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Universality in Network Dynamics Baruch Barzel1,2 and Albert-László Barabási1,2,3*

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Universality in Network Dynamics Baruch Barzel1,2 and Albert-László Barabási1,2,3* ARTICLES PUBLISHED ONLINE: 8 SEPTEMBER 2013 | DOI: 10.1038/NPHYS2741 Universality in network dynamics Baruch Barzel1,2 and Albert-László Barabási1,2,3* Despite significant advances in characterizing the structural properties of complex networks, a mathematical framework that uncovers the universal properties of the interplay between the topology and the dynamics of complex systems continues to elude us. Here we develop a self-consistent theory of dynamical perturbations in complex systems, allowing us to systematically separate the contribution of the network topology and dynamics. The formalism covers a broad range of steady-state dynamical processes and offers testable predictions regarding the system’s response to perturbations and the development of correlations. It predicts several distinct universality classes whose characteristics can be derived directly from the continuum equation governing the system’s dynamics and which are validated on several canonical network-based dynamical systems, from biochemical dynamics to epidemic spreading. Finally, we collect experimental data pertaining to social and biological systems, demonstrating that we can accurately uncover their universality class even in the absence of an appropriate continuum theory that governs the system’s dynamics. espite the profound diversity in the scale and purpose of the probability of infection23–25; biochemical dynamics (B), in 26–29 networks observed in nature and technology, their topology which xi represents the concentration of a reactant ; birth– 30–32 Dshares several highly reproducible and often universal death processes (BD), in which xi represents the population 1–8 characteristics : many real networks exhibit the small-world at site i; and regulatory dynamics (R), in which xi is the property9, are scale-free10, develop distinct community structure11, expression level of a gene33,34. and show degree correlations12,13. Yet, when it comes to the The traditional probing of the dynamics of a complex system is dynamical processes that take place on these networks, diversity achieved through perturbation experiments, which explore changes 14–16 wins over universality . To be sure, advances in our under- in the activity xi of node i in response to changes induced in the standing of synchronization17,18, spreading processes19–21 or spectral activity of node j. Hence, we focus on the system's linear response by 22 phenomena have offered important clues on the interplay between inducing a permanent perturbation dxj on the steady-state activity network topology and network dynamics. We continue to lack, xj and following the subsequent changes in all xi through the however, a general predictive framework that can treat a broad correlation matrix28 (Supplementary Section SII) range of dynamical models using a unified theoretical toolbox. dxi=xi Indeed, at present, each network-based dynamical process, from Gij D reaction dynamics in cellular metabolism to the spread of viruses dxj =xj in social networks, is studied on its own terms, requiring its In biology Gij represents the impact of a perturbed gene j on dedicated analytical formalism and numerical tools. This diversity a target gene i; in social systems Gij captures the influence of of behaviour raises a fundamental question: are there common an individual j on i. There is ample empirical evidence from patterns in the dynamics of various complex systems? Alternatively, gene expression35–39 to metabolism40 and neuronal systems41 that could the present diversity of modelling platforms and dynamical the distribution of pairwise node–node correlations, or P(Gij ), is characteristics reflect an inherent and ultimately unbridgeable gulf fat-tailed, a phenomenon that lacks quantitative explanation. Our between different dynamical systems? measurements support this: we obtained Gij for the four dynamical We illustrate the depth of this problem by focusing on the systems described in Table1 , in each case finding that P(G) ∼ G−ν dynamics of a system with N components (nodes), where each node (Fig.1 a1–a4 and Supplementary Figs S5a1–a3). We find systematic i is characterized by an activity xi(t), following differences in ν, however: for B and BD ν D 2 and for R and Eν D 3=2. We also find that the distribution P(G) is independent N dxi X of the nature of the underlying network (scale-free, Erdős–Rényi D W (x (t))C A Q(x (t);x (t)) (1) i ij i j or networks provided by experimental data), suggesting that ν is dt jD1 determined only by the dynamical laws that govern these systems. providing a rather general deterministic description of systems To obtain a more detailed understanding of a system's response governed by pairwise interactions. The first term on the right-hand to perturbations, we also explored several other frequently pursued side of equation( 1) describes the self-dynamics of xi, accounting for dynamical measures. processes such as influx, degradation or reproduction. The second captures the interactions of i with its neighbours, in which Aij is the Impact and stability adjacency matrix and Q(xi;xj ) describes the dynamical mechanism We define the impact of i as governing the pairwise interactions. With the appropriate choice of the nonlinear W (xi) and Q(xi;xj ), equation( 1) can be mapped N D X T exactly into several dynamical models explored in the literature Ii Aij