Behind any complex system in nature or engineering, there is an intricate network of interconnections that is often unknown. Using a control-theoretical approach, we study the problem of network reconstruction (NR): inferring both the network structure and the coupling weights based on measurements of each node’s activity. We derive two new methods for NR, a low-complexity reduced-order estimator (which projects each node’s dynamics to a one-dimensional space) and a full-order estimator for cases where a reduced-order estimator is not applicable. We prove their convergence to the correct network structure using Lyapunov-like theorems and persistency of excitation. Importantly, these estimators apply to systems with partial state measurements, a broad class of node dynamics and internode coupling functions, and in the case of the reduced-order estimator, node dynamics and internode coupling functions that are not fully known. The effectiveness of the estimators is illustrated using both numerical and experimental results on networks of chaotic oscillators.

Understanding the interplay between the network topology of interconnected units (or components) and their emergent behavior is a fundamental problem in network science.1–3 For instance, the topology of interactions between different institutions in financial networks is central to predicting behaviors that arise during a crisis.4 Network topology affects coordination in human ensembles5 and the stability of power networks.6 Moreover, recent developments on controllability,7 observability,8 and control9 of complex networks require the topology to be known. In many applications, however, including mobile sensor networks, biological networks, and social networks, the network topology may be unknown or time-varying.10–12 Uncovering the topology of such networks is fundamental to understanding and controlling their complex behavior.11,12 Using adaptive control theory, we study the problem of inferring the network topology of complex systems based on measurements of each node’s activity. In particular, we construct two different estimators, a reduced-order estimator and a full-order estimator, that can infer the network topology in real time.

Discovering the underlying network topology of complex systems is a fundamental problem in fields ranging from biology to engineering.11,12 Consider, for example, a gene regulatory network. Even if the individual dynamics of gene expression are well understood, determining the tangled network of interactions is still a significant challenge.13,14 As another example, the network topology of a power grid might change over time. For instance, renewable sources depending on weather conditions might be connected/disconnected or transmission failures might occur.15 A pressing challenge is to design appropriate methods to infer (or reconstruct) the topology of a network in real time.

We consider a general network model of N>1 nodes, each described by a set of nonlinear ODEs of the form

x˙i=fi(xi)+j=1Naijhij(xi,xj),yi=Cixi,yiRmi,for alliN,
(1)

where xiRn is the state variable of the ith node and N={1,,N} is the finite set of indexes. The state could represent the abundance of species in ecological networks16 or voltages and currents in electrical networks.17 The measurable “output” of a node is yi, a linear combination of the states specified by the output matrix CiRmi×n. The function fi:RnRn is continuous and represents the intrinsic dynamics of each node. The constants aij are the entries of a weighted adjacency matrix A=[aij], where aij0 indicates that there is an interconnection link from node j to node i through a continuous coupling function hij:R2nRn. We assume that the diagonal elements aii are zero, as such terms can be absorbed into fi.

We are interested in the network reconstruction (NR) problem, i.e., inferring the coupling weights aij in real time using measurements of each node’s activity yi.

Several methods have been proposed for NR, and they can be classified in two major categories: “model free” and “model based.” In the model-free approach, information theory can be exploited for NR without requiring full knowledge of the network model (1).12 In particular, interconnection links are inferred using metrics based on conditional,18 transfer,19 or causal entropies.20 These techniques are not able to precisely reconstruct the coupling weights associated with each link, however, and they typically require a large amount of data and computational resources,19 making their real-time implementation cumbersome. Similarly, the network reconstruction problem has also been addressed within the context of graphical models where the network topology is assumed to have locally treelike graphs21 or the node states are normally distributed.22 

In contrast, model-based techniques make use of model (1). For instance, in Ref. 13, the network is linearized around a structurally stable equilibrium point, and the network is reconstructed using classic regression techniques. Extensions of this strategy have been proposed for cases where the network solutions are periodic23 or can be driven to an equilibrium point by adding linear feedback controllers at every node of the network.24 (For a more comprehensive list of NR techniques, see Ref. 11 and references therein.)

More recently, Ref. 25 proposed different metrics inspired by the Kuramoto model to infer coupling links, while in Ref. 26, the NR problem is presented and solved as an underdetermined regression model based on the fact that all x˙i are known or can be computed from measurements xi. A related approach is described in Ref. 27. However, none of these approaches is able to exactly infer the interconnection links and their associated coupling weights. In fact, they lack a proper convergence analysis, and they often rely on strong assumptions like special steady-state solutions, local approximations (linearization), full state measurements (yi=xi), and perturbing or controlling all the nodes of the unknown network. The network reconstruction problem is also known as network tomography28,29 where particular diffusion models are considered such as the Combine-Then-Adapt (CTA) diffusion mechanism.29 A similar diffusion process is considered in Ref. 30 where the goal is to reconstruct the topology associated with the graph Laplacian.

Copy-synchronization31–34 is a model-based approach to the NR problem where exact convergence can be guaranteed. This method consists of an estimated “copy” (or slave) network with adaptive coupling weights, and the goal is to drive the error between the copy network and the unknown (or master) network to zero. Often, but not always, copy-synchronization strategies require full knowledge of the network dynamics (fi and hij), full state measurements (yi=xi), and particular coupling functions (e.g., linear diffusive coupling).31,32,34,35 In addition, the convergence of these strategies has not been fully understood within the context of complex network systems (see, for instance, the supplementary material in Ref. 36 and page 18 in Ref. 11).

To overcome these limitations, in this paper, we interpret network copy-synchronization as a control-theoretic adaptive observer technique, where the copy network is an adaptive observer that reconstructs the master network. This formulation allows us to derive novel NR methods guaranteeing exact convergence while making fewer assumptions about the network and its measurements, as described below. We wish to emphasize that the proposed adaptive estimators are similar to gradient-descent-based algorithms that are known to track slowly varying coupling weights.37–39 

Using a copy network for NR was originally proposed in Ref. 31. As pointed out in Ref. 32, however, convergence to the real network structure fails when all the trajectories of the unknown network converge to each other, i.e., synchronization. The authors state that a sufficient condition to guarantee exact convergence to the real network topology is that all the node trajectories should be linearly independent (LI) on the zero-error manifold, that is, when the error between the copy network and the real one is zero. More recently, extensions of the copy-synchronization approach satisfying the LI condition were given in the context of outer-synchronization34 (where the copy network can be an oscillator of lower dimension, but full state measurements are required), linearly coupled networks with partial state measurements,35 and systems with delays and uncertainties.40–42 The requirement of LI on the zero-error manifold is a significant limitation of these approaches, however, since it is only a sufficient condition that might be very difficult, if not impossible, to test for networks with a large number of components.34 Reference 33 instead uses the persistent excitation (PE) condition, which is simpler to test. Only networks with diffusive couplings and full state measurements were addressed, however.

