^{1}

^{*}

^{2}

^{1}

^{2}

Edited by: Dumitru Baleanu, University of Craiova, Romania

Reviewed by: Carla M. A. Pinto, Instituto Superior de Engenharia do Porto (ISEP), Portugal; Yilun Shang, Tongji University, China

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The relation between the behavior of a single element and the global dynamics of its host network is an open problem in the science of complex networks. We demonstrate that for a dynamic network that belongs to the Ising universality class, this problem can be approached analytically through a subordination procedure. The analysis leads to a linear fractional differential equation of motion for the average trajectory of the individual, whose analytic solution for the probability of changing states is a Mittag-Leffler function. Consequently, the analysis provides a linear description of the average dynamics of an individual, without linearization of the complex network dynamics.

The last decade has witnessed the blossoming of two quite different strategies for the mathematical modeling of the complex systems, which are network science [

Despite the simplicity of their basic building blocks, complex systems, such as cooperative animal behavior [

Typical description of the dynamics arising from the interaction of numerous basic elements over a complex network that focuses on the global behavior of the system (

In this paper, we address this issue by posing the inverse question. Rather than inferring the global dynamics by combining the behavior of single elements within the dynamical system, we ask whether it is possible to construct a description of the dynamics of the individual elements, provided information about the network's global behavior. We approach the problem by considering statistical properties of the global variable.

Frequently, the macrovariables observed in complex networks display emergent properties of spatial and/or temporal scale-invariance. These are manifested by, for example, the inverse power scaling of waiting-time probability density functions (PDFs) between events, such as communication instances in human interactions or occurrence of earthquakes. At the same time, the inverse power laws (IPLs) that characterize the emergent macroscopic behavior are reminiscent of particle dynamics near a critical point, where a dynamic system undergoes a phase transition [

Herein, we address the problem of quantifying the response of an individual unit to the dynamics of the collective. This is done by taking advantage of the fractional calculus apparatus, whose utility arises from its ability to seamlessly incorporate the IPL statistics into its dynamics. The phase transitions that characterize many complex systems suggest the wisdom of using a generic model from the Ising universality class to characterize system dynamics. It is then possible to demonstrate that the individual trajectory response to the collective dynamics of the system is described by a linear fractional differential equation. This is achieved through a subordination procedure without the necessity of linearizing the underlying dynamics. Following this procedure, it is shown that the analytic solution to the linear fractal differential equation retains the influence of the nonlinear network dynamics on the behavior of the individual. Moreover, the solution to the fractional equation of motion suggests a new direction for designing mechanisms to control the dynamics of complex networks.

In section 2, we sketch out the mathematics of the dynamical decision making model (DMM), introduce renewal events, and subordinate the behavior of the individual to the mean field behavior of the network. In section 2.2, the dynamics of the individual is determined from the subordination theory to be a tempered fractional differential equation. The exact solution to this equation is given by an attenuated Mittag-Leffler function, which is fitted to the numerical solution of the DMM equation. In section 4, we discuss some implications of the high quality convergence of the analytical and numerical results of this complex network.

As demonstrated by Grinstein et al. [_{i} of the model is a stochastic oscillator and can be found in either of the two states, +1 or −1. The dynamics are defined in terms of the probability of an individual to be in either state, and it is modeled by the coupled two-state master equation,

where _{0} < 1.

Positioning _{i} experiences due to the presence of its neighbors is expressed by a modification of its transition rate

which becomes a time dependent variable. Here, ^{(i)} denotes the degree of the node _{i}(_{i}(_{i} change their states, quantities ^{(i)} = 4 and

Time-dependent transition rates modify the two-state master (Equation 1) to take the form

where the matrix of rates _{i}(

and ^{(i)}(

Dynamics of an entire network is described by a system of

As depicted in Figure

shows a pronounced transition as a function of the control parameter

Behavior of a discrete, two-state dynamic unit on a two-dimensional lattice. Temporal evolution and corresponding survival probability Ψ(τ) for the transitions between two states for the single unit _{i}(_{0} = 0.01 and increasing values of the control parameter _{C} ≈ 1.72. Black dashed line on the plots of Ψ(τ) denotes an exponential distribution, with the decay rate _{0}.

To characterize the changes in the temporal properties of the micro- and macro-variables, we evaluate the survival probability function, Ψ(τ), of time intervals τ between consecutive events defined as changes of the state or crossing of the zero-axis, for the single element or the global variable, respectively. These calculations unveil modest deviations of Ψ(τ) for a single individual from the exponential form, Ψ(τ) = _{0}τ), that characterizes single non-interacting elements, as shown in Figure

Many physical processes, for example earthquakes, radioactive decay, and social processes, such as making a decision, can be viewed as particular events. A characteristic property of an event is that it's onset can be precisely localized in time, even if its occurrence has extended consequences in space. Thus, the dynamics of a process characterized by events is described in terms of the probability of an event occurring, rather than by a more traditional Hamiltonian approach.

