Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks

Introduction—

Complex dynamical systems are modeled by $𝑁$ degrees of freedom $𝑥 𝑖 (𝑡)$ ( $𝑖 = 1, \dots, 𝑁$ ) that evolve in time according to the differential equation

˙ 𝑥 𝑖 (𝑡) = - 𝑓 (𝑥 𝑖) + 𝑁 \sum 𝑗 = 1 𝐴 𝑖 𝑗 𝑔 (𝑥 𝑖, 𝑥 𝑗),

(1)

where ${𝐴 𝑖 𝑗} 𝑖, 𝑗 = 1, \dots, 𝑁$ defines the interaction network. The function $𝑓 (𝑥)$ governs the dynamics in the absence of interactions, and the kernel $𝑔 (𝑥, 𝑥')$ shapes the pairwise couplings. Equation models the nonequilibrium dynamics of neural networks and ecosystems , epidemic spreading , synchronization phenomena , opinion dynamics , and multivariate Ornstein-Uhlenbeck processes . Table specifies $𝑓 (𝑥)$ and $𝑔 (𝑥, 𝑥')$ for paradigmatic models of complex behavior.

TABLE I.

Explicit form of $𝑓 (𝑥)$ and $𝑔 (𝑥, 𝑥')$ in complex systems modeled by Eq. . Equation for the SIS model is derived through the quenched mean-field approximation .
Model	$𝑓 (𝑥)$	$𝑔 (𝑥, 𝑥')$
Ornstein-Uhlenbeck process	$𝑥$	$𝑥'$
SIS model of epidemic spreading	$𝑥$	$(1 - 𝑥) 𝑥'$
Lotka-Volterra (LV) model	$𝑥 (𝑥 - 1)$	$𝑥 𝑥'$
Neural network (NN) model	$𝑥$	$t a n h (𝑥')$
Kuramoto model	0	$s i n (𝑥' - 𝑥)$

The foremost problem in the study of complex systems is how to reduce the dynamics of many interacting elements to the dynamics of a few variables . Dynamical mean-field theory (DMFT) is a powerful method to tackle this problem in the limit $𝑁 \to \infty$ , yielding a solution in terms of the path probability for the effective dynamics of a single degree of freedom. The application of DMFT to models described by Eq. has been attracting an enormous interest , especially in the context of neural networks and ecosystems. Phase diagrams of these models reveal a rich phenomenology, including different types of phase transitions , chaotic behavior , and coexistence of multiple attractors . Despite this substantial theoretical progress, DMFT is currently limited to dense networks, where each dynamical variable is essentially coupled to all others by means of Gaussian interaction strengths. However, the interactions in real-world complex systems are known to be sparse and heterogeneous . Sparseness indicates that each element of the system interacts on average with a finite number of others, while heterogeneity refers to fluctuations in the local topology of the interaction network. How to integrate these more realistic features in the formalism of DMFT for systems modeled by Eq. remains an unresolved challenge, and even basic questions, such as deriving the phase diagram of sparse complex systems, are still out of reach.

On the other hand, the dynamics of complex systems on sparse networks has been extensively studied in the case of Ising spins . In this context, both the cavity method and DMFT provide an analytic solution in terms of the path probability for the effective dynamics of a single dynamical variable. However, the presence of bidirected edges induces a temporal feedback, which, in the particular case of sparse systems, leads to an exponential growth of the path-probability dimension , rendering numerical computations unfeasible. Approximation schemes, such as the one-time approximation , make these computations possible . When the interactions are directed or unidirectional, there is no temporal feedback and the path-probability equation can be efficiently solved .

Inspired by these results for Ising spins, in this Letter we solve the dynamics of models governed by Eq. on sparse directed networks with an heterogeneous topology. The solution represents a foundational step in the study of complex systems, as it ultimately incorporates the sparse and heterogeneous structure of complex networks into the formalism of DMFT. Directed networks are interesting because they model the nonreciprocal interactions in real-world complex systems , including the human cortex , food webs , gene regulatory networks , online social networks , and the World Wide Web . The formalism presented here thus opens the possibility to analytically investigate how realistic interactions impact the dynamics of complex systems.

By generalizing DMFT to sparse systems, we obtain an exact equation for the path probability describing the effective dynamics of a single variable for $𝑁 \to \infty$ , and we solve this equation for different models in Table by using the population dynamics algorithm . The excellent agreement between our theoretical results and numerical simulations of finite systems confirms that our solution applies to various nonlinear models interacting through different network topologies, including networks with power-law degree distributions .