Despite the fact that there is a direct relation between network copy-synchronization and adaptive observers, until now the relationship has not been made explicit, even though master-slave synchronization has been shown to be closely related to observers in the context of single chaotic systems.43,44

In this paper, we exploit adaptive observer theory to study the NR problem for a more general class of networks, including networks with nonlinear heterogeneous nodes and coupling functions, networks where full state sensing is not available, and networks where the node dynamics and coupling functions are not fully known. Compared to the existing literature, the contribution of the paper is threefold:

  1. Section III: We introduce the “reduced-order copy-synchronization estimator,” where the dynamics of each node are projected to a single state variable. It is always possible to design such an estimator if all state variables of the unknown network are measurable. Due to the smaller number of estimator state variables, the reduced-order estimator has lower computational complexity than existing copy-synchronization estimators, making the approach more suitable for large networks. In addition, reduced-order estimators can be designed for systems where only a subset of states can be measured, and it is not necessary to have full knowledge of the node dynamics or coupling functions. Using the PE condition, we derive necessary and sufficient conditions guaranteeing convergence of the copy network to the unknown network.

  2. Section IV: In cases where a reduced-order estimator cannot be constructed, we extend the “full-order copy-synchronization estimator” approach to (i) a more general class of networks than previously considered (e.g., networks with nonidentical coupling functions and non-Lipschitz node dynamics) and (ii) cases where only partial state measurements are available. Convergence to the unknown network structure is guaranteed using PE rather than LI. Preliminary results for networks with diffusive couplings were presented in Ref. 45.

  3. Sections V and VI: We validate the new real-time NR estimators on simulated networks as well as experimental networks of chaotic electrical oscillators (Chua’s circuits). We find that, in practice, there is a trade-off between the accuracy of the estimation and the convergence rate that depends on the estimator parameters.

The N×N identity matrix is written IN and 0 is a matrix of zeros of appropriate dimension. The notation denotes the Euclidean norm for vectors and the spectral L2-norm for matrices. A block diagonal matrix D with diagonal blocks d1,,dN is indicated by D={diag}{d1,,dN} and λmin(A) denotes the minimum eigenvalue of a square matrix A. A0 indicates that the matrix A is positive definite. AB indicates that the matrix AB is positive semidefinite.

Definition II.1
A matrix function ω:R+Rp×q is said to be persistently exciting (PE) if there exist nonnegative constants T,c1, and c2 such that
c1INQc2IN,Q=tt+Tω(τ)ω(τ)dτfor allt0.
(2)

A signal ω(t) satisfies the “PE condition” if it excites all the modes of the system.38 This condition has been widely used in the context of adaptive control and adaptive observers.39,44,46 (The reader is referred to Chap. 6 of Ref. 38 for more details about the PE condition.)

Let x:=[x1,,xN] be the stack vector of all network states. Then, (1) can be written as

x˙i=fi(xi)+Hi(x)ai,yi=Cixi,
(3)

where aiRN1 is the stack vector of unknown coupling weights and Hi(x)Rn×(N1) is the regressor matrix, i.e.,

ai:=[aik,,aij],
(4)
Hi(x):=[hik(xi,xk),,hij(xi,xj)]
(5)

for all i,k,jN such that k,ji and j>k. In this paper, we address the problem of designing adaptive strategies for updating an estimate a^i of the unknown network topology ai based on measurements yi. The network (3) can be seen as a dynamical system with linearly parametrized unknowns,37,38 so we can consider estimators of the form

a^˙i=χi(y,ζi),a^i(0)=a^0,i,a^iRN1,
(6)
ζ˙i=qi(ζi,y,a^i),ζi(0)=ζ0,i,ζiRs,
(7)

where a^i are the estimated coupling weights and a^0,i, ζ0,i are the initial conditions. Functions χi(y,ζi) and qi(ζi,y,a^i) are the “estimator functions,” where χi represents the adaptation law for reconstructing the coupling weights. In what follows, we will show that NR is guaranteed by properly choosing these functions.

Assumption II.1

We assume network (1)[or equivalently (3)] has bounded solutions and all outputs yi can be measured.

Definition II.2
Let a~i=a^iai be the coupling weight errors. Then, estimators (6)(7) are said to asymptotically reconstruct the network topology of (1) if for any initial states ζ0,iRm and a^0,iR, the error satisfies
limta~i(t)=0andlimtζi(t)c<
(8)
for alliN.

The original copy-synchronization approach,31 which this paper extends, consists of estimators (6) and (7) with functions

χi=μiHi(ζ)(ζixi),
(9)
qi=fi(ζi)+Hi(ζ,t)a^iκi(ζixi),
(10)

where μi and κi are positive constant parameters representing the adaptation gain and control strength, respectively. Note that this copy-synchronization approach requires (i) the dimension of each estimator state ζi to be the same as the node dynamics of the unknown network (s=n), (ii) full state measurements, i.e., Ci=In, and (iii) full knowledge of the intrinsic node dynamics fi and coupling functions hij.

In Sec. III, we show that the dimension of the estimator state ζi can often be reduced to one by using a one-dimensional projection. Partial rather than full state measurements are considered, and most importantly, full knowledge of the intrinsic node dynamics and coupling functions is not required.

To reconstruct the network structure (the coupling matrix A), it is often not necessary to estimate all the nodes’ states or to have full knowledge of the node dynamics and coupling functions. We can project each node’s dynamics to a one-dimensional space, meaning we only need to estimate this scalar value to reconstruct the coupling weights. The ability to do this relies on the existence of a particular kind of projection operator, which is discussed below.

Definition III.1
A continuous and differentiable function ϕi:RmiR is said to be output-compatible if the gradient ϕi(yi) is continuous and satisfies the following conditions:
ϕi(yi)Cifi(xi)=αi(yi),
(11)
ϕi(yi)Cihij(xi,xj)=βij(yi,yj),
(12)
i.e.,αi and βij are scalar functions depending only on the output measurements yi for all i,jN,ij.
Proposition III.1
Consider the projection zi:=ϕi(yi)R, where ϕi is an output-compatible function. Then, the dynamics (3) are projected onto a one-dimensional space whose dynamics are given by
z˙i=αi(yi)+wi(y)ai,
(13)
where wi(y)=[βik(yi,yk),,βij(yi,yj)]RN1 with i,k,jN such that k,ji and j>k.
Proof.

From zi=ϕi(yi), we have z˙i=ϕi(yi)Cix˙i. Using (3), and the fact that ϕi(yi) is output-compatible (see Definition III.1), yields (13).

Note that the projected dynamics of the ith node can be seen as a first-order system with N1 linearly parametrized unknowns. We can directly make use of classic observer-based estimators37,38 to obtain the following result.

Theorem III.1
Consider a network (3) satisfying Assumption II.1, an output-compatible function ϕi(yi), and the estimator,
a^˙i=μiwi(y)(ziϕi(yi)),
(14)
z˙i=αi(yi)+wi(y)a^iκi(ziϕi(yi)),
(15)
where αi and wi are calculated according to Definition III.1 and Proposition III.1 and μi,κi>0 are arbitrary constant parameters. Then,
(a) A necessary and sufficient condition for asymptotic NR is that there exist positive constants δi,Ti, and ci with a t2[t,t+δi] such that for any unit vector vRN1,
1δit2t2+Tiwi(τ)vdτc,forallt0,iN.
(16)

(b) A stronger sufficient but not necessary condition for asymptotic NR is that all ϕi(yi) have continuous second partial derivatives, all hij(xi,xj) have continuous Jacobians, and all wi are PE (see Definition II.1).

