagnets.

We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erdos-Rényi) or prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite because then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random independent and identically distributed degrees washes away the universality class with power-law critical exponents.

Engineering transient dynamics of artificial cells by stochastic distribution of enzymes.

Random fluctuations are inherent to all complex molecular systems. Although nature has evolved mechanisms to control stochastic events to achieve the desired biological output, reproducing this in synthetic systems represents a significant challenge. Here we present an artificial platform that enables us to exploit stochasticity to direct motile behavior. We found that enzymes, when confined to the fluidic polymer membrane of a core-shell coacervate, were distributed stochastically in time and space. This resulted in a transient, asymmetric configuration of propulsive units, which imparted motility to such coacervates in presence of substrate. This mechanism was confirmed by stochastic modelling and simulations in silico. Furthermore, we showed that a deeper understanding of the mechanism of stochasticity could be utilized to modulate the motion output. Conceptually, this work represents a leap in design philosophy in the construction of synthetic systems with life-like behaviors.

Trade-offs between mobility restrictions and transmission of SARS-CoV-2.

In their response to the COVID-19 outbreak, governments face the dilemma to balance public health and economy. Mobility plays a central role in this dilemma because the movement of people enables both economic activity and virus spread. We use mobility data in the form of counts of travellers between regions, to extend the often-used SEIR models to include mobility between regions. We quantify the trade-off between mobility and infection spread in terms of a single parameter, to be chosen by policy makers, and propose strategies for restricting mobility so that the restrictions are minimal while the infection spread is effectively limited. We consider restrictions where the country is divided into regions, and study scenarios where mobility is allowed within these regions, and disallowed between them. We propose heuristic methods to approximate optimal choices for these regions. We evaluate the obtained restrictions based on our trade-off. The results show that our methods are especially effective when the infections are highly concentrated, e.g. around a few municipalities, as resulting from superspreading events that play an important role in the spread of COVID-19. We demonstrate our method in the example of the Netherlands. The results apply more broadly when mobility data are available.

Long Paths in First Passage Percolation on the Complete Graph II. Global Branching Dynamics.

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdos-Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters ( s n ) n ≥ 1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where s n → ∞ with s n = o ( n 1 / 3 ) , which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately s n log ( n / s n 3 ) and variance s n 2 log ( n / s n 3 ) . Remarkably, our proof also applies to n-dependent edge weights of the form E s n , where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29-69, 2012). The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in Eckhoff et al. (Long paths in first passage percolation on the complete graph I. Local PWIT dynamics. Electron. J. Probab. 25:1-45, 2020. 10.1214/20-EJP484); the current paper focuses on the global branching dynamics.

Diameter in ultra-small scale-free random graphs.

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

Triadic Closure in Configuration Models with Unbounded Degree Fluctuations.

The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k), i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and the graph size n and eventually for k = Ω ( n ) settles on a power law c ( k ) ∼ n 5 - 2 τ k - 2 ( 3 - τ ) with τ ∈ ( 2 , 3 ) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

Cluster Tails for Critical Power-Law Inhomogeneous Random Graphs.

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299-2361, 2012). It was proved that when the degrees obey a power law with exponent τ ∈ ( 3 , 4 ) , the sequence of clusters ordered in decreasing size and multiplied through by n - ( τ - 2 ) / ( τ - 1 ) converges as n → ∞ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237-269, 2001) for the Erdos-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

Local clustering in scale-free networks with hidden variables.

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent τ∈(2,3), so that the degrees have infinite variance. The natural cutoff h_{c} characterizes the largest degrees in the hidden variable models, and a structural cutoff h_{s} introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient C scales with the network size N, as a function of h_{s} and h_{c}. For scale-free networks with exponent 2<τ<3 and the default choices h_{s}∼N^{1/2} and h_{c}∼N^{1/(τ-1)} this gives C∼N^{2-τ}lnN for the universality class at hand. We characterize the extremely slow decay of C when τ≈2 and show that for τ=2.1, say, clustering starts to vanish only for networks as large as N=10^{9}.

Clustering spectrum of scale-free networks.

Real-world networks often have power-law degrees and scale-free properties, such as ultrasmall distances and ultrafast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of c[over ¯](k), the probability that two neighbors of a degree-k node are neighbors themselves. We investigate how the clustering spectrum k↦c[over ¯](k) scales with k in the hidden-variable model and show that c[over ¯](k) follows a universal curve that consists of three k ranges where c[over ¯](k) remains flat, starts declining, and eventually settles on a power-law c[over ¯](k)∼k^{-α} with α depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks.

Power-law relations in random networks with communities.

Most random graph models are locally tree-like-do not contain short cycles-rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of HCM, we observe two previously undiscovered power-law relations: between the number of edges inside a community and the community sizes, and between the number of edges going out of a community and the community sizes. We also relate the power-law exponent τ of the degree distribution with the power-law exponent of the community-size distribution γ. In the case of extremely dense communities (e.g., complete graphs), this relation takes the simple form τ=γ-1.

Epidemic spreading on complex networks with community structures.

Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both enforce as well as inhibit diffusion processes. Our models further show that it is the mesoscopic set of communities that matters. The exact internal structures of communities barely influence the behavior of percolation processes across networks. This insensitivity is likely due to the relative denseness of the communities.

Probing exchange pathways in one-dimensional aggregates with super-resolution microscopy.

Supramolecular fibers are prominent structures in biology and chemistry. A quantitative understanding of molecular exchange pathways in these one-dimensional aggregates was obtained by a combination of super-resolution stochastic optical reconstruction microscopy and stochastic simulation. The potential of this methodology is demonstrated with a set of well-defined synthetic building blocks that self-assemble into supramolecular fibrils. Previous ensemble measurements hid all molecular phenomena underpinning monomer exchange, but the molecular pathway determined from single-aggregate studies revealed unexpected homogeneous exchange along the polymer backbone. These results pave the way for experimental investigation of the structure and exchange pathways of synthetic and natural supramolecular fibers.

Uncovering disassortativity in large scale-free networks.

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, and social and biological networks, are often characterized by degree-degree dependencies between neighboring nodes. In this paper, we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman's ρ. Our experiments convincingly show that Spearman's ρ produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.