Amplified signal response in scale-free networks by collaborative signaling
aa r X i v : . [ phy s i c s . c o m p - ph ] S e p . Amplified signal response in scale-free networks by collaborative signaling
Juan A. Acebr´on, Sergi Lozano, and Alex Arenas
Departament d’Enginyeria Inform`atica i Matem`atiques,Universitat Rovira i Virgili, 43007 Tarragona, Spain (Dated: November 14, 2018)Many natural and artificial two-states signaling devices are connected forming networks. Theinformation-processing potential of these systems is usually related to the response to weak externalsignals. Here, using a network of overdamped bistable elements, we study the effect of a heteroge-neous complex topology on the signal response. The analysis of the problem in random scale-freenetworks, reveals that heterogeneity plays a crucial role in amplifying external signals. We havecontrasted numerical simulations with analytical calculations in simplified topologies.
PACS numbers: 05.45.-a,05.45.Xt,89.75.Fb
Signaling devices are the building blocks of many nat-ural and artificial information-processing systems, for ex-ample, cells in living organisms respond to their environ-ment by means of an interconnected network of receptors,messengers, protein kinases and other signaling molecules[1, 2]. In artificial systems, we remark recent studies ofnetworks of fluxgate magnetometers, with potential waysto enhance the utility and sensitivity of a large class ofnonlinear dynamic sensors e.g. the magnetometers, fer-roelectric detectors for electric fields, or piezoelectric de-tectors for acoustics applications by careful coupling andconfiguration [3, 4, 5]. Their proper functioning alwaysimply high sensitivity to external signals.One of the most remarkable discoveries in non-linearphysics, over the last thirty years, was the phenomenonof stochastic resonance. It represents the surprising ef-fect manifested in nonlinear dynamics where a weak sub-threshold input signal can be amplified by the assis-tance of noise. The requirements for such amplifica-tion are three: (i) a non-linear dynamical system endow-ing a potential with energetic activation barriers, (ii) asmall amplitude (usually periodic) external signal, and(iii) a source of noise inherent or coupled to the system.Given these features, the response of the system under-goes resonance-like behavior as a function of the noisefloor; hence the name stochastic resonance [6, 7].The response of the system can be enhanced by cou-pling it through all-to-all [8] or in arrays configurations[9]. This has been done mainly resorting to a linear cou-pling among oscillators. The most intriguing part of thephenomenon is that noise (a source of disorder) can fa-vor the amplification of a signal, an effect that usuallyrequires a very precise synchrony (order). Recently in[11], it has been shown that even in absence of noise asimilar beneficial effects can be observed. There, differentsources of diversity plays the role of the noise, inducinga resonant collective behavior. Moreover, it was pointedout that such a resonance effect does not depend on thesource of disorder, and could be observed even in non- regular network of connectivities.Our goal here is to give evidences that an equivalentamplification of a external signal can be obtained in adeterministic dynamical system of signaling devices incomplex heterogeneous networks.The typical example, where the phenomenon has beeninvestigated, consists on a bistable potential with a pe-riodic signal in a thermal bath. A general dimensionlessequation describing this scenario is˙ x = x − x + A sin( ωt ) + η ( t ) (1)where the bistable potential is V ( x ) = − x / x / A sin( ωt ) and η ( t ) a Gaussian whitenoise with zero mean and autocorrelation h η ( t ) η ( t ′ ) i =2 Dδ ( t − t ′ ). Eq.(1) represents the overdamped motion ofa Brownian particle in a bistable potential and periodicforcing. In absence of forcing and noise, the system hastwo stable fixed points centered around ±
