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Emergence of growth, complexity threshold and structural tendencies during adaptive evolution of system
 EPNACS in ECCS’07
, 2007
"... Summary. Adaptive evolution of a functioning network (e.g. Kauffman network) may force growth of this network. We consider random addition and removing of nodes in a wide range of networks types, including scalefree BA networks. Growth of a network leads to reaching a certain complexity threshold w ..."
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Summary. Adaptive evolution of a functioning network (e.g. Kauffman network) may force growth of this network. We consider random addition and removing of nodes in a wide range of networks types, including scalefree BA networks. Growth of a network leads to reaching a certain complexity threshold which appears suddenly as phase transition to chaos. It can be observed in the distribution of damage size of system outputs after damage spreading initialised by a small modification of the system. Over this threshold the adaptive condition as condition of network growth is a source of structural tendencies which were observed e.g. in developmental biology as regularities of ontogeny evolution, but still were not explained. Using specific algorithm the simulation has shown mechanisms of these tendencies. Our model describes living and human designed systems. We remark that the widely used two equally probable variants of signal (Boolean network, Ising model, spin glasses) are in many cases (especially outside physics) not adequate. They incorrectly lead to extreme structural stability instead of chaos. Therefore more than two equally probable variants of signal should be used. We1 define fitness on systems outputs, which allows us to omit the local extremes and the Kauffman’s complexity catastrophe. 1
10 Complexity Threshold for Functioning Directed Networks in Damage Size Distribution
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, 801
"... We describe systems using Kauffman and similar networks. They are directed functioning networks consisting of finite number of nodes with finite number of discrete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of ‘structural tendencies’. ..."
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We describe systems using Kauffman and similar networks. They are directed functioning networks consisting of finite number of nodes with finite number of discrete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of ‘structural tendencies’. Along the way we expand Kauffman networks, which were a synonym of Boolean networks, to more than two signal variants and we find a phenomenon during network growth which we interpret as ‘complexity threshold’. For simulation we define a simplified algorithm which allows us to omit the problem of periodic attractors. We estimate that living and human designed systems are chaotic (in Kauffman sense) which can be named complex. Such systems grow in adaptive evolution. These two simple assumptions lead to certain statistical effects i.e. structural tendencies observed in classic biology but still not explained and not investigated on theoretical way. E.g. terminal modifications or terminal predominance of additions where terminal means: near system outputs. We introduce more than two equally probable variants of signal, therefore our networks generally are not Boolean networks. They grow randomly by additions and removals of nodes imposed on Darwinian elimination. Fitness is defined on external outputs of system. During growth of the system we observe a phase transition to chaos (threshold of complexity) in damage spreading. Above this threshold we identify mechanisms of structural tendencies which we investigate in simulation for a few different networks types, including scalefree BA networks.