As an application, we determine the phase diagram of the neural network model by Sompolinsky et al. in the sparse regime. We show that the phase diagram displays trivial and nontrivial fixed-point phases, a chaotic phase with zero mean activity, and a chaotic phase with nonzero mean activity . By calculating certain macroscopic observables, we determine the transition lines as functions of the mean degree and the variance of the coupling strengths, showing their consistency with the universal critical lines derived from random matrix theory . In particular, we provide numerical evidence that the transition between the chaotic phases coincides with the vanishing of the gap between the leading and the subleading eigenvalue of the interaction matrix.

Sparse directed networks—

Let $𝐴 𝑖 𝑗 = 𝐶 𝑖 𝑗 𝐽 𝑖 𝑗$ be the elements of the $𝑁 \times 𝑁$ interaction matrix $𝑨$ . The binary variables $𝐶 𝑖 𝑗 \in {0, 1}$ determine the network topology, while $𝐽 𝑖 𝑗 \in ℝ$ controls the interaction strengths. If $𝐶 𝑖 𝑗 = 1$ , there is a directed edge $𝑗 \to 𝑖$ pointing from node $𝑗$ to $𝑖$ , while $𝐶 𝑖 𝑗 = 0$ otherwise. The indegree $𝐾 𝑖 = \sum 𝑁 𝑗 = 1 𝐶 𝑖 𝑗$ and the outdegree $𝐿 𝑖 = \sum 𝑁 𝑗 = 1 𝐶 𝑗 𝑖$ count the number of links entering and leaving node $𝑖$ , respectively. The random variables ${𝐶 𝑖 𝑗} 𝑖 \neq 𝑗$ follow the distribution

ℙ ({𝐶 𝑖 𝑗}) = 1 𝒩 𝑁 \prod 𝑖 \neq 𝑗 = 1 [𝑐 𝑁 𝛿 𝐶 𝑖 𝑗, 1 + (1 - 𝑐 𝑁) 𝛿 𝐶 𝑖 𝑗, 0] \times 𝑁 \prod 𝑖 = 1 𝛿 𝐾 𝑖, \sum 𝑁 𝑗 = 1 𝐶 𝑖 𝑗 𝛿 𝐿 𝑖, \sum 𝑁 𝑗 = 1 𝐶 𝑗 𝑖,

(2)

where $𝒩$ is the normalization constant and $𝐶 𝑖 𝑖 = 0$ . The degrees ${𝐾 𝑖, 𝐿 𝑖} 𝑖 = 1, \dots, 𝑁$ are independent and identically distributed random variables drawn from $𝑝 𝑘, ℓ = 𝑝 i n, 𝑘 𝑝 o u t, ℓ$ , where $𝑝 i n, 𝑘$ and $𝑝 o u t, ℓ$ are the indegree and the outdegree distribution , respectively. The parameter $𝑐$ is the average degree

𝑐 = \infty \sum 𝑘 = 0 𝑘 𝑝 i n, 𝑘 = \infty \sum ℓ = 0 ℓ 𝑝 o u t, ℓ .

(3)

Equation defines a network ensemble where directed links are randomly placed between pairs of nodes with probability $𝑐 / 𝑁$ , subject to the prescribed degree sequences generated from $𝑝 𝑘, ℓ$ . In the limit $𝑁 \to \infty$ , network samples generated from Eq. are similar to those produced by the configuration model . The coupling strengths ${𝐽 𝑖 𝑗} 𝑖, 𝑗 = 1, \dots, 𝑁$ are independent and identically distributed random variables drawn from a distribution $𝑝 𝐽$ with mean $𝜇 𝐽$ and variance $𝜎 2 𝐽$ . The distributions $𝑝 𝑘, ℓ$ and $𝑝 𝐽$ fully specify the network ensemble, allowing for a systematic investigation of how network heterogeneities impact the dynamics of complex systems.

