Entropical Analysis of an Opinion Formation Model Presenting a Spontaneous Third Position Emergence
EEntropical Analysis of an Opinion Formation Model Presenting a Spontaneous Third Position Emergence
Marcos E. Gaudiano a,c , Jorge A. Revelli b,c a Centro de Investigación y Estudios de Matemática. Consejo Nacional de Investigaciones Científicas y Técnicas. b Instituto de Física Enrique Gaviola. Consejo Nacional de Investigaciones Científicas y Técnicas. c Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación.Av. Medina Allende s/n , Ciudad Universitaria, X5000HUA Córdoba, Argentina.
Abstract
Characterization of complexity within the sociological interpretation has resulted in a large number ofnotions, which are relevant in different situations. From the statistical mechanics point of view, these notionsresemble entropy.In a recent work, intriguing non-monotonous properties were observed in an opinion dynamics Sznajdmodel. These properties were found to be consequences of the hierarchical organization assumed for thesystem, though their nature remained unexplained. In the present work we bring an unified entropicalframework that provides a deeper understanding of those system features.By perfoming numerical simulations, the system track probabilistic dependence on the initial structures isquantified in terms of entropy. Several entropical regimes are unveiled. The myriad of possible systemoutputs is enhanced within a maximum impredictability regime. A mutual structural weakness of the initialparties could be associated to this regime, fostering the emergence of a third position.
Keywords: entropy, hierarchical patterns, fractal dimensions, emergence of third position, ideological map, Sznajd model
1. Introduction
Statistical mechanics of disordered systems is an apropriated field for the description of complex systems.This area has found applications in many interdisciplinary areas which can be related to biology, chemistry,economics, social sciences and so on [3, 4, 5].In particular, the high-throughput methods available for describing social dynamics have drasticallyincreased our ability to gather comprehensive social level information on an ever growing number ofsituations [5, 6, 7, 8, 9, 10]. These results show that such systems can be thought as a dense network ofnonlinear interactions among its components [11], and that this interconnectedness is responsible for theirefficiency and adaptability. At the same time, however, such interconnectedness poses significant challengesto researchers trying to interpret empirical data and to extract the deep social principles that will enable us tobuild new theories and to make new predictions [12].A central fact, which can be ubiquitously found in nature, is that systems are often organized forminghierarchical structures [13]. Indeed, this organization may play a determining role in system dynamics [1].Hierachical structure is usually related to a fractal dimension [2, 14, 15]. However, given a set of structuresand their interactions, there is not, on one hand, an objective manner to assess whether such hierarchicalorganization does indeed exist, or to objectively identify the different levels in the hierarchy. On the otherhand, by supposing the new structure has already been detected, maybe there is not an explicit natural way toproperly describe what is the underlying process behind this emergence.Entropy is an example of a quantity used in an extensive number of applications [16]. In physics, entropywas shown to be suitable for description of systems in and out of equilibrium. In certain situations, entropy isa measure of a system’s evolution in time [17].ecently, an entropical framework to study a general class of complex sytems was proposed in [2]. Thearticle provides a natural way for characterizing evolution regimes starting from an entropy functionassociated to hierarchical system configurations. Regimes having maximum incontrollability can be definedand have been already observed in real and different contexts [18, 19].Besides, in [1] it was shown how a simple opinion formation model develops statistically very differentresponses if initial conditions are supposed to be random or hierarchically organized. While random initialconditions produce expected results, mainly defined by the relative sizes between the parties, distinct andnovel results take place when initial conditions with fractal dimension are assumed. In this case non-monotonous behaviors are obtained. However, despite of the novel properties of the model, no explanationfor the source of non-monotonicity was given. In the present work, we show how the general framework of[2] could provide a way to understand not only the source of the non-monotonity underlying the system, butalso to explain several other system’s properties and behaviors that were so far taken for granted.The paper is organized as follows: In Section 2 we briefly describe the model studied in [1]. In Section 3 weperform the entropical analysis of the system. Under the framework described in [2], we address motivatingdiscussions about relative entropy productions and the connexion with party winning probabilities. Finally,general conclusions and further extensions of this work are described in Section 4.
