Growth and addition in a herding model with fractional orders of derivatives

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where 0<α⩽1 . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as nk=p(2p−1)(2−α)p(1−α)+1Γ(α+2−α1−p)Γ(k)Γ(k−1+α+2−α1−p)t for α∈(0,1] , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p, thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.


Introduction
The ubiquitous network characteristics in complex systems [1][2][3] has fascinated the scientific community for decades.This fascination stems from the challenge the field poses to the reductionism [4] and determinism commonly employed by scientists; that by analysing the parts, the whole can be understood.However, science has constantly defies this notion and there has been ample evidence that suggests that given sufficient complexity in a network of interacting agents, the system may display an emergent behavior.This is the 'more is different' paradigm [5].This emergent behavior has been studied in many different fields.Emergence is commonly believed to be a result of self-organized criticality (SOC) [6], as illustrated by the sand pile models.Systems at SOC, which are characterized by scale invariance [6], take a bottom-up approach where local interactions in an initially disordered system allow for the creation of an optimized solution to a given problem without requiring interference from an external agent, an attribute shared by the human brain [7].The presence of thresholds and an intermediate level of connectivity, which are responsible for the occurrence of SOC, draw resemblance to the synapses and neural networks in the brain respectively.As such, the results of SOC have been applied to research on neural networks, artificial intelligence and robotics.This self-organizing behavior also found its way into the natural sciences.In statistical physics, SOC occurs in the form of phase transitions and symmetry breaking in spontaneous magnetization [8] and Bose-Einstein condensate [9].In chemistry, molecular self-assembly is attributed to SOC where molecules are rearranged without the guidance from an outside source [10].In biology, animal flocking behavior, also known as herding effect [11][12][13] in other contexts, is a common example of individuals acting in a coherent manner without coordination amongst them.The theme is also at the centre of the research area of active matter.The herding effect does not limit itself to animals as humans have also been shown to display such behaviors.Common examples include the fat-tail return-size distribution in the financial markets [11] and the hospital waiting lists [14].Given the interest on the topic from the scientific community, this work, whilst using the growth and addition features from Rodgers and Yap [15], develops an original herding model with fractional orders of derivatives with the aim of generalizing previous results in herding models [11,13,15], allowing for a better understanding of the mechanics behind a SOC model of herding in complex networks.Previous models are frequently based on statistical mechanics and classical calculus with integer orders.The use of fractional calculus on models, such as the studies on non-linear impurity problems in one and two dimensional lattices [16,17], will yield important and interesting results which are not otherwise seen in models with differentiation with integer orders.Fractional calculus and its applications to problems in physics, engineering and technology have attracted much attention in recent years [18][19][20][21].The definition and a few properties of Riemann-Liouville (R-L) fractional derivatives which will be used in the following sections are presented below.More details on fractional calculus can be found, for example, in Kilbas et al [18].The R-L fractional integral I α y of order α > 0 is defined by where Γ(α) is the Gamma Function, and the Caputo fractional derivative D α y of order α > 0 is defined by where, n − 1 < α < n, n ∈ N. It follows that and We note that for 0 < α ⩽ 1, we have

Model
The agent-based numerical herding model of Rodgers and Yap [15] incorporates the phenomena of growth and addition.This section starts with an introduction to the algorithm of the model.One of two possible events occur at every time step.The system may grow by introducing a new agent who remains free.This is the growth event.Alternatively, the system may allow an existing free agent in the system to join a group of size k randomly at a rate which is proportional to the group size.This is the addition event.The probability of growth and addition events are denoted by p and q = 1 − p respectively.Let n k (t) denote the number of groups in the system with size k at time t.The total number of groups is and the total number of agents in the system is The time-evolution of n k>1 and n 1 are governed by [15] dn and Analysing the dynamics revealed a power-law group-size distribution and all moments of sizes of groups scale linearly in t when the latter are large for p > 1/2, while the p = 1/2 and p < 1/2 cases correspond to a static system and a system which is devoid of free agents in t respectively.Due to a greater interest placed in the dynamical, as opposed to static, version of the model, we will restrict ourselves to studying the p > 1/2 version of our generalized model.
To extend the study, we invoke fractional calculus and generalize the governing dynamical equations to and From equation ( 8), we have giving Therefore, using equation ( 9) and thus The case of α = 1 gives D 1 N(t) = dN dt = 2p − 1, as reported [15].In the original work, M = pt for p > 1/2.It follows from equation (7) that in the original model.Considering the fractional calculus generalization, we have Using equation (13), D α M(t) and D α n 1 can be related as which is analogous to equation (14).
To determine N(t) in the generalized model, we solve equation ( 13) with the initial conditions N(0) = 0. Applying equation (5) to equation ( 13), we have The results for N(t) is the same as in the model in [15].In particular, the prefactor (2p − 1) suggests that the criterion for the number of groups to be increasing with time remains p > 1/2.Although the criterion is independent of the value of α, the group-size distribution for p > 1/2 does depend on α as we now show.As in the original model, we assume that for p > 1/2, the solution for k = 1, 2, 3 . . .for n k is linear in t for t → ∞, so that n k (t) = c k t, and using this together with equation ( 8) we obtain, for k > 1, It follows that which can be further simplified to upon using M(t) = pt.This is a recursive relation between the coefficients c k and c k−1 for the group sizes n k (t) and n k−1 (t).Equation ( 20) includes the previous result for α = 1 [15] as a special case.
The remaining coefficient c 1 can be solved by a similar procedure.From equation ( 9), we have It follows that which can be solved for c 1 to give Finally, the recursive relation (equation ( 19)) gives the coefficients c k for k > 1 as and the group-size distribution for α ∈ (0, 1] as For p > 1/2, there will be many groups of various sizes in the long time limit.Apart from the small groups, the distribution of the group sizes follows the form asymptotically as k → ∞, which is reminiscent of the power law.This result generalized the previous result c k ∼ k − 1 1−p when α = 1 [15].From equation (26), it can be seen that the exponent is from 2 to ∞, which agrees with the requirement in the power law for a meaningful mean.The power law in (26) is the generalized power law.
Figures 1-6 show the plots of n k as given by equation ( 25) for different values of α and p.The α = 1 case corresponds to the result reported previously [15], which has the form of n k ∼ k −β , with β → 2 as p → 1/2 and β → ∞ as p → 1. Invoking fractional calculus with 0 < α ⩽ 1, the asymptotic behavior becomes n (α) k ∼ k −β , with β → 3 − α as p → 1/2 and β → ∞ as p → 1.Therefore, the group sizes n k drop more rapidly with k when the order of differentiation is less than unity and there will be more groups of smaller sizes.As α → 0, most of the free agents introduced remain free.This dependence on the value of α is more noticeable in the results of p = 0.6 in figures 1, 3 and 5 for which n k drops with increasing k more rapidly for small values of α.For p = 0.9, n k drops rapidly with increasing k when the exponent β is large for all values of α.

