key: cord-0613062-ifldtz92
authors: Ivanova, Inga A.
title: Information exchange, meaning and redundancy generation in anticipatory systems: self-organization of expectations -- the case of Covid-19
date: 2021-05-25
journal: nan
DOI: nan
sha: d972f7beb0e695fb3825e29af066568af5ff12de
doc_id: 613062
cord_uid: ifldtz92

When studying the evolution of complex systems one refers to model representations comprising various descriptive parameters. There is hardly research where system evolution is described on the base of information flows in the system. The paper focuses on the link between the dynamics of information and system evolution. Information, exchanged between different system's parts, before being processed is first provided with meaning by the system. Meanings are generated from the perspective of hindsight, i.e. against the arrow of time. The same information can be differently interpreted by different system's parts (i,e,provided with different meanings) so that the number of options for possible system development is proliferated. Some options eventually turn into observable system states. So that system evolutionary dynamics can be considered as due to information processing within the system. This process is considered here in a model representation. The model under study is Triple Helix (TH) model, which was earlier used to describe interactions between university, industry and government to foster innovations. In TH model the system is comprised of three interacting parts where each part process information ina different way. The model is not limited to the sphere of innovation and can be used in a broader perspective. Here TH is conceptualized in the framework of three compertment model used to describe infectious disease. The paper demonstrates how the dynamics of information and meaning can be incorporated in the description of Covid-19 infectious propagation. The results show correspondence of model predictions with observable infection dynamics.

Complex biological and social systems can be considered as set of interacting agents (actors or sub-systems). Actors are connected with each other via communications. Actors, as nodes, and communications, as links, form structural network. Structural network carry system evolution on the top of structural layer in terms of functions, such as supply, demand and control (Leydesdorff, Ivanova & Meyer, 2019) . Interaction between two actors occasionally generates change from the system's previous state. When the system comprises three or more actors each third actor disturbs the communication between other two actors. This disturbance can reinforce or constrain this change. Thus autocatalytic (reinforcing) or stabilizing (constraining) cycles may arise. Mixed structures in addition to cycles are also possible (Fig.1) . In stabilizing mode the system keeps to its previous historical states generated with arrow of time and evolves along the trajectory. While in autocatalytic mode the system relies on available options, which have not yet been realized and self-organizes. In a mixed mode the system temporary deviates from historical trajectory.

When the actors are of different nature they also span a correlation network. Correlation means that actors are different or similar only to some degree with respect to processing the information. In other words they are positionally differentiated. Latent structures organize different meanings into structural components. These structures are driven by coding rules (Leydesdorff, 2010) . That is, the same information may be supplied with different meanings by different actors. E.g. the information about some technology may be treated differently with respect to its scientific importance, industrial applicability or patentability. The structural differences among the coding and decoding algorithms provide a source of additional options in reflexive and anticipatory communications, meaning generating structures act as selection environments (Leydesdorff, 2021) .

Meanings produce expectations about possible system states which are generated with respect to future moments (i.e. against the arrow of time). Expectations can be considered as options. The more options possess the system, the more probable the system is to change when self-organization is present.

An example of a system comprising positionally differentiated actors is the Triple Helix of university-industry-government relations (Etzkowitz and Leydesdorff, 1998) . A TH is expected to reveal non-linear interactions among the helices since it is intrinsically non-linear system. Non-linear interactions mean that system behavior is described by non-linear equations. Ivanova, Leydesdorf (2014a) showed that the origin of non-linearity lies in TH triadic structure.

TH actors can be mapped as components of three dimensional vector and their interactions presented as rotations of this vector. Non-linearity stems from non-commutativity of vector rotation the in a three dimensional space.

Actors not only exchange information, but can also share meanings provided by partially overlapping perspectives, Partial overlaps generate redundancy since the same information can be provided with different meanings and meanings are used to generate new redundant options.

This redundancy can be considered as the number of new and not yet realized options in a system of innovations. Redundancy can be measured quantitatively using the synergy indicator (Leydesdorff, Dolfsma, & van der Panne, 2006; Leydesdorff, Ivanova, 2014) .