2. Third position emergence from structured initial conditions
Continuing the ideas explored in [1], a comunity debate is modelled as the time evolution of a squared matrixM . Every entry M ij = ± represents the ideological preference of individuals participating actively in thedynamics, while M ij =0 corresponds to persons that neither envolve into debates nor ultimately determine thefinal opinion state of the system. As in [1], we remark that ideas may be connected to individuals or group ofpeople, in a way that it is possible to image the whole system’s dynamics as an idea contend. This makes Mto be visualized as an ideological map. Figure 1: An example of an ideologicalmap as the ones studied in[1]. This is a pattern composed of yellow(Y ) and light blue (L)pixels representing the active agents,while the rest (i.e. the darkblue ones) are associated to the apathicpolytical fraction (whichdoes not participate of the system’sdebate).
On Fig.(1) we show an example. We name party L (Y ) to the set of light blue (yellow) pixels. The dark bluepixels represents the apathy. In addition, it is worth remarking here that Y and L pixels are not randomlydistributed. Specifically, they initially adopt hierarchically organized structures, which are characterized byfractal dimensions D Y and D L , respectively [1, 15].The meaning associated to fractal dimensions D will be a kind of organization level of the followers of aparticular (Y or L) ideological position [1]. The more organized the party, the higher D value. Higher Dmeans that supporters of an idea conform a more compact pattern, in which an important number of highlyconvinced people can be found. For low dimensionality patterns, it happens the opposite.he initial fractions of supporters associated to each party will be equal between each other and denoted byr (0 ≤ r ≤ 1). As in [1], this is done in order to remark the model dependence with respect to the fractaldimension .In addition, the dimension D of every pattern has to run within the following range:J r = {D : D min ≤ D ≤ D max } (1)where Dmin and Dmax are monotonous increasing function of r defined in [1] (see also [2]). Thus, the levelof participation (or parties’ initial sizes) determine the possible organizational structures of the parties.The patterns are generated without any further constraints, except for the system size which is defined by Mside length λ. Throughout this work, it is asumed that λ = 64 for computational simplicity . We consider, as in [1], a slightly modified Sznajd model [10, 20] for the iteraction among the ideas .At time t+ ∆t just one active site (i, j) is randomly chosen for update . If Mij is the no-neutral positionassociated to the site, it may change according toM ij (t + ∆t) = M īj̄ (t) (2)where Mīj̄ ( t )≠ + (j − j̄) ≥ (n − ī) + (m − j̄) . (3) Figure 2: Left: a possible ideological final pattern assuming Fig.(1)’s initial configuration. Right: the chess-board partof the final state of above is coloured in green (emergent party G). L and r Y will be studied in [22].2 for further details about the generation of the patterns see [1]3 despite being Sznajd’s model widely known, the aim of [1] was to be minimalistic and such model was assumed because it has isotropic interactions.4 we model a system that is not externally driven or forced, so it is plausible to assume an asynchronic update [21]. he system evolves under the above mentioned dynamics. On Fig.(2 , left) it is shown an example of astationary ultimate state, starting from the initial pattern of Fig.(1). Another stable chess-board-like patternarises, which is considered as a third emergent party. This new party, composed by a regular mixture of Yand L supporters, is named G. On Fig.(2, right) G supporters have been green colored .
