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2. [MATH] if and only if [MATH] is deterministic, i.e. we know beforehand the value of the process. 3. [MATH] if and only if [MATH] has a uniform distribution, i.e. any state of the process is equally likely. |
4. [MATH] has a negative correlation with emergence [MATH] We propose as the measure [EQUATION] It is straightforward to check that this function fulfills the axioms stated. Nevertheless it is not unique. However, it is the only affine (linear) function which fulfills the axioms. For simplicity, we propose the use of a... |
[MATH] means that there is maximum order, i.e. no new information is produced ( [MATH] ). On the other extreme, [MATH] when there is no order at all, i.e. when any random variable becomes known, information is produced/emerges ( [MATH] ). When [MATH] , maximum order, dynamics do not produce novel information, so the fu... |
Note that equation makes no distinction on whether the order is produced by the system (self) or by its environment. Thus, [MATH] would have a high value in systems with a high organization, independently on whether this is a product of local interactions or imposed externally. This distinction can be easily made descr... |
3.3 Complexity Following Lopez-Ruiz et al 1995 , we can define complexity [MATH] as the balance between change (chaos) and stability (order). We have just defined such measures: emergence and self-organization. The complexity function [MATH] should have the following properties: |
1. The range is the real interval [MATH] 2. [MATH] if and only if [MATH] 3. [MATH] if and only if [MATH] or [MATH] It is natural to consider the product of [MATH] and [MATH] to satisfy the last two requirements. We propose: |
[EQUATION] Where the constant 4 is added to normalize the measure to [MATH] , fulfilling the first axiom. [MATH] can also be represented in terms of [MATH] as: |
[EQUATION] Figure plots the measures proposed so far for different values of [MATH] . It can be seen that [MATH] is maximal when [MATH] and minimal when [MATH] or [MATH] . The opposite holds for [MATH] : it is minimal when [MATH] and maximal when [MATH] or [MATH] [MATH] is minimal when [MATH] or [MATH] are minimal, i.e... |
Shannon information can be seen as a balance of zeros and ones (maximal when [MATH] ), while [MATH] can be seen as a balance of [MATH] and [MATH] (maximal when [MATH] ). |
3.4 Homeostasis The previous three measures ( [MATH] [MATH] , and [MATH] ) study how single variables change in time. To calculate the measures for a system, one can plot the histogram or simply average the measures for all variables in a system. For homeostasis [MATH] , we are interested on how all variables of a syst... |
Let [MATH] represent the state of a system of [MATH] variables (i.e. a row in Table ). If the system has a high homeostasis, we would expect that its states do not change too much in time. The homeostasis function [MATH] should have the following properties: |
1. The range is the real interval [MATH] 2. [MATH] if and only if for states [MATH] and [MATH] [MATH] , i.e. there is no change in time. |
3. [MATH] if and only if [MATH] , i.e. all variables in the system changed. A useful function for comparing strings of equal length is the Hamming distance. The Hamming distance [MATH] measures the percentage of different symbols in two strings [MATH] and [MATH] . For binary strings, it can be calculated with the XOR f... |
[EQUATION] [MATH] measures the fraction of different symbols between [MATH] and [MATH] . For the Boolean case, [MATH] and [MATH] , while [MATH] and [MATH] are uncorrelated [MATH] |
We can use the inverse of [MATH] to define [MATH] [EQUATION] which clearly fulfills the desired properties of homeostasis between two states. |
To measure the homeostasis of a system in time, we can generalize: [EQUATION] where [MATH] is the total number of time steps being evaluated. [MATH] will be simply the average of different [MATH] from [MATH] to [MATH] . As well as the previous measures based on [MATH] [MATH] is a unitless measure. |
When [MATH] is measured at higher scales, it can capture periodic dynamics. For example, let us have a system with [MATH] variables and a cycle of period 2: [MATH] [MATH] for base 2 will be minimal, since every time step all variables change, i.e. ones turn into zeros or zeros turn into ones. However, if we measure [MA... |
3.5 Autopoiesis Let [MATH] represent the trajectories of the variables of a system and [MATH] represent the trajectories of the variables of the environment of the system. A measure of autopoiesis [MATH] should have the following properties: |
1. [MATH] 2. [MATH] should reflect the independence of [MATH] over [MATH] . This implies: (a) [MATH] produces more of its own information than [MATH] for a given [MATH] |