The synergy indicator measures the generation of redundancy as a result of a three-way interaction among TH actors (Leydesdorff, Ivanova, & Meyer, 2019) . TH also incorporates anticipatory dynamics based on expectations. The theory and computation of anticipatory systems can help numerically evaluate and predict the incursive and recursive dynamics in a model representation (Rosen, 1985; Dubois, 1998) .

The measurement of expectations with synergy indicator allows estimate the synergy in interaction among the actors which in some way can help to predicts system evolution. However two major issues that require further research can be distinguished: 1) redundancy is evaluated as a static measure without direct reference to system's evolutionary dynamics and indeed one should develop a dynamical equivalent to TH indicator (Leydesdorff, 1991) , also interactions in complex systems tend to involve circularities which can not be described by static products of probabilities (Krippendorff, 2009) ; and 2) what way redundancy is linked to the transformation of the system? The answer to these questions can open up additional opportunities in studying TH like systems of different origin, including economic, biological, and social ones. The relations between evolutionary theory and systems theory can further be specified using communication theory (Theil, 1972) .

The first research question of the present paper is how TH redundancy dynamics arises from interactive relationships among historical and anticipated states? I argue that the dynamics of redundancy in these interactions is subject to self-organization provided by non-linear mechanisms, run with non-linear evolutionary differential equation. This dynamics manifests itself in recognizable longitudinal patterns. The patterns in turn can be used to predict future system evolution and are linked to logistic function.

As a second research question I study whether there is a link between generated redundancy and TH-like system transformation. I argue that model predictions are in good agreement with Covid-19 epidemic spread data. The obtained results show that the unfoldment of Covid-19 infection matches model predictions in terms of successive infection wave amplitudes ratios. This suggests a link between redundancy dynamics and system transformation dynamics.

In a TH model the communication between each of two selection environments is shaped by the third selection environment (Sun & Negishi, 2010) . This mechanism drives the system transformation and is known as "triadic closure" (Granovetter, 1973; Bianconi et al., 2014) Two cycles with positive and negative feedback and feedforward loops are possible: autocatalytic and stabilizing the dynamic in organizational formats one. A positive cycle amplifies a change from previous system state and can be considered as system self-organization, while negative cycle corrects this change and stabilizes the system along a historical trajectory so that system evolution is driven by two opposing tendencies (Ulanowicz, 2009 ). These cycles can be modeled as two three dimensional vectors Q and P which rotate in opposite directions (Ivanova & Leydesdorff, 2014b) . Redundancy is the result of a balance between historical stabilization and selforganization (Ivanova & Leydesdorff, 2014a) :

Also mutual redundancy 123 in a TH system can be measured with help of synergy indicator (Ulanowicz, 1986; Leydesdorff, Dolfsma, & van der Panne, 2006) :

where H is an entropy of one, two and three variables distribution

..corresponding probabilities.

A calculus of redundancy is a complement to Shannon's calculus of information (Bar-Hillel, 1955) . TH redundancy can be either positive or negative, depending on the nature of actor's interaction. Negative redundancy is the result of self-organization in the communications which provides more opportunities and positive redundancy indicates the historical organization in the instantiations by exploiting existing opportunities.

The first term in Eq. (1) can be considered as due to historical organization (which adds to positive redundancy) and the second term corresponds to self-organization and which augments negative redundancy. Historical organization relates to historically realized options which are generated via recursive mode and self-organization bears on new, not yet realized options, generated via incursive mode 2 . Comparing two expressions for redundancy provided by Eqs. (1) and (2) For non-harmonic oscillations, one can derive (see Appendix A):

The probability density function is the solution of a non-linear evolutionary equation:

and can be written in the form of a solitary wave, otherwise named as soliton (see Appendix B for the derivation):

Or in a more general form:

2 Recursive systems use their past states to modulate the present ones, while hype-incursive systems employ future anticipated states to shape their present states (e.g. Leydesdorff & Dubois, 2004) 

An impulse in the form as in Eq. 6 eventually evolves in a train of n solitary waves with amplitudes 2 2 , 8 2 , 18 2 …2 2 2 and the corresponding velocities 4 2 , 16 2 , 32 2 , … 4 2 2 3 (Miura, 1976 ) .