Solution through DMFT—

We solve the coupled dynamics of Eq. on directed networks by using dynamical mean-field theory (DMFT) . We consider the sparse regime, where the mean degree $𝑐$ is finite, independent of $𝑁$ . DMFT is based on the generating functional

𝒵 [𝝍] = \int (𝑁 \prod 𝑖 = 1 𝐷 𝑥 𝑖) 𝒫 [𝒙] 𝑒 𝑖 \int 𝑑 𝑡 \sum 𝑁 𝑖 = 1 𝑥 𝑖 (𝑡) 𝜓 𝑖 (𝑡)

(4)

of the probability density $𝒫 [𝒙]$ of observing a dynamical path of states $𝒙 (𝑡) = [𝑥 1 (𝑡), \dots, 𝑥 𝑁 (𝑡)]$ in a fixed time interval. The correlation functions of ${𝑥 𝑖 (𝑡)} 𝑖 = 1, \dots, 𝑁$ follow from the derivatives of $𝒵 [𝝍]$ with respect to the external sources $𝝍 (𝑡) = [𝜓 1 (𝑡), \dots, 𝜓 𝑁 (𝑡)]$ . The $𝑛$ th moment of the local variable $𝑥 𝑖 (𝑡)$ ,

⟨ 𝑥 𝑛 𝑖 (𝑡) ⟩ = \int (𝑁 \prod 𝑖 = 1 𝐷 𝑥 𝑖) 𝑥 𝑛 𝑖 (𝑡) 𝒫 [𝒙] = (- 𝑖) 𝑛 𝛿 𝑛 𝒵 [𝝍] 𝛿 𝜓 𝑛 𝑖 (𝑡) | 𝝍 = 0,

(5)

yields the time evolution of the macroscopic quantities

𝑚 (𝑡) = l i m 𝑁 \to \infty 1 𝑁 𝑁 \sum 𝑖 = 1 ⟨ 𝑥 𝑖 (𝑡) ⟩, 𝑞 (𝑡) = l i m 𝑁 \to \infty 1 𝑁 𝑁 \sum 𝑖 = 1 ⟨ 𝑥 2 𝑖 (𝑡) ⟩ .

(6)

Clearly, $𝒵 [𝝍]$ fulfills $𝒵 [0] = 1$ .

In , we calculate the average of $𝒵 [𝝍]$ over the network ensemble defined by Eq. for finite $𝑐$ , recasting the problem in terms of the solution of a saddle-point integral. More importantly, we give a clear physical interpretation of the order parameters, simplifying the saddle-point equations and obtaining a feasible solution for the dynamics in the sparse regime. In the limit $𝑁 \to \infty$ , the microscopic dynamical variables decouple, and the path probability $𝒫 [𝑥]$ for the effective dynamics of a single variable $𝑥 (𝑡)$ is determined from

𝒫 [𝑥] = \infty \sum 𝑘 = 0 𝑝 i n, 𝑘 \int (𝑘 \prod 𝑗 = 1 𝐷 𝑥 𝑗 𝒫 [𝑥 𝑗]) \int (𝑘 \prod 𝑗 = 1 𝑑 𝐽 𝑗 𝑝 𝐽 (𝐽 𝑗)) \times 𝛿 𝐹 [˙ 𝑥 (𝑡) + 𝑓 [𝑥 (𝑡)] - 𝑘 \sum 𝑗 = 1 𝐽 𝑗 𝑔 [𝑥 (𝑡), 𝑥 𝑗 (𝑡)]],

(7)

where $𝛿 𝐹$ is the functional Dirac- $𝛿$ . The macroscopic observables are computed from $𝑚 (𝑡) = ⟨ 𝑥 (𝑡) ⟩ *$ and $𝑞 (𝑡) = ⟨ 𝑥 2 (𝑡) ⟩ *$ , where

⟨ 𝑥 𝑛 (𝑡) ⟩ * = \int 𝐷 𝑥 𝑥 𝑛 (𝑡) 𝒫 [𝑥] (𝑛 = 1, 2)

(8)

is the average over the effective dynamics governed by $𝒫 [𝑥]$ . The self-consistent Eq. is the exact solution of a broad class of models (see Table ) on sparse directed networks with a local treelike structure and arbitrary distributions $𝑝 𝐽$ and $𝑝 𝑘, ℓ = 𝑝 i n, 𝑘 𝑝 o u t, ℓ$ . In , we address the more general case of dynamical models with Gaussian additive noise on networks with correlated indegrees and outdegrees. The solution of Eq. determines the time evolution of the full probability distribution of the microscopic variables in the limit $𝑁 \to \infty$ .