(b) [MATH] produces more of its own information in [MATH] than in [MATH] (c) [MATH] produces as much of its own information than [MATH] for a given [MATH] |
(d) [MATH] produces as much of its own information in [MATH] than in [MATH] (e) [MATH] if all of the information in [MATH] is produced by [MATH] |
Following the classification of types of information transformation proposed in Gershenson ( 2012b , dynamic and static transformations are internal (a system producing its own information), while active and stigmergic transformations are external (information produced by another system). |
It is problematic to define in a general and direct way how some information depends on other information, as causality can be confounded with co-occurrence. For this reason, measures such as mutual information are not suitable for measuring [MATH] |
As it has been proposed, adaptive systems require a high [MATH] in order to be able to cope with changes of its environment while at the same time maintaining their integrity (Langton, 1990 ; Kauffman, 1993 . If [MATH] had a high [MATH] , then it would not be able to produce the same patterns for different [MATH] . Wit... |
[EQUATION] If [MATH] , then either [MATH] is static ( [MATH] ) or pseudorandom ( [MATH] ). This implies that any pattern (complexity) which could be observed in [MATH] (if any) should come from [MATH] . This case gives a minimal [MATH] . On the other hand, if [MATH] , it implies that any pattern (if any) in [MATH] shou... |
Since [MATH] represents a ratio of probabilities, it is a unitless measure. [MATH] , although it could be mapped to [MATH] using a function such as [MATH] . We do not normalize [MATH] because it is useful to distinguish [MATH] and [MATH] (see Section 5.8 ). |
3.6 Multi-scale profiles Bar-Yam 2004 proposed the “complexity profile”, which plots the complexity of systems depending on the scale at which they are observed. This allows to compare how a measure changes with scale. For example, the [MATH] profile compares the “satisfaction” of systems at different scales to study o... |
In a similar way, multi-scale profiles can be used for each of the measures proposed, giving further insights about the dynamics of a system than measuring them at a single scale. This is clearly seen, for example, with different types of elementary cellular automata (Gershenson and Fernández, 2012 |
Results In this section we apply the measures proposed in the previous section to two case studies: random Boolean networks and an aquatic ecosystem. A further case, elementary cellular automata, can be found in Gershenson and Fernández ( 2012 |
4.1 Random Boolean Networks Results show averages of 1000 RBNs, where 1000 steps were run from a random initial state and [MATH] [MATH] [MATH] and [MATH] were calculated from data generated in 1000 additional steps. |
(R Project Contributors, 2012 was used with packages BoolNet (Müssel et al 2010 and entropy (Hausser and Strimmer, 2012 Figure shows results for RBNs with 100 nodes, as the connectivity [MATH] varies. For low [MATH] , there is high [MATH] and [MATH] , and a low [MATH] and [MATH] . This reflects the ordered regime of RB... |
As for autopoiesis, to model a system and its environment, we coupled two RBNs: One “internal” RBN with [MATH] nodes and [MATH] average connections and one “external” with [MATH] nodes and [MATH] average connections. A “coupled” RBN is considered with [MATH] nodes and [MATH] connections. At every time step, the externa... |
Figure and Table show results for [MATH] and [MATH] for different combinations of [MATH] and [MATH] As it was shown in Figure [MATH] changes with [MATH] , so it is expected to have [MATH] when [MATH] . When [MATH] is high ( [MATH] or [MATH] ), then the environment dominates the patterns of the system, yielding [MATH] .... |
[MATH] does not try to measure how much information emerges internally or externally, but how much the patterns are internally or externally produced. A high [MATH] means that there is no pattern, as there is constant change. A high [MATH] implies a static pattern. A high [MATH] reflects complex patterns. We are intere... |
4.2 An Ecological System: An Arctic Lake The data from an Artic lake model used in this section was obtained using The Aquatic Ecosystem Simulator (Randerson and Bowker, 2008 |
In general, Arctic lake systems are classified as oligotrophic due to their low primary production, represented in chlorophyll values of 0.8-2.1 mg/m3. The lake’s water column, or limnetic zone, is well-mixed; this means that there are no stratifications (layers with different temperatures). During winter (October to M... |