The process of event occurrence is characterized by the waiting-time PDF ψ(τ), which specifies the distribution of times between consecutive events. The probability for an event to occur in the short time interval [

where τ is measured from the occurrence of the previous event. Consequently, one can define the survival probability Ψ(τ) as the probability that no event occurs up to the time since the last event as

As a consequence of this integral, the waiting-time PDF can be written as

and the PDF ψ(τ) is a properly normalized function,

since it is assumed that an event occurs somewhere within the time interval (0, ∞). It is also true that no event occurs at time

A particular class of events can be defined,

The renewal character of events is captured by the probability of _{0}(_{1}(

Frequently, experimentally observed waiting-time PDFs are exponential, but quite often in complex networks they are IPLs. For the purpose of this paper, we define the waiting-time PDF in terms of the hyperbolic distribution

If the events are generated by an ergodic process, then μ > 2, and the first moment of the hyperbolic PDF is

In the framework of renewal theory, Equation (11) denotes the average time that one would have to wait between successive events. However, when μ < 2, the process is non-ergodic, and the mean value of the distribution diverges. In the non-ergodic case,

The notion of different clocks associated with different physical systems arises naturally in physics; the linear Lorentz transformation in relativistic physics being probably the most familiar example. Thanks to the recent availability of time-resolved data, biological, and social sciences have also started adopting the notion of multiple clocks, distinguishing between cell-specific and organ-specific clocks in biology and person-specific and group-specific clocks in sociology. Of course, the notion of subjective and objective time dates back to the middle of the nineteenth century with the introduction of the empirical Weber-Fechner law [

However, the striking difference between the clocks of classical physics and natural sciences is that the relations between the latter clocks are nonlinear. While the global activity of an organ, such as the brain or the heart, might be characterized by quite regular, often periodic fluctuations, the activity of single neurons demonstrates burstiness and noisiness. Similarly, in a society, people operate according to their individual schedules, not always being able to perform particular actions in the same global time frame. Thus, owing to the stochastic behavior of one or both clocks, a probabilistic transformation between times is necessary. An example of such a transformation is the subordination procedure.

We begin by defining two clocks. The first clock records a discrete operational time

Every advancement of the operational clock is an event, which in the chronological time occurs at time intervals drawn from the renewal waiting-time PDF. Because of this randomness, one needs to sum over all events, and the result is an average over many realizations of the transformation.

As an example, consider the behavior of a two-state operational clock, whose evolution is shown in Figure

The upper curve is the regular transition between the two states of the individual in operational time. The lower curve is the subordination of the transition times to an IPL PDF to obtain chronological time.

We note that the time subordination procedure can also be used to model communication delays in the system. However, contrary to frequently used approaches, where individual units of the system are subordinated to model the interaction delay, here, we adopt the statistics of the macroscopic variable to derive the behavior of the interacting individual units. The coupling between units causes them to deviate from the Poisson behavior of an individual non-interacting unit. However, as illustrated in Figure

To determine the network's influence on the dynamics of the individual, we adapt the subordination argument of the preceding section and relate the time scale of the macro-variable ξ(_{i}(

where the notations φ(_{1} − _{2} depict the difference in probabilities for the typical individual to assume one of the two states. The solution to this discrete equation is

which, in the limit _{0} Δ τ < < 1, becomes an exponential. However, when the individual is a part of a network, the dynamics are not so simple.

Adopting the subordination interpretation, we define the discrete index

Here, the time

The dominant behavior of the empirical survival probability is an IPL as indicated in Figure

Using a renewal theory argument, Pramulkkul et al. [

where λ_{0} ≡ _{0} Δ τ and

Note that

so that Equation (17) reduces to

The inverse Laplace transform of Equation (20) yields the tempered rate equation

where the operator

Note that owing to the dichotomous nature of the states, 〈φ(_{i}(

The solution of the asymptotic fractional master equation (Equation 21) for a randomly chosen unit within the network is given by an exponentially attenuated Mittag-Leffler function (MLF):

and the MLF is defined by the series

The MLF is a stretched exponential at early times and an IPL at late times, with α = μ − 1 being the IPL index in both domains.

We test the above analysis with numerical simulations of the dynamic network on a two-dimensional lattice with nearest-neighbor interactions in all three regions of DMM dynamics: subcritical, critical, and supercritical. The time-dependent average opinion of a randomly chosen individual is presented in Figure ^{4} independent realizations of the dynamics in the subcritical, critical, and supercritical regimes.

The probability difference 〈φ(^{4} independent realizations of single element trajectories. Each trajectory corresponds to evolution of a randomly selected node within a _{0} = 0.01 and the same initial condition _{i}(0) = 1. The parameter values for the numerical data are given in Figure

A comparison with the exponential form of 〈φ(

The probability difference 〈φ(

μ | 1.8920 | 1.8050 | 1.5580 |

λ | 0.0147 | 0.0206 | 0.0293 |

ϵ | 4.00 × 10^{−3} |
1.40 × 10^{−11} |
5.58 × 10^{−12} |

^{2}^{1} |
0.9910 | 0.9667 | 0.9725 |

_{0} = 0.01, the parameters of an analytical solution are μ = 3/2 and λ = 0.0318

Herein, the subordination procedure provides an equivalent description of the average dynamics of a single individual within a complex network, in terms of a linear fractional differential equation. The fractional rate equation is solved exactly, determining the Poisson statistics of the isolated individual becomes attenuated Mittag-Leffler statistics, owing to the interaction of that individual with the other members of a complex dynamic network.

Consequently, an individual's simple random behavior, when isolated, is replaced with behavior that might serve a more adaptive role in social networks. We conjecture that the behavior of the individual is generic, given that the DMM network dynamics belong to the Ising universality class. Members of this universality class share the critical temporal behavior [

As pointed out by Liu et al. [

BW developed the theoretical formalism and performed the analytic calculations. MT performed numerical simulations. Both authors discussed the results and contributed to the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge the US Army Research Laboratory for supporting this research.

_{α, β}, (

^{1}Adjusted goodness of fit, _{reg}) and for the fit to the average value of data points (_{tot}), where