Equation is formally similar to other distributional equations appearing in the study of sparse disordered systems . Therefore, we can numerically solve this equation using the population dynamics algorithm . In the standard version of this algorithm , a probability density is parametrized by a population of stochastic variables. Here, we generalize the algorithm to calculate $𝒫 [𝑥]$ by introducing a population of dynamical trajectories. At each iteration step, a single path is chosen randomly from the population and updated according to the differential equation imposed by the Dirac- $𝛿 𝐹$ in Eq. . After sufficient iterations, the population of paths converges to a stationary distribution, providing a numerical solution for $𝒫 [𝑥]$ . A detailed account of the algorithm is in .

In Fig. , we compare the solutions of Eq. with numerical simulations of the original coupled dynamics, Eq. , for finite $𝑁$ . The panels showcase the time evolution of the mean $𝑚 (𝑡)$ and the standard deviation $\sqrt 𝑞 (𝑡) - 𝑚 2 (𝑡)$ of the microscopic variables for different network topologies and three models of Table : the NN model of , the LV model, and the SIS model of epidemic spreading. The results in Fig. are for Poisson and geometric indegrees, where $𝑝 i n, 𝑘$ is given by

𝑝 i n, 𝑘 = 𝑐 𝑘 𝑒 - 𝑐 𝑘! and 𝑝 i n, 𝑘 = 𝑐 𝑘 (𝑐 + 1) 𝑘 + 1,

(9)

respectively. In , we compare the solutions of Eq. with numerical simulations for two additional cases: the NN model on networks with power-law indegree distributions and the susceptible-infected-susceptible (SIS) model at the epidemic threshold. In all cases, the agreement between our theoretical results for $𝑁 \to \infty$ and numerical simulations for large $𝑁$ is excellent, confirming the exactness of Eq. .

FIG. 1.

Dynamics of the mean $𝑚 (𝑡)$ and the standard deviation $\sqrt 𝑞 (𝑡) - 𝑚 2 (𝑡)$ for the SIS model, LV model, and NN model (see Table ). Results are shown for directed random networks with average degree $𝑐 = 5$ and two indegree distributions [Eq. ]: Poisson (left column) and geometric (right column). The distribution $𝑝 𝐽$ of the coupling strengths has mean $𝜇 𝐽$ and standard deviation $𝜎 𝐽 = 0.1$ . For the LV and NN models, $𝑝 𝐽$ is Gaussian; for the SIS model, $𝑝 𝐽$ is uniform. Solid lines are solutions of Eq. using the population dynamics algorithm with $5 \times 104$ paths and initial condition $𝑥 𝑖 (0) = 10 - 3$ . Symbols denote numerical simulations of Eq. for an ensemble of 10 random networks generated from the configuration model with $𝑁 = 4000$ nodes. Vertical bars are the standard deviation of the macroscopic observables.

An important question is whether Eq. recovers the analytic results of fully connected models as $𝑐 \to \infty$ . By taking the limit $𝑐 \to \infty$ after the thermodynamic limit $𝑁 \to \infty$ , we are effectively considering the scaling regime where $𝑐 \propto 𝑁 𝑎$ , with $0 < 𝑎 < 1$ . We show in that, in the limit $𝑐 \to \infty$ , degree fluctuations remain significant, and $𝒫 [𝑥]$ depends on the full distribution $𝜈 i n (𝜅)$ of rescaled indegrees $𝜅 𝑖 = 𝐾 𝑖 / 𝑐$ ( $𝑖 = 1, \dots, 𝑁$ ). The well-known effective dynamics on fully connected networks is only recovered when $𝜈 i n (𝜅) = 𝛿 (𝜅 - 1)$ . This breakdown in the universality of fully connected models due to degree fluctuations was anticipated in .

Phase diagram of neural networks—

To demonstrate the strength of Eq. and its concrete applications, we derive the phase diagram of the NN model on sparse directed networks. In this context, $𝑥 𝑖 (𝑡) \in ℝ$ represents the synaptic current at neuron $𝑖$ , $𝑓 (𝑥) = 𝑥$ , and $𝑔 (𝑥, 𝑥') = t a n h (𝑥')$ . Let us order the complex eigenvalues ${𝜆 𝛼} 𝛼 = 1, \dots, 𝑁$ of $𝑨$ according to their real parts as $R e 𝜆 1 \geq R e 𝜆 2 \geq \dots \geq R e 𝜆 𝑁$ . A linear stability analysis of Eq. shows that $𝒙 = 0$ is stable if $R e 𝜆 1 < 1$ . Based on analytic results for the spectra of directed networks for $𝑁 \to \infty$ , we find that the trivial fixed point is stable provided $𝑐 < 𝑐 s t a b$ , where

𝑐 s t a b = {1 / 𝜇 𝐽 if 𝑐 > 1 + 𝜎 2 𝐽 / 𝜇 2 𝐽, 1 / (𝜎 2 𝐽 + 𝜇 2 𝐽) if 𝑐 \leq 1 + 𝜎 2 𝐽 / 𝜇 2 𝐽 .