1, which corre-sponds to the minimum of the potential energy function V ( x ). When the amplitude of the forced signal is sub-threshold, each node i oscillates around the minimum ofits potential with the same frequency of the forcing sig-nal.The deterministic system we will study corresponds toa network of elements obeying Eq.(1) in absence of noise.Noise has been explicitly excluded to avoid a possible un-controlled superposition of effects, that would make dif-ficult to determine the influence of the topology. Thenetwork is expressed by its adjacency matrix A ij withentries 1 if i is connected to j , and 0 otherwise. Forsimplicity, from now on we consider only undirected un-weighted networks. Mathematically the system reads˙ x i = x i − x i + A sin( ωt ) + λL ij x j i = 1 , ..., N, (2)where L ij = k i δ ij − A ij is the Laplacian matrix of thenetwork, being k i = P j A ij the degree of node i .We have conducted numerical simulations of the sys-tem above for the Barabasi-Albert (BA) network model, λ 〈 G 〉 BA scale-free ( 〈 k 〉 = 3 )BA scale-free ( 〈 k 〉 = 5 )All-to-all FIG. 1: Average amplification h G i as a function of the cou-pling λ , for both topologies, scale-free and all-to-all. Resultscorresponding to two different values of the average degree( h k i = 3, marked with circles, and h k i = 5 with triangles)for the BA model network are plotted. Here N = 500, and ω = 2 π − . The simulations were averaged over 1000 ran-domly chosen initial conditions. While the scale-free networksshow a clear amplification of the external signal, for a signi-ficative range of values of λ that depends on the average de-gree, the all-to-all connectivity does not amplifiy the signal. and all-to-all connectivity networks. While BA has be-come the paradigm of a growing model that provides witha scale-free (power-law) degree distribution [10], all-to-allconnectivity is a representative of homogeneous in degreenetworks.The results are depicted in Figure 1. Here the aver-age amplification h G i over different initial conditions hasbeen computed as function of the coupling λ , being theamplification defined as G = max i x i /A . The frequencyof the forcing signal ω , and the number of nodes N werehere kept fixed. The obtained results show that in BAnetworks an amplification of the signal occurs, while inall-to-all networks no amplification of the external signalis observed for any value of the coupling λ .This difference of behavior is attributed to the pres-ence of hubs in the BA networks, which are the mainresponsible for the heterogeneity in degree of the net-work. Keeping this in mind, we study a topology con-sisting on a star-like network (one hub and N-1 periph-eral leaves). Such a topology is simple enough to bemathematically tractable, and simultaneously capable tocapture the main trait of heterogeneity found in scale-free networks. Indeed, the resulting dynamical systemdecomposes in two parts: the dynamics of the hub (thehighly connected node in the network with N − x H , and the leaves, y i , linked to the hub. Then, thedynamical system (2) becomes˙ x H = [1 − λ ( N − x H − x H + A sin( ωt ) + λ N − X i =1 y i (3) ˙ y i = (1 − λ ) y i − y i + A sin( ωt ) + λx H , i = 1 , ..., N (4)Now we make the following hypothesis: for a coupling λ sufficiently small, the dynamics of the leaves can be de-coupled from that of the hub, obtaining then from Eq.(4)the well known equation of an overdamped bistable oscil-lator. Moreover, when the signal is weak enoughcompared with the potential barrier, this can beconveniently linearized around one of the poten-tial minima . Then, Eq.(4) can be solved for the i thnode, and asymptotically for long time yields, y i ( t ) t →∞ ∼ ξ i − Aω + 4 [ ω cos( ωt ) − ωt )] (5)with ξ i = ± x H = − V ′ H ( x H ) + A ′ sin( ωt ) + B ′ cos( ωt ) + λη i (6)where V H ( x ) = − [1 − λ ( N − x / x / η i = P N − i =1 ξ i , and A ′ = A [1+2 λ ( N − / ( ω +4), B ′ = − λ ( N − / ( ω +4). No-tice that, the problem has been reduced to the motion ofan overdamped oscillator, in an effective potential drivenby a reamplified forcing signal coming from the globalsum of the leaves. The two possible solutions for thenodes Eq.(5), namely oscillations around ±
1, affect thedynamics of the hub as a quenched disorder representedby λη i , and is equivalent to the diversity studied in [11].Strictly speaking, the central limit theorem allows us tostate that in the limit N → ∞ , η i behaves as a randomvariable governed by a gaussian probability distributionwith variance σ = N −
1. This is so because the initialconditions were randomly chosen. The height of the po-tential barrier for the hub is now h = [1 − ( N − λ ] / λ via the factor( N − λ = 1 / ( N −