Proof.
From Proposition III.1, we have that ri:=ϕi(yi) projects the network (3) onto a one-dimensional space (13). Next, defining the estimation errors a~i:=a^iai and ei:=ziri yields
a~˙i=μiwi(y,t)ei,e˙i=κiei+wi(y,t)a~i.
(17)

From Assumption II.1, we have that all xi are bounded. Then, by continuity of the functions ϕi and hij, we have that all wi are bounded as well. Using Theorem A.1 in the  Appendix with A=κi and the fact that all wi satisfy (16), we can conclude that limtei(t)=a~i(t)=0 for all iN.

For the sufficient condition in the last sentence of the theorem, we need to show that all w˙i are bounded. To do so, we differentiate βij in (12) with respect to time, yielding
β˙ij=x˙iCiHϕi(yi)hij(xi,xj)+ϕi(yi)CiDhij(xi,xj)[x˙ix˙j],
(18)
where Hϕi(xi) is the Hessian matrix of ϕi(yi) and Dhij(xi,xj)Rn×2n denotes the Jacobian matrix of hij(xi,xj). From the continuity of functions fi and hij, we have that all x˙i in (1) are bounded. Because we have further assumed that both Hϕi and Dhij are continuous, we finally have that all β˙ij and, thus, w˙i are all bounded along the network trajectories. Using the second condition of Theorem A.1 and the fact that all wi are PE (see Definition II.1) concludes the proof.

In contrast to the original full-order copy-synchronization approach, the reduced-order estimators (14) and (15) use partial state measurements and the dimension of the estimator state zi is one. Indeed, each estimator i only needs N state variables to reconstruct the entire network, rather than n+(N1) as in (10). In addition, both the node dynamics fi and coupling functions hij are not required to be fully known, as illustrated in the following example.

Example III.1
Consider the case where each node in (3) is a three-dimensional system (n=3) with xi=[pi,qi,ri]R3,Ci=C for all iN, and the node dynamics
f(xi)=[pi+qi+riφi(pi,qi,ri)ripiqi],C=[100001],
(19)
where φi(pi,qi,ri) is an unknown smooth scalar function. Note that only two state variables,pi and ri, can be measured. Choosing ϕi(yi)=(1/2)pi2+ri yields ϕi(yi)=[pi,1], and αi(yi) is calculated by Eq. (11) to be αi(yi)=[pi,1]Cif(xi)=pi2+(pi1)ri, which depends only on the output variables. To implement the estimators (14) and (15), the unknown term φi(pi,qi,ri) in the node dynamics is not needed; we just need αi(yi). Similarly, the full coupling functions hij(xi,xj) are not required; all that is needed are the projections βij(yi,yj).
Remark III.1

Theorem III.1 provides two conditions guaranteeing asymptotic NR using (16) and (2), respectively. Condition (16) is necessary to guarantee convergence to the real network topology. Similar to the PE condition,(16) measures the richness of the signals wi. In practice, however, the PE condition (2) is easier to verify than (16),38 but extra conditions on the projection function ϕi(yi) and coupling functions hij are required. Indeed, in our experimental results, we show that condition (2) can be easily tested [e.g., see Fig. 8(c) ].

Remark III.2

In the case of full state measurements (Ci=In for all iN), as usually considered in the NR literature,31–34,41,42 a one-dimensional projection that is output-compatible (see Definition III.1) always exists. In some cases of partial state measurements, however, such a projection might not exist. In such cases, a projection onto a pi-dimensional space may be possible instead, i.e., a function ϕi:RmiRpi with 1<pi<n. This also results in a reduced-order estimator.

In Sec. III, we showed that the state of each estimator can be reduced to one by using the reduced-order observer (14) and (15) under Assumption III.1. In some cases, however, there may not exist a valid projection of the dynamics to a lower-dimensional space (see also Remark III.2), and full-order estimation is required. In this section, we derive a full-order copy-synchronization estimator that is convergent for a broader class of networks than previously studied in the copy-synchronization literature.

We assume that the intrinsic node dynamics can be written as the sum of linear and nonlinear terms, i.e.,

fi(xi)=Aixi+ξi(xi),iN,
(20)

where AiRn×n and ξi:ΩRnRn. Many nonlinear systems can be written in this form, e.g., Lorenz oscillators, Chua’s circuit, and others.47 Moreover, we rewrite the coupling functions as hij(xi,xj)=Γih~ij(xi,xj), with ΓiRn×n being a square matrix. For instance, in the example discussed in Sec. V B, the matrix Γi represents the fact that nodes are coupled though the first state variable only. This matrix is often called the “inner coupling matrix.”2 We will show that Γi must satisfy “the matching condition” [see Eq. (23)] to guarantee stability of the full-order copy-synchronization estimator.

With these definitions, classic adaptive estimators37,38 can be used for NR,

a^˙i=μiH~i(ζ)Mi(Ciζiyi),
(21)
ζ˙i=Aiζi+ξi(ζi)+ΓiH~i(ζ)a^iLi(Ciζiyi),
(22)

where H~i(ζ):=[h~ik(ζi,ζk),,h~ij(ζi,ζj)] for all i,k,jN such that k,ji and j>k. The n×mi matrices Li and Mi should be properly designed according to the following result.

Theorem IV.1

Consider the unknown network (1) where fi(xi) can be written as in (20), Assumption II.1 holds, and the elements of the network adjacency matrix A satisfy |aij|amax. Then, estimators (21) and (22) asymptotically reconstruct the network topology if the following conditions are satisfied:

(C1) there exist matrices Gi0,Ri:=LiPi, and Mi such that for all iN,
PiΓi=CiMi,
(23)
Gi=PiAi+AiPiRiCiCiRi;
(24)
(C2) there exist constants ρiR such that for all iN and v,gΩRn,
(vg)Pi(ξi(v)ξi(g))ρi(vg)2;
(25)
(C3) the coupling functions h~ij are differentiable and their Jacobians are continuous. In addition, there exist positive constants Lij such that for all v1,v2,g1,g2ΩRn and i,jN with ij,
h~ij(v1,g1)h~ij(v2,g2)Lij[v1v2g1g2];
(26)
(C4)γ>ρ+amaxMmaxL(1+N),ρ:=maxiN{ρi},γ:=(1/2)maxiN{λmin(Qi)},Lmax:=maxi,jN{Lij}, and Mmax:=maxiNCiMi; and

(C5) all matrices Hi(ζ)=H~i(ζ)Γi are PE (see Definition II.1).

Proof.

For the sake of clarity, we split the proof in three steps as follows:

Step 1 (Stability): Letting ei=ζixi and a~i:=a^iai, we find from (3) and (21)(22) that the error dynamics are given by
a~˙i=μiH~i(ζ)MiCiei,
(27)
e˙i=(AiLiCi)ei+ξi(ζi)ξi(xi)+ΓiH~i(ζ)a~i+(Hi(ζ)Hi(x))ai.
(28)
Consider the Lyapunov candidate function
V=12i=1N12μia~ia~i+eiPiei,
(29)
where all matrices Pi are symmetric and positive definite. Differentiating (29) along the trajectories of (27)(28) yields
V˙=i=1NeiPi(AiLiCi)ei+eiPi(ξi(ζi)ξi(xi))+eiPiΓij=1,jiNaij(h~ij(ζi,ζj)h~ij(xi,xj))12i=1Nei(PiAi+AiPiPiRiRiPi)ei+i=1NeiPi(ξi(ζi)ξi(xi))+j=1N|aij|eiPiΓih~ij(ζi,ζj)h~ij(xi,xj).
(30)
Next, from (25) and (26), we have (x^ixi)Pi(fi(x^i)fi(xi))ρieiei, and hij(ζi,ζj)hij(xi,xj)Lij(ei+ej). Then, (30) can be upper bounded by
V˙12i=1NxiQixi+ρeiei+amaxMmaxj=1NLijei(ei+ej)i=1Nj=1N(ρ+amaxMmaxLmaxγ)ei2+amaxMmaxLmaxeiej,
(31)
where we have used the assumption PiΓi=CiMi. Letting e~:=[e1,,eN], we have
V˙(ρ+amaxMmaxLmaxγ)e~e~+amaxMmaxLmaxe~1N1Ne~ce~e~,c=(ρ+amaxMmaxLmax(1+N)γ).
(32)

Condition (C4) of Theorem IV.1 guarantees V˙0, implying ei and a~i are bounded for all iN, and limtV(t)=V<.