Table and Figure show the variables and daily data we obtained from the Arctic lake simulation. The model used is deterministic, so there is no variation in different simulation runs. Figure depicts a high dispersion for the following variables: temperature ( [MATH] ) and light ( [MATH] ) at the three zones of the Arct... |
Observing [MATH] and [MATH] in logarithmic scale, we can see that their values are located at the extremes, but their range is not long. Consequently, these variables have considerable variability in a short range. However, the ranges of the other variables do not reflect large changes. This situation complicates the i... |
[EQUATION] where [MATH] is the floor function of [MATH] Once all variables are in transformed into a finite alphabet, in this case, base 10 ( [MATH] ), we can calculate emergence, self-organization, complexity, homeostasis and autopoiesis. Figure depicts the number of points in each of the ten classes and shows the dis... |
4.2.1 Emergence, Self-organization, and Complexity Figure shows the values of emergence, self-organization, and complexity of the physiochemical subsystem. Variables with a high complexity [MATH] reflect a balance between change/chaos (emergence) and regularity/order (self-organization). This is the case of benthic and... |
Since [MATH] , these measures can be categorized into five categories as shown in Table . These categories are described on the basis of the range value, the color and the adjective in a scale from very high to very low. This categorization is inspired on the categories for Colombian water pollution indices. These indi... |
Table shows results of [MATH] [MATH] , and [MATH] using the categories just mentioned. From Table and a principal component analysis (not shown), we can divide the values obtained in complexity categories as follows: |
Very High Complexity [MATH] . The following variables balance self-organization and emergence: benthic and planktonic pH ( [MATH] [MATH] ), inflow and outflow ( [MATH] ), and retention time ( [MATH] ). It is remarkable that the increasing of the hydrological regime during summer is related in an inverse way with the di... |
High Complexity [MATH] . This group includes 11 of the 21 variables and involves a high [MATH] and a low [MATH] . These 11 variables that showed more chaotic than ordered states are highly influenced by the solar radiation that defines the winter and summer seasons, as well as the hydrological cycle. These variables we... |
Very Low Complexity [MATH] . In this group, [MATH] is very high, and [MATH] is very low. This category includes the inflow conductivity ( [MATH] ) and water mixing variance ( [MATH] ). Both are high and directly correlated; it means that an increase of the mixing percentage between planktonic and benthic zones is assoc... |
4.2.2 Homeostasis The homeostasis was calculated by comparing the daily values of all variables, representing the state of the Arctic subsystem. The temporal timescale is very important, because [MATH] can vary considerably if we compare states every minute or every month. |
The [MATH] values have a mean ( [MATH] ) of 0.95739726 and a standard deviation of 0.064850247. The minimum [MATH] is 0.60 and the maximum [MATH] is 1.0. In an annual cycle, homeostasis shows four different patterns, as shown in Figure , which correspond with the seasonal variations between winter and summer. These fou... |
As it can be seen, using [MATH] , periodic or seasonal dynamics can be followed and studied. 4.2.3 Autopoiesis Autopoiesis was measured for three components (subsystems) at the planktonic and benthic zones of the Arctic lake. These were physiochemical, limiting nutrients and biomass. They include the variables and orga... |
According to the complexity categories established in Table , the planktonic and benthic components have been classified in the following categories: limiting nutrient variables in the low complexity category ( [MATH] ; orange color), planktonic physiochemical variables in the high complexity category ( [MATH] ; green ... |
In order to compare the autonomy of each group of variables, equation 10 was applied to the complexity data, as shown in Figure 10 . For the planktonic and benthic zones, we calculated the autopoiesis of the biomass elements in relation to limiting nutrient and physiochemical variables. All [MATH] values are greater th... |
4.2.4 Multiple scales The previous analysis of the Arctic lake was performed using base ten. We obtained the measures for the same data using bases [MATH] , as shown in Figures 11 and 12 |
For base 2 (Figure 11 ), there is a very high [MATH] for all variables, as the richness of the dynamics cannot be captured by only two values. Thus, [MATH] and [MATH] are low. Base 8 (Figure 11 ) gives results very similar to those of base 10 (Figure ), indicating that the measures are not sensitive to slight changes o... |