Result analysis
As in [22], the left-sided R-L fractional integral in (1) can be rewritten as where From equations ( 27) and ( 28), it can be seen that mathematically, the physical interpretation of fractional differentiation and fractional integration is based on the introduction of a third dimension, g τ (t), to the classical pair of t and y(t).It can be seen from ( 28) that if t is changed by a factor of k, the g τ t is changed by a factor of k α .If we multiply this factor to the t in n k = c k t, this factor will be propagated to (26), thereby reproducing c k ∼ k − 1 1−p on the condition that α = p.This carries two interesting implications.Firstly, it implies that if the fractional order is α, an attempted change in the time t by a factor of k results in an actual change of the time by a factor of k α .Thus, for k = 1, which is the most common group size, time is counted in unit steps, and there is no apparent change in the time scale.Time scale also remains unchanged for α = 1.However, for 0 < α < 1 and k > 1, there is a deformation of the time t by a factor of k α .
Secondly, it implies that in order to observe the same phenomenon, and in this case the group size-distribution depicted in (26), the rate of growth p of the system must bring about a change in the original time frame by a factor of k p where p = α.The physical implication of this is explored in the following subsection.This scaling property is reminiscent of complex systems with self-similar characteristics.

Physical interpretation of fractional differentiation and fractional integration
Based on the discussion in the previous subsection, we deduce that there are two time scales [22]; the individual time scale and the cosmic time scale corresponding to the t and g τ (t) respectively.The individual time scale has equally spaced time intervals.It is the set of time intervals observed by an individual after taking into account the last recorded time τ , and since the latter is the last observation of time, the person can only assume that it is constant and that it does not affect his future calculation of time changes.However, in reality, the τ changes continuously, which affects subsequent calculations of time intervals, and the actual time scale is thus inhomogeneous and corresponds to g τ (t).

Conclusion
The time dependence of size distribution of groups of agents in [15] is re-modelled in this work through the proposal of a general version of the model in the original work by including derivatives of order α where 0 < α ⩽ 1.For the case with growth and addition, which is the main focus of the original work, we have obtained the generalized result for the number of groups of size k as t. for α ∈ (0, 1].We have also shown that the result in the [15] is reproduced when we set α = 1.The results corresponding to various values of α and p have been presented.A power-law decay of the group-size distribution in the long time limit is extracted.The exponent depends on both α and p. In particular, we have found that the group sizes n k drop more rapidly with k when the order of differentiation is less than unity.This results in having more groups with smaller sizes.As α → 0, most of the free agents introduced remain free, and this dependence on the value of α is more noticeable in the results of p farther away from unity, e.g.p = 0.6, than p closer to unity.This leads to an implication of a system consisting largely of free agents when the differential order of the rate of change of the number of groups with size k approaches zero.This effect of a fractional exponent on the size distribution of groups is reminiscent of the said effect on the bandwidth in [23].In addition, the use of fractional calculus in this work, just like the case of Newton's Law of Cooling in [24], allows for a better representation of the system.More specifically, the real value fractional model we have developed has demonstrated, through equation ( 26), that the scale invariance property of a complex system holds regardless of the time scale of the observations.A generalized power law has also been developed as a result.
In summary, we have developed a the herding model with fractional orders of derivatives which provides insights on a fractional form of the system with growth and addition.We have also presented the effect of the fractional exponent on the group-size distribution, and demonstrated the universality of scale invariance in a complex system.The physical interpretation of fractional calculus is also explored based on the results of this work.

Figure 1 .
Figure 1.Plot of n k against k for α = 0.1 and p = 0.6.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.

Figure 2 .
Figure 2. Plot of n k against k for α = 0.1 and p = 0.9.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.

Figure 3 .
Figure 3. Plot of n k against k for α = 0.5 and p = 0.6.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.

Figure 4 .
Figure 4. Plot of n k against k for α = 0.5 and p = 0.9.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.

Figure 5 .
Figure 5. Plot of n k against k for α = 1.0 and p = 0.6.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.

Figure 6 .
Figure 6.Plot of n k against k for α = 1.0 and p = 0.9.The number of time-steps used for the plotting of all graphs is 100 000.A log-log plot is shown in the inset.