Step 2 (Uniform Continuity): Next, we show that ei vanishes asymptotically. Integrating both sides of (32) and rearranging terms, we find that e(t) is squared integrable, i.e., 0e(t)dt(V(0)V)/c<. Next, we note that eiPie˙i=12ddteiPiei, and from the fact that all Pi are assumed to be symmetric and positive definite, we have 12ddteiPiei(λmin(Pi)/2)ddtei2. Then, multiplying both sides of (28) by eiPi, we have that error for the ith node is given by
λmin(Pi)2ddtei2eiPi(ξi(ζi)ξi(xi))+pi,
(33)
where pi=eiPi(AiLiCi)ei+eiPiΓiH~i(ζ)a~i+eiPiΓi(H~i(ζ)H~i(x))ai. Because ei and xi are bounded (xi is bounded from Assumption II.1), we have that all ζi=ei+xi are bounded as well. Then, we can conclude that the right-hand side of (33) is bounded provided ξi satisfies (25) and the functions h~ij(ζi,ζj) are continuous along the bounded trajectories ζi,ζj. This implies that dei2/dt is bounded and from the fact that ei is square integrable; from Barbalat’s lemma (Lemma 5.1 in Ref. 39), we have that ei(t) is uniformly continuous, so ei(t)=0 as t.
Step 3 (Convergence): From (27) and (28), we can write
a~˙i=ψi(t),e˙i=wi(t)a~i+φi(t),iN,
(34)
with ψi(t)=μiH~i(ζ)MiCiei, φi(t)=(AiLiCi)ei+ξi(ζi,t)ξi(xi,t)+(Hi(ζ)Hi(x))ai, and wi=ΓiH~i(ζ). Note that limtφi(t)=limtψi(t)=0 as limtei(t)=0, satisfying condition (i) of Lemma A.1. From the continuity of all functions h~ij [the Lipschitz condition (26)], we know that all functions h~ij are bounded along the bounded trajectories so that the wi(t) are bounded as well. Next, we note that h˙ij=Dhij(xi,xj)[x˙i,x˙j], with Dhij(xi,xj) being the Jacobian matrix. Because all fi(xi) are continuous in xi, we have that the w˙i are bounded for all iN. In addition, from condition (C2), we have that all the Jacobian matrices Dhij(xi,xj) are continuous and thus they are bounded along bounded trajectories xi,xj. This implies that all h˙ij, and thus w˙i, are bounded as well. This together with all wi=Hi(ζ) being PE guarantees that all a~i asymptotically converge to zero, thus completing the proof.
Remark IV.1

Note that the linear part of the vector field is crucial for stabilizing the estimator, as with the classical Luenberger and adaptive observers with Lipschitzian nonlinearities.48–50 Finding appropriate matrices satisfying condition (C1), however, might not be an easy task. This is a common problem in the literature of adaptive observers.50,51 The existence of matrices Li can be studied using the Kalman-Yakubovich-Popov lemma on strictly positive real functions,52 and its solutions can be found using standard software for Linear Matrix Inequalities (LMI) (see the example in Sec. VIC2).

Remark IV.2

Condition (25) is known as a QUAD (quadratic) condition, encompassing a wider class of nonlinear systems than Lipschitz systems.53 

Remark IV.3

Within the context of adaptive control systems, estimators in the form of (15) are similar to gradient-descent-based parameter identification algorithms,37–39 which are able to track slowly varying couplings weights aij(t).

We consider the network reconstruction problem in (1) where the intrinsic node dynamics are Lorenz systems given by2 

xi=[vipiqi],fi(xi)=[σ(pivi)vi(ρqi)pivipiωqi],
(35)

where σ=10, ω=8/3, and ρ is an unknown constant. We consider the nonlinear coupling functions in (1) to be given by hij(xi,xj)=[tanh((vivj)/εi),0,0], with εi being a positive constant for all iN. In addition, we assume that only the states vi and qi are accessible for measurements for each node iN; that is, yi=Cxi, where CR2×3 is given in (19).

To implement the reduced-order estimators (14) and (15), we need to find functions αi and βij satisfying Assumption III.1. We start by choosing the one-dimensional function ϕi(yi)=(1/2)vi210qi, whose gradient is given by ϕi(yi)=[vi,10]. Then, using Eqs. (11) and (12), we find that

ϕi(yi)Cifi(xi)=αi(yi)=10vi2+(80/3)qi,
(36)
ϕi(yi)Cihij(xi,xj)=βij(yi,yj)=vitanhvjviεi.
(37)

Note that the functions αi and βij depend only on the output states, satisfying Assumption III.1. Also, similar to Example III.1, there is no need to know the unknown constant ρ.

To test the reduced-order estimator, we consider the unknown network topology depicted in Fig. 1 where N=8 (N={1,2,3,4,5,6,7,8}). We choose ρ=28 and εi=1.5i/(i+1) for all iN for the node dynamics and coupling functions. In addition, we select κi=κ=100 and μi=μ=10 for all iN.

FIG. 1.

Network of eight Lorenz oscillators.

FIG. 1.

Network of eight Lorenz oscillators.

Close modal

To verify the PE condition of Theorem III.1, we define Qi=tt+Twi(yi(τ))wi(yi(τ))dτ, and we calculate the minimum eigenvalue λmin(Qi) setting T=10s for all iN. The time evolution of the estimated network topology and λmin(Qi) are shown in Figs. 2(a) and 2(b), respectively. Note that λmin(Qi) is always positive, thus fulfilling the PE condition of Theorem III.1 and guaranteeing convergence to the unknown coupling weights.

FIG. 2.

Time trajectories of (a) estimated coupling weights a^i for κ=100 (the blue dashed lines represent the values of the unknown coupling weights). (b) λmin(Qi(t)).

FIG. 2.

Time trajectories of (a) estimated coupling weights a^i for κ=100 (the blue dashed lines represent the values of the unknown coupling weights). (b) λmin(Qi(t)).

Close modal

As a second example, we consider networks of chaotic circuits interconnected through identical resistors.

