I go deep into mathematical philosophy, i.e. into the kind of philosophy, where I spare some of my words and use mathematical notation instead, or, if you want, the kind of maths where I am sort of freestyling. I go back to my mild obsession about using artificial neural networks as representation of collective intelligence, and more specifically to one of my networks, namely that representing social roles in the presence of an exogenous disturbance.

I am currently encouraging my students to come up with ideas of new businesses, and I gently direct their attention towards the combined influence of a long-term trend – namely that of digital technologies growing in and into the global economy – with a medium-term one which consists in cloud computing temporarily outgrowing other fields of digital innovation, and finally a short-term, Black-Swan-type event, namely the COVID-19 pandemic.  I want to re-use, and generalize the interpretation of the neural network I presented in my update from May 25th, 2020, titled ‘The perfectly dumb, smart social structure’, in order to understand better the way that changes of various temporal span can overlap and combine.

I start with introducing a complex, probabilistic definition of, respectively, continuity and change. Stuff happens all the time and happening can be represented mathematically as the probability thereof. Let’s suppose that my life is really simple, and it acquires meaning through the happening of three events: A, B, and C. I sincerely hope there are no such lives, yet you never know, you know. In Poland, we have that saying that a true man needs to do three things in life: build a house, plant a tree and raise a son. See, just three important things over the whole lifetime of a man. I hope women have more diverse existential patterns. 

Over the 365 days of a year, event A happened 36 times, event B happened 12 times, and event C took place 120 times. As long as I keep the 365 days of the year as my basic timeline, those three events have displayed the following probabilities of them happening: P(A) = 36/365 = 0.0986, P(B) = 12/365 = 0.0329, and P(C) = 12/365 = 0.3288. With these three probabilities computed, I want to make reasonable expectations as for the rest of my life, and define what is normal, and what is definitely weird, and therefore interesting. I define three alternative versions of the normal and the weird, using three alternative maths. Firstly, I use the uniform distribution, and the binomial one, to represent the most conservative approach, where I assume that normal equals constant, and anything else than constant is revolution. Therefore NORMAL = {P(A) = 0.0986 and P(B) = 0.0329 and P(C) = 0.3288} and WEIRD = {P(A) ≠ 0.0986 or P(B) ≠ 0.0329 or P(C) ≠ 0.3288}. Why do I use ‘and’ in the definition of NORMAL and ‘or’ in that of WEIRD? That’s sheer logic. My NORMAL is all those three events happening exactly at the probabilities computed for the base year. All the 3 of them need to be happening precisely at those levels of incidence. All of them means A and B and C. Anything outside that state is WEIRD, therefore WEIRD happens when even one out of three goes haywire. It can be P(A) or P(B) or P(C), whatever.

Cool. You see? Mathematics don’t bite. Not yet, I mean. Let’s go a few steps further. In mathematical logic, conjunction ‘and’ is represented as multiplication, therefore with symbols ‘x’ or ‘*’.  On the other hand, logical alternative ‘or’ is equivalent to arithmetical addition, i.e. to ‘+’. In other words, my NORMAL = {P(A) = 0.0986 * P(B) = 0.0329 * P(C) = 0.3288} and WEIRD = {P(A) ≠ 0.0986 + P(B) ≠ 0.0329 + P(C) ≠ 0.3288}. The NORMAL is just one value, namely  0,0986 * 0,0329 * 0,3288 = 0,0011, whilst the WEIRD is anything else.

From one thing you probably didn’t like in elementary school, i.e. arithmetic, I will pass to another thing you just as probably had an aversion to, i.e. to geometry. I have those three things that matter in my life, A, B, and C, right? The probability of each of them happening can be represented as an axis in a 3-dimensional, finite space. That space is finite because probabilities are shy and never go above and beyond 100%. Each of my three dimensions maxes out at 100% and that’s it. My NORMAL is just one point in that manifold, and all other points are WEIRD. I computed my probabilities at four digits after the decimal point, and my NORMAL is just one point, which represents 0,0001 out of 1,00 on each axis. Therefore, on each axis I have 1 – 0,0001 = 0,9999 alternative probability of WEIRD stuff happening. I have three dimensions in my existence, and therefore the total volume of the WEIRD makes 0,9999 * 0,9999 * 0,9999 = 0,99993 = 0,9997.

