Degree-based goodness-of-fit tests for heterogeneous random graph models : independent and exchangeable cases

The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness-of-fit for a given random graph model. In this paper we introduce goodness-of-fit tests for two classes of models. First, we consider the case of independent graph models such as the heterogeneous Erd\"os-R\'enyi model in which the edges have different connection probabilities. Second, we consider a generic model for exchangeable random graphs called the W-graph. The stochastic block model and the expected degree distribution model fall within this framework. We prove the asymptotic normality of the degree mean square under these independent and exchangeable models and derive formal tests. We study the power of the proposed tests and we prove the asymptotic normality under specific sparsity regimes. The tests are illustrated on real networks from social sciences and ecology, and their performances are assessed via a simulation study

A degree-based goodness-of-fit test for heterogeneous random graph models

The degree variance has been proposed for many years to study the topology of a network. It can be used to assess the goodness-of-fit of the Erdös-Renyi model. In this paper, we prove the asymptotic normality of the degree variance under this model which enables us to derive a formal test. We generalize this result to the heterogeneous Erdös-Renyi model in which the edges have different respective probabilities to exist. For both models we study the power of the proposed goodness-of-fit test. We also prove the asymptotic normality under specific sparsity regimes. Both tests are illustrated on real networks from social sciences and ecology. Their performances are assessed via a simulation study

Motif-based tests for bipartite networks

Bipartite networks are a natural representation of the interactions between entities from two different types. The organization (or topology) of such networks gives insight to understand the systems they describe as a whole. Here, we rely on motifs which provide a meso-scale description of the topology. Moreover, we consider the bipartite expected degree distribution (B-EDD) model which accounts for both the density of the network and possible imbalances between the degrees of the nodes. Under the B-EDD model, we prove the asymptotic normality of the count of any given motif, considering sparsity conditions. We also provide close-form expressions for the mean and the variance of this count. This allows to avoid computationally prohibitive resampling procedures. Based on these results, we define a goodness-of-fit test for the B-EDD model and propose a family of tests for network comparisons. We assess the asymptotic normality of the test statistics and the power of the proposed tests on synthetic experiments and illustrate their use on ecological data sets

Goodness of fit of logistic models for random graphs

Logistic models for random graphs are commonly used to study binary networks when covariate information is available. After estimating the logistic parameters, one of the main questions which arises in practice is to assess the goodness of fit of the corresponding model. To address this problem, we add a general term, related to the graphon function of W-graph models, to the logistic function. Such an extra term aims at characterizing the residual structure of the network, that is not explained by the covariates. We approximate this new generic logistic model using a class of models with blockwise constant residual structure. This framework allows to derive a Bayesian procedure from a model based selection context using goodness-of-fit criteria. All these criteria depend on marginal likelihood terms for which we do provide estimates relying on two series of variational approximations. Experiments on toy data are carried out to assess the inference procedure. Finally, two real networks from social sciences and ecology are studied to illustrate the proposed methodology

Rare Pulmonary Neuroendocrine Cells Are Stem Cells Regulated by Rb, p53, and Notch

Pulmonary neuroendocrine (NE) cells are neurosensory cells sparsely distributed throughout the bronchial epithelium, many in innervated clusters of 20–30 cells. Following lung injury, NE cells proliferate and generate other cell types to promote epithelial repair. Here, we show that only rare NE cells, typically 2–4 per cluster, function as stem cells. These fully differentiated cells display features of classical stem cells. Most proliferate (self-renew) following injury, and some migrate into the injured area. A week later, individual cells, often just one per cluster, lose NE identity (deprogram), transit amplify, and reprogram to other fates, creating large clonal repair patches. Small cell lung cancer (SCLC) tumor suppressors regulate the stem cells: Rb and p53 suppress self-renewal, whereas Notch marks the stem cells and initiates deprogramming and transit amplification. We propose that NE stem cells give rise to SCLC, and transformation results from constitutive activation of stem cell renewal and inhibition of deprogramming