1. Node dynamics

The dynamics of each node are given by the electrical oscillator depicted in Fig. 3(a). The element D represents a modified nonlinear Chua’s diode whose voltage-current relation is a linear function as shown in Fig. 3(b). The circuit realization of D is depicted in Fig. 3(c). Using Kirchhoff’s laws, we find the circuit dynamics are given by54 

C1v˙C1=1R(vC2vC1)ID(vC1)+INet,
(38)
C2v˙C2=1R(vC1vC2)+IL,
(39)
LI˙L=vC2,
(40)

where vC1 and vC2 are the voltages across capacitors C1 and C2, respectively, IL is the current through the inductor L, and INet is an incoming current from interconnections with neighboring circuits. In addition, ID(vC1) is the diode current given by

ID(vC1)={m0vC1E1vC1E1,m1vC1+b1E1<vC1E2,m1vC1b1E2vC1<E1,m2vC1+b2VsatvC1<E2,m2vC1b2E2<vC1Vsat,0{otherwise},
(41)

where

E0=R6R6+R5Vsat,E1=R3R2+R3Vsat,
(42)
m0=R3+R6R3R6,m1=1R41R3,m2=1R1,
(43)
b1=Vsat/R4,b2=Vsat/R1.
(44)
FIG. 3.

(a) Schematic diagram of Chua’s circuit. (b) Current-voltage function of Chua’s diode. (c) An op amp implementation of Chua’s diode.

FIG. 3.

(a) Schematic diagram of Chua’s circuit. (b) Current-voltage function of Chua’s diode. (c) An op amp implementation of Chua’s diode.

Close modal

All parameter values are reported in Table I.

TABLE I.

Parameter values of a single Chua’s circuit.

SymbolParameterValueUnit
C1 Capacitance 10.05 nF 
C2 Capacitance 100.65 nF 
L Inductance 18.5 mH 
R Resistance 1796 Ω 
R1 Resistance 219.79 Ω 
R2 Resistance 219.83 Ω 
R3 Resistance 2200 Ω 
R4 Resistance 22 kΩ 
R5 Resistance 22 kΩ 
R6 Resistance 3296.25 Ω 
Vsat Voltage 14 
SymbolParameterValueUnit
C1 Capacitance 10.05 nF 
C2 Capacitance 100.65 nF 
L Inductance 18.5 mH 
R Resistance 1796 Ω 
R1 Resistance 219.79 Ω 
R2 Resistance 219.83 Ω 
R3 Resistance 2200 Ω 
R4 Resistance 22 kΩ 
R5 Resistance 22 kΩ 
R6 Resistance 3296.25 Ω 
Vsat Voltage 14 

2. Network model

We consider N>1 identical interconnected Chua’s circuits. The variables vC1,i and vC2,i represent the voltages across C1 and C2, and IL,i represents the current through the inductor L for the ith circuit. The interconnections between circuits are through identical resistors Rlink. The overall dynamics of the electrical network are given by

C1v˙C1,i=1R(vC2,ivC1,i)ID(vC1,i)+INet,i,
(45)
C2v˙C2,i=1R(vC1,ivC2,i)+IL,i,
(46)
LI˙L,i=vC2,i,
(47)
INet,i=j=1NSij(vC1,jvC1,i)Rlink,
(48)

where Sij=1 if there is an interconnection between circuits i and j, while Sij=0 otherwise. Letting xi=[vC1,i,vC2,i,IL,i], the network of Chua’s circuits can be rewritten as in (1) with fi(xi)=Axi+ξi(xi), where

A=[1RC11RC101RC21RC2101L0],ξ(xi)=[ID,i(vC1)C100].
(49)

The unknown coupling weights are given by aij=Sij/Rlink and the coupling functions hij(xi)=(vC1,jvC1,i)/C1. We assume that both vC1,i and vC2,i for all iN are accessible for measurements, i.e., yi=Cxi, where C is given in (19).

1. Case a: Reduced-order estimator

We first design the reduced-order estimators (14) and (15) by choosing the function ϕi(yi)=vC1,i for all iN. Using (49) and (11) and (12), we find that α(yi)=(1/(RC1))(vC2,ivC1,i) and βij=(1/C1)ID(vC1,i), and Assumption III.1 is satisfied. We assume that the coupling resistor Rlink is unknown but belongs to the interval Rlink[20,70]kΩ, implying |aij|1/(20kΩ)=50μS. Note that all estimates of aij=Sij/Rlink take very small values (on the order of 105) if there is an interconnection link between nodes i and j. Although this is not an issue for numerical simulations, in practice, there is not exact convergence due to unavoidable model mismatches and noise (see Sec. VI). This induces oscillations around the true value and would make difficult to distinguish between links that are zero or those which are in the order of 105. To overcome this issue, we replace the state a^i in the left hand of (15) by a^i/ar, where ar is a positive constant representing the rescaling of the estimate. The estimate in (14) will asymptotically converge to a^iarai=arSij/Rlink. By setting ar=100kΩ, we have that the rescaled estimates would converge to either zero or values between (100/70,100/20)=(1.4286,5) for nonexisting (Sij=0) or existing (Sij=1) links, respectively. This makes it easier to distinguish between links even if there are oscillations.

For the numerical simulations, we consider the case where the unknown network topology is a line [see Fig. 7(a)] and Rlink=50kΩ. The time evolution of the reduced-order estimator using the rescaled variable with ar=100kΩ is shown in Fig. 4(a) for κi=κ=50000 and μi=μ=20000. Note that network reconstruction is attained provided the PE condition of Theorem III.1 is satisfied. Indeed, the minimum eigenvalue of Qi=tt+Twi(yi(τ))wi(yi(τ))dτ for T=3μs depicted in Fig. 4(b) is always positive. We have chosen these values of κ and μ in order to compare the simulations with the full-order copy-synchronization approach and with experimental results. We show in Sec. VI that there is a trade-off between the rate of convergence and accuracy of the estimation.

FIG. 4.

Time evolution of (a) the estimated network topology and (b)λmin(Qi) for the reduced-order estimator. The blue dashed lines represent the unknown values aij that are either zero or the rescaled value 100kΩ/50kΩ=2. The red dashed line represents the minimum value of λmin(Qi).

FIG. 4.

Time evolution of (a) the estimated network topology and (b)λmin(Qi) for the reduced-order estimator. The blue dashed lines represent the unknown values aij that are either zero or the rescaled value 100kΩ/50kΩ=2. The red dashed line represents the minimum value of λmin(Qi).

Close modal

2. Case b: Full-order estimator

To design the estimators (21) and (22), we use Theorem IV.1. We first start by noticing that the coupling functions can be rewritten as hij(xi)=(vC1,jvC1,i)/C1=Γh~ij with Γ={diag}{1,0,0} and h~ij=(1/C1)(xjxi) for all i,jN. Because the coupling functions h~ij are linear, the Lipschitz condition (C3) of Theorem IV.1 is easily verified with Lij=Lmax=1/C1.

Next, assuming all nodes share identical parameters, we need to find matrices Pi=P, Ri=R, and Mi=MiN such that the first two conditions of Theorem IV.1 are satisfied. Using the software cvx, we solve the LMI of condition (C1),

M=[100000],R=k~C,k~=50000

and

P=[10009.660.01400.0145394],L=[k~000.1035k~00].
(50)

Next, we show that ξ() satisfies the QUAD condition (25). Using the fact that ξ() is a piecewise smooth linear function, we have that (z1z2)P(ξ(z1)ξ(z2))ρ(z1z2)(z1z2), for all z1,z2R3, with ρ=m0/C1=7.5415×104. From the earlier calculation, we have that Lmax=1/C1 and |aij|amax=50×106. Plugging Mmax=1, L=Lmax, and N=4 into the right-hand side of the inequality in condition (C4) of Theorem IV.1, we get γ>ρ+amaxMmaxL(1+N)=1.003×105. It follows that γ=1.3672×105 and condition (C4) is fulfilled. As also done in the previous case, we rescale the estimation variable a^i in (22) by substituting a^i/100 kΩ. The time evolution of the estimated network topology is shown in Fig. 5(a) for μi=μ=20000. Similar to the previous examples, we calculate λmin(Qi(t)), where Qi=tt+THi(τ)Hi(τ)dτ. We find that setting T=3μs, all λmin(Qi(t)) are lower-bounded by a positive constant as shown in Fig. 5(b), satisfying the PE condition of Theorem IV.1.