Let’s check with arithmetic. My NORMAL = {P(A) = 0.0986 * P(B) = 0.0329 * P(C) = 0.3288} = 0,0011, right? This is the arithmetical probability of all those three probabilities happening. If I have just two events in my universe, the NORMAL and the WEIRD, the probability of WEIRD is equal to the arithmetical difference between 1,00 and the probability of NORMAL, thus P(WEIRD) = 1,00 – 0,0011 = 0,9989. See? The arithmetical probability of WEIRD is greater than the geometrical volume of WEIRD. Not much, I agree, yet the arithmetical probability of anything at all happening in my life outgrows the geometrical volume of all the things happening in my life by 0,9989 – 0,9997 = 0,0008. Still, there are different interpretations. If I see the finite space of my existence as an arithmetical product of three dimensions, it means I see it as a cube, right? That, in turn, means that I allow my universe to have angles and corners. Yet, if I follow the intuition of Karl Friedrich Gauss and I perceive my existence as a sphere around me (see The kind of puzzle that Karl Friedrich was after), that sphere should have a diameter of 100% (whatever happens happens), and therefore a radius of 50%, and a total volume V = (4/3)*π*(r3) = (4/3)*π*(0,53) =  0,5236. In plain, non-math human it means that after I smooth my existence out by cutting all the corners and angles, I stay with just a bit more than one half of what can possibly, arithmetically happen.

WTF? Right you would be to ask. Let’s follow a bit in the footsteps of Karl Friedrich Gauss. That whole story of spherical existence might be sensible. It might be somehow practical to distinguish all the stuff that can possibly happen to me from the things which I can reasonably expect to happen. I mean, volcanoes do not erupt every day, right? Most planes land, right? There is a difference between outliers of the possible and the mainstream of existential occurrence. The normal distribution is a reasonably good manner of partitioning between those two realms, as it explicitly distinguishes between the expected and all the rest. The expected state of things is one standard deviation away from the average, both up and down, and that state of things can be mathematically apprehended as mean-reverted value. I have already messed around with it (see, for example ‘We really don’t see small change’ ).

When I contemplate my life as the normal distribution of what happens, I become a bit more lucid, as compared to when I had just that one, privileged state of things, described in the preceding paragraphs. When I go Gaussian, I distinguish between the stuff which is actually happening to me, on the one hand, and the expected average state of things. I humbly recognize that what I can reasonably expect is determined by the general ways of reality rather than my individual expectations, which I just as humbly convert into predictions. Moreover, I accept and acknowledge that s**t happens, as a rule, things change, and therefore what is happening to me is always some distance from what I can generally expect. Mathematically, that last realization is symbolized by standard deviation from what is generally expected.

All that taken into account, my NORMAL is an interval: (μ – σ) ≤  NORMAL ≤ (μ + σ), and my WEIRD is actually two WEIRDS, the WEIRD ≤ (μ – σ) and the WEIRD ≥  (μ + σ).

The thing about reality in normal distribution is that it is essentially an endless timeline. Stuff just keeps happening and we normalize our perception thereof by mean-reverting everything we experience. Still, life is finite, our endeavours and ambitions usually have a finite timeframe, and therefore we could do with something like the Poisson process, to distinguish the WEIRD from the NORMAL. Besides, it would still be nice to have a sharp distinction between the things I want to happen, and the things that I can reasonably expect to happen, and the Poisson process addresses this one, too.

Now, what all that stuff about probability has to do with anything? First of all, things that are happening right now can be seen as the manifestation of a probability of them happening. That’s the deep theory by Pierre Simon, marquis de Laplace, which he expressed in his ‘Philosophical Essay on Probabilities’: whatever is happening is manifesting an underlying probability of happening, which, in turn, is a structural aspect of reality. Thus, when I run my simplistic perceptron in order to predict the impact of Black-Swan-type disruptions on human behaviour ( see ‘The perfectly dumb, smart social structure’), marquis de Laplace would say that I uncover an underlying structural proclivity, dormant in the collective intelligence of ours. Further down this rabbit hole, I can claim that however we, the human civilization, react to sudden stressors, that reaction is always a manifestation of some flexibility, hidden and available in our collective cultural DNA.  

