Mauricio Barahona

Synchronization in small-world systems

We quantify the dynamical implications of the small-world phenomenon by considering the generic synchronization of oscillator networks of arbitrary topology. The linear stability of the synchronous state is linked to an algebraic condition of the Laplacian matrix of the network. Through numerics and analysis, we show how the addition of random shortcuts translates into improved network synchronizability. Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes. However, the small-world property does not guarantee synchronizability: the synchronization threshold lies within the boundaries, but linked to the end of the small-world region.

Transcriptome-wide noise controls lineage choice in mammalian progenitor cells

Phenotypic cell-to-cell variability within clonal populations may be a manifestation of ‘gene expression noise’, or it may reflect stable phenotypic variants. Such ‘non-genetic cell individuality’ can arise from the slow fluctuations of protein levels in mammalian cells. These fluctuations produce persistent cell individuality, thereby rendering a clonal population heterogeneous. However, it remains unknown whether this heterogeneity may account for the stochasticity of cell fate decisions in stem cells. Here we show that in clonal populations of mouse haematopoietic progenitor cells, spontaneous ‘outlier’ cells with either extremely high or low expression levels of the stem cell marker Sca-1 (also known as Ly6a; ref. 9) reconstitute the parental distribution of Sca-1 but do so only after more than one week. This slow relaxation is described by a gaussian mixture model that incorporates noise-driven transitions between discrete subpopulations, suggesting hidden multi-stability within one cell type. Despite clonality, the Sca-1 outliers had distinct transcriptomes. Although their unique gene expression profiles eventually reverted to that of the median cells, revealing an attractor state, they lasted long enough to confer a greatly different proclivity for choosing either the erythroid or the myeloid lineage. Preference in lineage choice was associated with increased expression of lineage-specific transcription factors, such as a >200-fold increase in Gata1 (ref. 10) among the erythroid-prone cells, or a >15-fold increased PU.1 (Sfpi1) (ref. 11) expression among myeloid-prone cells. Thus, clonal heterogeneity of gene expression level is not due to independent noise in the expression of individual genes, but reflects metastable states of a slowly fluctuating transcriptome that is distinct in individual cells and may govern the reversible, stochastic priming of multipotent progenitor cells in cell fate decision.

Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks

Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dy namical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computedMost methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process provides a dynamical sweeping across all scales in the graph, and the time scale is an intrinsic parameter that uncovers communities at different resolutions. This dynamic-based community detection leads to a compound optimization, which favours communities of comparable centrality (as defined by the stationary distribution), and provides a unifying framework for spectral algorithms, as well as different heuristics for community detection, including versions of modularity and Potts model. Our dynamic framework creates a systematic link between different stochastic dynamics and their corresponding notions of optimal communities under distinct (node and edge) centralities. We show that the Markov Stability can be computed efficiently to find multi-scale community structure in large networks.

Stability of graph communities across time scales

The complexity of biological, social, and engineering networks makes it desirable to find natural partitions into clusters (or communities) that can provide insight into the structure of the overall system and even act as simplified functional descriptions. Although methods for community detection abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We introduce here the stability of a partition, a measure of its quality as a community structure based on the clustered autocovariance of a dynamic Markov process taking place on the network. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser communities. Our dynamical definition provides a unifying framework for several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler’s spectral clustering emerges at long times. We apply our method to characterize the relevance of partitions over time for constructive and real networks, including hierarchical graphs and social networks, and use it to obtain reduced descriptions for atomic-level protein structures over different time scales.

SC3: consensus clustering of single-cell RNA-seq data

Single-cell RNA-seq enables the quantitative characterization of cell types based on global transcriptome profiles. We present single-cell consensus clustering (SC3), a user-friendly tool for unsupervised clustering, which achieves high accuracy and robustness by combining multiple clustering solutions through a consensus approach (http://bioconductor.org/packages/SC3). We demonstrate that SC3 is capable of identifying subclones from the transcriptomes of neoplastic cells collected from patients.