(10)

The two distinct regimes in Eq. result from the gap-gapless transition in the spectrum of $𝑨$ . For $𝑐 > 1 + 𝜎 2 𝐽 / 𝜇 2 𝐽$ , the spectral gap $| 𝜆 1 - 𝜆 2 |$ remains finite as $𝑁 \to \infty$ , while it vanishes for $𝑐 \leq 1 + 𝜎 2 𝐽 / 𝜇 2 𝐽$ . Equation holds for $𝜇 𝐽 > 0$ and $𝑐 > 1$ . The condition $𝑐 > 1$ ensures that directed networks with degree distribution $𝑝 𝑘, ℓ = 𝑝 i n, 𝑘 𝑝 o u t, ℓ$ contain a giant strongly connected component , implying the existence of a continuous part in the eigenvalue distribution of $𝑨$ .

We emphasize that the linear stability analysis of the trivial solution provides no information about the relaxation dynamics or the stationary solutions that emerge when $𝒙 = 0$ becomes unstable. Therefore, we study the dynamics and the stationary states of the NN model by solving Eq. . Figure presents the resulting phase diagrams for different regimes of $𝜇 𝐽$ . In Phase I, $𝑚 (𝑡)$ relaxes exponentially fast to the trivial solution $𝑚 = 0$ , while in Phase II, $𝑚 (𝑡)$ evolves to a nonzero fixed point. In Phases III and IV, the neural network exhibits chaotic activity, characterized by slow and aperiodic oscillations of $𝑚 (𝑡)$ . The stability lines that delimit Phase I are universal, as they depend only on the first and second moments of $𝑝 𝐽$ and $𝑝 i n, 𝑘$ .

FIG. 2.

Phase diagrams $(𝑐, 𝜎 𝐽)$ of the NN model for $𝜇 𝐽 = 0$ , $𝜇 𝐽 \geq 1$ , and $0 < 𝜇 𝐽 < 1$ . The average synaptic current $𝑚 (𝑡)$ relaxes to trivial and nontrivial fixed points in Phases I and II, respectively. In the chaotic Phases III and IV, $𝑚 (𝑡)$ exhibits aperiodic oscillations around zero and nonzero values, respectively (see Fig. ). The dash-dotted line at $𝑐 = 1$ marks the percolation threshold, while the dashed curve identifies the gap-gapless transition. Circles and diamonds represent numerical results for the transition lines delimiting Phase IV (see the main text), obtained from solutions of Eq. using population dynamics with $5 \times 104$ paths, Poisson indegrees, and two values of $𝜇 𝐽$ ( $𝜇 𝐽 = 1 / 3$ and $𝜇 𝐽 = 3 / 2$ ). The red dot marks $(𝑐 *, 𝜎 * 𝐽) = [𝜇 - 1 𝐽, \sqrt 𝜇 𝐽 (1 - 𝜇 𝐽)]$ .

Figure characterizes the transition between the fixed-point Phases I and II. As $𝑐$ approaches the critical mean degree $𝑐 * = 𝜇 - 1 𝐽$ from above, the nontrivial fixed point $𝑚 = 𝑚 (𝑡)$ vanishes continuously. The critical point $𝑐 *$ is independent of $𝑝 i n, 𝑘$ and $𝑚 (𝑡)$ relaxes exponentially fast inside Phases I and II. Because of critical slowing down, the numerical solution of Eq. becomes computationally more demanding near $𝑐 *$ . In Phase II, the neuronal firing rates evolve to a stationary distribution with a finite variance. Figure shows the dynamics of $𝑚 (𝑡)$ within the chaotic phases for several initial conditions. After a transient time $𝑇 t r$ , $𝑚 (𝑡)$ stabilizes into an attractor, oscillating around zero in Phase III and around a nonzero value in Phase IV. The inset in Fig. demonstrates the sensitivity of $𝑚 (𝑡)$ to small perturbations in the initial conditions, characteristic of deterministic chaos . By solving Eq. and numerically computing the temporal averages

𝑀 = 1 𝑇 - 𝑇 t r \int 𝑇 𝑇 t r 𝑑 𝑡 𝑚 (𝑡),

(11)

