Similar impact of topological and dynamic noise on complex patterns

Marr, Carsten; Hütt, Marc‐Thorsten

doi:10.1016/j.physleta.2005.08.096

Cited by 30 publications

(25 citation statements)

References 15 publications

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“…On the contrary, for chains of regularly connected nodes, a topological perturbation in general leads to a decreasing dynamic complexity. In a previous study [40], we showed this behaviour for synchronous and asynchronous update schemes with different observables. Moreover we showed, that both topological perturbation (i.e., the rewiring of links) and dynamic noise increase the regularizing capacity of an initially undisturbed cellular automaton.…”

Section: Resultssupporting

confidence: 57%

Regularizing capacity of metabolic networks

2007

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Despite their topological complexity almost all functional properties of metabolic networks can be derived from steady-state dynamics. Indeed, many theoretical investigations (like flux-balance analysis) rely on extracting function from steady states. This leads to the interesting question, how metabolic networks avoid complex dynamics and maintain a steady-state behavior. Here, we expose metabolic network topologies to binary dynamics generated by simple local rules. We find that the networks' response is highly specific: Complex dynamics are systematically reduced on metabolic networks compared to randomized networks with identical degree sequences. Already small topological modifications substantially enhance the capacity of a network to host complex dynamic behavior and thus reduce its regularizing potential. This exceptionally pronounced regularization of dynamics encoded in the topology may explain, why steady-state behavior is ubiquitous in metabolism.

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Section: Resultssupporting

confidence: 57%

Regularizing capacity of metabolic networks

2007

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“…CA have been used in a vast number of investigations to explore the emergence of complex patterns from simple dynamic rules. Originally defined on regular lattices [27], they have also been studied on more complex topologies [15,16,36] and in noisy environments [32,37]. It should be noted that due to the diverse neighborhood sizes (compared to a regular 1D or 2D lattice) and the lack of ordering of neighbors, only a very small set of rules (from classical CA) can be plausibly transferred to general graphs.…”

Section: Cellular Automata On Graphsmentioning

confidence: 99%

“…Both measures have been successfully used for the quantification of pattern complexity in CA-on-graph dynamics (see Section 2.1) [15,16,32,35]. The Shannon entropy E S serves as a measure for the asymmetry between zeros and ones in the time series of each node and, when averaged over all nodes, of the "spatio"-temporal patterns.…”

Section: Entropy Measuresmentioning

confidence: 99%

“…The principal goal of discussing CA on graphs [15] is to explore the relationship between network architecture and dynamics from the perspective of pattern formation (and, more specifically, the Wolfram classes [27,28], as a way of characterizing observed dynamic behaviors). In a series of numerical studies we have employed this framework and related systems to analyze discrete dynamical processes on graphs [16,[32][33][34]. Furthermore, the concept of probing real networks with binary dynamics has been formulated for assessing the regularizing capacity of networks and, conversely, its ability to display complex dynamics [16].…”

Section: Introductionmentioning

confidence: 99%

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Cellular Automata on Graphs: Topological Properties of ER Graphs Evolved towards Low-Entropy Dynamics

Marr

Hütt

2012

Entropy

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Cellular automata (CA) are a remarkably efficient tool for exploring general properties of complex systems and spatiotemporal patterns arising from local rules. Totalistic cellular automata, where the update rules depend only on the density of neighboring states, are at the same time a versatile tool for exploring dynamical processes on graphs. Here we briefly review our previous results on cellular automata on graphs, emphasizing some systematic relationships between network architecture and dynamics identified in this way. We then extend the investigation towards graphs obtained in a simulated-evolution procedure, starting from Erdős-Rényi (ER) graphs and selecting for low entropies of the CA dynamics. Our key result is a strong association of low Shannon entropies with a broadening of the graph's degree distribution.

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“…On graphs, this classification inevitably fails because of a lacking natural node order. Instead, we apply two entropy-like measures, the Shannon entropy S and the word entropy W , which we have previously shown to provide a feasible framework for the quantification of pattern complexity [10,11,12]. The Shannon entropy S serves as a measure for the homogeneity of the spatio-temporal pattern, by averaging over all nodes: We compare 10 3 random initial conditions on the regular architecture with 10 3 samples of a randomized graph, where the number of incoming and outgoing links of every node is preserved and kept to d = 2, but the link architecture has been randomized [13].…”

mentioning

confidence: 99%

Outer-totalistic cellular automata on graphs

Marr

Hütt

2009

Physics Letters A

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We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.Introduction-Cellular automata (CA) on graphs in principle provide the possibility to monitor systematic changes of dynamics under variation of network topology. In practice, however, unambiguously studying the relation between topology and dynamics with CA is conceptually difficult, since changes in topology inevitably induce changes in the rule space. Proposed by von Neumann [1] as a model system for biological selfreproduction, a surge of research activity from the 80's onwards [2] established them as the standard tool of complex systems theory and spatio-temporal pattern formation [3] on regular grids. Another discrete and binary modeling approach for complex biological systems are random Boolean networks (RBNs), introduced by Kauffman [4]. While the CA framework introduces one rule for all regularly ordered cells with bi-directional links, the original RBNs consist of randomly and directionally linked nodes with individual rules. Here, we present a formalism that generalizes CA to arbitrary architectures. It allows (i) the establishment of a general correspondence between CA and isotropic RBNs and (ii) the comparison of the potential of different topologies to generate complex dynamics. As applications we examine the topological sensitivity of elementary CA, monitor the number of complex rules of CA under increasing neighborhood size, and compare the dynamic potential of scale-free graphs and representations of metabolism as substrate graphs.

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Similar impact of topological and dynamic noise on complex patterns

Cited by 30 publications

References 15 publications

Regularizing capacity of metabolic networks

Regularizing capacity of metabolic networks

Cellular Automata on Graphs: Topological Properties of ER Graphs Evolved towards Low-Entropy Dynamics

Outer-totalistic cellular automata on graphs

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