The baseline mechanism of collective learning in a social structure can be represented as a network of conscious agents moving from one complex state to another inside a state space organized as a Markov chain of states, i.e. each current state of the social structure is solely the outcome of transformation in the preceding state(s). This transformation is constrained by two sets of exogenous phenomena, namely a vector of desired social outcomes to achieve, and a vector of subjectively aleatory stressors acting like Black-Swan events (i.e. both impossible to predict accurately by any member of the society, and profoundly impactful).

The current state of the social structure is a matrix of behavioural phenomena (i.e. behavioural states) in individual conscious agents comprised in that structure. Formal definition of a conscious agent is developed, and then its application to social research is briefly discussed. Consistently with Hoffman et al. 2015[1] and Fields et al. 2018[2], conscious existence in the world is a relation between three essential, measurable spaces: states of the world or W, conscious experiences thereof or X, and actions, designated as G. Each of these is a measurable space because it is a set of phenomena accompanied by all the possible transformations thereof. States of the world are a set, and this set can be recombined through its specific σ-algebra. The same holds for experiences and actions. Conscious existence (CE) consists in consciously experiencing states of the world and taking actions on the grounds of that experience, in a 7-tuple defined along the following dimensions:

  1. States of the world W
  2. Experiences X
  3. Actions G
  4. Perception P defined as a combination of experiences with states of the world, therefore as a Markovian kernel P: W*X → X
  5. Decisions D defined as a Markovian kernel transforming experiences into actions, or D: X*G → G
  6. Consequences A of actions, defined as a Markovian kernel that transforms actions into further states of the world, or A: G*W →W.
  7. Time t

Still consistently with Hoffman et al. 2015 (op. cit.) and Fields et al. 2018 (op. cit.) it is assumed that Conscious Agents (CA), are individuals autonomous enough to align idiosyncratically their perception P, decisions D, and actions G in the view of maximizing the experience of positive payoffs among the consequences A of their actions. This assumption, i.e. maximization of payoffs, rather than quest for truth in perception, is both strongly substantiated by the here-cited authors, and pivotal for the rest of the model and for the application of artificial neural networks discussed further. Conscious Agent perceive states of the world as a combination of rewards to strive for, threats to avoid, and neutral states, not requiring attention. The capacity to maximize payoffs is further designated as Conscious Agents’ fitness to environment, and is in itself a complex notion, entailing maximization of rewards strictly spoken, minimization of exposure to threats, and a passive attitude towards neutral states. Fitness in Conscious Agents is gauged, and thus passed onto consecutive generations against a complex environment made of rewards and threats of different recurrence over time. The necessity to eat is an example of extremely recurrent external stressor. Seasonal availability of food exemplifies more variant a stressor, whilst a pandemic, such as COVID-19, or a natural disaster, are incidental stressors of subjectively unpredictable recurrence, in the lines of Black-Swan events (Taleb 2007[3]; Taleb & Blyth 2011[4]). 

Conscious Agents are imperfect in their maximization of payoffs, i.e. in the given population, a hierarchy of fitness emerges, and the fittest CA’s have the greatest likelihood to have offspring. Therefore, an evolutionary framework is added, by assuming generational change in the population of Conscious Agents. Some CA’s die out and some new CA’s come to the game. Generational change does not automatically imply biological death and birth. It is a broader phenomenological category, encompassing all such phenomena where the recombination of individual traits inside a given population of entities contributes to creating new generations thereof. Technologies recombine and give an offspring in the form of next-generation solutions (e.g. transistors and printed circuits eventually had offspring in the form of microchips). Business strategies recombine and thus create conditions for the emergence of new business strategies. Another angle of theoretical approach to the issue of recombination in the fittest CA’s is the classical concept of dominant strategy, and that of dynamic equilibrium by John Nash (Nash 1953[5]). When at least some players in a game develop dominant strategies, i.e. strategies that maximize payoffs, those strategies become benchmarks for other players.

The social structure, such as theoretically outlined above, learns by trial and error. It is to stress that individual Conscious Agents inside the structure can learn both by trial and error and by absorption of pre-formed knowledge. Yet, the structure as a whole, in the long temporal horizon, forms its own knowledge by experimenting with itself, and pre-formed knowledge, communicated inside the structure, is the fruit of past experimentation. Collective learning occurs on the basis of a Markov-chain-based mechanism: the structure produces a range of versions of itself, each endowed with a slightly different distribution of behavioural patterns, expressed in the measurable space of actions G, as formalized in the preceding paragraphs. Following the same logic, those behavioural patterns loop with states of the world through consequences, perception, experience, and decisions.