Gij (2) (Table1 ), such as epidemic processes (E), where xi represents jD1 1Center for Complex Network Research and Departments of Physics, Computer Science and Biology, Northeastern University, Boston, Massachusetts 02115, USA, 2Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Harvard Medical School, Boston, Massachusetts 02115, USA, 3Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA. *e-mail: [email protected] NATURE PHYSICS j VOL 9 j OCTOBER 2013 j www.nature.com/naturephysics 673 © 2013 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2741 Table 1 j Dynamical models. Dynamics Model Rate equation W(xi) Q(xi;xj) f(x) g(x) δ ' β ν ! N dxi X Rx B MAK D F −Bx − A Rx x F −Bx −Rx x x 0 0 0 2 0 dt i ij i j i i j F −Bx jD1 N dxi X R 3 3 BD PD D −Bxb C A Rxa −Bxb Rxa xa 0 0 2 dt i ij j i j b 2 2 jD1 Bx N h h h dxi X xj xj R x 3 1 R MM D −Bx C A R −Bx R 0 0 1 dt i ij C h i C h Bx C h 2 2 jD1 1 xj 1 xj 1 x N dxi X R(1−x) 3 E SIS D −Bx C A R(1−x )x −Bx R(1−x )x x 1 1 1 1 dt i ij i j i i j Bx 2 jD1 Summary of four frequently explored network-based dynamical models analysed in the paper. Biochemical dynamics (B): the dynamics of protein–protein interactions is captured by mass-action kinetics26–29 (MAK). In the model, proteins are produced at rate F, degraded at rate B and generate hetero-dimers at rate R (Supplementary Section SVII.C, where we also account for the hetero-dimer dissociation). Birth–death processes (BD): this equation, emerging in queuing theory32, population dynamics30,31 (PD) and biology26, describes the population density at site i. The first term describes the local population dynamics and the second term describes the coupling between adjacent sites. Here we set b D 2, representing pairwise depletion, and a D 1, describing a linear flow from the sites neighbouring i (Supplementary Section SVII.D, where we solve for a general choice of a and b). Regulatory dynamics (R): the dynamics of gene regulation is captured by the Michaelis–Menten (MM) equation33,34. Here, the parameter h is the Hill coefficient, which we set to h D 1 (Supplementary Section SVII.B, where we solve for general h). Epidemic dynamics (E): the spread of infectious 23–25 diseases/ideas is captured by the susceptible–infected–susceptible (SIS) model (Supplementary Section SVII.A). For each of the four models, the table lists the associated dynamical functions W(xi) and Q(xi;xj) in equation (1) and f(x) and g(x) as defined in equation (5), the exponents δ, ' and β determined from the Laurent expansions in equations (6) and (7), and the exponents ν and ! in equations (14) and (16) respectively, derived from δ, ' and β. capturing the average response of the neighbourhood of i to the changed following a genetic perturbation or all individuals who perturbation of i. Similarly, we define the stability of i as adopt an innovation. The distribution of cascade sizes induced by perturbations, P(C), is frequently measured in social42–44, 1 technological45,46 and biological41,47 systems, finding that P(C) is Si D (3) PN often fat-tailed, an observation whose origins remain unclear. Our Aij Gij jD1 simulations (Fig.1 e1–e4 and Supplementary Fig. S5e1–e3) indicate that P(C) depends on the interplay between the topology and in which the denominator captures the magnitude of the response dynamics: for BD, R and E, P(C) is driven by P(k); hence, these of i to individual perturbations of the nearest neighbours of i. If i systems develop heterogeneous cascades with a fat-tailed P(C) on a responds strongly to neighbouring perturbations, then Si is small, scale-free network but a bounded P(C) on a random network. Pro- indicating that node i is unstable. Hence, Ii captures the influence of tein dynamics (B), however, has uniform cascades, characterized by node i on its neighbourhood, and Si captures the inverse process, the a bounded P(C), independent of the network topology.
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    (PHUJLQJ5HVHDUFKLQWKH $QDO\VLVDQG0RGHOLQJ RI*HQH5HJXODWRU\ 1HWZRUNV ,YDQ9,YDQRY 7H[DV$ 08QLYHUVLW\86$ ;LDRQLQJ4LDQ 7H[DV$ 08QLYHUVLW\86$ 5DQDGLS3DO 7H[DV7HFK8QLYHUVLW\86$ $YROXPHLQWKH$GYDQFHVLQ 0HGLFDO7HFKQRORJLHVDQG &OLQLFDO3UDFWLFH $07&3 %RRN6HULHV Published in the United States of America by Medical Information Science Reference (an imprint of IGI Global) 701 E. Chocolate Avenue Hershey PA 17033 Tel: 717-533-8845 Fax: 717-533-8661 E-mail: [email protected] Web site: http://www.igi-global.com Copyright © 2016 by IGI Global. All rights reserved. No part of this publication may be reproduced, stored or distributed in any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark. Library of Congress Cataloging-in-Publication Data Names: Ivanov, Ivan V., 1962- | Qian, Xiaoning, 1975- | Pal, Ranadip, 1980- Title: Emerging research in the analysis and modeling of gene regulatory networks / Ivan V. Ivanov, Xiaoning Qian, and Ranadip Pal, editors. Description: Hershey, PA : Medical Information Science Reference, [2016] | Includes bibliographical references and index. Identifiers: LCCN 2016005966| ISBN 9781522503538 (hardcover) | ISBN 9781522503545 (ebook) Subjects: LCSH: Genetic regulation--Computer simulation. | Gene regulatory networks. Classification: LCC QH450 .E44 2016 | DDC 572.8/65--dc23 LC record available at https://lccn. loc.gov/2016005966 This book is published in the IGI Global book series Advances in Medical Technologies and Clinical Practice (AMTCP) (ISSN: 2327-9354; eISSN: 2327-9370) British Cataloguing in Publication Data A Cataloguing in Publication record for this book is available from the British Library.
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