Additional term at the right hand side of Eq.5 attenuates the initial impulses amplitudes with time. Wave, described by Eq. 5, moves forward and after time span: One can obtain the probability function from the probability density function according the formula:

The resulting probability function is a sigmoid curve which is frequently used to describe the behavior of a dynamic system. Often this behavior resembles a series of logistic waves.

Growth and diffusion patterns can be decomposed in one or more sigmoid curves.

One of simple models depicting infectious disease dynamics is the SIR model originally Daily data were used to distinguish different periods presented by infection surge. Data in each period were processed with MathCad software to determine the parameters of the logistic curve:

Finally these curves were merged to approximate the whole dataset. 

The function ( ) determines parameters of empirically observed infection waves, such as amplitude, width and position. These parameters are further compared with model predicted ones.

Descriptive statistics comprising parameters of the data fitting (Figs 2a-9a) is provided in Table 1 . Infection outspread is country-specific and unfolds in two, three or four waves. , , -parameters of logistic curves (Eq. 9),time shifts. In order to test the model I compare the ratio of the amplitudes of subsequent waves to the amplitude of the first wave and the ratio of the time of the peak of subsequent waves to the time of the peak of the first wave for the set of eight countries. The results are summarized in Table 2 . The values , , , are obtained from empirical data fit (Table 1)   Table 2 The ratio of the amplitudes of subsequent waves to the amplitude of the first wave and the ratio of the time of the peak of subsequent waves to the time of the peak of the first wave country time ratios amplitude ratios

One can mention the proportionality between the ratios of amplitudes and ratios of time.

According to the model predictions (Eq. 7), time ratios should exactly equal amplitude ratios. In case the model predictions hold there should be significant correlation between sets of these quantities. And really in the above example, one obtains substantial linear correlation between these sets. The Pearson correlation coefficient is r = .898.

Here is proposed a model which describes evolution of a TH system as a result of information processes within the system. The major finding is that redundancy density function can be presented as the solution of non-linear evolutionary equation and evolves in recognizable patterns so that summary redundancy growth can be mapped as overlapping logistic curves. real data, which means that the information processes in the system can be considered as drivers of system evolution. Model predicted recognizable patterns can be used to forecast future system evolution, which may also be of interest for practitioners and policy makers.

Thus, Shannon's information theory can be embedded in the theory of evolution of complex self-organizing systems described in the framework of the TH model.

The subject of future research includes the study of the applicability of this model to other areas besides those considered here and the extension of the theory to the case of higher order helices.

Shannon informational entropy for temporal cyclic systems can be written as:

Dubois showed (2019) that taking into account temporal cyclic systems:

with entropy and normalization conditions:

in case = 2 one obtains harmonic oscillator equation:

where 0 = 1 ∫ ( ) 0 and is any function of , :

Following Dubois one can define the state of reference:

and develop informational entropy in Taylor's series around the reference state: 

The function * corresponds to non-linear residue in (A6) which is a truncated version of (A5).

Using non-truncated equations (A5) we obtain: we obtain non-linear evolutionary equation:

which corresponds to Korteweg de Vries (KdV) equation (Gibbon, 1985) :

with additional term 1 . By substitution: → 1 6 ⁄ ; → √ ; → √ Eq. (B7) is reduced to the form:

Taking into account that equation (B8) possesses a soliton solution:

= 2

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The author is grateful to Loet Leydesdorff for valuable comments on an earlier draft of this paper.The paper was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (NRU HSE) and with support of the Russian Academic Excellence Project '5-100'.

Redundancy (Eq. 1) is a balance between two dynamics -evolutionary self-organization and historical organization, (Leydesdorff, 2010) or, in other words, between recursion on a previous state along the historical axis as opposed to meaning provided to the events from the perspective of hindsight (Dubois, 1998) . Redundancy dynamics drives corresponding probabilities dynamics with recursive and incursive perspectives. Provided that probabilities oscillate in non-harmonic mode (Eq. A15) one can write: and keeping the terms up to ℎ 4 order of magnitude one obtains (Fermi, Pasta, Ulam, 1955) :