the modern world is an interconnected mesh of networks satisfying a myriad of functions : transportation , electrical power , food distribution , finance , and health care to name a few .
the interoperability of these networks developed as part of urban evolution over the past century such that these and other webs connect to national and/or global networks ( national research council of the national academies , 2005 ) .
this is the engineered webbing of humanity , but there are comparable natural structures in the spheres of biology , ecology , sociology , and physiology .
this modernity is manifest in the military through the development of network - centric warfare ( ncw ) which takes cognizance of human behavior in a networked environment of organized actions directed toward political , social , and military ends and is the basis of a new theory of war ( office of force transformation , 2004 ; garstka and alberts , unpublished ) .
thus , ncw has at its core a shift in focus from military platforms such as ships and tanks to networks having platforms as members .
army scientists need to understand the dynamics , controllability , and predictability of generic non - linear complex networks in order to realize their goal of supporting both society and the soldier through research and the development of new technologies .
it is not only our external world that is cluttered with networks , but our internal world as well .
the neuronal network carrying the brain s signal to the body s physiological networks is even more complex than the modern city or a typical ecological network .
thus , the basic research into network science must span and encompass a multitude of disciplines ; understanding each sheds light on the others .
the problem addressed within this program is to develop the basic research foundation of a science of networks that supports the linkage between the cognitive and social domains as they relate to decision making .
this approach is not directed at the totality of developing a network science , but has the more modest goal of understanding the deeply interdependent human networks of crucial importance to society as a whole and to the army in particular .
even such a restricted problem is a significant challenge due to the multiply interconnecting networks buttressing the common decision making objective . on the one hand ,
the military is proactive in that networked forces can operate in an agile manner to promote decision making superiority . on the other hand
, the army is reactive in the need to respond to enemies who are also using the power of networks against united states interests .
the research program provides insight to allow the army to anticipate the enemy s use of network strategy and thereby reduce the reactive mode of operation . in grigolini and west ( 2011 )
we reviewed what is presently known about complex networks , regardless of the disciplinary context and adapted that understanding to the decision making paradigm .
moreover , the barriers to further understanding and to filling the gaps in knowledge of the linkages between social and human decision making networks were addressed .
the research strategy of the arl program is based on theory , computation / simulation , and experiment / observation .
this is a cyclic interactive process in which new theory stimulates unique simulations , yielding insight into parameter values and network configurations , which in turn suggests specific experiments , whose outcome guides the refinement and development of theory . this modern approach to scientific research is applied to the phenomenon of human decision making with a view for eventual application to ncw .
the core group of army scientists is the focal point for external researchers requiring militarily relevant challenges and internal arl efforts .
one of the mysteries of human social interaction is how agreements are reached and cooperative alliances are made .
individuals become part of social groups or networks in a number of ways : choice , peer pressure , and subliminal seduction ; but always through a sequence of decisions , either conscious or not .
network characteristics can not be deduced from the properties of individuals ; they emerge during the formation and growth of the network .
consequently we need mathematics to quantify the strength of the interactions between the network components , as well as to describe how a network develops in time and responds to perturbations ( stimulation ) .
this has been done through the construction of the decision making model ( dmm ) that for very weak coupling is much like the ising model of cooperative behavior , but for strong coupling can be very different ( turalska et al . , 2009 )
ubiquitous aspects of complex networks are the appearance of non - stationary , non - ergodic , and renewal statistical processes .
law statistical distributions that not only challenge traditional understanding of complexity in physical networks , but require new strategies for understanding how information is exchanged between networks ( west et al . , 2008 ; west and grigolini , 2011 ) , as in the case of interest here among human networks including cognitive networks .
the approach is to adapt the methods of non - equilibrium statistical physics that have been used to characterize the dynamics of complex phenomena and phase transitions .
these methods were extended to the study of such social phenomena as linguistics , biofeedback techniques , and the brain s response to music ( bianco et al . , 2007 ) and to further develop them to model decision making with incomplete information in an uncertain environment .
the mathematics of complex networks has been examined using the newly developed dmm to understand consensus ( turalska et al . , 2009 ) ; a psychophysical model of how individuals make decision and then irrationally change their minds shows agreement between theory and experiments ( west and grigolini , 2010a ) ; renewal statistics reveal how we habituate to the familiar ( west and grigolini , 2010b ) and forget the uninteresting ( west and grigolini , 2010c ) ; 1/f variability captures how the brain processes tasks of increasing complexity during decision making ( grigolini et al . , 2009 ) ; and finally we have determined how all these various pieces fit into the overall picture of exchanging information between complex networks ( west et al . , 2008 ; grigolini and west , 2011 ; west and grigolini , 2011 ) .