3. Entropical Analysis
Figure 3: Average population (left) of followers and dimension (right) of the parties, everytime that either Y , L or Gfinally wins (Eq.(4)). The total number n of realizations is assumed to be 1000. The initial conditions are: D Y = 1.70 yD L = 1.60 and r = 0.15. Every winning party increases its area. When party Y turns out to be the winner, it alwaysreduces its dimensionality (on average). In order to develop our ideas, we start by considering the particular case shown in Fig.(3). As mentionedabove, Y and L are supossed to have initially each one the same number of suporters, but D Y > D L .According to [1], this corresponds to a typical situation in which Y wins most of times because D Y > D L ,despite both initial parties had the same number of supporters. The total number n of realizations was always1000 for every statisticallly studied issue troughout this work. The figure depicts the corresponding averagetime evolution of the parties sizes (A) and dimensions (D) considering just the cases in which either Y , L orG turned out to win. Specifically, the averages are computed with the formula: ⟨ X ⟩= n K ∑ i = nK X i K = Y, L, G (4)where index i runs within the class of n K realizations in which party K finally turned out to win (note that n K ≤ n), and X is any given observable variable under consideration (i.e. D(t), A(t) of any particular party, at agiven time t, etc.). For the cases in which Y finally wins, it initially shrinks mainly because of the arising ofG. The same will in average happen for the cases in which L turns out to win. Whatever be the case, thewinner party (L, Y or G) naturally needs to increase their population at the end. But Y is the party that winsmost of times, and according to Fig.(3, right), it does it with an overall reduction of its dimensionality. Wewonder if this corresponds with the general idea that many times, in order to win the election, a partyincorporates as much new members as possible, loosing likely ideological purity. On Figs.(4) and (5), theabove observation looks more general. It depicts overall average (Eq.(4)) variation of A and D as a function of the mean initial dimensionality of Y and L : ij =−M i′ j ′ or M i′ j ′ , for every active neighbor (i′ , j ′) for which(i- i′ ) + (j- j′ ) is equal to 1 or 2, respectively.6 Throughout this work, lines help to the reader to follow many numerically computed quantities. They do not represent model curves.7 We consider D Y − D L = const. = δD = 0.1 as in [1].D = 0.1 as in [1]. D = D L + D Y (5)within its correspondent possible range J r (Eq.(1)), for participation ratios r = 0.10, 0.15 and 0.20. Again, areaincreases for whatever the winning party be, but only Y always decreases its dimensionality. Figure 4: Area variations (X = ∆A = A K (t = ∞)− A K (t = 0), K = Y, L, G in Eq.(4)), every timethat either Y (◦), L (•) or G (×) party wins forthree partipation ratios: r = 0.10 (top), r =0.15 (middle) and r = 0.20 (bottom). Figure 5: Dimension variations (X = ∆D = D K (t= ∞) − D K (t =0), K = Y, L, G in Eq.(4)), everytime that either Y (◦), L (•) or G (×) party winsfor three partipation ratios: r = 0.10 (top), r =0.15 (middle) and r = 0.20 (bottom). Thus, Y seems to incorporate new members by relaxing at the same time its ideology because itsdimensionality decreases when it wins. According to [1] we can recognize Y as the most sophisticatedideology party (D Y > D L ), and it seems to be stronger than L for not only retaining its own less convincedsupporters, but also for undergoing the welcome of people from other parties.ne may wonder why the political ideas commented just above fit very well between each other. There arenot many parameters involved in the model at all. We claim that this may be a consequence of some generalprinciples, rather than a particular dynamics shown by a particular system. In particular, we point out theexistence of entropical reasons driving this issue. We will focus on the ideas developed in [2], in which an entropical framework for describing a general classof complex systems is proposed. The reasons for this choice will look clearer below. By now, we start bysaying that from that paper, it is possible to deduce (see Appendix A) a formula for the log number S(D, A)of all possible pattern configurations (of pixels) having fractal dimension D and area A: S ( D , A )= AU ( D ) s ( D ) (6)where U (D) and s(D) are positive functions defined in Eqs. (A.1) and (A.5) of Appendix A. Entropy S willnot be associated to the total system. The formula will be applied to every particular party having size A anddimension D. Indeed, Eq.(6) will just be applied to the winner party. Figure 6: Averaged (Eq.(4)) timedependence of entropy < S(t) >for the cases in which either Y , L or Gturns out to win (r = 0.15, D Y = 1.7 andD L = 1.6, as in Fig.(3)). Fig.(6) was made by setting X = S(t) = S(D(t), A(t)) in Eq.(4). It shows that entropy increases in time forevery winner party, assuming the same initial conditions of Fig.(3). In general, winner parties overall entropyvariations will be always positive across their possible dimensional range J r , independently of r. This can beobserved on Fig.(7), which was made by taking X equal to the difference∆S K = S(D K (t=∞), A K (t=∞)) - S(D K (t=0), A K (t=0)) (7) for K = Y, L, G in Eq.(4). One can see that every winner party evolves in time toward a more probableconfiguration, increasing its entropy analogously to a gas which irreversibly expands into a box.Now, taking into account the above mentioned over Y and the fact that: ∂ S ( D , A )∂ A > (8) ∂ S ( D , A )∂ D < (9)(see Fig.(11)), it is important to mention two points. First, the relative initial entropy values clearlydetermines the most probable winner (of the contend) between parties Y and L. In other words, Eq.(9) sayshat party Y is initially more organized (has a lower entropy) than L because D Y > D L . Secondly, Eqs. (8) and(9) say that the entropy gradient points out a natural evolution tendency:mostly increasing A, which is an obvious requirement for winning for every party, and decreasing D,something that just Y party turns out to do it. L party does not work in the same way as Y ’s because itsdimension variation is either positive or less negative than Y ’s (Fig.(5)) .In addition, assuming entropy gradient as one of the main ingredients for determining the track the mostprobably winner party will follow, we can see why Y entropy variation is higher than L’s (e.g. Fig.(7), despiteY started from a lower entropy state, i.e.:< ∆SY > > < ∆SL > . (10)We remark again this inequality is computed in the sense of Eq.(4). As a consequence, Y is more probable towin than L, reinforcing the gas thermodynamic analogy mentioned above. Figure 7: Top: Entropy variations everytimethat either Y (◦), L (•) or G (×) wins for r =0.10. Bottom: Corresponding L’s and Y ’s < ∆S >’s for r = 0.15 (note the apparent non-monotonous behaviors with respect to D̄).