The social structure experiments and learns by producing many variations of itself and testing their fitness against the aggregate vector of external stressors, which, in turn, allows social evolutionary tinkering (Jacob 1977[6]) through tacit coordination, such that the given society displays social change akin to an adaptive walk in rugged landscape (Kauffman & Levin 1987[7]; Kauffman 1993[8]). Each distinct state of the given society is a vector of observable properties, and each empirical instance of that vector is a 1-mutation-neighbour to at least one other instance. All the instances form a space of social entities. In the presence of external stressor, each such mutation (each entity) displays a given fitness to achieve the optimal state, regarding the stressor in question, and therefore the whole set of social entities yields a complex vector of fitness to cope with the stressor. The assumption of collective intelligence means that each social entity is able to observe itself as well as other entities, so as to produce social adaptation for achieving optimal fitness. Social change is an adaptive walk, i.e. a set of local experiments, observable to each other and able to learn from each other’s observed fitness. The resulting path of social change is by definition uneven, whence the expression ‘adaptive walk in rugged landscape’.

There is a strong argument that such adaptive walks occur at a pace proportional to the complexity of social entities involved. The greater the number of characteristics involved, the greater the number of epistatic interactions between them, and the more experiments it takes to have everything more or less aligned for coping with a stressor. Formally, with n significant epistatic traits in the social structure, i.e. with n input variables in the state space, the intelligent collective needs at least m ≥ n +1 rounds of learning in order to develop adaptation. A complete round of learning occurs when the intelligent collective achieves two instrumental outcomes, i.e. it measures its own performance against an expected state, and it feeds back, among individual conscious agents, information about the gap from expected state. For the purposes of the study that follows it is assumed that temporization matters, for a social structure, to the extent that it reflects its pace of collective learning, i.e. the number of distinct time periods t, in the 7-tuple of conscious existence, is the same as the number m of experimental rounds in the process of collective learning.    

Epistatic traits E of a social structure are observable as recurrent patterns in actions G of Conscious Agents. Given the formal structure of conscious existence such as provided earlier (i.e. a 7-tuple), it is further assumed that variance in actions G, thus in behavioural patterns, is a manifestation of underlying variance in experiences X, perception P, decisions D, and consequences A.

A set of Conscious Agents needs to meet one more condition in order to be an intelligent collective: internal coherence, and the capacity to modify it for the purpose of collective learning. As regards this specific aspect, the swarm theory is the main conceptual basis (see for example: Stradner et al. 2013[9]). Internal coherence of a collective is observable as the occurrence of three types in behavioural coupling between Conscious Agents: fixed, random, and correlated. Fixed coupling is a one-to-one relationship: when Conscious Agent performs action Gi(A), Conscious Agent B always responds by action Gi(B). Fixed coupling is a formal expression of what is commonly labelled as strictly ritual. By opposition, random coupling occurs when the Conscious Agent B can have any response to action in Conscious Agent A, without any pattern. Across the spectrum that stretches between fixed coupling and random coupling, correlated coupling entails all such cases when Conscious Agent B chooses from a scalable range of behaviours when responding to action performed by Conscious Agent A, and coincidence in the behaviour of conscious agents A and B explains a significant part of combined variance in their respective behaviour.

It is to note that correlation in behavioural coupling, such as provided in the preceding paragraph, is a behavioural interpretation of the Pearson coefficient of correlation, i.e. it is statistically significant coincidence of local behavioural instances. Another angle is possible, when instead of correlation strictly speaking, we think rather about cointegration, thus about functional connection between expected states (expected values, e.g. mean values in scalable behaviour) in Conscious Agents’ actions. 

A social structure dominated by fixed behavioural coupling doesn’t learn, as behavioural patterns in Conscious Agents are always the same. Should random coupling prevail, it is arguable whether we are dealing with a social structure at all. A reasonably adaptable social structure needs to be dominated by correlated behavioural coupling between conscious agents, and its collective learning can be enhanced by the capacity to switch between different strengths of correlation in behaviours.  

