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3. Finally, calculate [MATH] according to formula ( ). The calculation is applicable for most EAs on both continuous and discrete optimization. We take an example to illustrate the average convergence rate. Consider the problem of minimizing Ackley’s function: |
[EQUATION] where [MATH] . The optimum is [MATH] and [MATH] We compare the Multi-grid EA (MEA) he2009mixed with the Fast Evolutionary Programming (FEP) yao1999evolutionary under the same experiment setting (where [MATH] is 30 and population size is 100). Run the two EAs for 1500 generations and 100 times. Calculate [MAT... |
The average convergence rate is different from the progress rate such as [MATH] or logarithmic rate [MATH] used in salomon1998evolutionary . The progress rate measures the fitness change; but the convergence rate measures the rate of the fitness change. We demonstrate this difference by an example. Let [MATH] . In term... |
III Analysis for Discrete Optimization Looking at Fig. again, two questions may be raised: what is the lower bound or upper bound on [MATH] ? Does [MATH] converge or not? For discrete optimization, a theoretical answer is provided to these questions in this section. For continuous optimization, its analysis is left for... |
In the rest of the paper, we analyze EAs for discrete optimization and assume that their genetic operators do not change with time. Such an EA can be modeled by a homogeneous Markov chain he2003towards with transition probabilities |
[MATH] where populations [MATH] denote states of [MATH] and [MATH] denotes the set of populations (called the population space ). Let [MATH] denote the transition matrix with entries [MATH] |
A population is called optimal if it includes an optimal solution; otherwise called non-optimal . Let [MATH] denote the set of optimal populations, and [MATH] . Because of the stopping criterion, the optimal set is always absorbing, [EQUATION] Transition matrix [MATH] can be split into four parts: [EQUATION] |
# Source: arxiv 1505.04518 # Title: Emergence-focused design in complex system simulation # Sections: all # Downloaded: 2026-03-03T02:00:38.061215+00:00 |
Emergence-focused design in complex system simulation Abstract Emergence is a phenomenon taken for granted in science but also still not well understood. We have developed a model of artificial genetic evolution intended to allow for emergence on genetic, population and social levels. We present the details of the curr... |
Introduction Many of life’s mysteries are systems resulting from the interactions between heterogeneous components in a lengthy flow of behavior, yet the results can be predicted due to linear interactions. A mechanical clock is a great example and Rube Goldberg perfected this idea in art with his famous machines (Wolf... |
Other mysteries of life are systems resulting from the interactions between homogeneous or heterogeneous components interacting such that the overall behavior is more than just the sum of its parts (Gleick,, 1987 ; Johnson,, 2002 . Such systems are called complex or non-linear. Herding is an example of a complex popula... |
Emergence, as it is currently understood, plays a privileged role in the current organization of science (Morowitz,, 2002 ; Ellis,, 2006 . As Fodor, ( 1974 famously argued, while we commonly think of psychology, biology, chemistry and the like to be reducible to physics (technically the respective subjects of these stu... |
Kim, ( 1992 helped bring some metaphsyical peace to reductionism by helping to explain how on the one hand everything is physics, but on the other hand no clear reduction exists from one level of science to others. The concept introduced was supervenience the idea that the higher order objects and processes were the fu... |
(Chalmers,, 2006 ). If this story is accurate, then the world of chemistry emerged out of physics, and the world of biology emerged out of chemistry sometime later, and now the worlds of psychology, culture, economics, politics, art, and many others have emerged out of biology. The objects of each level organize themse... |
Analytical science has made some progress attempting to handle emergence. This is especially true in the field of evolution. Many analytical models have been developed to predict evolutionary trajectories and their accuracies have been successfully verified in simulation (Hansen and Wagner,, 2001 ; Guillaume and Otto,,... |
Modern simulations of complex systems have led to many breakthroughs in understanding the mechanisms of emergence. However in most of these simulations the same simple methodology is used. In simulation the phenomena of study is treated as emergent, so the underlying phenomena is modeled as a system of interacting part... |
It is our desire to study emergence of complex adaptive systems at multiple levels and in particular we are interested in the co-evolution of culture and biology in humans and other organisms. We believe that one necessary next step in complex system simulation is to incorporate more complicated (heterogeneous, multi-l... |
For our purposes we want to study the emergence of culture. Commonly agent-based models choose to model agents in culture as simple linear systems ( Reynolds, ( 1987 ); Bonabeau, ( 2002 ); Macy and Willer, ( 2002 ). Our desire is to start at least one level lower so we begin with agents with behaviors determined by ric... |
Model Our model has incorporated several properties that enrich the dynamics of evolutionary simulation. First, our agents exist in a structured environment following a principle that complexity in an agent must reflect complexity in its environmental interaction. Our agents live in a random geometric network of foragi... |