The probabilities of winning pK for every party (K = Y, L, G) are shown on Fig.(8). As it was studied in [1],these winning probabilities depend on D̄, which in turn varies across a given Jr . We conveniently condensethe probabilities per party by plotting them in terms of ¯ D − D min (Eq.(1)). igure 8: The probability of winning ofevery party plotted with respect to D min fordifferent participation ratios (r). The probabilities of winning of the parties show several types of regimes with respect to both dimensionalityand participation ratio r (see [1]). As above, let us explain them in terms of entropy production. Actually, inorder to explore the mechanisms that drive a given party to victory, we can delve into the meaning of Eq.(10)and study the relative entropy production between two any given possiblewinning parties (Y , L or G). We quantify this idea as ∆S
KK ′ > =p K < ∆S K > −p K′ < ∆S K ′ > K, K ′=Y, L, G (11)where unlike Eq.(10), every entropy production is now weighed according to the probabilities of winning ofthe parties.
Figure 9: Relative overall entropyvariations (Eq.(11)) with respectto D min for different ratios (r). n Fig.(9) we show every possible < ∆S
KK ′ > for three partipation ratios r, as a function of ¯ D − D min .Note there is a general increasing behavior with respect to the participation ratio r, which clearly correspondsto the fact that S increases with A (see Eq.(6)). Secondly, it is remarkable relative entropy productions of G look clearly different in shape and magnitude,reinforcing its emergent nature. Only initial party relative entropy productions show a strong dependencewith the dimensionalities.For extremely low participation ratios (× - curves of Fig.(8) and (9)), the dynamics is known to be stuck (seealso [1]). This clearly corresponds to comparatively lower relative entropy productions between the parties.In other words, every pair of parties rearranges for winning, contributing to disorder in almost the same way.Consequently, system’s dynamics shows a weak dependence on the initial dimensionalities of Y and L. Thisseems to explain why p Y and p L features look similar between each other. In addition, Fig.(9, middle, bottom)suggests that G has to rearrange comparatively more than Y or L, per unit of relative entropy production.This seems to explain why the emergent party G has a remarkably lower probability of winning than the rest.In contrast, for medium low participation ratios (r = 0.15), the system shows a remarkable ¯ D -dependence (• - curves of Figs. (8) and (9)). There are substantially more dynamics and more probableemergence of new 3rd ideas ([1]), etc. that correspond to comparatively higher relative entropy productions.The existence of non-montonous behaviours on J points out a kind of maximum in the myriad of possibleinitial configurations and subsequent system’s evolution tracks.It is possible to observe on Fig.(9) that every relative entropy variation is almost constant for low dimension-alities, while on the right part of J , they present a maximum (∆S Y L ) or minima (∆S GL and ∆S GY ).Moreover, the increasing and decreasing regimes of • -curves of Figs.(9, top) and Fig.(8, top) clearlycorrespond between each other. The same happens between the • -curves of Fig.(9, middle, bottom) and Fig.(8, bottom).Finally, for the highest low participation ratios (r = 0.20), system’s ¯ D - dependence decreases back (◦ -curves of Figs. (8) and (9)). Non-monotonicity features disappear. The approximately linear relative entropyproductions produces an almost constant ¯ D - dependence for the winning probabilities. This means thatthe most important fact here is that D Y > D L and not the value of ¯ D . The latter also explains why p G becomes comparable to p Y . Actually, despite the natural increments of relativeentropy productions, r = 0.20 lies at the limit of the high apathy regimes, for which the assumptions aboutrandom or hierarchically organized initial configurations become less relevant [1].In summary, dynamics’ ¯ D - dependence remarkably lessens at both extreme of the high apathy regimebecause of different already mentioned reasons. Only for intermediate partipation ratios, dimensionalitydependence seems to produce a kind of optimal and diverse set of system outputs. The just above mentioned regimes clearly depends on the participation parameter r. This idea in principlesounds different from the regimes related to hierarchical behaviours that have been associated with D so far.In order to the study the role that r plays in system dynamics, we will associate an averaged dimension D r toeach interval J r (Eq.