1) the barrier forthe hub disappears, leading to a unique single fixed point X H = 0. Note that both mechanisms, the quenched dis-order and the decrement of the potential barrier, rootedon the heterogeneity of the network, may cooperate toallow the hub to surmount the potential barrier. When λ = 1 / ( N − and for coupling and signal ampli-tude sufficiently small , Eq.(6) can be solved in theasymptotic long-time limit and yields x H ( t, η ) t →∞ ∼ x (0) H − ω + a H [( B ′ ω − A ′ a H ) sin( ωt ) − ( B ′ a H + A ′ ω ) cos( ωt )] (8) λ 〈 G 〉 TheoreticalNumerical
100 1000 10000 1e+05 N 〈 G 〉 m a x
100 1000 10000 1e+051.922.12.22.32.4
FIG. 2: Average amplification h G i as a function of the cou-pling λ . We compare the numerical simulation results of thedynamics of the star-like network Eqs.(3-4), and the theoret-ical result Eq.(10). Inset: Maximum average amplification h G i max as a function of the size of the network N . We com-pare again, the numerical simulation results of the dynamicsof the star-like network Eqs.(3-4), and the theoretical approx-imation Eq.(11). Parameters are as in Fig. 1 where a H = V ′′ H ( x (0) H ) being x (0) H the equilibrium points inabsence of forcing. Note that x H ( t, η ) depends also im-plicitly on the random variable η through the equilibriumpoint x (0) H to which the hub dynamics relaxes. Knowingthe long time evolution for the hub, its amplification canbe readily evaluated, and the result is G ( η ) = 1 A a H + ω p ( B ′ ω − A ′ a H ) + ( B ′ a H + A ′ ω ) (9)In practice, we must average over initial conditions tocancel out the dependence on them. However, this turnsout to be equivalent to averaging over η i . Therefore, theaverage amplification is given by h G i = 1 p π ( N − Z ∞−∞ dη e − η N − G ( η ) (10)The influence on the size of the network, N can beanalyzed resorting to the equation above. In particu-lar, we are interested in the case of N large, and for λ ≈ / ( N − N → ∞ , which is h G i = G (0) + 1 N d Gdη (cid:12)(cid:12)(cid:12)(cid:12) η =0 η + O ( N − ) , (11) ω 〈 G 〉 m a x Star-like ( theoretical )Star-like ( numerical )BA scale-free ( 〈 k 〉 = 3 ) FIG. 3: Maximum average amplification h G i max obtained forthe star-like network and BA network as a function of thefrequency of the external signal. The coupling is kept fixedto the value λ = 0 .
04 for the BA network, and λ = 0 .
002 forthe star-like network. These coupling values are those thatexhibit the maximum of h G i in Fig. 1 and Fig. 2, respectively.the solid line corresponds to the theoretical prediction for thestar-like network given by Eq.(10). Other parameters are asin Fig. 1. Therefore, the maximum average amplification can beextracted from a network, turns out to be bounded fromabove by G (0).We have performed extensive numerical simulations forthe dynamical system (3)-(4), varying N and λ . In Fig. 2,the results of the amplification in Eq.(10) and the numer-ical simulations of the system equations are compared fordifferent values of the coupling λ , keeping fixed N and ω . The amplification grows monotonically with λ untilreaching a maximum value. Such a value coincides ap-proximately with λ = 1 / ( N − topological resonance . The inset shows themaximum amplification obtained for different values of N . Notice that for large N , the solution tends asymp-totically to a constant value as expected from Eq.(11).Finally, in Fig. 3, we compare the maximum amplifica-tion in different networks as a function of the frequencyof the external signal. The amplification decreases mono-tonically with the frequency as predicted by the model.The remarkable agreement between simulations andanalytical results validates our hypothesis of decouplingthe hub from leaves. We are now in the situation of in-terpreting the results in the BA scale-free networks. Aneat picture is given by considering that each highly con-nected node acts locally as it were the hub of a star-likenetwork, with a degree k picked up from the degree dis-tribution. Therefore, for a given coupling λ , we can findseveral star-like networks in different stages, dependingon the degree of its local hubs. Recall that when λ = 1 /k ,the maximum signal amplification is attained for a hubwith degree k . Furthermore, increasingly larger values of λ would activate local hubs with smaller