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We apply modified diffusion entropy analysis (MDEA) to assess multifractal dimensions of ON time series (ONTS) and complexity synchronization (CS) analysis to infer information transfer among ONs that are part of a network of organ networks (NoONs). The purpose of this paper is to advance the validation, standardization, and repeatability of MDEA and CS analysis of heterogeneous neurophysiological time series data. Results from processing these datasets show that the complexity of brain, heart, and lung ONTS significantly co-vary over time during cognitive task performance but that certain principles, guidelines, and strategies for the application of MDEA analysis need consideration.

We introduce a dynamic model for complexity control (CC) between systems, represented by time series characterized by different temporal complexity measures, as indicated by their respective inverse power law (IPL) indices. Given the apparent straightforward character of the model and the generality of the result, we formulate a hypothesis based on the closeness of the scaling measures of the model to the empirical complexity measures of the human brain. CC is a proper model for describing the recent experimental results, such as the rehabilitation in walking arm in arm and the complexity synchronization effect. The CC effect can lead to the design of mutual-adaptive signals to restore the misaligned complexity of maladjusted organ networks or, on the other hand, to disrupt the complexity of a malicious system and lower its intelligent behavior., Comment: 12 pages

The analysis of glioblastoma (GB) cell locomotion and its modeling inspired by Levy random walks is presented herein. We study such walks occurring on a two-dimensional plane where the walk is similar to the motion of a bird flying with a constant velocity, but with random changes of direction in time. The intelligence of the bird is signaled by the instantaneous changes of flying direction, which become invisible in the time series obtained by projecting the 2D walk either on the x axis or the y axis. We establish that the projected 1D time series share the statistical complexity of time series frequently used to monitor physiological processes, shedding light on the role of crucial events (CE-s) in pathophysiology. Such CE-s are signified by abrupt changes of flying direction which are invisible in the 1D physiological time series. We establish a connection between the complex scaling index \delta generated by the CE-s through \mu_{R} = 2 - \delta , where \mu_{R} is the inverse power law index of the probability density function of the time interval between consecutive failures of the process of interest. We argue that the identification of empirical indices along with their theoretical relations afford important measures to control cancer., Comment: 30 pages, 44 references, 13 figures

In this work, we use a simple multi-agent-based model (MABM), implementing selfish algorithm (SA) agents, to create an adaptive environment and show, using modified diffusion entropy analysis (MDEA), that the mutual-adaptive interaction between the parts of such a network manifests complexity synchronization (CS). CS has been experimentally shown to exist among organ-networks (ONs) of the brain (neurophysiology), lungs (respiration), and heart (cardiovascular reactivity) and to be explained theoretically as a synchronization of the multifractal scaling parameters characterizing each time series. Herein, we find the same kind of CS in the emergent intelligence (i.e., without macroscopic control and based on self-interest) between two groups of agents playing an anti-coordination game, thereby suggesting the potential for the same CS in real-world social phenomena and in human-machine interactions., Comment: 28 pages, 12 Figures

Noise-induced phase transitions are common in various complex systems, from physics to biology. In this article, we investigate the emergence of crucial events in noise-induced phase transition processes and their potential significance for understanding complexity in such systems. We utilize the first-passage time technique and coordinate transformations to study the dynamics of the system and identify crucial events. Furthermore, we employ Diffusion Entropy Analysis, a powerful statistical tool, to characterize the complexity of the system and quantify the information content of the identified events. Our results show that the emergence of crucial events is closely related to the complexity of the system and can provide insight into its behavior. This approach may have applications in diverse fields, such as climate modeling, financial markets, and biological systems, where understanding the emergence of crucial events is of great importance.