the major accomplishment of the present research program has been the identification of the first universal principle in the science of networks , that being , the principle of complexity management ( pcm ) discussed in grigolini and west ( 2011 ) , west and grigolini ( 2011 ) .
pcm states that the maximum information is exchanged between two complex networks when there is compatibility of the complexity of the two networks .
a mathematical proof of this principle has been constructed over the past year ( aquino et al . , 2010 , 2011 ) .
the mathematician norbert wiener speculated that the transfer of influence from a complex network high in information to one low in information even though the latter may be higher in energy represents a new kind of interaction ( wiener , 1985 ) , which we called wiener s rule ( grigolini and west , 2011 ) .
his insight was vindicated a half century later ( aquino et al . , 2010 , 2011 ) and required the generalization of a number of concepts from statistical physics ( allegrini et al .
, 2007 , 2011 ; aquino et al . , 2007 ; budini and grigolini , 2009 ) resulting in the pcm ( west et al . , 2008 ; west and grigolini , 2011 ) as we discussed last year ( grigolini and west , 2011 ) .
one measure of the information content of a network is provided by the probability density most often used in the determination of the negative entropy of shannon and wiener .
an apparently ubiquitous distribution in the description of empirical complex networks is the hyperbolic , having the survival probability : which asymptotically becomes an inverse power  law .
the average time between events in complex webs such as power grid blackouts , heartbeats , time between earthquakes ( west and grigolini , 2011 ) of a given magnitude can be determined using the probability density (t )  =  d(t)/dt , to be it is interesting that when the power 
  <  3 the distribution has a finite first moment and the statistics are ergodic , meaning that the time average and ensemble averages yield the same result .
law index is   <  2 there are no finite integer moments and the time and ensemble averages are not the same , that is , the process is non - ergodic .
one measure of the information transfer between two complex networks is the cross - correlation between a complex network p and a complex network s being perturbed by p with  the strength of the perturbation . for our purposes
it is sufficient to apply the generalized linear response theory ( lrt ; aquino et al . , 2010 , 2011 )
we previously developed the normalized cross - correlation function : the perturbing complex network p is characterized by the non - stationary autocorrelation function p ( t , t ) , which depends separately on the time of the last perturbation t and time of the measurement t. the function rs(t ) is the rate of generating perturbing events at time t within the network being perturbed and is based on renewal theory ( west and grigolini , 2011 ) .
the perturbed network s is characterized by the stationary autocorrelation function s ( t    t ) , which depends only on the difference in times from the last perturbation to the measurement . in figure 1
the asymptotic cross - correlation function normalized to the strength of the perturbation is graphed as a function of the power 
the cube displays a number of remarkable properties : ( 1 ) when the power  law indices
are both equal to two there is an abrupt jump from zero correlation in region ii to perfect consensus in region iii ; ( 2 ) the upper plateau region iii indicates that when p is non - ergodic 1  <  p  <  2 and s is ergodic 2  <  s  <  3 there is an information response in which the perturbed network tracks the perturbing network exactly and the information transfer is maximal ; ( 3 ) when p is ergodic 2  <  p  <  3 and s is non - ergodic 1  <  s  <  2 there is no response asymptotically and the information transfer is minimal as shown in region ii .
how a complex network responds to a perturbation by another complex network is determined by the kind of mismatch that exists in the complexity of the fluctuations in the two networks . the cross - correlation cube . the asymptotic cross - correlation function defined by eq .
law indices of the perturbed network s and the perturbing network p. ( from west and grigolini , 2011 , with permission . )
wiener s rule describes the influence of the perturbing network outside the lower plateau region of the cross - correlation cube .
in all regions except this one the weak perturbation significantly modifies the properties of the complex network being perturbed . in the upper plateau region the perturbation by network p actually dominates the properties of the perturbed network s and reorganizes it , just as wiener anticipated .
the pcm embodied in the cross - correlation cube therefore subsumes wiener s rule as we reviewed in west et al .
in addition we showed the application of pcm to the phenomenon of habituation and other activities involving the human brain .
aquino et al . ( 2011 ) observe that the growing interest in the dynamics of complex networks is shifting research attention from the synchronization of two stochastic units ( pecora and carroll , 1990 ) to the synchronization of large numbers of units ( wang , 2002 ) , an interesting phenomenon that is closely related to the very popular model of kuramoto ( 1984 ) .
the single units of the processes of chaos synchronization are chaotic and they surprisingly synchronize while maintaining the erratic dynamics that they have in isolation .
although the single units of the kuramoto model are regular , it is becoming increasingly evident that the emergence of a global synchronization is a condition independent of whether the single units are regular or stochastic .
