^{1}, Claudius Gros

^{1}and André Schuelein

^{2}

### Abstract

Conserved dynamical systems are generally considered to be critical. We study a class of critical routing models, equivalent to random maps, which can be solved rigorously in the thermodynamic limit. The information flow is conserved for these routing models and governed by cyclic attractors. We consider two classes of information flow, Markovian routing without memory and vertex routing involving a one-step routing memory. Investigating the respective cycle length distributions for complete graphs, we find log corrections to power-law scaling for the mean cycle length, as a function of the number of vertices, and a sub-polynomial growth for the overall number of cycles. When observing experimentally a real-world dynamical system one normally samples stochastically its phase space. The number and the length of the attractors are then weighted by the size of their respective basins of attraction. This situation is equivalent, for theory studies, to “on the fly” generation of the dynamical transition probabilities. For the case of vertex routing models, we find in this case power law scaling for the weighted average length of attractors, for both conserved routing models. These results show that the critical dynamical systems are generically not scale-invariant but may show power-law scaling when sampled stochastically. It is hence important to distinguish between intrinsic properties of a critical dynamical system and its behavior that one would observe when randomly probing its phase space.

Power law scaling is observed in many real-world phenomena, like neural avalanches in the brain. In statistical physics, all critical systems, at the point of a second-order phase transition, show power law scaling. Power law scaling is hence commonly attributed to criticality, but it is an open question to which extend this relation is satisfied for complex dynamical systems. There is, in addition, a difference between the distribution an observer may be able to sample and the exact properties of the underlying dynamical system. An observer will sample in general the number and the size of attractors as weighted by size of their respective basins of attraction. Here, we investigate the critical models for information routing and show that the number and the length of attractors does not obey power law scaling, while, on the other hand, an external observer, sampling the weighted distribution, would find power law scaling. Hence when drawing conclusions from experimentally observed power law scaling one needs to take into account the implicitly employed sampling procedures.

I. INTRODUCTION

II. MODELS

III. CYCLE LENGTH DISTRIBUTION

IV. RESULTS

A. Quenched dynamics

B. Stochastic sampling of phase space

V. DISCUSSION

### Key Topics

- Phase space methods
- 42.0
- Attractors
- 19.0
- Numerical modeling
- 16.0
- Markov processes
- 14.0
- Networks
- 7.0

## Figures

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.

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.

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.

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.

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.

## Tables

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.

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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