The observational ubiquity of inverse power law spectra (IPL) in complex phenomena entails theory for dynamic fractal phenomena capturing their fractal dimension, dynamics, and statistics. These and other properties are consequences of the complexity resulting from nonlinear dynamic networks collectively summarized for biomedical phenomena as the Network Effect (NE) or focused more narrowly as Network Physiology. Herein we address the measurable consequences of the NE on time series generated by different parts of the brain, heart, and lung organ networks, which are directly related to their inter-network and intra-network interactions. Moreover, these same physiologic organ networks have been shown to generate crucial event (CE) time series, and herein are shown, using modified diffusion entropy analysis (MDEA), to have scaling indices with quasiperiodic changes in complexity, as measured by scaling indices, over time. Such time series are generated by different parts of the brain, heart, and lung organ networks, and the results do not depend on the underlying coherence properties of the associated time series but demonstrate a generalized synchronization of complexity. This high order synchrony among the scaling indices of EEG (brain), ECG (heart), and respiratory time series is governed by the quantitative interdependence of the multifractal behavior of the various physiological organs' network dynamics. This consequence of the NE opens the door for an entirely general characterization of the dynamics of complex networks in terms of complexity synchronization (CS) independently of the scientific, engineering, or technological context.

We propose a model for demonstrating spontaneous emergence of collective intelligent behavior from selfish individual agents. Agents' behavior is modeled using our proposed selfish algorithm ($SA$) with three learning mechanisms: reinforced learning ($SAL$), trust ($SAT$) and connection ($SAC$). Each of these mechanisms provides a distinctly different way an agent can increase the individual benefit accrued through playing the prisoner's dilemma game ($PDG$) with other agents. The $SA$ provides a generalization of the self-organized temporal criticality ($SOTC$) model and shows that self-interested individuals can simultaneously produce maximum social benefit from their decisions. The mechanisms in the $SA$ are self-tuned by the internal dynamics and without having a pre-established network structure. Our results demonstrate emergence of mutual cooperation, emergence of dynamic networks, and adaptation and resilience of social systems after perturbations. The implications and applications of the $SA$ are discussed.

Dunbar hypothesized that $150$ is the maximal number of people with whom one can maintain stable social relationships. We explain this effect as being a consequence of a process of self-organization between $N$ units leading their social system to the edge of phase transition, usually termed criticality. Criticality generates events, with an inter-event time interval distribution characterized by an inverse power law (IPL) index $\mu_{S}<2$. These events break ergodicity and we refer to them as crucial events. The group makes decisions and the time persistence of each decision is given by another IPL distribution with IPL index $\mu_{R}$, which is different from $\mu_{S}$ if $N\neq 150$. We prove that when the number of interacting individuals is equal to $150$, these two different IPL indexes become identical, with the effect of generating the Kardar Parisi Zhang (KPZ) scaling $\delta =1/3$. We argue this to be an enhanced form of intelligence, which generates efficient information transmission within the group. We prove the inflrmation transmission efficiency is maximal when $N=150$, the Dunbar number., Comment: Main text 5 pages with 4 figures; supplementary material 7 pages with 5 figures

Complexity matching characterizes the role of information in interactions between systems and can be traced back to the 1957 Introduction to Cybernetics by Ross Ashby. We argue that complexity can be expressed in terms of crucial events, which are generated by the processes of spontaneous self-organization. Complex processes, ranging from biological to sociological, must satisfy the homeodynamic condition and host crucial events that have recently been shown to drive the information transport between complex systems. We adopt a phenomenological approach, based on the subordination to periodicity that makes it possible to combine homeodynamics and self-organization induced crucial events. The complexity of crucial events is defined by the waiting-time probability density function (PDF) of the intervals between consecutive crucial events, which have an inverse power law (IPL) PDF $\psi (\tau )\propto 1/(\tau )^{\mu }$ with $1<\mu <3$. We show that the action of crucial events has an effect compatible with the shared notion of complexity-induced entropy reduction, while making the synchronization between systems sharing the same complexity different from chaos synchronization. We establish the coupling between two temporally complex systems using a phenomenological approach inspired by models of swarm cognition and prove that complexity matching, namely sharing the same IPL index $\mu $, facilitates the transport of information, generating perfect synchronization. This new form of complexity matching is expected to contribute significantly to progress in understanding and improving biofeedback therapies., Comment: 21 pages, 8 figures