Definition: An Intelligent Collective (IC) is a set of z Conscious Agents, which, over a sequence of m distinct time periods, understood as experimental rounds of learning, whilst keeping significant correlation in behavioural coupling between Conscious Agents’ actions and thus staying structurally stable, produces m such different instances of itself that the last instance in the sequence displays a vector of n epistatic traits significantly different from that observable in the first instance, with the border condition m ≥ n + 1.  

When a set of z Conscious Agents behaves as an Intelligent Collective, it produces a set of n significant epistatic traits, and m ≥ n + 1 instances of itself, over a continuum of m time periods and w distinct and consecutive states of the world. Collective intelligence is observable as the correlation between variance in local states of the world W, on the one hand, and variance in epistatic traits of the social structure. The same remarks, as those made before, hold as regards the general concept of correlation and the possibility of combining it with cointegration.

It is to notice that the border condition m ≥ n +1 has another interesting implication. If we want n epistatic traits to manifest themselves in a population of Conscious Agents, we need at least m ≥ n +1 experimental rounds of learning in that population. The longer is the consciously, and culturally owned history of a social structure, the more complex vector of epistatic traits can this structure develop to cope with external stressors.

Now, we enter the more epistemological realm, namely the question of observability. How are Intelligent Collectives observable, notably in their epistatic traits and in their evolutionary tinkering? From the point of view of a social scientist, observability of the strictly speaking individual behaviour in Conscious Agents, would they be individual persons or other social entities (e.g. businesses) is a rare delicacy, usually accessible at the price of creating a tightly controlled experimental environment. It is usually problematic to generalize observations made in such a controlled setting, so as to make them applicable to the general population. Working with the concept of Intelligent Collective requires phenomenological bridging between the data we commonly have access to, and the process of collectively intelligent evolutionary social tinkering.

Here comes an important, and sometimes arguable assumption: that of normal distribution in the population of Conscious Agents. If any type of behaviour manifests as an epistatic trait, i.e. as important for the ability of the social structure to cope with external stressors, then it is most likely to be an important individual trait, i.e. it is likely to be significantly correlated with the hierarchical position of social entities inside the social structure. This, in turn, allows contending that behavioural patterns associated with epistatic traits are distributed normally in the population of Conscious Agents, and, as such, display expected values, representative thereof. With many epistatic traits at work in parallel, the population of Conscious Agents can be characterized by a vector (a matrix) of mean expected values in scalable and measurable behavioural patterns, which, in turn, are associated with the epistatic traits of the whole population.

This assumption fundamentally connects individual traits to those of the entire population. The set of epistatic traits in the population of Conscious Agents is assumed to be representative for the set of mean expected values in the corresponding epistatic traits at the individual level, in particular Conscious Agents in the population. There are 3 σ – algebras, and one additional structural space, which, together, allow mutual transformation between 3 measurable and structurally stable spaces, namely between: the set of behavioural patterns BP = {bp1, bp2, …, bpn}, the set PBP = {p(bp1), p(bp2), …, p(bpn)} of probabilities as regards the occurrence of those patterns, and the set μBP = {μ(bp1), μ(bp2), …, μ(bpn)} of mean expected values in scalable and observable manifestations of those behavioural patterns.

The 3 σ – algebras are designated as, respectively:

  • the σ – algebra SB, transforming BP into PBP, and it represents the behavioural state of the intelligent collective IC
  • the σ – algebra SE, which transforms BP into μBP and is informative about the expected state of the intelligent collective IC
  • the σ – algebra SD, allowing the transition from PBP to μBP and representing the internal distribution of behavioural patterns inside the intelligent collective IC 

The additional measurable space is V = {v1, v2, …, vn} of observable Euclidean distances between measurable aspects of epistatic traits, thus between probabilities PBP, or between μBP mean expected values. Three important remarks are to make as regards the measurable space V. Firstly, as the whole model serves to use artificial neural networks in an informed manner as a tool of virtual social experimentation, the ‘between’ part in this definition is to be understood flexibly. We can talk about Euclidean distances between probabilities, or distances between mean expected values, yet it is also possible to compute Euclidean distance between a probability and a mean expected value. The Euclidean distance per se does not have a fixed denominator, and, therefore, can exist between magnitudes expressed on different scales of measurement.