(1)): D r = D min + D max (12)The yellow curve of Fig.(10) shows that D r increases with r. The relation between D r and r is apparently non-linear as the ones of D min and D max with respect to r [1]. Thus, we can in some way see the participationparameter as a dimensionality too and viceversa. Let us see how further this supposition could take us.As commented in the Introduction, the source of D- non-monotonicity underlying many aspects of [1] hadnot been analyzed in that paper. We proppose the non-monotonicity is just a counterpart of the fact thatentropy function s(D) developed in [2] has a non-monotonous behavior with respect to D. Moreover, in [2] it
9 Recall that as in [1], we always move within high apathy regimes. s shown how the self-similarity of the succesive derivatives of s(D) with respect to D, provides a naturalway to define different regimes regarding the notion of controllability of the (complex) system. In thecontext of [1] and the present work, we can associate the latter with the notion of predictability/previsibilityabout the final result of the election (or ideas contend) and/or the third party emergence and victory, etc.. Ofcourse, due to the fact that the work [2] is too general and abstract, we cannot expect that all meanings of theregimes exactly fit to the present particular context. However, we claim there are many aspects of theopinion dynamic system considered in our work that clearly corresponds to those regimes. For instance,according to [2], it is possible to define a rangeI = {D : D ≤ D ≤ D }, (13)which can be associated with a kind of most uncontrolable regime in which not only the system’simpredictability is higher but it is also increasing. For λ = 64, it turns out that D = 1.77 and D = 1.51. Figure 10: Relation between the participationdegree r and dimension D. Note that thispicture sums up the existence of an unpre-dictability regime D < D < D . Let us consider again the curve of p G for r = 0.15 shown on Fig.(8, bottom) . The very fact that (D + D )/2≈1.64 is significant because it almost coincides with the local maximum of G probability of winning locatedat ¯ D ≈ 1.63 (it corresponds to ¯ D − D min ≈ 0.18 on the figure). Thus, there is a local weaknessassociated to Y and L initial structures that seems to trigger the occurrence of a higher G probability ofwinning. In other words, the fact that D Y and D L initially lie in I interval, makes the system to bear a kind ofhidden indefinition that fosters an enhancement of G.In addition we see from Figs. (8, bottom) and (10) that the non-monotonicity of p G is apparent when D r isclearly inside of the imprevisibility range I (r = 0.15). If D r is closer to D or D or outside of I , p G willlook more monotonous or constant (r = 0.075 and r = 0.20).Figs. (7, bottom), (8) and (9) also show that non-monotonous behaviors arise when D r is inside of theimprevisibility range (r ≈ 0.15). All these non-monotonous features could be explained with the ideas ofSection 3.1 as follows. The norm of the gradient of S(D, A) presents a minimum with respect to D that is almost independent ofA (Fig.(11)). Remarkably, the minimum is located at D = D ∗ ≈ 1.61, i.e almost at the center of the imprevis-ibility range. This fact clearly should have consequences on the entropy production of the winning party Y ,which evolves in the direction of the gradient of S. Let us consider the r-dependent average
