The dynamics of an agreement protocol interacting with a disagreement process over a common random network is considered. The model can represent the spreading of true and false information over a communication network, the propagation of faults in a large-scale control system, or the development of trust and mistrust in a society. At each time instance and with a given probability, a pair of network nodes are selected to interact. At random each of the nodes then updates its state towards the state of the other node (attraction), away from the other node (repulsion), or sticks to its current state (neglect). Agreement convergence and disagreement divergence results are obtained for various strengths of the updates for both symmetric and asymmetric update rules. Impossibility theorems show that a specific level of attraction is required for almost sure asymptotic agreement and a specific level of repulsion is required for almost sure asymptotic disagreement. A series of sufficient and or necessary conditions are then established for agreement convergence or disagreement divergence. In particular, under symmetric updates, a critical convergence measure in the attraction and repulsion update strength is found, in the sense that the asymptotic property of the network state evolution transits from agreement convergence to disagreement divergence when this measure goes from negative to positive. The result can be interpreted as a tight bound on how much bad action needs to be injected in a dynamic network in order to consistently steer its overall behavior away from consensus.
Probabilistic models for networks such as random graphs, provide an important and convenient means for modeling large-scale systems, and have found numerous applications in various fields of science. The classical Erd " o s--R ' e nyi model, in which each edge exists randomly and independently of others with a given probability, was studied in @cite_1 . The degree distribution of the Erd " o s--R ' e nyi graph is asymptotically Poisson. Generalized models were proposed in @cite_53 and @cite_20 , for which the degree distribution satisfies certain power law that better matches the properties of real-life networks such as the Internet. A detailed introduction to the structure of random networks can be found in @cite_57 @cite_54 .
@cite_53 @cite_54 @cite_1 @cite_57 @cite_20
2112090702 2147824439 2908457301 1627599966 2008620264
Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices. A process for finishing keratinous material, especially rendering it shrink-resistant or imparting to it durably pressed effects, comprises 1. TREATING THE MATERIAL WITH A POLYTHIOL ESTER OF THE FORMULA [OH]q(s) ¦ R1-(CO)rO(CO)sR2SH]p ¦ [COOH]q(r) WHERE R1 represents an aliphatic or araliphatic hydrocarbon radical of at least 2 carbon atoms, which may contain not more than one ether oxygen atom, R2 represents a hydrocarbon radical, P IS AN INTEGER OF FROM 2 TO 6, Q IS ZERO OR A POSITIVE INTEGER OF AT MOST 3, SUCH THAT (P + Q) IS AT MOST 6, AND R AND S EACH REPRESENT ZERO OR 1 BUT ARE NOT THE SAME, AND 2. CURING THE POLYTHIOL ESTER ON THE MATERIAL BY MEANS OF A POLYENE CONTAINING, PER AVERAGE MOLECULE, AT LEAST TWO ETHYLENIC DOUBLE BONDS EACH beta TO AN OXYGEN, NITROGEN, OR SULFUR ATOM, THE SUM OF SUCH ETHYLENIC DOUBLE BONDS IN THE POLYENE AND OF THE MERCAPTAN GROUPS IN THE POLYTHIOL ESTER BEING MORE THAN 4 AND THE COMBINED WEIGHT OF THE POLYENE AND THE POLYTHIOL ESTER BEING FROM 0.5 TO 15 BY WEIGHT OF THE KERATINOUS MATERIAL TREATED. Only recently did mankind realise that it resides in a world of networks. The Internet and World Wide Web are changing our life. Our physical existence is based on various biological networks. We have recently learned that the term "network" turns out to be a central notion in our time, and the consequent explosion of interest in networks is a social and cultural phenomenon. The principles of the complex organization and evolution of networks, natural and artificial, are the topic of this book, which is written by physicists and is addressed to all involved researchers and students. The aim of the text is to understand networks and the basic principles of their structural organization and evolution. The ideas are presented in a clear and a pedagogical way, with minimal mathematics, so even students without a deep knowledge of mathematics and statistical physics will be able to rely on this as a reference. Special attention is given to real networks, both natural and artifical. Collected empirical data and numerous real applications of existing theories are discussed in detail, as well as the topical problems of communication networks. Available in OSO: http: www.oxfordscholarship.com oso public content physics 9780198515906 toc.html Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.