( 2011 ) are poisson processes and if one of them drives the other , they would obey the principle of aperiodic stochastic resonance ( lukovic et al . , 2008 ) .
if the two units are bi - directionally coupled they are expected to undergo a condition of perfect synchronization if the coupling is sufficiently intense .
when the number of interacting units is very large a phase transition occurs from the non - cooperative to the cooperative behavior ( bianco et al . , 2008 ; turalska et al . , 2011 ) .
it is important to stress that at criticality no permanent consensus is reached , and the mean value of the global field vanishes . yet
(  no  ) state for an extended time before making a transition to the  no  (  yes  ) state .
it is surprising that the phase transition literature seems to have overlooked , with only a few exceptions ( contoyiannis and diakonos , 2000 ; bianco et al .
, 2008 ; turalska et al . , 2011 ) , that the transitions from the  yes 
in other words , the time interval between two consecutive transitions is derived from a pdf that has the asymptotic time structure of eq .
<    <  3 . some authors ( bianco et al . , 2008 ; turalska et al . , 2011 ) argue that   =  1.5 and others ( frantsuzov et al . , 2009 ) , releasing the condition that all the units share the same poisson rate , generate a global condition with crucial events characterized by   <  2 , but significantly departing from the value   =  1.5 .
( 2009 ) , might yield the misleading impression that the crucial value of  is a consequence of ordinary statistical physics .
( 2011 ) is the discovery of a promising road to settle the problem of information transmission from one to another complex network .
in fact , if the inner synchronization corresponds to a criticality condition and criticality generates crucial events with a power 
  <  3 , then a complex network at criticality is a generator of 1/f noise , with a power spectrum s(f  )    1/f .
thus , the problem of information transmission from one to another complex network becomes equivalent to the phenomenon of 1/f resonance illustrated in aquino et al .
the experiments ( onsager , 1944 ; silvestri et al . , 2009 ) support the dynamical rather than the phenomenological lrt .
it is important to stress that phenomenological lrt is a natural consequence of adopting the asymptotic time perspective replacing the waiting - times pdf ( ) of eq . 1 with ( ) 
  1/. this way of proceeding , although generating the elegant mathematics of fractional derivatives , has as an ultimate effect the misleading discovery of the death of linear response .
this is not a unique way of connecting the longtime to the short - time regime .
however , whatever form we adopt we are convinced that there will be a parameter corresponding to the parameter t of eq .
it is reasonable to assume that an external perturbation may perturb either t or  , or both .
however , the perturbation of  is incompatible with the assumption of a weak stimulus .
in fact ,  is a consequence of the cooperation among the units of the network , and a perturbation may affect  only if its strength is large enough to influence the interaction among the units of the network .
thus , an external weak perturbation can only have an effect on t , thereby making the dynamical lrt become the proper way to study the response of a complex network to a weak external stimulus , in accordance with the experimental results ( onsager , 1944 ; allegrini et al .
, 2009 ; silvestri et al . , 2009 ) . for these reasons , we can conclude that figure 1 is an original , and important , result of this research program .
we hope that the application of the pcm may open the door to solving the problem of information transmission from one complex network to another , a research topic that is still in its infancy .
in order to better understand the transfer of information between complex networks last year we numerically analyzing networks consisting of a large number of non - linearly interacting nodes .
( 2009 ) using a master equation formalism ( turalska et al . , 2011 ) were discussed where each element of the network is a two state oscillator and is described by a master equation of the form and pj(t ) is the probability of being in the state j  =  1 ; p1  +  p2  =  1 .
the coupling functions are time - independent for non - interacting members of the network .
subsequently , we considered a network consisting of l discrete variables located at the nodes of a 2d square lattice .
each unit si is a stochastic oscillator and can be found in either of the above two states .
for the dynamic complex lattice each element si interacts with each of its nearest neighbors and is updated in an elementary time step with transition rate g : here m is the total number of nearest neighbors ; m  +  1 and m    1 are the number nearest neighbors that have made the decision  yes  and  no ,  respectively . the single individual changes opinion , and as a consequence these numbers are variables fluctuating in time , while the total number of elements is constant .
(  no  ) makes a transition to the state  no  (  yes  ) faster or slower according to whether the majority of the elements are in the state  no  (  yes  ) or  yes  (  no  ) , respectively .
2011 ) do all calculations on a l    l lattice with periodic boundary conditions .
they characterize the network in terms of the global order parameter whose variability is not dichotomous . in figure 2 an example of the temporal evolution for a single unit is compared with that of the global order parameter .