The battlefield of the future will be densely populated by a variety of entities ("things") -- some intelligent and some only marginally so -- performing a broad range of tasks: sensing, communicating, acting, and collaborating with each other and human warfighters. We call this the Internet of Battle Things, IoBT. In some ways, IoBT is already becoming a reality, but 20-30 years from now it is likely to become a dominant presence in warfare. To become a reality, however, this bold vision will have to overcome a number of major challenges. As one example of such a challenge, the communications among things will have to be flexible and adaptive to rapidly changing situations and military missions. In this paper, we explore this and several other major challenges of IoBT, and outline key research directions and approaches towards solving these challenges., Comment: This is a version of the article that appears in IEEE Computer as: Kott, Alexander, Ananthram Swami, and Bruce J. West. "The Internet of Battle Things." Computer 49.12 (2016): 70-75

Posturography is routinely used to qualitatively assess one of the cardinal symptoms of Parkinson's disease -- postural instability. While most measures of balance control are derived from displacement of the center of pressure there is evidence that such control is more likely to be velocity-based. We performed static posturographic tests (eyes open and eyes closed) during quiet standing in narrow stance for n=30 patients with Parkinson's disease (PDP) in the ON state and compared the results with those of n=30 age-matched senior controls (HSC) and n=60 young controls (HYC). We used differentiator filters to generate time series of low-frequency fluctuations of sway velocity and calculated their Lempel-Ziv complexity (LZC). With eyes closed, the mediolateral LZC of HSC 0.21 (0.02) was significantly higher than those of HYC 0.19 (0.02) and PDP 0.18 (002). Thus aging and PD have opposite effects on mediolateral LZC which strongly differentiates between HSC and PDP (92% sensitivity and 87% specificity).

We present a technique to search for the presence of crucial events in music, based on the analysis of the music volume. Earlier work on this issue was based on the assumption that crucial events correspond to the change of music notes, with the interesting result that the complexity index of the crucial events is mu ~ 2, which is the same inverse power-law index of the dynamics of the brain. The search technique analyzes music volume and confirms the results of the earlier work, thereby contributing to the explanation as to why the brain is sensitive to music, through the phenomenon of complexity matching. Complexity matching has recently been interpreted as the transfer of multifractality from one complex network to another. For this reason we also examine the mulifractality of music, with the observation that the multifractal spectrum of a computer performance is significantly narrower than the multifractal spectrum of a human performance of the same musical score. We conjecture that although crucial events are demonstrably important for information transmission, they alone are not suficient to define musicality, which is more adequately measured by the multifractality spectrum.

The crucial aspect of this demonstration is the discovery of renewal events, hidden in the computed dynamics of a multifractal metronome, which enables the replacement of the phenomenon of strong anticipation with a time delayed cross-correlation between the driven and the driving metronome. We establish that the phenomenon of complexity matching, which is the theme of an increasing number of research groups, has two distinct measures. One measure is the sensitivity of a complex system to environmental multi-fractality; another is the level of information transfer, between two complex networks at criticality. The cross-correlation function is evaluated in the ergodic long-time limit, but its delayed maximal value is the signature of information transfer occurring in the non ergodic short-time regime. It is shown that a more complex system transfers its multifractality to a less complex system while the reverse case is not possible.

We study the connection between multi-fractality and crucial events. Multi-fractality is frequently used as a measure of physiological variability. Crucial events are known to play a fundamental role in the transport of information between complex networks. To establish a connection we focus on the special case of heartbeat time series and on the search for a diagnostic prescription to distinguish healthy from pathologic subjects. Over the last twenty years two apparently different diagnostic techniques have been established: the first is based on the observation that the multi-fractal spectrum of healthy patients is broader than the multi-fractal spectrum of pathologic subjects; the second is based on the observation that heartbeat dynamics are a superposition of crucial and Poisson events, with pathologic patients hosting Poisson events with larger probability than the healthy patients. In this paper, we prove that increasing the percentage of Poisson events hosted by heartbeats has the effect of making their multi-fractal spectrum narrower, thereby establishing that the two different diagnostic techniques are compatible with one another and, at the same time, establishing a dynamic interpretation of multi-fractal processes that has been previously overlooked., Comment: 9 pages and 11 figures. Submitted to PRE