As more diversity is possible with higher bases, homeostasis values decrease with base. Still, the different periods of the year can be identified at all scales, with different levels of detail. |
In the case of the Arctic lake model, studying the dynamics with a single base, i.e. at a single scale, can be very informative. However, studying the same phenomena at multiple scales can give further insights, independently on whether the measures change or not with scale. |
Discussion 5.1 Measures The proposed measures characterize the different configurations and dynamics that elements of complex systems acquire through their interactions. Just like temperature averages the kinetic energy of molecules, much information is lost in the averaging, as the description of phenomena changes sca... |
5.2 Complexity as balance or entropy? Some approaches relate complexity with a high entropy, i.e. information content (Bar-Yam, 2004 ; Delahaye and Zenil, 2007 . Just as chaos should not be confused with complexity (Gershenson, 2013 , a very high entropy (high emergence [MATH] ) implies too much change, where complex p... |
It might seem contradictory to define emergence as the opposite of self-organization, as they are both present in several complex phenomena. However, when one takes one to the extreme (emergence or self-organization), the other is negligible. It is precisely when both of them are balanced that complexity occurs, but th... |
5.3 Fisher Information [MATH] is correlated with Fisher information, which has been shown to be related to phase transitions (Prokopenko et al 2011 . Following the view of high complexity as a balance, it is natural that [MATH] is maximal at phase transitions, which is the case for both [MATH] and Fisher information. H... |
5.4 Tsallis entropy Tsallis 1988 proposed a generalized measure of Shannon’s information for non-ergodic systems. This measure has been correlated with complexity (Tsallis, 2002 ; Gell-Mann and Tsallis, 2004 . On the one hand, it would be interesting to compare Tsallis entropy with [MATH] for different systems. On the ... |
5.5 Guided Self-organization The measures proposed have several implications for GSO, beyond providing a measure of self-organization. In order to guide a complex system, one has to detect what kind of dynamical regime it has. Depending on this, and on the desired configuration for the system, different interventions c... |
For example, if we want to have a system with a high complexity, first we need to measure what is its actual complexity. If it is not the desired one, then the dynamics can be guided. But we also have to measure the complexity during the guiding process, to evaluate the effectiveness of the intervention. |
5.6 Scales The proposed measures can be applied at different scales, with drastic outcomes. For example, the string ’ [MATH] ’ will have [MATH] in base 2, as [MATH] . However, in base 4, each symbol pair is transformed into a single symbol, so the string is transformed to ’ [MATH] ’, and thus [MATH] and [MATH] , giving... |
5.7 Normalization For treating continuous data, we used equation 11 to normalize to a finite alphabet, which is equally distributed. Clustering methods could also be used to process data into finite categories. Still, an issue might arise for either case: if the available data does not represent the total range of poss... |
5.8 Autopoiesis and Requisite Variety Ashby’s Law of Requisite Variety (Ashby, 1956 states that an active controller requires as much variety (number of states) as that of the controlled system to be stable. For example, if a system can be in four different states, its controller must be able to discriminate between th... |
The proposed measure of autopoiesis is related to the law of requisite variety, as a system with a [MATH] must have a higher complexity (variety) than its environment, also reflecting its autonomy. Thus, a successful controller should have [MATH] (at multiple scales (Gershenson, 2011 ), although the controller will be ... |
Conclusions We reviewed measures of emergence, self-organization, complexity, homeostasis, and autopoiesis based on information theory. Axioms were postulated for each measure and equations were derived form them. Having in mind that there are several different measures already proposed (Prokopenko et al 2009 ; Gershen... |
The generality and usefulness of the proposed measures will be evaluated gradually, as these are applied to different systems. These can be abstract (e.g. Turing machines (Delahaye and Zenil, 2007 2012 [MATH] -machines (Shalizi and Crutchfield, 2001 ; Görnerup and Crutchfield, 2008 ), biological (ecosystems, organisms)... |
