Spontaneous organizations of diverse network structures in coupled logistic maps with a delayed connection change

In this study, we performed comprehensive morphological investigations of the spontaneous formations of effective network structures among elements in coupled logistic maps, specifically with a delayed connection change. Our proposed model showed ten states with different structural and dynamic features of the network topologies. Based on the parameter values, various stable networks, such as hierarchal networks with pacemakers or multiple layers, and a loop-shaped network were found. We also found various dynamic networks with temporal changes in the connections, which involved hidden network structures. Furthermore, we found that the shapes of the formed network structures were highly correlated to the dynamic features of the constituent elements. The present results provide diverse insights into the dynamics of neural networks and various other biological and social networks.

coupled chaotic map systems with Hebbian-like rules may result in systems that could exhibit spontaneous hierarchical network structures with asymmetric couplings, as well as the emergence of pacemakers [21,22].
Such hierarchical structure formations, with various types of pacemaker-like elements, have also been found in various other social systems inspired models [23][24][25]. However, the universal features and diversity of such self-organized network structures have not been sufficiently investigated.
In this study, we focus on the behaviors of a simple coupled chaotic map system that involves temporal coupling changes inspired by Hebb's rule [26], with a time delay proposed by Ito and Ohira [21]. This model was proposed as one of the simplest models of dynamical network systems that exhibits an obvious hierarchical network structure with a pacemaker element. Furthermore, we found that the simple extension of this model involved the potential to spontaneously form a wider variety of network structures, as mentioned below. In the following arguments, we consider the morphology of self-organized network structures and reveal the relationships between the formed network structures and dynamic features of each element in this model.

II. MODEL
We consider a simple extension of the Ito-Ohira model [21], which is a coupled logistic map system with a temporal change in the connections (couplings), among elements described by strength. Note that the present dynamical system fully involves the original Ito-Ohira model, and corresponds when = 1 [21]. In this model, !! = 0 (no self-connecting) was assumed, and ! !" = 1 ! ! always satisfied for .
For the dynamics of ! !" , the present model employs a simple extension of Hebb's rule [26] where the state of the -th element influences the connection from the -th to -th elements with a one-step time delay. Furthermore, !!! !" tends to be large when ! ! and ! !!! are within close proximity to each other. Moreover, the "normalization" of connections strengths over all elements, , can be regarded as a simple representation of the global competition among the connection strength.
In the following simulations, we consider the case = 30 ( = 0, 1, 2, … , 29) as in the recent study by Ito and Ohira. For the initial condition of ! ! and !!!
uniform probability (we reorder the indices of all elements to satisfy !"!# connected to all the elements, except itself.
We performed simulations of the model for each set of , and from 20 different initial conditions and focused on the most frequently obtained type of network structures at = 10 ! ∼ 10 ! as the typical network structure. Thus, we confirmed that = 10 ! is sufficiently large for the relaxation to the state with its typical network structure for each parameter set. This is supported by the autocorrelation functions of the connection strength among the elements, as discussed below and in Appendix. We classified the typical network structures obtained by various parameter sets according to their connection profiles and the temporal changes in these profiles through the following procedures.
In the present model, the set ! !" represents the directional network structures of the elements at time . Here, the connection from the -th to -th element is considered strong (weak) when ! !" is large (small). Now, similar to the recent studies by Ito and Ohira, etc. [9,10,21,22], we define the connection from -th to -th elements as existing The dynamic features of connection profiles in typical networks were estimated by the where < > was assumed as < >=

III. RESULTS
A. Phase diagram and typical dynamical properties of network structures.  Table 1. for all (Appendix), indicated that the network structures were stable and unchanged once they were formed. We call these stable networks. We also found that, in other cases, decreased but for a relaxed finite value for → ∞ (Appendix). This indicated that the networks of these observed states are not time stable, but hidden basic network structures do exist and ! !" changes temporally around such basic structures. We call these dynamic networks.
Based on these facts, we next focused on the structural and dynamical features of the typical stable and dynamic network structures, where > 0 for → ∞, as described by ! !" ! !! , and the dynamic properties of elements, as described by { ! ! } ! .