Second, our agents have a structured genome following a principle that genetic dynamics emerge from the physical interaction of an actual genetic sequence. Each agent has a genome represented by a path of foraging attempts at sites in the network. |
Third, the genome must interact with the environment to express a phenotype following a principle of decoupling the phenotype from the genotype (a.k.a. ontogenetic development and learning). The path of foraging sites in the agent’s genome together with the current site of the agent determine the agent’s daily behavior... |
Fourth, the agent’s survival and reproductive fecundity are functions of their interaction with their environment following a principle of natural selection as opposed to artificial selection by a fitness function. A day’s foraging costs the agent energy and this energy is replenished by consuming the resources gathere... |
Each of these principles is intended to take a component of the evolutionary system that is often modeled as a linear system and allow it instead to have the potential for emergent dynamics. |
Environment A random geometric network is created by randomly generating [MATH] points on a plane and then connecting the close points (Penrose,, 2003 ; Antonioni et al.,, 2014 . For our network we generated points in the range [MATH] and connected points that had a Euclidean distance of at most [MATH] . Figure shows a... |
Each site in the network has a foraging task for gathering resources. We extend the model of Marriott and Chebib, ( 2014 . Each site will have a fixed resource reward for any agent that forages at the site. For typical settings our sites may have a reward of one, two or three resources. Every agent receives the same re... |
Foraging at a site will cost an agent energy. Energy in our model is temporal energy so corresponds to an amount of time spent on activities. The energy cost is determined by the degree of success of the agent’s foraging strategy at the foraging task. The foraging task is to find the rewards that are hidden in five pos... |
Moving in the environment also costs the agent energy. The cost is proportional to the length of the edge they travel to get to the new site. In our simulation agents are not allowed to forage at the same site twice in a row, though they may return if they forage at at least one other site first. This is because their ... |
The total energy that an agent may spend in a day cannot exceed its current energy reserves or the maximum daily energy expenditure. This means that agents with less than the maximum energy are free to use all of their energy to gather resources. Agents with more than the maximum energy can only spend up to this maximu... |
At the moment we only test static environments so these features and parameters do not change over the course of the simulation. However, because of the depleted resource sites the agents do experience some minor variation in their environment over the course of the day and between days. |
Agent The agents in our simulation have their behaviors determined by the interaction of their genomes with the environment. The genome of an agent is represented by a path of foraging attempts at sites in the network. This path in the network is a single unbroken path though it may and often does loop back on itself r... |
A seed agent for our simulation has a randomly generated genome. We build the path by simulating a random walker on the network starting at a random site (Lovász,, 1993 . At each site we must select a random foraging strategy for the agent to use. This means selecting an order in which the agent will search the hidden ... |
Once a seed agent is generated we select a gene from its genome at random and start the agent in the site indicated by that gene. Then the agent must select its actions for the day. An agent selects its daily activities by searching its genome forwards and backwards for a subpath beginning at its current site and that ... |
Reproduction An agent can reproduce if it has stored enough energy. How much energy is necessary is determined by what type of reproduction the agent engages in. First, the total cost of reproduction is equal to the daily maximum energy expenditure. This is because the parent agent(s) create a new agent with maximum en... |
Sexual reproduction is carried out between two agents. For this to occur they must both be spending time seeking a mate at the same foraging site for overlapping time periods during the day. This may seem a rare occurrence but due to forces of common descent and convergence the agents develop social organizations like ... |
When a new agent is created from reproduction, its new genome may mutate, recombine with itself, or recombine with another genome (in the case of sexual reproduction). The mutation rate is set at five percent but can be altered. In the simulation there are three kinds of random alterations of the genome. |
First, each gene has a mutation probability. Specifically, the foraging strategy in the gene (the permutation) is allowed to mutate. A permutation mutates by swapping two of its entries. This may change the energy cost of that gene. We consider each possible strategy at the same site a different allele of the same gene... |
Second, the path at either end of the genome may grow with some probability. When this occurs, a random walker is simulated as described for the seed agent above. The random walk begins at the site indicated by the last (or first) gene in the genome. The length of growth is bound by a constant (this is a simulation par... |
Third, the path at either end of the genome may shrink with some probability. This is done by selecting a length and an end at random and deleting that many genes from the path (the length of the deletion is bound by the same parameter as growth of a genome). |