10 D and D are given by s(D ) = max D s(D) and s′(D ) =max D s′ (D), respectively. There is an explicit formula for s(D)in Appendix A.11 see also [1] ∆ S K ⟩= ∫ D min + δDD / D max − δDD / ⟨ ∆ S K ⟩ d ¯ DD max − D min − δDD K =Y, L, G, (14)where the bracket inside of the integral is computed according to Eq.(4) varying ¯ D in a way that D L andD Y lie always within J r (Eq.(1)).In correspondence to Section 3.2, Fig.(12, top) shows that every entropy variation has an increasing behaviorwith respect to r (or D r (Fig.(10)). The minimum gradient effect cannot be observed at glance. Actually, it isa 2nd order effect that can be unveiled after derivating with respect to r, as it is shown on Fig.(12, bottom): d < ∆S K > r /dr (K = Y, L) present either maxima (K = Y, L) or singular points (K = G) arround r ≈ 0.14, i.e atthe center of the imprevisibility range I (Fig.(10)). However, note that the gradient acts primarily over themost structured initial party (Figs. (8, top) and (9, top)). To a less degree, the effect is also present in theemergent party which is a mixture of the initial parties (Figs. (8, bottom) and (9, center and bottom)). Thegradient effects over the less structured initial party are weak and cannot be directly observed in terms ofwinning probability (Figs. (7, bottom)) and (8, center)).Note that in correspondence to the already mentioned fact that p Y > p L , the quantities < ∆S Y > r and< ∆S L > r increase in different ways (Fig.(12, bottom)). Indeed for r ≥ 0.15, < ∆D L > r is positive, so L has byfar less chances of being driven by the gradient than Y (Fig.(13)). Figure 11: The norm of the gradient of S(D, A) hasa minimum as a function of D that is almostindependent of A (see Appendix B). Note these plotsalso show that ∂S(D, A)/∂D = Ad(s/U )/dD < 0 (Eq.(9)).igure 12: < ∆S K > r and its derivative withrespect to the participation ratio r, K = Y, L,G.
4. Conclusions
The entropical analysis turned out to provide an explanation for not only the non-monotonous behaviorspointed out in [1], but for many others D-dependent regimes shown in the present work. The connexionbetween the winning probabilities of the parties and their relative entropy productions is paradigmatic (Figs.(8) and (9)).
Figure 13: r − dependent averages < ∆D > r (asin Eq.(14)) for Y (◦), L (•) and G (×). he different entropical regimes point out an apparent track evolution dependence on the initial structure ofthe parties. This fact can clearly be associated to a kind of system’s memory about both relative and absoluteinitial dimensionalities. Indeed, the most structured initial party is the one that wins most of the time, but thereasons seem to be more profound than the ones exposed in [1]. First, the higher initial dimension, the lowerinitial entropy. Secondly, the entropy gradient mostly drives the winning party divergent temporal evolution(see also Fig.(13)). Its positive relative entropy production with respect to L can be interpreted as a clearobservable of that fact (Fig.(9, top)).Conversely, as it was shown throughout this work, the less structured initial party has an associated narrowbunch of winning tracks that correspond both to less diverse victory set of possibilities and lower entropyproductions. Even in the less probable cases in which L defeats Y by relaxing its ideology (r < 0.15 on Fig.(13)), it does it to a less degree than Y ’s. The correspondence to sociological situation looks clear. In order togain supporters, more organized parties are always able to relax their ideals. In contrast, less organizedpositions necessarily have to modify their speeches according to the apathy level.In turn, G party is different and inevitably has to organize for winning. This entropical analysis also enablesto distinguish its emergent nature from the rest (Sections 3.2 and 3.3).It was observed how the incontrollability ideas mentioned in [2] show up in the form of imprevisibility forpresent opinion dynamics model. The fact that the most diverse regimes in entropy production and winningprobabibilities approximately lie within the range D ≤ D ≤ D is clearly significant, being in plentyagreement to [2].For other interaction frameworks, the rising of incontrolability properties might be also plausible but justinside of I . It will be studied in [22].We think that the moral of this study has to do with ubiquitous non-classical thermodynamical propertiesfound in Complex Systems. These systems’ hierarchical structures drive entropy production in anomalousways. The assumptions of this work (and [1]) have a meaningful connexion with those hierarchical ideas. Acknowledgments
This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET,Argentina). Authors thank Dr. H. S. Wio and Dra. D. A. Pedernera for stimulating and fruitful discussions.