The potential benefits of general measures as the ones proposed here are manifold. Even if with time more appropriate measures are found, aiming at the goal of finding general measures which can characterize complexity, emergence, self-organization, homeostasis, autopoiesis, and related concepts for any observable syst... |
Acknowledgements We should like to thank Nihat Ay, Ragnar Behncke, Paul Bourgine, Niccolo Capanni, Wilfried Elmenreich, Tom Froese, Virgil Griffith, Joseph Lizier, Luis Miramontes-Hercog, Roberto Murcio, Oliver Obst, Daniel Polani, Mikhail Prokopenko, Robert Ulanowicz, Rosalind Wang, and an anonymous referee for intere... |
# Source: arxiv 1304.4051 # Title: Coordinating metaheuristic agents with swarm intelligence # Sections: all # Downloaded: 2026-03-03T02:01:58.420060+00:00 |
Coordinating metaheuristic agents with swarm intelligence Abstract Coordination of multi agent systems remains as a problem since there is no prominent method to completely solve this problem. Metaheuristic agents are specific implementations of multi-agent systems, which imposes working together to solve optimisation ... |
Keywords: metaheuristic agents, swarm intelligence, particle swarm optimization, simulated annealing Introduction Metaheuristic agents are collaborating agents to solve large scale optimisation problems in the manner of multi agent systems in which metaheuristic algorithms are adopted by the agents as the problem solve... |
In this paper, the coordination problem of multi-agent systems has been tackled once again, but, with swarm intelligence algorithms this time. It is observed as expected that swarm intelligence algorithms help for better interactions and information/experience exchange. We illustrated the idea in coordinating simulated... |
Previously, a couple of multi agent coordination approaches applied to metaheuristic agent teams to examine their performance in coordinating them (Aydin 2007; Hammami and Ghediera 2005). Obviously, each one provides with different benefits in tackling search and problem solving. However, swarm intelligence has not bee... |
Multidimensional knapsack problem is one of the most tackled combinatorial optimisation problems due to its flexibility in convertibility into the real world problems. The problem briefly is to maximise the total weighted p index subject to the constraints where x is a binary variable and r is a matrix of coefficients ... |
[EQUATION] Subject to: [EQUATION] [EQUATION] Equation (1) is the objective function which measures the overall capacity of the knapsacks used while Equation (2) and (3) provide the hard constraints where (2) declares the upper limit of each knapsack and (3) makes sure that the decision variable, x, can only take binary... |
The rest of the paper is organised as follows. The second section is to briefly introduce the notions of metaheuristic agents and swarm intelligence with short presentation of considered metaheuristics within the study; they are particle swarm optimisation (PSO), bee colony optimisation (BCO), and simulated annealing (... |
Metaheuristic Agents and Swarm Intelligence The concept of metaheuristic agents is identified to describe multi agent systems equipped with metaheuristics to tackle hard optimisation problems. The idea of multi agency is to build up intelligent autonomous entities whose form up teams and solve problems in harmony. The ... |
Metaheuristic applications have been implemented as mostly standalone systems in an ordinary sense and examined under the circumstances of their own standalone systems. Few multi agent implementations in which metaheuristics have been exploited are examined in the literature. Various implementations of metaheuristic ag... |
Swarm intelligence is referred to artificial intelligence (AI) systems where an intelligent behaviour can emerge as the outcome of the self-organisation of a collection of simple agents, organisms or individuals. Simple organisms that live in colonies; such as ants, bees, bird flocks etc. have long fascinated many peop... |
The swarm intelligence approaches are to reveal the collective behaviour of social insects in performing specific duties; it is about modelling the behaviour of those social insects and use these models as a basis upon which varieties of artificial entities can be developed. In such a way, the problems can be solved by... |
2.1 Bee colonies Bee colonies are rather recently developed sort of swarm intelligence algorithms, which are inspired of the social behaviour of bee colonies. This family of algorithms has been successfully used for various applications such as modelling oh communication networks (Farooq 2008), manufacturing cell forma... |
2.2 Particle swarm optimisation (PSO) PSO is a population-based optimization technique inspired of social behaviour of bird flocking and fish schooling. PSO inventors were implementing such scenarios based on natural processes explained below to solve the optimization problems. Suppose the following scenario: a group o... |
