Variational principle for scale-free network motifs

For scale-free networks with degrees following a power law with an exponent $\tau\in(2,3)$, the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations

Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models

W-graph refers to a general class of random graph models that can be seen as a random graph limit. It is characterized by both its graphon function and its motif frequencies. In this paper, relying on an existing variational Bayes algorithm for the stochastic block models along with the corresponding weights for model averaging, we derive an estimate of the graphon function as an average of stochastic block models with increasing number of blocks. In the same framework, we derive the variational posterior frequency of any motif. A simulation study and an illustration on a social network complete our work

Networks with communities and clustering

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Scale-free network clustering in hyperbolic and other random graphs

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model

Hierarchical network structure as the source of power-law frequency spectra (state-trait continua) in living and non-living systems: how physical traits and personalities emerge from first principles in biophysics

What causes organisms to have different body plans and personalities? We address this question by looking at universal principles that govern the morphology and behavior of living systems. Living systems display a small-world network structure in which many smaller clusters are nested within fewer larger ones, producing a fractal-like structure with a power-law cluster size distribution. Their dynamics show similar qualities: the timeseries of inner message passing and overt behavior contain high frequencies or 'states' that are nested within lower frequencies or 'traits'. Here, we argue that the nested modular (power-law) dynamics of living systems results from their nested modular (power-law) network structure: organisms 'vertically encode' the deep spatiotemporal structure of their environments, so that high frequencies (states) are produced by many small clusters at the base of a nested-modular hierarchy and lower frequencies (traits) are produced by fewer larger clusters at its top. These include physical as well as behavioral traits. Nested-modular structure causes higher frequencies to be embedded in lower frequencies, producing power-law dynamics. Such dynamics satisfy the need for efficient energy dissipation through networks of coupled oscillators, which also governs the dynamics of non-living systems (e.g. earthquake dynamics, stock market fluctuations). Thus, we provide a single explanation for power-law frequency spectra in both living and non-living systems. If hierarchical structure indeed produces hierarchical dynamics, the development (e.g. during maturation) and collapse (e.g. during disease) of hierarchical structure should leave specific traces in power-law frequency spectra that may serve as early warning signs to system failure. The applications of this idea range from embryology and personality psychology to sociology, evolutionary biology and clinical medicine

High-resolution temporal profiling of transcripts during Arabidopsis leaf senescence reveals a distinct chronology of processes and regulation

Leaf senescence is an essential developmental process that impacts dramatically on crop yields and involves altered regulation of thousands of genes and many metabolic and signaling pathways, resulting in major changes in the leaf. The regulation of senescence is complex, and although senescence regulatory genes have been characterized, there is little information on how these function in the global control of the process. We used microarray analysis to obtain a highresolution time-course profile of gene expression during development of a single leaf over a 3-week period to senescence. A complex experimental design approach and a combination of methods were used to extract high-quality replicated data and to identify differentially expressed genes. The multiple time points enable the use of highly informative clustering to reveal distinct time points at which signaling and metabolic pathways change. Analysis of motif enrichment, as well as comparison of transcription factor (TF) families showing altered expression over the time course, identify clear groups of TFs active at different stages of leaf development and senescence. These data enable connection of metabolic processes, signaling pathways, and specific TF activity, which will underpin the development of network models to elucidate the process of senescence