Vertex routing dynamics for a N = 4 complete graph (a) A realization of the routing tables. Routing through the first vertex follows , with all other vanishing. There are three cyclic attractors, namely (123), (243), and (1342). (b) Enumeration of all N(N – 1) = 12 directed edges, the phase-space elements. (c) The corresponding phase-space graph. (d) The same realization of the routing table as in (a), now in terms of the phase-space graph.
Random walks through configuration space for the Markovian model (left) and for the vertex routing model (right). In order to find an attractor independent of the size of their basins of attraction (light color) one needs to close the path at the respective starting points. The probability to find a given attractor is, on the other side, proportional to the size of its basin of attraction for stochastic ‘on the fly’ sampling of phase space.
The cycle length distributions , rescaled by , for the vertex routing model. The dashed line, 2/ L, represents the large- N and small- L limiting behavior. In the inset two quantities are plotted as a function of the phase space volume . The average number of cycles (see Eq. (9) , filled blue circles, log-linear plot) and the expected total cycle length (see Eq. (10) , green filled diamonds, log-log plot). Also included are fits using (red dashed line), with a = –0.345(3) and b = 0.4988(2), andusing (black dashed line) with and . The coefficient of determination is in both cases, within the numerical precision.
Log-log plot, as a function of the phase space volume , of the mean cycle lengths , see Eq. (11) , for the vertex routing with quenched dynamics ( , blue circles) and the vertex routing with on the fly dynamics ( , green diamonds). The dotted and dashed lines are fits using and , respectively, with a = 8.1(8), b = 2.6035(9), c = –69(9), and . The coefficient of determination is in both cases, within the numerical precision.
Scaling relations, as a function of the number of vertices N, for the number of cycles and for the mean of the cycle length distribution, respectively, for vertex routing (v) and the Markovian (m) model. The routing table distribution is either quenched (exact result) or generated on the fly, as it corresponds to a stochastic sampling of phase space. Only relative quantities can be evaluated for on the fly dynamics.
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