Recombination, another type of mutation that is less random, also has three forms: cycle copy, cycle deletion, and sexual recombination. When an agent mutates it has equal chance of a random growth or deletion (as above) or a cycle copy or cycle deletion. That is, when a mutation occurs, one of a random growth, a rando... |
When recombination occurs all cycles in the genome are found (this can be done in a single pass over the genome). A cycle occurs when the path returns to a site it has been to before. Assuming at least one cycle exists (there are often many) we select one uniformly. If a cycle deletion has been selected we delete the c... |
During sexual reproduction we also have the likelihood of sexual recombination. The recombination mechanism first computes the maximum number of crossover positions between the two paths by computing the longest common subsequence (similar to synapsis). Every entry in the longest common subsequence is a site that the t... |
More formally the offspring genome is created first by selecting one of the two parents to begin copying from. We copy genes from this agent to the offspring agent until we reach the first crossover location. Whenever we reach a crossover location we flip a coin to determine which parent to read from next. We then copy... |
After carrying out sexual recombination the new genome has probabilities for mutation or recombination with itself as well. A new agent is born on the next day into the site where the parents conceived it. There is the chance that a mutation or recombination has lead to a deletion that left the agent with no strategies... |
Observations Our agent-based model results in very rich emergent genetic dynamics and social structure. This includes the evidence for genotype-phenotype divergence (e.g. phenotypic plasticity), non-coding regions of the genome (Hardison,, 2000 , pseudogenes (Mighell et al.,, 2000 , genetic drift (Hartl and Clark,, 199... |
Analytically, we can argue how these features have emerged in our model. For example, consider the divergence of the phenotype from the genotype. One dimension of divergence is phenotypic plasticity, the ability to alter the phenotype in response to environmental features. Two agents in our model that have an identical... |
Evidence for these behaviors can be gathered based on a few advantages of our model. First since genomes in our model are sequences we can use standard genome comparison methods like the Levenshtein edit-distance (Levenshtein,, 1966 to calculate mutation distance between agents. A relevant phenotype of our agents is th... |
We know that the regions of DNA serve different roles roughly grouped as coding and non-coding regions. Coding regions code for functional proteins whereas non-coding regions range from coding for other functional elements like RNA to a catch all “junk DNA” for regions that have no known function. Some non-coding seque... |
An important factor of the pseudogene sequences in our model is that they are neutral to selection but have potential functionality or even sub-functionality (Qian et al.,, 2010 ; Guillaume and Otto,, 2012 . Mutations in the pseudogene sequence are commonly neutral because these regions are not used. Assuming that muta... |
This drift occurs in two ways. First, the alleles in the pseudogene sequence will mutate (though these changes are not reflected in the phenotype and thus are not selected for or against). As a result there is no trend towards optimization in the alleles of the pseudogene sequence, whereas we do see such optimization i... |
The cycle copying recombination can play a different role when copying a region of the expressed gene sequence of the genome. All cycle copying recombination can be seen as a type of gene duplication because the duplication of what we are calling genes (we are copying the specific alleles in these cases). When this occ... |
Consider now the child agent with two such regions. Mutation that occurs to one will not affect the other. Thus if a negative allelic mutation occurs on one of the regions but not the other the agent is insulated from this mutation. Its phenotype is unaltered as it can use the non-mutated region. Being insulated from n... |
Extending our Model The genetic and population level dynamics we have encountered in our model, which may be among others we have not yet identified, are due to the commitment to emergent dynamics in as many aspects of our model as possible. For instance the genetic dynamics discussed here are largely due to evolving a... |
Too often steps are made to simplify the model computationally. This typically means reducing one or many of these genetic sub-systems to linear dynamics. This then removes possibilities for synergistic emergent dynamics among these systems. Even so, there are many aspects of our model that still rely on simple rules a... |
The sequential properties of paths in a network allow for natural mechanisms of mutation and recombination to be employed. These in turn lead to dynamics similar to those observed in nature and alow for the application of many biological concepts through analogy to our artificial agents. Our agents make use of a foragi... |
In the short term, we plan to explore in more details the nature of the genetic dynamics displayed by our model. The mechanisms of genetic drift and gene duplication deserve further study. At present we are developing means of detecting and tracking events of recombinant duplication. We are also currently developing me... |
The goal of our long term research is to understand the origins of social behavior and social learning in artificial and biological populations. We feel it is critical to these goals that the manner of social interaction in our models is not scripted as is common in multi-agent systems and agent based models in the soc... |