Appendix A. Formula of S(D, A)
According to [2], every pattern of A pixels and fractal dimension D lying in a λ × λ matrix, has to satisfy: 2 mD ≤ A ≤ U = U (D) = λ (2 m ) D−2 , (A.1)where m = log(λ/2). In addition, the number Ω of pixel configurations having dimensionality D and area Ais given by Eq.(5) of [2]: Ω = ∏ k = m f ( A − kD , D A −kD ) , (A.2)where f (x, y, z) turns out to be equal to the number of ways of setting z balls into y boxes with xcompartments, leaving at least one ball per box. Eq.(8) of [2] shows that f ( x , y , z )≈ log ( xyz ) and thatStirling’s approximation can be used (y and z z can be considered large numbers) to arrive at: f ( x , y , z )≈ xy H ( zxy ) (A.3)
12 As in [2], log will be used as a shorthand for log 2. here H is the Shannon’s entropy function: H(t) = −t log t − (1 − t) log(1 − t). Then, taking log in Eq.(A.2) itfollows that S ( D , A )≈ AU ∑ k = m log f ( U − kD , D U − kD )≈ AU s (A.4)where s = s(D) is the log number of pixels configurations having dimensionality D (see Eqs.(16-18) of [2]): s ( D )≈( λ m − ) H ( D − ) mD − D − . (A.5) Appendix B. || S(D, A)|| approximation ∇S(D, A)|| approximation
From Eqs. (A.1) and (A.4) one has that:|| S(D, A)|| ∇S(D, A)|| = A (d(s/U )/dD) + (s/U ) = (A/U) [(s’ − sm ln 2) + (s/A) ]. (B.1)Now, s and s′ are shown to have the same order of magnitude in [2]. In turn, if A represents the initial sizeparty (Y or L), one will have that A -2 =(rλ) -2 << 1 for every reasonable λ and r varying in the high apathyregime [1]. Consequently, the right-side term envolving A -2 of Eq.(B.1) can be neglected arriving at:|| S(D, A)|| ≈ (A/U) |s − sm ln 2| = A|d(s/U )/dD|. ∇S(D, A)|| References [1] M. E. Gaudiano and J. A. Revelli, Physica A 521, 501 (2019).[2] M. E. Gaudiano, Physica A 440, 185 (2015).[3] Crossing in Complexity: Interdisciplinary Application of Physics in Biological and Social Systems, I.Licata and A. Sakaji (Nova Science Publishers, New York, 2010).[4] Econophysics and Sociophysics: Trends and Perspectives, B. K. Chakrabarti, A. Chakraborti, and A.Chatterjee (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006).[5] S. Galam, Physica A 274, 132 (1999).[6] S. Galam, Physica A 336, 56 (2004).[7] S. Galam, Internat. J. Modern Phys. C 19, 409 (2008).[8] W. WeidLich, Physics Reports 204, No. 1, 1 (1991).[9] C. Castellano, S. Fortunato, and V. Loreto, Rev. Modern Phys. 81, 591 (2009).[10] K. Sznajd-Weron, J. Sznajd and T. Weron, Physica A 565 (2021) 125537.[11] Network Science: Complexity in Nature and Technology, E. Estrada, M. Fox, D. J. Higham, and G-L.Oppo (Springer-Verlag, London, 2010).[12] S. Y. Auyang, Foundations of Complex-System Theories in Economics, evolutionary Biology andStatistical Physics (Cambridge Univ. Press, Cambridge, 1998).[13] Y. Bar-Yam, Dynamics of Complex Systems (Perseus Books, New York, 1997).[14] Hierarchy in Natural and Social Sciences, D. Pumain (Springer, Dordrecht, 2006).[15] B. Mandelbrot, The Fractal Geometry of Nature (Macmillan, New York, 1983).[16] Non-equilibrium Thermodynamics and the Production of Entropy, A. Kleidon and R. Lorenz (Springer-Verlag, Heidelberg, 2005).[17] Entropy and Entropy Generation: Fundamentals and Applications, J. Shiner (Kluwer AcademicPublishers, New York, 2002).[18] S. Encarnação, M. Gaudiano, F. C. Santos, J. A. Tenedório, and J. M. Pacheco, Scientific Reports 2, 527(2012).[19] J. Sun, Z. Huang, Q. Zhen, J. Southworth, and S. Perz, Appl. Geogr. 52, 204 (2014).[20] K. Sznajd-Weron and J. Sznajd, International Journal of Modern Physics C 11, No. 06, 1157 (2000).[21] B. Schönfisch and A. de Roos, BioSystems 51, 123 (1999).[22] J. A. Revelli and M. Gaudiano: paper in preparation (2020).
13 (rλ) -2-2