There is mounting evidence of a link between the properties of electroencephalograms (EEGs) of depressive patients and the outcome of pharmacotherapy. The goal of this study was to develop an EEG biomarker of antidepressant treatment response which would require only a single EEG measurement. We recorded resting, 21-channel EEG in 17 inpatients suffering from bipolar depression in eyes closed and eyes open conditions. The EEG measurement was performed at the end of the short washout period which followed previously unsuccessful pharmacotherapy. We calculated the normalized wavelet power of alpha rhythm using two referential montages and an average reference montage. In particular, in the occipital (O1, O2, Oz) channels the wavelet power of responders was up to 84% higher than that of nonresponders. Using a novel classification algorithm we were able to correctly predict the outcome of treatment with 90% sensitivity and 100% specificity. The proposed biomarker requires only a single EEG measurement and consequently is intrinsically different from biomarkers which exploit the changes in prefrontal EEG induced by pharmacotherapy over a given time., Comment: 19 pages, 3 figures

Optimization of energy cost determines average values of spatio-temporal gait parameters such as step duration, step length or step speed. However, during walking, humans need to adapt these parameters at every step to respond to exogenous and/or endogenic perturbations. While some neurological mechanisms that trigger these responses are known, our understanding of the fundamental principles governing step-by-step adaptation remains elusive. We determined the gait parameters of 20 healthy subjects with right-foot preference during treadmill walking at speeds of 1.1, 1.4 and 1.7 m/s. We found that when the value of the gait parameter was conspicuously greater (smaller) than the mean value, it was either followed immediately by a smaller (greater) value of the contralateral leg (interleg control), or the deviation from the mean value decreased during the next movement of ipsilateral leg (intraleg control). The selection of step duration and the selection of step length during such transient control events were performed in unique ways. We quantified the symmetry of short-term control of gait parameters and observed the significant dominance of the right leg in short-term control of all three parameters at higher speeds (1.4 and 1.7 m/s)., Comment: 11 pages, 4 figures

We consider the coupling between two networks, each having N nodes whose individual dynamics is modeled by a two-state master equation. The intra-network interactions are all to all, whereas the inter-network interactions involve only a small percentage of the total number of nodes. We demonstrate that the dynamics of the mean field for a single network has an equivalent description in terms of a Langevin equation for a particle in a double-well potential. The coupling of two networks or equivalent coupling of two Langevin equations demonstrates synchronization or antisynchronization between two systems, depending on the sign of the interaction. The anti-synchronized behavior is explained in terms of the potential function and the inter-network interaction. The relative entropy is used to establish that the conditions for maximum information transfer between the networks are consistent with the Principle of Complexity Management and occurs when one system is near the critical state. The limitations of the Langevin modeling of the network coupling are also discussed.

A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the fractional Riccati equation, the fractional logistic equation and a fractional cubic equation. The solutions reduce to those of the ordinary nonlinear differential equations, when the order of the fractional derivative is $\alpha =1$. The exact analytic solutions to the fractional nonlinear differential equations are not known, so we evaluate how well the derived solutions satisfy the corresponding fractional dynamic equations. In the three cases we find a small, apparently generic, systematic error that we are not able to fully interpret.

A cooperative network model of sociological interest is examined to determine the sensitivity of the global dynamics to having a fraction of the members behaving uncooperatively, that is, being in conflict with the majority. We study a condition where in the absence of these uncooperative individuals, the contrarians, the control parameter exceeds a critical value and the network is frozen in a state of consensus. The network dynamics change with variations in the percentage of contrarians, resulting in a balance between the value of the control parameter and the percentage of those in conflict with the majority. We show that the transmission of information from a network $B$ to a network $A$, with a small fraction of lookout members in $A$ who adopt the behavior of $B$, becomes maximal when both networks are assigned the same critical percentage of contrarians., Comment: 5 pages, 3 figures, 1 supplement

A two-state master equation based decision making model has been shown to generate phase transitions, to be topologically complex and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

