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Journal articleLambiotte R, Sinatra R, Delvenne J-C, et al., 2011,
Flow graphs: Interweaving dynamics and structure
, PHYSICAL REVIEW E, Vol: 84, ISSN: 1539-3755The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics, and show how our framework improves our understanding of these processes.
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Journal articleWu J, Barahona M, Tan Y-J, et al., 2011,
Robustness of regular ring lattices based on natural connectivity
, International Journal of Systems Science, Vol: 42, Pages: 1085-1092-1085-1092, ISSN: 0020-7721It has been recently proposed that natural connectivity can be used to efficiently characterise the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating the weighted number of closed walks of all lengths and can be seen as an average eigenvalue obtained from the graph spectrum. In this article, we explore both analytically and numerically the natural connectivity of regular ring lattices and regular random graphs obtained through degree-preserving random rewirings from regular ring lattices. We reformulate the natural connectivity of regular ring lattices in terms of generalised Bessel functions and show that the natural connectivity of regular ring lattices is independent of network size and increases with K monotonically. We also show that random regular graphs have lower natural connectivity, and are thus less robust, than regular ring lattices.
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Journal articleStrelkowa N, Barahona M, 2011,
Transient dynamics around unstable periodic orbits in the generalized repressilator model
, CHAOS, Vol: 21, ISSN: 1054-1500We study the temporal dynamics of the generalized repressilator, a network of coupled repressing genes arranged in a directed ring topology, and give analytical conditions for the emergence of a finite sequence of unstable periodic orbits that lead to reachable long-lived oscillating transients. Such transients dominate the finite time horizon dynamics that is relevant in confined, noisy environments such as bacterial cells (see our previous work [Strelkowa and Barahona, J. R. Soc. Interface 7, 1071 (2010)]), and are therefore of interest for bioengineering and synthetic biology. We show that the family of unstable orbits possesses spatial symmetries and can also be understood in terms of traveling wave solutions of kink-like topological defects. The long-lived oscillatory transients correspond to the propagation of quasistable two-kink configurations that unravel over a long time. We also assess the similarities between the generalized repressilator model and other unidirectionally coupled electronic systems, such as magnetic flux gates, which have been implemented experimentally. (C) 2011 American Institute of Physics. [doi:10.1063/1.3574387]
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Journal articleAugust E, Barahona M, 2011,
Obtaining certificates for complete synchronisation of coupled oscillators
, Physica D: Nonlinear Phenomena, Vol: 240, Pages: 795-803, ISSN: 0167-2789In this paper, we provide a novel reformulation of sufficient conditions that guarantee global complete synchronisation of coupled identical oscillators to make them computationally implementable. To this end, we use semidefinite programming techniques. For the first time, we can efficiently search for and obtain certificates for synchronisability and, additionally, also optimise associated cost functions. In this paper, a Lyapunov-like function (certificate) is used to certify that all trajectories of a networked system consisting of coupled dynamical systems will eventually converge towards a common one, which implies synchronisation. Moreover, we establish new conditions for complete synchronisation, which are based on the so called Bendixson’s Criterion for higher dimensional systems. This leads to major improvements on the lower bound of the coupling constant that guarantees global complete synchronisation. Importantly, the certificates are obtained by analysing the connection network and the model representing an individual system only. In order to illustrate the strength of our method we apply it to a system of coupled identical Lorenz oscillators and to coupled van der Pol oscillators. (C) 2010 Elsevier B.V. All rights reserved.
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Journal articleThomas P, Straube AV, Grima R, 2010,
Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models
, Journal of Chemical Physics, Vol: 133, ISSN: 1089-7690Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.
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Journal articleWu J, Barahona M, Tan Y, et al., 2010,
Robustness of Random Graphs Based on Natural Connectivity
Recently, it has been proposed that the natural connectivity can be used toefficiently characterise the robustness of complex networks. Naturalconnectivity quantifies the redundancy of alternative routes in a network byevaluating the weighted number of closed walks of all lengths and can beregarded as the average eigenvalue obtained from the graph spectrum. In thisarticle, we explore the natural connectivity of random graphs both analyticallyand numerically and show that it increases linearly with the average degree. Bycomparing with regular ring lattices and random regular graphs, we show thatrandom graphs are more robust than random regular graphs; however, therelationship between random graphs and regular ring lattices depends on theaverage degree and graph size. We derive the critical graph size as a functionof the average degree, which can be predicted by our analytical results. Whenthe graph size is less than the critical value, random graphs are more robustthan regular ring lattices, whereas regular ring lattices are more robust thanrandom graphs when the graph size is greater than the critical value.
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Journal articleWu J, Barahona M, Tan Y-J, et al., 2010,
Natural Connectivity of Complex Networks
, CHINESE PHYSICS LETTERS, Vol: 27, ISSN: 0256-307XThe 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.
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Journal articleDelvenne J-C, Yaliraki SN, Barahona M, 2010,
Stability of graph communities across time scales
, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, Vol: 107, Pages: {12755-12760}-{12755-12760}, ISSN: 0027-8424The 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.
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Journal articleGrima R, Yaliraki SN, Barahona M, 2010,
Crowding-Induced Anisotropic Transport Modulates Reaction Kinetics in Nanoscale Porous Media
, JOURNAL OF PHYSICAL CHEMISTRY B, Vol: 114, Pages: {5380-5385}-{5380-5385}, ISSN: 1520-6106We quantify the emergence of persistent anisotropy in the diffusion of spherical tracer particles through a nanoscale porous medium composed of a uniform distribution of purely symmetric crowding particles. We focus on the interior of a biological cell as an example of such a medium and find that diffusion is highly directional for distances comparable to the size of some organelles. We use a geometrical procedure that avoids the standard orientational averaging to quantify the anisotropy of diffusive paths and show that the point source distributions are predominantly of prolate ellipsoidal shape as a result of local volume exclusion. This geometrical symmetry breaking strongly skews the distribution of kinetic rates of diffusion-limited reactions toward small values, leading to the result that, for short to intermediate times, almost 80% of the rates measured in an ensemble of heterogeneous media are smaller than the expected rate in an ideal homogeneous medium of similar excluded volume fraction. This crowding-induced modulation may have implications for our understanding and measurement of diffusion-controlled intracellular reaction kinetics and for experimental nanotechnology applications, such as nanoparticle-based bioimaging and drug delivery, where diffusion plays an important role.
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Journal articleAnastassiou CA, Montgomery SM, Barahona M, et al., 2010,
The Effect of Spatially Inhomogeneous Extracellular Electric Fields on Neurons
, JOURNAL OF NEUROSCIENCE, Vol: 30, Pages: {1925-1936}-{1925-1936}, ISSN: 0270-6474The 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.
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