The pure PSO algorithm builds each particle based on, mainly, two key vectors; position [MATH] , and velocity [MATH] . Here, [MATH] , denotes the [MATH] |
position vector in the swarm, where [MATH] , is the position value of the [MATH] particle with respect to the [MATH] dimension [MATH] , while [MATH] denotes the [MATH] velocity vector in the swarm, where [MATH] is the velocity value of the [MATH] |
particle with respect to the [MATH] dimension. Initially, the position and velocity vectors are generated as continuous sets of values randomly uniformly. Personal best and global best of the swarm are determined at each iteration following by updating the velocity and position vectors using : |
[EQUATION] where [MATH] is the inertia weight used to control the impact of the previous velocities on the current one, which is decremented by |
[MATH] , decrement factor, via [MATH] [MATH] is constriction factor which keeps the effects of the randomized weight within the certain range. In addition, [MATH] and |
[MATH] are random numbers in and [MATH] and [MATH] are the learning factors, which are also called social and cognitive parameters. The next step is to update the positions in the following way. |
[EQUATION] After getting position values updated for all particles, the corresponding solutions with their fitness values are calculated so as to start a new iteration if the predetermined stopping criterion is not satisfied. For further information, Kennedy and Eberhart (1995) and Tasgetiren et al (2007) can be seen. |
PSO has initially been developed for continuous problems not for discrete ones. As MKP is a discrete problem, we use one of discrete PSO, which is proposed by Kennedy and Eberhart (1997). The idea is to create a binary position vector based on velocities as follows: |
[EQUATION] where equation (5) is replaced with (6) so as to produce binary values for position vectors. 2.3 Simulated annealing Simulated annealing (SA) is one of the most powerful metaheuristics used in optimisation of many combinatorial problems, which relies on a stochastic decision making process in which a control... |
[MATH] , and a particular state in search space, [MATH] , a neighbourhood function, [MATH] , conducts a move from [MATH] , to [MATH] , where the decision to promote the state is made subject to the following stochastic rule:- |
[EQUATION] where [MATH] [MATH] is the iteration index, [MATH] is the random number generated for making a stochastic decision for the new solution and [MATH] is the level of temperature (at the [MATH] iteration), which is controlled by a particular cooling schedule, [MATH] . This means that, in order to make the new so... |
SA agents collaborating with swarm intelligence As explained above, simulated annealing (SA) is one of the most commonly used metaheuristic approaches that offer a stochastic problem solving procedure. It is used for numerous and various successful applications (Kolonko 1999; Aydin and Fogarty 2004) in combinatorial an... |
The original idea of swarm intelligence is to form up populations of enabled individuals for collaboratively problem solving spurposes. However, due to computational complexity and the hardship in furnishing the enabled individuals with multiple advanced functionalities, swarms are usually designed as population of ind... |
This study has aimed to find out a better way of organising agents in a more proactive collaboration so that the agents are to be enabled with contributing problem solving whilst coordinating. For this purposes, few algorithms have been examined; evolutionary simulated annealing, bee colony optimization and particle sw... |
Figure 1 sketches the progress of searching for optimum solution through generations reflecting how each agent plays its role and how the collaboration algorithm merges the intelligence produced by each agent. First of all, a swarm of SA agents is created, where each agent starts searching with a randomly generated pro... |
The interaction of the SA agents in this way reminds the idea of variable neighbourhood search (Hansen et al 2004; Sevkli and Aydin 2006) where a systematic switch-off between search algorithms is organised in order to diversify the solutions. In an overall point of view, the swarm of SA agents sounds borrowing this id... |
The multidimensional knapsack problem is represented in a binary way to be inline with the integer programming model in which a decision variable of [MATH] plays the main role in process of optimisation, where [MATH] is a vector of [MATH] binary variables. This is also the way how to present a problem state. Here, once... |
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