Δ 2 = 1 𝑇 - 𝑇 t r \int 𝑇 𝑇 t r 𝑑 𝑡 [𝑀 - 𝑚 (𝑡)] 2,

(12)

for $𝑇 ≫ 1$ , we estimate the transition lines delimiting Phase IV . For $𝑇 \to \infty$ , the transition between Phases II and IV is determined by $Δ$ , since $Δ = 0$ in the fixed-point phases and $Δ > 0$ in the chaotic phases. The parameter $𝑀$ distinguishes the chaotic phases: $𝑀 = 0$ in Phase III and $𝑀 \neq 0$ in Phase IV (see Fig. ). Our numerical results for the transition between Phases III and IV are consistent with the dashed line in Fig. , suggesting that this transition is governed by the gap $| 𝜆 1 - 𝜆 2 |$ in the network spectrum. Nevertheless, for $𝑐 \to \infty$ this transition to the chaotic Phase III slightly deviates from the gap-gapless transition , particularly for large $𝜇 𝐽$ .

FIG. 3.

Continuous transition between the fixed-point phases for the NN model on directed random networks. The coupling strengths follow a Gaussian distribution with mean $𝜇 𝐽 = 1 / 3$ and standard deviation $𝜎 𝐽 = 0.1$ . The results are derived from Eq. using population dynamics with $5 \times 104$ paths. (a) Fixed-point solution $𝑚 (𝑡) = 𝑚$ as a function of $𝑐$ for Poisson and geometric indegrees [Eq. ]. (b) Relaxation dynamics of $𝑚 (𝑡)$ across the transition for Poisson indegrees and initial condition $𝑥 𝑖 (0) = 1$ . For $𝑐 < 𝜇 - 1 𝐽$ , $𝑚 (𝑡)$ relaxes exponentially to $𝑚 = 0$ as $𝑡 \to \infty$ .

FIG. 4.

Dynamics of $𝑚 (𝑡)$ inside the chaotic phases for directed networks with Poisson indegrees and a Gaussian distribution $𝑝 𝐽$ with mean $𝜇 𝐽 = 1 / 3$ . The results follow from the solutions of Eq. using population dynamics with $5 \times 104$ paths and several initial conditions. The dashed lines indicate the value of $𝑀$ , Eq. , characterizing the chaotic attractor in each phase. (a) $𝑐 = 2.5$ and $𝜎 𝐽 = 2$ . Inset: dynamics of $𝑚 (𝑡)$ obtained from the *deterministic* Eq. for a single network instance ( $𝑁 = 4000$ ), initial condition $𝑥 𝑖 (0) = 0.15 - 𝜀$ , and different $𝜀$ . (b) $𝑐 = 5$ and $𝜎 𝐽 = 0.62$ .

Conclusions—

We have developed a dynamical mean-field theory of complex systems on sparse directed networks, deriving an exact path-probability equation for the effective dynamics in the limit $𝑁 \to \infty$ . Unlike sparse models with bidirected edges , our path-probability equation can be numerically solved using population dynamics . We confirmed the exactness of our general solution by comparing it with numerical simulations of fundamental models in the study of epidemic spreading, neural networks, and ecosystems. Finally, we applied our solution to determine the complete phase diagram of the sparse and directed version of the canonical neural network model of .

The numerical solution of Eq. does not require generating networks from the configuration model , providing moderate computational advantages over large-scale simulations of Eq. on networks . Beyond its numerical applications, Eq. provides a foundational framework for studying the stationary states , computing correlation functions , deriving approximate dynamical equations for macroscopic order parameters, and developing systematic perturbative approaches .

Our work paves the way for exploring the role of sparse heterogeneous networks on the dynamics of ecosystems, coupled oscillators, epidemic spreading, and beyond. Future works include determining the phase diagram of the sparse Lotka-Volterra model , the influence of external noise on phase diagrams , and the role of network heterogeneities on the critical exponents of complex systems . Lastly, it would be interesting to connect the present formalism with the cavity method in .

Acknowledgments—

F. L. M. thanks Tuan Minh Pham and Alessandro Ingrosso for many stimulating discussions. The author also thanks Thomas Peron and Tuan Minh Pham for their valuable comments on the manuscript. F. L. M. acknowledges support from CNPq (Grant No 402487/2023-0) and from the ICTP through the Associates Program (2023-2028).

Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks

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