We consider a dual model of decision making, in which an individual forms its opinion based on contrasting mechanisms of imitation and rational calculation. The decision making model (DMM) implements imitating behavior by means of a network of coupled two-state master equations that undergoes a phase transition at a critical value of a control parameter. The evolutionary spatial game (EGM), being a generalization of the Prisoner's dilemma game, is used to determine in objective fashion the cooperative or anti-cooperative strategy adopted by individuals. Interactions between two sources of dynamics increases the domain of initial states attracted to phase transition dynamics beyond that of the DMM network in isolation. Additionally, on average the influence of the DMM on the game increases the final observed fraction of cooperators in the system.

We use a Cooperative Decision Making (CDM) model to study the effect of committed minorities on group behavior in time of crisis. The CDM model has been shown to generate consensus through a phase-transition process that at criticality establishes long-range correlations among the individuals within a model society. In a condition of high consensus, the correlation function vanishes, thereby making the network recover the ordinary locality condition. However, this state is not permanent and times of crisis occur when there is an ambiguity concerning a given social issue. The correlation function within the cooperative system becomes similarly extended as it is observed at criticality. This combination of independence (free will) and long-range correlation makes it possible for very small but committed minorities to produce substantial changes in social consensus.

We study the transport of information between two complex networks with similar properties. Both networks generate non-Poisson renewal fluctuations with a power-law spectrum 1/f^(3-\mu), the case \mu= 2 corresponding to ideal 1/f-noise. We denote by \mu_S and \mu_P the power-law indexes of the network "system" of interest S and the perturbing network P respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal condition of 1/f-noise for both networks corresponds to maximal information transport. We prove that to make the network S respond when \mu_S < 2 we have to set the condition \mu_P < 2. In the latter case, if \mu_P < \mu_S, the system S inherits the relaxation properties of the perturbing network. In the case where \mu_P > 2, no response and no information transmission occurs in the long-time limit. We consider two possible generalizations of the fluctuation-dissipation theorem and show that both lead to maximal information transport in the condition of 1/f-noise., Comment: 14 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1007.2917

Non-ergodic renewal processes have recently been shown by several authors to be insensitive to periodic perturbations, thereby apparently sanctioning the death of linear response, a building block of nonequilibrium statistical physics. We show that it is possible to go beyond the ``death of linear response" and establish a permanent correlation between an external stimulus and the response of a complex network generating non-ergodic renewal processes, by taking as stimulus a similar non-ergodic process. The ideal condition of 1/f-noise corresponds to a singularity that is expected to be relevant in several experimental conditions., Comment: 4 pages, 2 figures, 1 table, in press on Phys. Rev. Lett

In many applications expectation values are calculated by partitioning a single experimental time series into an ensemble of data segments of equal length. Such single trajectory ensemble (STE) is a counterpart to a multiple trajectory ensemble (MTE) used whenever independent measurements or realizations of a stochastic process are available. The equivalence of STE and MTE for stationary systems was postulated by Wang and Uhlenbeck in their classic paper on Brownian motion (Rev. Mod. Phys. 17, 323 (1945)) but surprisingly has not yet been proved. Using the stationary and ergodic paradigm of statistical physics -- the Ornstein-Uhlenbeck (OU) Langevin equation, we revisit Wang and Uhlenbeck's postulate. In particular, we find that the variance of the solution of this equation is different for these two ensembles. While the variance calculated using the MTE quantifies the spreading of independent trajectories originating from the same initial point, the variance for STE measures the spreading of two correlated random walkers. Thus, STE and MTE refer to two completely different dynamical processes. Guided by this interpretation, we introduce a novel algorithm of partitioning a single trajectory into a phenomenological ensemble, which we name a threshold trajectory ensemble (TTE), that for an ergodic system is equivalent to MTE. We find that in the cohort of healthy volunteers, the ratio of STE and TTE asymptotic variances of stage 4 sleep electroencephalogram is equal to 1.96 \pm 0.04 which is in agreement with the theoretically predicted value of 2., Comment: 4 pages, 3 figures