B. Morphology of typical structures of stable networks
We considered the morphology of typical states with stable network structures that respectively. Here, is the Heviside function that yields = 1 for > 0 and   (a-c). Here, the lifetime order of the tentative networks was expected to be to as shown by the relaxed constant in FIG. 9 (f).
As a result of such temporal connection changes among elements, in the cases of  . 13 (f). However, some ⟨w ! !" ⟩ remained larger than 1/( − 1). Each element involved five to ten of ! and ! with, ! ~ ! as shown in FIG. 13 (g-j), by which the system exhibited hidden network structures with (not sparse but also not dense) random connections among the elements without any pacemaker-like specific elements or significant directionalities.

D. Relationships between formed network structures and the dynamics of the elements
Then, we focused on the relationship between the formed network structures and To estimate such relations, we calculated the split exponent (tangential Lyapunov exponent) of the -th element [27] as We focused on the maximum split exponent !"# = max ! = 0, 1, … − 1} in the following arguments: __ !"# were measured for = 3.6, 3.61, 3.62, ___, 4 and = 0, 0.01, 0.02, ___, 0.6 to compare them to the phase diagram of the dynamics of ! ! ! , as reported in a recent study [21]. Notably, we evaluated the !"! using ! ! ! for n = 5×10 ! ∼ 10 ! from five different initial conditions, but the results were almost the same, independent of the initial conditions.
In the case of δ = 1.0 the phase diagram of the and dependent dynamics of ! ! ! were reported, where their dynamics were classified as the coherent phase, ordered phase, partially ordered phase, and desynchronized phase [21]. As shown in Fig. 14 (a), the boundary curves among these phases in the phase diagram were closely related to the !"# of the landscape, as follows. The boundary between the coherent phase and the partially ordered phase was obtained as a curve located at !"# = −0.1 ∼ −0.05 for larger than ∼ 0.2. Furthermore, the boundary between the partially ordered phase and the ordered phase was obtained as a curve, which grew along some contours, and was from !"# ∼ 0 to between ∼ 0.15 and ∼ 0.2. Finally, the boundary between the ordered phase and the desynchronized phase was obtained as a curve, which grew along some contours from !"# ∼ 0 at < 0.15. were found in cases where the elements showed ordered motions. In the cases where the elements exhibited desynchronized motions, states with distinct hierarchical networks (red stars) appeared in the region near the boundary between the coherent phase and partially ordered phase, and dynamic networks with hidden paired layers network (white squares) were formed. Furthermore, in the region near the boundary between the desynchronized phase and the ordered motions, dynamic networks with hidden randomly connected networks (purple squares) were formed.
Similar correlations were formed between typical network structures, and the dynamics of elements are also found in the cases of δ = 0.1 and 0.01. Figure 15 (a-b) shows !"# for δ = 0.1 and 0.01 where three curves are drawn according to the same criteria as those in FIG. 14 (a), and FIG. 15 (c-d) show the superposition of !"# and the phase diagram of the network structure in FIG. 1 in these cases. We found almost the same relationships between the formed typical network structures and the dynamic properties of the elements, as shown in FIG. 14 (b), except for the following cases. In the region near the boundary between the two regions corresponding to the desynchronized phase and the ordered phase in Fig. 15 (a) and (b), the loop network structure (red diamonds) was observed differently from the case in δ = 1. Furthermore, in the region near the boundary between the two regions corresponding to the partially ordered motions and the ordered phase in Fig. 15 (a), a dynamic network with a hidden modular network (yellow laying triangle) appeared in the case of δ = 0.01.

IV. DISCUSSION
In this study, we considered the behaviors of a simple coupled map system involving temporal changes, which had connections among elements with a time delay proposed by Ito and Ohira [21]. We found that the present system has the potential to exhibit a of spike-timing-dependent-plasticity that is observed in neural network systems [28].
Still, we need more detailed qualitative and quantitative considerations. Recently, various dynamical system models with a change in the connections to the simple Hebb's like rules were studied, and various hierarchical networks were observed. Still, the formed network structures were dependent on the properties of the models [24,25,29].
On the other hand, our discussed model in the present arguments showed not only various hierarchical networks but also other types of networks that were not model-dependent but also parameter-dependent. We expect that the time delay in the rule of connection changes plays a key role in creating such various network structure formations.
A more detailed parameter-dependent study is still needed that observes the behaviors of the present model in the future since this model is expected to involve further potential, specifically to emerge a richer variety of network structures. We also need to