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.
The structure of complex networks, and the dynamics of the internal states of the nodes in these networks, are two fundamental issues in the study of network science @cite_54 @cite_17 .
@cite_54 @cite_17
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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. The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system’s entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network’s degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes. Control theory can be used to steer engineered and natural systems towards a desired state, but a framework to control complex self-organized systems is lacking. Can such networks be controlled? Albert-Laszlo Barabasi and colleagues tackle this question and arrive at precise mathematical answers that amount to 'yes, up to a point'. They develop analytical tools to study the controllability of an arbitrary complex directed network using both model and real systems, ranging from regulatory, neural and metabolic pathways in living organisms to food webs, cell-phone movements and social interactions. They identify the minimum set of driver nodes whose time-dependent control can guide the system's entire dynamics ( http: go.nature.com wd9Ek2 ). Surprisingly, these are not usually located at the network hubs.