Titration of chaos with added noise

Deterministic chaos has been implicated in numerous natural and man-made complex phenomena ranging from quantum to astronomical scales and in disciplines as diverse as meteorology, physiology, ecology, and economics. However, the lack of a definitive test of chaos vs. random noise in experimental time series has led to considerable controversy in many fields. Here we propose a numerical titration procedure as a simple “litmus test” for highly sensitive, specific, and robust detection of chaos in short noisy data without the need for intensive surrogate data testing. We show that the controlled addition of white or colored noise to a signal with a preexisting noise floor results in a titration index that: (i) faithfully tracks the onset of deterministic chaos in all standard bifurcation routes to chaos; and (ii) gives a relative measure of chaos intensity. Such reliable detection and quantification of chaos under severe conditions of relatively low signal-to-noise ratio is of great interest, as it may open potential practical ways of identifying, forecasting, and controlling complex behaviors in a wide variety of physical, biomedical, and socioeconomic systems.

The Effect of Spatially Inhomogeneous Extracellular Electric Fields on Neurons

The cooperative action of neurons and glia generates electrical fields, but their effect on individual neurons via ephaptic interactions is mostly unknown. Here, we analyze the impact of spatially inhomogeneous electric fields on the membrane potential, the induced membrane field, and the induced current source density of one-dimensional cables as well as morphologically realistic neurons and discuss how the features of the extracellular field affect these quantities. We show through simulations that endogenous fields, associated with hippocampal theta and sharp waves, can greatly affect spike timing. These findings imply that local electric fields, generated by the cooperative action of brain cells, can influence the timing of neural activity.

Spectral Measure of Structural Robustness in Complex Networks

We introduce the concept of natural connectivity as a measure of structural robustness in complex networks. The natural connectivity characterizes the redundancy of alternative routes in a network by quantifying the weighted number of closed walks of all lengths. This definition leads to a simple mathematical formulation that links the natural connectivity to the spectrum of a network. The natural connectivity can be regarded as an average eigenvalue that changes strictly monotonically with the addition or deletion of edges. We calculate both analytically and numerically the natural connectivity of three typical networks: regular ring lattices, random graphs, and random scale-free networks. We also compare the proposed natural connectivity to other structural robustness measures within a scenario of edge elimination and demonstrate that the natural connectivity provides sensitive discrimination of structural robustness that agrees with our intuition.

Markov Dynamics as a Zooming Lens for Multiscale Community Detection: Non Clique-Like Communities and the Field-of-View Limit

In recent years, there has been a surge of interest in community detection algorithms for complex networks. A variety of computational heuristics, some with a long history, have been proposed for the identification of communities or, alternatively, of good graph partitions. In most cases, the algorithms maximize a particular objective function, thereby finding the ‘right’ split into communities. Although a thorough comparison of algorithms is still lacking, there has been an effort to design benchmarks, i.e., random graph models with known community structure against which algorithms can be evaluated. However, popular community detection methods and benchmarks normally assume an implicit notion of community based on clique-like subgraphs, a form of community structure that is not always characteristic of real networks. Specifically, networks that emerge from geometric constraints can have natural non clique-like substructures with large effective diameters, which can be interpreted as long-range communities. In this work, we show that long-range communities escape detection by popular methods, which are blinded by a restricted ‘field-of-view’ limit, an intrinsic upper scale on the communities they can detect. The field-of-view limit means that long-range communities tend to be overpartitioned. We show how by adopting a dynamical perspective towards community detection [1], [2], in which the evolution of a Markov process on the graph is used as a zooming lens over the structure of the network at all scales, one can detect both clique- or non clique-like communities without imposing an upper scale to the detection. Consequently, the performance of algorithms on inherently low-diameter, clique-like benchmarks may not always be indicative of equally good results in real networks with local, sparser connectivity. We illustrate our ideas with constructive examples and through the analysis of real-world networks from imaging, protein structures and the power grid, where a multiscale structure of non clique-like communities is revealed.

Natural Connectivity of Complex Networks

The concept of natural connectivity is reported as a robustness measure of complex networks. The natural connectivity has a clear physical meaning and a simple mathematical formulation. It is shown that the natural connectivity can be derived mathematically from the graph spectrum as an average eigenvalue and that it changes strictly monotonically with the addition or deletion of edges. By comparing the natural connectivity with other typical robustness measures within a scenario of edge elimination, it is demonstrated that the natural connectivity has an acute discrimination which agrees with our intuition.