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By Noah E. Friedkin and Eugene C. Johnsen

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Extra resources for ATTITUDE CHANGE, AFFECT CONTROL, AND EXPECTATION STATES IN THE FORMATION OF INFLUENCE NETWORKS

Sample text

Ours is the simplest case, when the discontinuity is present in the initial condition. Indeed, for r > 0, the characteristics starting at r and -r have speed (1 - 2A) and (1 - 2p) respectively and meet at time t(r) = r/(A - p). Calling feu) = u(I - u) and using the conservation law of the equation it is not difficult to show that the discontinuity propagates at velocity v := (I(A) - f(p))/(A - p) = 1 - A - p. In Figure 2 time is going down. We have drown the characteristics starting at rand -r that go at velocity 1 - 2A and 1 - 2p respectively.

This means that a configuration 7J picked from the distribution Va can be constructed in the following way: at each site of 7Z put a particle with probability a. Do this independently for each site. Define 7J:(r) := 31 SHOCKS IN THE BURGERS EQUATION 17e-1t(c:-1r), where c:-1r is an abuse of notation for integer part of c-1r. 1). 1) where TxTJ is the configuration defined by TxTJ(Z) = TJ(z + x) and Txf(TJ) = f(TxTJ). Eq. 1) has been proven first for initial profiles uo(r) that have only one discontinuity by Rost [r] in the non increasing case and extended by Benassi and Fouque [bfl] and Andjel and Vares [av] to the non decreasing case and for more general jump probabilities.

Intuitively the coupling works in the following way. Particles at site x of the configurations "It and "I; use the same arrows, so that they try to jump at the same time. If the destination site x + 1 is empty in both configurations the jump is realized in both marginals, but if it is occupied for one of the marginals, then the jump is realized only for the other marginal. ~ •x • before the arrow '" x • after the arrow Figure 5. Coupling. In Figure 5 we see the configurations of "I and "I' before and after an arrow.

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