The next major step is to allow those emergent social behaviors to create a new domain of interaction that can lead to higher level emergence. We call this a cascade of emergence. Complex life and especially the social behavior of humans is due to such a cascade. To accomplish this we are developing coupled emergent sy... |
# Source: arxiv 1507.07403 # Title: Requirements for Open-Ended Evolution in Natural and Artificial Systems # Sections: all # Downloaded: 2026-03-03T01:55:45.771757+00:00 |
Requirements for Open-Ended Evolution in Natural and Artificial Systems Abstract Open-ended evolutionary dynamics remains an elusive goal for artificial evolutionary systems. Many ideas exist in the biological literature beyond the basic Darwinian requirements of variation, differential reproduction and inheritance. I ... |
Introduction If there is one lesson to be learned from the first 60 years of research into the evolution of digital organisms, it is that the classic Darwinian ingredients of variation differential reproduction and inheritance are not, in themselves, sufficient for producing open-ended dynamics in which new, surprising... |
Taylor et al., 2014 Most evolutionary artificial life systems tend to rather quickly reach a quasi-stable state beyond which no qualitatively new innovations are seen to appear Taylor, 2013 . None has displayed dynamics which might be regarded as the holy grail of artificial life, where the continued evolution of novel... |
Various artificial life researchers have started to look at different aspects of the biological world for the missing ingredients. At the same time, our understanding of processes important in biological evolution has been greatly supplemented by new research in many areas, including epigenetics Jablonka et al., 2005 ,... |
Comfort, 2015 , neutral evolutionary networks Wagner, 2011 facilitated variation Gerhart and Kirschner, 2007 , niche construction |
Odling-Smee et al., 2003 , and others. While these new research directions are exciting and promise new insights into the important ingredients of biological evolution, the underlying simplicity of the Darwinian picture of variation, differential reproduction and inheritance soon disappears in the panoply of new ideas.... |
In the following section, I suggest that there are five fundamental requirements for a system to exhibit open-ended evolution. I show how the various ideas mentioned above fit into this picture, discuss how they relate to past work in artificial life, and suggest various directions that are indicated for future researc... |
Requirements At a very general level, the following five features are necessary, and I claim sufficient, for a system to exhibit open-ended evolutionary dynamics: |
Robustly reproductive individuals. A medium allowing the possible existence of a practically unlimited diversity of individuals and interactions, at various levels of complexity. |
Individuals capable of producing more complex offspring. An evolutionary search space which typically offers mutational pathways from one viable individual to other viable (and potentially fitter) individuals. |
Drive for continued evolution. Each of these features is discussed below. 2.1 Robustly reproductive individuals The basic components of any evolutionary system are individual entities that can catalyse the production of (sometimes imperfect) copies of themselves. Successful individuals must be robust enough to survive ... |
While this may appear to be a fairly basic statement, the question of what are the appropriate ways to achieve robustness in artificial life systems has not often received the attention it deserves. Von Neumann’s self-reproducing automata von Neumann, 1966 , and other systems of self-reproduction in 2D cellular automat... |
Digital organism systems such as Tierra and Avida hard-wire robustness into the system by not granting individuals write-access to other parts of memory (except in the special case where some new memory has been allocated for reproduction). This was a critical design decision that allowed prolonged evolution to happen ... |
Ray, 1991 . However, by hard-wiring write protection into the system, programs in Tierra and Avida become relatively isolated from each other, with consequences for what kinds of interactions are possible. |
Biological organisms need to actively maintain their organisation against the disruptive pull of the second law of thermodynamics. Concerns of entropy increase are not immediately applicable to digital organisms, unless entropy is intentionally built into the digital physics of the system. |
If entropy was built into an artificial life system, it would mean that the digital organisms would have to concern themselves with self-maintenance, and that most structures would naturally decay without the need for arbitrary mechanisms like reaper queues. This would entail the organisation of digital organisms more ... |
An significant open question for artificial life research is understanding the importance of topics such as entropy and self-maintenance for open-ended evolution. |
2.2 A medium allowing the possible existence of a practically unlimited diversity of individuals and interactions, at various levels of complexity |
A clear requirement for open-ended evolution is that many different types of organism must be conceivable within the system. The medium in which the evolutionary process is unfolding must allow the possibility of a practically unlimited diversity of organism organisations, processes and interactions. |
Much previous work within artificial life has concentrated on the ability of organisms to evolve complex computational and information processing capabilities, such as the ability of digital organisms in Avida to solve logic functions Lenski et al., 2003 or the evolution of complex neural network-driven behaviour in sy... |