degrees. Sincethe network has several hubs, it should exist a wide rangeof values of λ for which the maximum amplification isachieved following a cascade of amplification. In Fig. 1such a behavior can be clearly observed. This contrastswith the results found for the star-like network, and an-ticipates that the scale-free network consists on a muchmore robust topology where such a new phenomena oc-curs.The mechanism described above remains valid untilfull synchronization occurs. For large values of λ , thenodes become strongly coupled, and then they spatiallysynchronize behaving as a single node. We define the de-gree of spatial synchronization as the fraction of nodes | n + − n − | /N where n + and n − are the number of ele-ments at the positive or negative well, respectively. InFig. 4 we show how full synchronization emerges once acritical coupling is attained. In addition, numerical sim-ulations depicted in the same figure reveal that the pathto synchronization is more pronounced for scale-free net-works with larger average degree. Then, for this casea smaller range of values for which maximum amplifica-tion is sustained should be observed in accordance withFig. 1. Similar curves have been found in the prototypeKuramoto model [12], which describes synchronizationphenomena in nonlinear coupled oscillators, on top ofcomplex networks [13]. However, the relevance of thepresent case stems from the fact that synchronization,contrary to which often happens in nonlinear coupleddynamics, has a negative effect in the amplification pro-cess. The all-to-all configuration in Fig. 4 presents fullsynchronization for any λ different from 0. The reasonfor this observed behavior is that no source of disorderexist in this case. The spatial symmetry of the problemis unstable, and any small difference in concentration ofelements in one of the two wells produces an asymmetryin the potential that evolves towards a monostable (syn-chronized) system. Note that this mechanism explainsalso the absence of amplification in Fig. 1.Summarizing, in this work we have shown that a topo-logical resonant-like effect emerges in scale-free topolo-gies of signaling devices, exemplified by a deterministic λ | n + - n - | / N BA scale-free ( 〈 k 〉 = 3 )BA scale-free ( 〈 k 〉 = 5 )All-to-all FIG. 4: Degree of synchronization of the BA and all-to-allnetworks versus the coupling. Two different average degreevalues for the BA network are plotted. The larger the averagedegree, the lower the critical coupling needed to achieve fullsynchronization is. The all-to-all network synchronizes forany value of λ = 0. Parameters are as in Fig. 1 overdamped bistable dynamical system. The coopera-tive interaction of nodes connected to the hub due tothe topology, has been shown to be the key feature foramplifying external signals. We have analytically provedthat the phenomenon can be totally characterized in theframework of a simple topology consisting of a star-likenetwork. The agreement between simulations and ana-lytical predictions allow us to understand the amplifica-tion curves as a function of the parameters of the system:the coupling, the size of the system and the frequency ofthe periodic external signal. At the light of the currentresults, we speculate that the ubiquity of scale-free topo-logical structures in biological systems can be related toits ability to amplify sensitivity to weak external signals.This work is supported by Spanish Ministry of Scienceand Technology Grant FIS2006-13321-C02. J.A.A. ac-knowledges support from the Ministerio de Ciencia y Tec-nolog´ıa (MEC) through the Ram´on y Cajal programme.We acknowledge for the usage of the resources, technicalexpertise and assistance provided by BSC-CNS super-computing facility. [1] D. Bray, J. Theor. Biol. 143, 215-231 (1990).[2] U Alon, Science , 1866 (2003).[3] B. Ando, S. Baglio, A.R. Bulsara and V. Sacco, IEEEInst. & Meas. Mag. , 64 (2005).[4] V. In, A. R. Bulsara, A. Palacios, P. Longhini and A.Kho, Phys. Rev. E , 045104(R) (2005).[5] A. R. Bulsara, J. F. Lindner, V. In, A. Kho, S. Baglio,V. Sacco, B. Ando, P. Longhini, A. Palacios and W.-JRappel, Physics Letters A , 4 (2006).[6] R. Benzi, G. Parisi, A. Sutera, A. Vulpiani,Tellus ,10(1982)[7] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys.
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