FIG. 5.

Network reconstruction using the full-order copy-synchronization approach. Time evolution of (a) the estimated network (21) and (b) λmin(Qi(t)).

FIG. 5.

Network reconstruction using the full-order copy-synchronization approach. Time evolution of (a) the estimated network (21) and (b) λmin(Qi(t)).

Close modal

To carry out the experimental validation of the network reconstruction strategies, we use the experimental setup of interconnected Chua’s circuits shown in Fig. 6. The setup was developed at the University of Naples Federico II, Naples, Italy, and is fully described in Ref. 55. It consists of (1) an electrical module of Chua’s circuits, (2) a data acquisition system, (3) software developed in MATLAB for real-time monitoring of the network states, and (4) an external storage unit to record all the network output signals. The experimental parameters of the real circuits are those reported in Table I.

FIG. 6.

Experimental setup. Adapted with permission from Petrarca et al., “Experimental dynamics observed in a configurable complex network of chaotic oscillators,” in International Conference on Nonlinear Dynamics of Electronic Systems (Springer, Cham, 2014), pp. 203–210. Copyright 2014 Springer International Publishing.57 

FIG. 6.

Experimental setup. Adapted with permission from Petrarca et al., “Experimental dynamics observed in a configurable complex network of chaotic oscillators,” in International Conference on Nonlinear Dynamics of Electronic Systems (Springer, Cham, 2014), pp. 203–210. Copyright 2014 Springer International Publishing.57 

Close modal
FIG. 7.

Different network topologies: (a) line, (b) ring, and (c) star.

FIG. 7.

Different network topologies: (a) line, (b) ring, and (c) star.

Close modal
FIG. 8.

(a) Estimated network topology using the reduced-order observer with κ=100000 and μ=200. (b) Time trajectory of the norm of the error a~=a^a. The black dashed line represents the average steady-state error. (c) Time evolution of λmin(Qi).

FIG. 8.

(a) Estimated network topology using the reduced-order observer with κ=100000 and μ=200. (b) Time trajectory of the norm of the error a~=a^a. The black dashed line represents the average steady-state error. (c) Time evolution of λmin(Qi).

Close modal

For the sake of simplicity, the experiments were conducted on networks of four nodes with three different topologies, as depicted in Fig. 7. The unknown coupling resistance is Rlink=49.027kΩ. We obtained measurements of vC1,i and vC2,i for each node with a sampling rate of 13.3μs. We use those signals to drive the estimators designed in Sec. V.

We first consider the case where the unknown network topology is a line [see Fig. 7(a)]. We use the reduced-order estimator designed in Sec. V C where ϕi(yi)=vC1,i and the rescaled variable a^i/100kΩ.

The time trajectories of the estimated network topology and their corresponding λmin(Qi) are shown in Figs. 8(a) and 8(b), respectively. Note that λmin(Qi) is always positive, satisfying the PE condition of Theorem III.1 and suggesting that the signals are rich enough to reconstruct the network. Indeed, the estimates a^i converge with a low residual error [0.2472, see Fig. 8(b)]. This error is due to unavoidable mismatches between the mathematical model and the real experimental model and also from small delays and quantization errors, which are added during the signal acquisition.

Unlike the numerical simulations reported in Sec. V, we use a larger value of κ and a smaller value of the adaptation gain μ. In Fig. 4(a), convergence is attained faster (in 0.005 s) by setting μ=20000. In practice, however, the measurements yi have small quantization errors, and high values of μ amplify them, adversely affecting the network estimation. To illustrate this trade-off between accuracy and rate of convergence, we plot the estimation error a~, varying the adaptation gain μ (see Fig. 9) when the unknown network topology is a ring. Note that for lower values of μ, convergence is slower and the estimation error is small, while the error increases for larger values of μ.

FIG. 9.

Estimation error of the reduced-order estimator for the ring network topology [see Fig. 7(c)], κ=100000, and different values of μ.

FIG. 9.

Estimation error of the reduced-order estimator for the ring network topology [see Fig. 7(c)], κ=100000, and different values of μ.

Close modal

A similar conclusion can be made for all three network topologies (see Fig. 7) as shown in Fig. 10(a). Note that the steady-state error increases for larger values of μ. Moreover, we also observe that the error can be decreased by choosing larger values of κ as shown in Fig. 10(b).

FIG. 10.

Steady-state estimation error of the reduced-order estimator for the three network topologies in Fig. 7 with (a) κ=100000 and μ variable and (b) μ=200 and κ variable.

FIG. 10.

Steady-state estimation error of the reduced-order estimator for the three network topologies in Fig. 7 with (a) κ=100000 and μ variable and (b) μ=200 and κ variable.

Close modal

Finally, we also use the full-order estimator designed in Sec. V C to reconstruct the network topologies shown in Fig. 7. Similar to the previous case, we also find that the low estimation error is attained for low values of μ and high values of κ~, as shown in Fig. 11.

FIG. 11.

Steady-state estimation error of the full-order estimator for the three network topologies in Fig. 7 with (a) κ~=1000000 and μ variable and (b) μ=2500 and κ~ variable.

FIG. 11.

Steady-state estimation error of the full-order estimator for the three network topologies in Fig. 7 with (a) κ~=1000000 and μ variable and (b) μ=2500 and κ~ variable.

Close modal

Using a control theoretical approach, we have studied the inference problem of an arbitrary network structure based on measurements of each node’s activity. By recasting the problem in the context of adaptive observers, full-order and reduced-order estimators were proposed. Moving beyond existing methods in the literature, these estimators are effective for a broader class of node dynamics and coupling functions, arbitrary coupling weights, and partial state measurements. We derived necessary and sufficient conditions guaranteeing asymptotic network reconstruction, and we showed that under appropriate projections, the number of estimator variables can be substantially reduced, making the estimation suitable for large networks. Numerical and experimental results validate the effectiveness of the proposed strategies.

In the examples in this paper, the chaotic dynamics of the network nodes ensure that the PE condition is satisfied. For some complex network dynamics, the PE condition (see Definition II.1) may not be satisfied, as in the case that all node trajectories converge to each other (synchronization). The node trajectories are linearly dependent and the PE condition is not satisfied, making it impossible to reconstruct the network.32 Recent studies indicate that network symmetries represent a fundamental obstacle to network reconstruction.36 This issue can be sidestepped if the transient dynamics before synchronization are rich enough56 or if we break the network symmetry by injecting perturbation signals on a fraction of nodes. Finding the minimal number and location of such inputs is not a trivial task, however, since the only available information is the node state trajectories. We are currently investigating this issue.

