We analyze a model of interacting agents (e.g. prebiotic chemical species) which are represended by nodes of a network, whereas their interactions are mapped onto directed links between these nodes. On a fast time scale, each agent follows an eigendynamics based on catalytic support from other nodes, whereas on a much slower time scale the network evolves through selection and mutation of its nodes-agent. In the first part of the paper, we explain the dynamics of the model by means of characteristic snapshots of the network evolution and confirm earlier findings on crashes an recoveries in the network structure. In the second part, we focus on the aggregate behavior of the network dynamics. We show that the disruptions in the network structure are smoothed out, so that the average evolution can be described by a growth regime followed by a saturation regime, without an initial random regime. For the saturation regime, we obtain a logarithmic scaling between the average connectivity per node $\mean{l}_{s}$ and a parameter $m$, describing the average incoming connectivity, which is independent of the system size $N$.
Deep Dive into Aggregate Dynamics in an Evolutionary Network Model.
We analyze a model of interacting agents (e.g. prebiotic chemical species) which are represended by nodes of a network, whereas their interactions are mapped onto directed links between these nodes. On a fast time scale, each agent follows an eigendynamics based on catalytic support from other nodes, whereas on a much slower time scale the network evolves through selection and mutation of its nodes-agent. In the first part of the paper, we explain the dynamics of the model by means of characteristic snapshots of the network evolution and confirm earlier findings on crashes an recoveries in the network structure. In the second part, we focus on the aggregate behavior of the network dynamics. We show that the disruptions in the network structure are smoothed out, so that the average evolution can be described by a growth regime followed by a saturation regime, without an initial random regime. For the saturation regime, we obtain a logarithmic scaling between the average connectivity per
Many evolutionary processes in physical, biological or economic systems involve elements of selfreproduction and catalytic interactions. In his pioneering work, Eigen [3] pointet out their relevance for the prebiotic evolution of macromolecules, which lead to the theory of the hypercycle (see also [4]). The hypercycle can be seen as a paragon of a network of cooperating agents [7] (e.g. chemical or biological species), which counterbalances the effect of aggressive self-replication. While the latter one just leads to the survival of only one species -"survival of the fittest" -the dependence on catalytic interaction with other species also ensures the survival of the others and, hence, a coexistence of agents with very different "fitness" levels.
Recently, the hypercycle concept has been investigated in a modified setting, which combines the original idea of catalytic interactions with an external dynamics of the network representing the interaction structure. Inspired by earlier work [5,26] Jain and Krishna [9] have focussed on the emergence of so-called autocatalytic sets (ACS) among agents, which do not self-replicate individually, but only replicate by means of the help of others. An ACS is then a cooperative structure, where different agents interact in such a way that the links representing these interactions form a closed cycle in terms of the network structure. Once an ACS appears, it boosts the replication of the agents involved, which leads to a larger growth or “output” of those agents involved in the ACS. It further allows other agents not directly part of the ACS but only linked to it, to still benefit from it as freeloaders.
Because such a catalytic replication dynamics eventually leads to a stationary state, Jain and Krishna [9] have added a disturbance of the interaction network in terms of a so-called “extremal dynamics” [1]. There, the least performing agent, i.e. the one with the lowest output, is -together with its links to other agents -replaced by a new agent that is linked to the existing interaction network in a random way. This network dynamics occurs on a much slower time scale compared to the agent dynamics itself. It ensures (i) that the dynamics of the system of agents does not get stuck in an equilibrium state, and (ii) may allow for “evolutionary” scenarios towards a better performance of the whole system.
Our work, discussed in the following, is based on the model of Jain and Krishna (JK) described above (see also [10]). In Chapter 2 we explain the dynamics and our numerical implementation of the JK model in more detail. In Chapter 3 we reproduce some important features of the model behavior, such as the emergence of the ACS and the crashes and recoveries in the network structure, by means of computer simulations that elucidate the network evolution. In Chapter 4 we extend our investigations to the aggregate behavior of the system, which to our knowledge was not investigated before. In particular, we show that the crashes and recoveries in single network realizations are smoothed out, so that the average evolution can be described by a saturation dynamics. We further obtain, by means of computer simulations, a logarithmic scaling function for the average connectivity per node dependent on the average incoming connectivity (which is a measure for the catalytic interaction). In Chapter 5 we summarize these findings and point to further interesting extensions of the model. In particular, we already mention the relation to recent network models for social and economic applications. [22,8,14,12,18] 2 The Model
The model discussed in the following was originally developed in the context of the “origin of life problem”: the observation that something as structurally complex as a living cell was able to form, parting from a random mix of chemical components in a prebiotic “broth” [3,26] on earth four billion years ago.
For a modelling approach, we consider a set of N prebiotic chemical species, each of them characterized by a population y i ≥ 0 (i = 1, …, N). The dynamics of the variables y i shall be governed by the following equation:
φ is assumed to be a constant dilution flux, resulting e.g. from a natural movement of raw material out of the system (say through flood or tides). The c ij are the kinetic coefficients that describe the replication of species i resulting from binary interactions with other species j. For simplicity, only c ij ∈ {0, 1} is assumed. c ij = 1 represents a growth process of species i due to the presence of species j that acts as a catalysor only. Negative values of c ij would indicate inhibitory processes that are neglected here. Further, self-replicating species are not allowed, which means c ii = 0 for all i.
In a first approximation φ can be set to zero. This results in a linear dynamical system of coupled first-order differential equations in the populations y i . In vector notation this reads:
where C is the matrix containing all kinetic co
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