Secondly, for the sake of keeping mathematical complexity of the problem at hand within the limits of reasonable, Euclidean distance is further understood as mean Euclidean distance of the given epistatic trait from all the other k = n – 1 epistatic traits, i.e. as

It is also assumed that structural stability of the Intelligent Collective can be measured as, respectively, the mean and the variance in vi, both across the n epistatic traits and m ≥ n +1 experimental rounds. Thirdly, the averaging of Euclidean distances could be, technically, considered as an σ – algebra, as we are in the conceptual construct of state space. Still, it is always the same operation, and it would be always the same σ – algebra, and, as such, logically redundant.

Definition: Collective Intelligence is a two-dimensional σ–algebra CI = {n, m}, which transforms the 7-dimensional state space CA = {W, X, G, P, D, A, t} of individual conscious existence (in Conscious Agents) into the 7-dimensional state space IC = {BP, PBP, μBP, SB, SE, SD, V} of Intelligent Collective, and the transformation occurs by wrapping experience X, actions G, perception P, and decisions D into n epistatic traits of the Intelligent Collective, and by structuring states of the world W, and consequences A over the timeline t observable in the individual conscious existence into m ≥ n + 1 experimental instances of the Intelligent Collective IC so as the last instance IC(m) = {BP(m), PBP(m), μBP(m), SB(m), SE(m), SD(m), v(m)} is significantly different from the first instance IC(1) = {BP(1), PBP(1), μBP(1), SB(1), SE(1), SD(1), v(1)}.    

This definition of Collective Intelligence stays mathematically in the world of Markov chains. Each 7-dimensional state IC = {BP, PBP, μBP, SB, SE, SD, v}of the Intelligent Collective is a transformation of the previous state. Such as formulated above, Collective Intelligence can be referred to and pegged on exogenous phenomena, yet, as such, it can be observed as a phenomenon sui generis.

I got carried away, again. I mean, intellectually. Happens all the time, actually. Time to cool down. If you really want, you can watch, on the top of reading this update, you can watch those videos of mine on the philosophy of science:

The video recorded around 2:30 p.m., August 22nd, 2020, regards the Philosophy of Science. It is both extra-curricular content for all those among my students who want to develop their scientific edge, and my auto-reflection on the general issue of collective intelligence, and the possibility to use artificial neural networks for the study thereof. I dive into three readings: ‘Civilisation and Capitalism’ by Fernand Braudel, ‘Philosophical Essay on Probabilities’ by Pierre Simon, marquis de Laplace, and finally ‘Truth and Method’ by Hans Georg Gadamer. I focus on fundamental distinctions between reality such as it is, on the one hand, our perception, and our understanding thereof. The link is here: ( ).

In the second video, recorded on August 24th, 2020 (  ), I am investigating the nature of truth, with three basic readings: Philosophical Essay on Probabilities’ by Pierre Simon, marquis de Laplace, ‘Truth and Method’ by Hans Georg Gadamer, and an article entitled ‘Conscious agent networks: Formal analysis and application to cognition’, by Chris Fields, Donald D. Hoffman, Chetan Prakash, and Manish Singh. I briefly discuss the limitations we, humans, encounter when trying to discover truth about reality.

[1] Hoffman, D. D., Singh, M., & Prakash, C. (2015). The interface theory of perception. Psychonomic bulletin & review, 22(6), 1480-1506.

[2] Fields, C., Hoffman, D. D., Prakash, C., & Singh, M. (2018). Conscious agent networks: Formal analysis and application to cognition. Cognitive Systems Research, 47, 186-213.

[3] Taleb, N. N. (2007). The black swan: The impact of the highly improbable (Vol. 2). Random house.

[4] Taleb, N. N., & Blyth, M. (2011). The black swan of Cairo: How suppressing volatility makes the world less predictable and more dangerous. Foreign Affairs, 33-39.

[5] Nash, J. (1953). Two-person cooperative games. Econometrica: Journal of the Econometric Society, 128-140.

[6] Jacob, F. (1977). Evolution and tinkering. Science, 196(4295), 1161-1166

[7] Kauffman, S., & Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. Journal of theoretical Biology, 128(1), 11-45

[8] Kauffman, S. A. (1993). The origins of order: Self-organization and selection in evolution. Oxford University Press, USA

[9] Stradner, J., Thenius, R., Zahadat, P., Hamann, H., Crailsheim, K., & Schmickl, T. (2013). Algorithmic requirements for swarm intelligence in differently coupled collective systems. Chaos, Solitons & Fractals, 50, 100-114

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