This work was supported by the Office of Naval Research (ONR) under Grant No. N00014-13-1-0331 and by the Army Research Lab. In addition, we wish to thank the Department of Electrical Engineering and Information Technology, University of Naples Federico II, Italy, and, in particular, to Professor Mario di Bernardo for providing hosting in his lab, and Professor Massimiliano de Magistris and Professor Carlo Petrarca for providing access to their experimental setup. Additionally, we acknowledge the Young Investigator Training Program (YITP)—ISCAS2018 for providing the scholarship that funded the stay of Daniel Alberto Burbano Lombana in Italy. The dataset and codes used in the examples will be provided upon request.

(Theorem 2.17 and Corollary 2.3 in Ref. 38)

Theorem A.1
(Theorem 2.17 and Corollary 2.3 in Ref. 38)
Consider a dynamical system of the form
x˙1=cw(t)x2,x˙2=Ax2+w(t)x1,
(A1)
where x1Rm,x2Rn,ARn×n is a Hurwitz matrix,c is a positive constant, and wRm×n. Then, (i) a necessary and sufficient condition to guarantee limtx1(t)=x2(t)=0 is that w is bounded and there exist positive constants T0,T, and c with a t2[t,t+T0] such that for any unit vector vRm(1/T0)t2t2+Tw(τ)vdτc, for all t0. (ii) A sufficient but not necessary condition guaranteeing that limtx1(t)=x2(t)=0 is that both w(t) and w˙(t) are bounded and w(t) is PE (see Definition II.1).

(Ref. 46)