( a ) temporal evolution of a single unit and ( b ) of the global order parameter for the dmm realized on a square lattice with l  =  50 , g0  =  0.01 , and k  =  1.70 .
) note that the amplitude of the global order parameter depends on the value of the coupling constant k. when k  =  0 , single units of the network are independent .
when k  >  0 , single units are less and less independent , resulting in a non - zero average .
the quantity kc is the critical value of the control parameter k , at which point a phase transition to a global majority state occurs . in numerical calculations
they use the time average eq    |(t)| as a measure of the global majority .
more precisely after an initial million time steps , which is sufficient time to suppress any transients , an average is taken over the same number of consecutive time steps in the dmm .
they find that in the special case when m is the same for all the nodes and g0    1 , that dmm generates the same kind of phase transition as is observed in the 2d ising model discussed in onsager s seminal paper ( onsager , 1944 ) .
the phase transition for the global variable eq is indicated in figure 3 under various conditions .
it is evident that the dmm phase transition on a lattice is only equivalent to the ising model under very restricted conditions .
the apparent equivalence between the ising and dm models is merely formal , since the dmm does not have a hamiltonian origin and its elements are not in contact with a thermal bath ( turalska et al . , 2011 ) .
these differences explain why the equivalence requires the transition rate to vanish , so as to freeze the dynamics of the single units , in the absence of cooperation .
the phase diagram for the global variable eq . the thin solid line and dashed line are theoretical predictions for the fully connected and 2d regular networks , respectively . in both cases
l  =   and the latter case is the onsager prediction for a 2d regular lattice .
the thick solid line corresponds to the global states observed for a 2d regular lattice with l  =  100 and g0  =  0.01 .
when the transition rate assumes a finite value the equivalence between dmm and the ising model is lost .
( 2011 ) investigate the parameter phase space to determine the domain of phase transitions and find that they can occur for values of k below that of the theoretical kc .
there is also a situation for relative high transition rates in which every unit is surrounded by nearest neighbors in the opposite state , yielding an update of its state at every time step and generating the condition in which the order parameter is exactly zero at all times .
( 2011 ) conjecture that the crossings of the origin by the global order parameter are the significant events to observe .
as illustrated in figure 2 they interpret the time interval  between two consecutive crossings as the time duration of a given decision , even if this decision may rest on a slight and fluctuating majority .
they evaluate the distribution density of decision  time duration  , ( ) , and the corresponding survival probability ( ) , where although emerging from a simple regular lattice , that is , one with no structural complexity , the survival probability presented in figure 4 shows a scale - free property that extends over more than four orders of magnitude in time for k    kc a further increase in the coupling strength does not affect the power  law region .
survival probability function ( ) for the global order parameter evaluated on a 2d lattice with l  =  50 , g0  =  0.01 and increasing values of the coupling constant k. the straight line corresponds to a slope of 0.50 , namely   =  1.5 since ()1/. ( from turalska et al .
wiener s rule maintains that a network with high information can organize one with low information .
for example a tightly coupled organization , with rules and policies to cover all contingencies , changes little over time and therefore is low in information .
cpm quantifies wiener s rule by introducing a measure of complexity allowing us to compare the level of information in interacting complex networks .
law index of the hyperbolic distribution and a generalization of lrt enabled us to construct the cross - correlation cube to determine the degree of asymptotic influence one network has on another . in this way
the 1/f variability of stimuli is found to resonate with the human brain ( grigolini et al . , 2009 ) , as when we are entranced by music or irritated by a dripping faucet ( grigolini and west , 2011 ; west and grigolini , 2011 ) .
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 army research laboratory program on the network science of human decision making brings together researchers from a variety of disciplines to work on a complex research problem that defies confinement within any single discipline . 
 consequently , new and rewarding solutions have been obtained for a problem of importance to society and the army , that being , the human dimension of complex networks . 
 this program investigates the basic research foundation of a science of networks supporting the linkage between the cognitive and social domains as they relate to human decision making . 
 the research strategy extends recent methods of non - equilibrium statistical physics to non - stationary , renewal stochastic processes characteristic of the interactions among nodes in complex networks . 
 the theoretical analyses of complex networks , although mathematically rigorous , often elude analytic solutions and require simulation and computation to analyze the underlying dynamic process . the information transfer between two complex networks 
 is calculated using the principle of complexity management as well as direct numerical calculation of the decision making model developed within the project . 
Introduction
Principle of Complexity Management
Synchronization and Information Exchange
Decision Making Model and Phase Transitions
Conclusion
Conflict of Interest Statement