Channon, 2006 However, it is restrictive to only consider the evolution of information processing capabilities. Some of the most remarkable events in biological evolutionary history have involved the evolution of new ways of interacting with the environment via new sensors and effectors. The geochemical-physical medium... |
The need for complex environments for the production of interesting evolution in artificial life systems has been recognised right back to the earliest work in the area. Barricelli, 1963 spoke in terms of adding “toy bricks” to the environment to allow his digital organisms to evolve interesting behaviours. |
In addition, the major transitions in evolution identified by Maynard Smith and Szathmáry, 1995 involve changes in the organisation of individuals over evolutionary time. Hence, open-ended artificial life systems should allow the organisation of individual organisms to evolve as well. |
Many issues arise when designing complex virtual environments in which organisms can evolve to access and exploit that complexity for their own ends. These include questions such as whether the medium should have “messy” processes with side effects, to allow for the serendipity often apparent in biological evolution, a... |
2.3 Individuals capable of producing more complex offspring Beyond having a medium in which a wide variety of organism designs could possibly exist, in order for complex adaptations to |
evolve from simple progenitors, it must be possible for an individual (or multiple individuals) to produce offspring that are more complicated than their parent(s). |
There are (at least) two ways in which this may occur: A single individual is capable of producing an offspring of greater complexity than itself. |
Two or more individuals are jointly capable of producing an offspring of greater complexity than any one of its parents. The first solution is exactly the issue addressed by |
von Neumann, 1966 in his Theory of Self-Reproducing Automata . The fundamental requirement identified by von Neumann is that the inherited information-bearing structures must be involved in two distinct processes: (1) they are interpreted by the phenotype’s machinery as instructions to guide the construction of an indi... |
Seen in this general light, we can say that von Neumann’s requirements are satisfied by biological cells (in 3D), by his proposed self-reproducing cellular automata (in 2D), and by digital organisms such as those in Tierra (in 1D). Note, however, that in the case of Tierra, the interpretation machinery is hard-coded in... |
Biological examples of the second solution include horizontal gene transfer (HGT) and symbiogenesis. These processes are much less well explored in the artificial life literature, despite their significance in biological evolution and the fact that they provide a feasible complementary (or alternative) route to increas... |
2.4 Mutational pathways to other viable individuals For an open-ended evolutionary process, it is insufficient for individuals to have the theoretical capacity for producing more complicated offspring. The fitness landscape of the system must be such that there are often viable mutational pathways open to individuals, ... |
While this has been understood for a long time—e.g. Rensch, 1947 discussed the need for “improvements allowing further improvement”—the task of understanding the requirements for a fitness landscape to have this property is now a very active area of research. |
A wide variety of work can be seen as contributing to this topic, including Wagner, 2011 ’s work on evolutionary innovations and neutral networks, a wide range of work on the evolution of evolution, e.g. Hindré et al., 2012 , evolvable genotype–phenotype mappings, e.g. Gerhart and Kirschner, 2007 Wagner and Altenberg, ... |
1995 . Also relevant is work on understanding how complex structures can evolve from simpler components in modular, hierarchical and nearly-decomposable systems, e.g. Simon, 1962 Watson, 2006 Calcott, 2008 , and related work on semiosis in the origin of modular and loosely coupled systems, e.g. Auletta et al., 2008 Con... |
The importance of exaptation—where an existing phenotypic structure becomes selected for a different function—is well recognised in biology |
Gould and Vrba, 1982 Whitacre, 2010 A challenge for achieving open-ended evolution in artificial systems is to work with structures that potentially have multi-functional properties, perhaps in different domains of interaction |
Taylor, 2013 All of the topics mentioned here (and many others too) provide us with ideas of how to create artificial evolutionary systems in which individuals have room to move as they explore the evolutionary landscape. |
2.5 Drive for continued evolution Even with the first four requirements in place, a continued drive is required to force the system to explore new states. |
To create any drive in the system at all, selection pressure must exist. In general, this can be brought about by competition for some kind of limited resource (which may be matter, energy, space), or through environmental conditions, etc. Selection creates an adaptive landscape in which some variations of organism are... |
In order to achieve continued drive, the individuals must experience a changing adaptive landscape Waddington, 1969 . In biological populations this is brought about by other individuals being part of the ecological environment—those individuals are also evolving, and can alter the fitness landscape by direct interacti... |
Jones et al., 1997 and niche construction Odling-Smee et al., 2003 . Changes can also come about through (passive or active) diffusion of species to new environments (e.g. migration). |
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