Lemma A.1
(Ref. 46)
Consider the system
x˙1=ψ(t),x˙2=w(t)x1+φ(t),
(A2)
where x1Rm,x2Rn,w:R+Rn×m,φ:R+Rn, and ψ:R+Rm. Assume that (i)limtx1(t)=limtφ(t)=limtψ(t)=0, and (ii)w,w˙ are bounded with w being PE (see Definition II.1). Then, limtx2(t)=0.
1.
S. H.
Strogatz
, “
Exploring complex networks
,”
Nature
410
,
268
276
(
2001
).
2.
S.
Boccaletti
,
V.
Latora
,
Y.
Moreno
,
M.
Chavez
, and
D.-U.
Hwang
, “
Complex networks: Structure and dynamics
,”
Phys. Rep.
424
,
175
308
(
2006
).
3.
G.
Chen
,
X.
Wang
, and
X.
Li
,
Fundamentals of Complex Networks: Models, Structures and Dynamics
(
John Wiley & Sons
,
2014
).
4.
G.
Caldarelli
,
A.
Chessa
,
F.
Pammolli
,
A.
Gabrielli
, and
M.
Puliga
, “
Reconstructing a credit network
,”
Nat. Phys.
9
,
125
126
(
2013
).
5.
F.
Alderisio
,
G.
Fiore
,
R. N.
Salesse
,
B. G.
Bardy
, and
M.
Di Bernardo
, “
Interaction patterns and individual dynamics shape the way we move in synchrony
,”
Sci. Rep.
7
,
6846
(
2017
).
6.
F.
Dörfler
,
M.
Chertkov
, and
F.
Bullo
, “
Synchronization in complex oscillator networks and smart grids
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
2005
2010
(
2013
).
7.
F.
Pasqualetti
,
S.
Zampieri
, and
F.
Bullo
, “
Controllability metrics, limitations and algorithms for complex networks
,”
IEEE Trans. Control Netw. Syst.
1
,
40
52
(
2014
).
8.
Y.-Y.
Liu
,
J.-J.
Slotine
, and
A.-L.
Barabási
, “
Observability of complex systems
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
2460
2465
(
2013
).
9.
Y.-Y.
Liu
and
A.-L.
Barabási
, “
Control principles of complex systems
,”
Rev. Mod. Phys.
88
,
035006
(
2016
).
10.
K. M.
Lynch
,
I. B.
Schwartz
,
P.
Yang
, and
R. A.
Freeman
, “
Decentralized environmental modeling by mobile sensor networks
,”
IEEE Trans. Robot.
24
,
710
724
(
2008
).
11.
M.
Timme
and
J.
Casadiego
, “
Revealing networks from dynamics: An introduction
,”
J. Phys. A Math. Theor.
47
,
343001
(
2014
).
12.
W.-X.
Wang
,
Y.-C.
Lai
, and
C.
Grebogi
, “
Data based identification and prediction of nonlinear and complex dynamical systems
,”
Phys. Rep.
644
,
1
76
(
2016
).
13.
T. S.
Gardner
,
D.
Di Bernardo
,
D.
Lorenz
, and
J. J.
Collins
, “
Inferring genetic networks and identifying compound mode of action via expression profiling
,”
Science
301
,
102
105
(
2003
).
14.
Y. H.
Chang
,
J. W.
Gray
, and
C. J.
Tomlin
, “
Exact reconstruction of gene regulatory networks using compressive sensing
,”
BMC Bioinformatics
15
,
400
(
2014
).
15.
F.
Basiri
,
J.
Casadiego
,
M.
Timme
, and
D.
Witthaut
, “
Inferring power-grid topology in the face of uncertainties
,”
Phys. Rev. E
98
,
012305
(
2018
).
16.
G.
Sugihara
,
R.
May
,
H.
Ye
,
C.-H.
Hsieh
,
E.
Deyle
,
M.
Fogarty
, and
S.
Munch
, “
Detecting causality in complex ecosystems
,”
Science
338
,
496
500
(
2012
).
17.
M.
de Magistris
,
M.
di Bernardo
,
S.
Manfredi
,
C.
Petrarca
, and
S.
Yaghouti
, “
Modular experimental setup for real-time analysis of emergent behavior in networks of Chua’s circuits
,”
Int. J. Circuit Theory Appl.
44
,
1551
(
2015
).
18.
R.
Vicente
,
M.
Wibral
,
M.
Lindner
, and
G.
Pipa
, “
Transfer entropy: A model-free measure of effective connectivity for the neurosciences
,”
J. Comput. Neurosci.
30
,
45
67
(
2011
).
19.
A. F.
Villaverde
,
J.
Ross
,
F.
Morán
, and
J. R.
Banga
, “
MIDER: Network inference with mutual information distance and entropy reduction
,”
PLoS One
9
,
e96732
(
2014
).
20.
J.
Sun
,
D.
Taylor
, and
E. M.
Bollt
, “
Causal network inference by optimal causation entropy
,”
SIAM J. Appl. Dyn. Syst.
14
,
73
106
(
2015
).
21.
A.
Anandkumar
,
R.
Valluvan
et al., “
Learning loopy graphical models with latent variables: Efficient methods and guarantees
,”
Ann. Stat.
41
,
401
435
(
2013
).
22.
A.
Anandkumar
,
V. Y.
Tan
,
F.
Huang
, and
A. S.
Willsky
, “
High-dimensional Gaussian graphical model selection: Walk summability and local separation criterion
,”
J. Mach. Learn. Res.
13
,
2293
2337
(
2012
), ISSN: 15324435.
23.
M.
Timme
, “
Revealing network connectivity from response dynamics
,”
Phys. Rev. Lett.
98
,
224101
(
2007
).
24.
D.
Yu
, “
Estimating the topology of complex dynamical networks by steady state control: Generality and limitation
,”
Automatica
46
,
2035
2040
(
2010
).
25.
F.
Alderisio
,
G.
Fiore
, and
M.
di Bernardo
, “
Reconstructing the structure of directed and weighted networks of nonlinear oscillators
,”
Phys. Rev. E
95
,
042302
(
2017
).
26.
G.
Li
,
X.
Wu
,
J.
Liu
,
J.-A.
Lu
, and
C.
Guo
, “
Recovering network topologies via Taylor expansion and compressive sensing
,”
Chaos
25
,
043102
(
2015
).
27.
Y.
Chen
,
Z.
Zhang
,
T.
Chen
,
S.
Wang
, and
G.
Hu
, “
Reconstruction of noise-driven nonlinear networks from node outputs by using high-order correlations
,”
Sci. Rep.
7
,
44639
(
2017
).
28.
A.
Santos
,
V.
Matta
, and
A. H.
Sayed
, “Consistent tomography over diffusion networks under the low-observability regime,” in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, 2018), pp. 1839–1843.
29.
V.
Matta
and
A. H.
Sayed
, “
Consistent tomography under partial observations over adaptive networks
,”
IEEE Trans. Inf. Theory
65
,
622
646
(
2019
).
30.
B.
Pasdeloup
,
V.
Gripon
,
G.
Mercier
,
D.
Pastor
, and
M. G.
Rabbat
, “
Characterization and inference of graph diffusion processes from observations of stationary signals
,”
IEEE Trans. Signal Inf. Process. Netw.
4
,
481
496
(
2018
).
31.
D.
Yu
,
M.
Righero
, and
L.
Kocarev
, “
Estimating topology of networks
,”
Phys. Rev. Lett.
97
,
188701
(
2006
).
32.
L.
Chen
,
J.-A.
Lu
, and
K. T.
Chi
, “
Synchronization: An obstacle to identification of network topology
,”
IEEE Trans. Circuits Syst. II Express Briefs
56
,
310
314
(
2009
).
33.
J.
Zhao
,
Q.
Li
,
J.-A.
Lu
, and
Z.-P.
Jiang
, “
Topology identification of complex dynamical networks
,”
Chaos
20
,
023119
(
2010
).
34.
S.
Zhang
,
X.
Wu
,
J.-A.
Lu
,
H.
Feng
, and
J.
, “
Recovering structures of complex dynamical networks based on generalized outer synchronization
,”
IEEE Trans. Circuits Syst. I Regul. Pap.
61
,
3216
3224
(
2014
).
35.
C.-X.
Fan
,
Y.-H.
Wan
, and
G.-P.
Jiang
, “
Topology identification for a class of complex dynamical networks using output variables
,”
Chin. Phys. B
21
,
020510
(
2012
).
36.
M. T.
Angulo
,
J. A.
Moreno
,
G.
Lippner
,
A.-L.
Barabási
, and
Y.-Y.
Liu
, “
Fundamental limitations of network reconstruction from temporal data
,”
J. R. Soc. Interface
14
,
20160966
(
2017
).
37.
S.
Sastry
and
M.
Bodson
,
Adaptive Control: Stability, Convergence and Robustness
(
Courier Corporation
,
2011
).
38.
K. S.
Narendra
and
A. M.
Annaswamy
,
Stable Adaptive Systems
(
Courier Corporation
,
2012
).
39.
K. J.
Aström
and
B.
Wittenmark
,
Adaptive Control
(
Courier Corporation
,
2013
).
40.
H.
Liu
,
J.-A.
Lu
,
J.
, and
D. J.
Hill
, “
Structure identification of uncertain general complex dynamical networks with time delay
,”
Automatica
45
,
1799
1807
(
2009
).
41.
X.
Wei
,
S.
Chen
,
J.-A.
Lu
, and
D.
Ning
, “
Reconstruction of complex networks with delays and noise perturbation based on generalized outer synchronization
,”
J. Phys. A Math. Theor.
49
,
225101
(
2016
).
42.
Z.
Tang
,
J. H.
Park
, and
T. H.
Lee
, “
Topology and parameters recognition of uncertain complex networks via nonidentical adaptive synchronization
,”
Nonlinear Dyn.
85
,
2171
2181
(
2016
).
43.
H.
Nijmeijer
and
I. M.
Mareels
, “
An observer looks at synchronization
,”
IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
44
,
882
890
(
1997
).
44.
A.
Lorıa
,
E.
Panteley
, and
A.
Zavala-Rıo
, “
Adaptive observers with persistency of excitation for synchronization of chaotic systems
,”
IEEE Trans. Circuits Syst. I
56
,
2703
2716
(
2009
).
45.
D. A.
Burbano-L
,
R.
Freeman
, and
K.
Lynch
, “Inferring the network topology of interconnected nonlinear units with diffusive couplings,” in American Control Conference (ACC) (IEEE, 2018), pp. 3398–3403.
46.
G.
Besançon
, “
Remarks on nonlinear adaptive observer design
,”
Syst. Control Lett.
41
,
271
280
(
2000
).
47.
H.
Dimassi
and
A.
Loria
, “
Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication
,”
IEEE Trans. Circuits Syst. I Regul. Pap.
58
,
800
812
(
2011
).
48.
R.
Rajamani
, “
Observers for Lipschitz nonlinear systems
,”
IEEE Trans. Automat. Contr.
43
,
397
401
(
1998
).
49.
A.
Zemouche
and
M.
Boutayeb
, “
On LMI conditions to design observers for Lipschitz nonlinear systems
,”
Automatica
49
,
585
591
(
2013
).
50.
Y. M.
Cho
and
R.
Rajamani
, “
A systematic approach to adaptive observer synthesis for nonlinear systems
,”
IEEE Trans. Automat. Contr.
42
,
534
537
(
1997
).
51.
Z.
Zhang
and
S.
Xu
, “
Observer design for uncertain nonlinear systems with unmodeled dynamics
,”
Automatica
51
,
80
84
(
2015
).
52.
F.
Zhu
, “
Full-order and reduced-order observer-based synchronization for chaotic systems with unknown disturbances and parameters
,”
Phys. Lett. A
372
,
223
232
(
2008
).
53.
P.
DeLellis
,
M.
di Bernardo
, and
G.
Russo
, “
On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks
,”
IEEE Trans. Circuits Syst. I Regul. Pap.
58
,
576
583
(
2011
).
54.
L.
Fortuna
,
M.
Frasca
, and
M.
Xibilia
, Chua’s Circuit Implementations: Yesterday, Today and Tomorrow, World Scientific Series on Nonlinear Science Series A Vol. 65 (World Scientific, 2009).
55.
M.
de Magistris
,
M.
di Bernardo
,
E.
Di Tucci
, and
S.
Manfredi
, “
Synchronization of networks of non-identical Chua’s circuits: Analysis and experiments
,”
IEEE Trans. Circuits Syst. I Regul. Pap.
59
,
1029
1041
(
2012
).
56.
F.
Sun
,
H.
Peng
,
J.
Xiao
, and
Y.
Yang
, “
Identifying topology of synchronous networks by analyzing their transient processes
,”
Nonlinear Dyn.
67
,
1457
1466
(
2012
).
57.
C.
Petrarca
,
S.
Yaghouti
, and
M.
de Magistris
, “Experimental dynamics observed in a configurable complex network of chaotic oscillators,” in International Conference on Nonlinear Dynamics of Electronic Systems (Springer, Cham, 2014), pp. 203–210.