Reconstructing network structure and the underlying dynamics from temporal data is a fundamental problem in network science and has various applications in biological, climate, financial and economic systems. However, conventional reconstruction methods, including compressing sensorshen_reconstructing_2014, maximum entropymatin2017, granger causal inferenceching_reconstructing_2017 etc. require the underlying dynamical rules having specific forms, or even to be linear. Although the method presented in casadiego_model-free_2017 do not have these limitations, it can only be applied to continuous dynamical systems so that the discrete derivative can be calculated. Recently, Kipf et al. kipf_neural_2018 proposed an Encoder-Decoder framework called NRI (Neural Relation Inference) based on graph network modelbattaglia2018relational and deep learning techniques to reconstruct the network structure and dynamics simultaneously, however, the sizes of the graphs to be reconstructed are very small (smaller than 40 nodes), and the dynamics are always continuous. A general framework for reconstructing network structure and dynamics from the time series data which can be applied for various dynamics including continuous, discrete or even binary ones is necessary.
In this work, we present a deep learning framework for dynamics learning and network reconstruction from the observed time series data. This model is composed of two parts, network generator, and dynamics learner. In the first part, we use Gumbel-softmaxjang_categorical_2016 method to build the graph directly. While, for the dynamics learner, we adopt a graph network modelbattaglia2018relational with five layers. The network generator and dynamics learner alternate to operate such that the whole model converge. The whole framework is named as Gumbel Graph Network (GGN). We applied GGN on three representative dynamical models, the Kuramoto oscillator model representing for continuous dynamics, the coupled map lattices model representing for discrete dynamics, and the boolean network representing for binary dynamics. Our experimental results show that our model can not only reconstruct the network structure in high accuracy, but also can learn the various dynamics efficiently, and all the graph network structures are same for those three models.
Next, we will briefly introduce our method, experiments, and results.
In our method, the input consists of trajectories generated by a dynamical model of interacting objects which can be modelled as an unweighted directed graph. The target is to learn the network structure (Specifically, adjacency matrix) and the dynamical model simultaneously in an unsupervised way. Our algorithm consists of two jointly trained parts: A network generator that generates a discrete network by using Gumbel Softmax trickjang_categorical_2016; and a dynamics learner that can utilise the network generated by the generator and one-step trajectory value to predict the value in the next one or multiple steps. We alter the network generator for steps and the dynamics learner for steps. The and are two hyper-parameters which take different values for different models. The objective of our model is to minimise the errors between the predictions and the time series data. When the time series to be predicted is a discrete sequence with finite countable symbols, the cross-entropy objective function is adopted otherwise the mean square errors are used. Figure 1. outlines the basic structure of our framework.
2.2 Network Generator
One of the difficulties for reconstructing a network from the data is the discreteness of the graph data such that the back-propagation technique which is widely used in deep learning and artificial neural networks cannot be applied on networks.
To conquer this problem, Gumbel-softmax trick is used to reconstruct the adjacency matrix of the network directly. This technique uses a continuous distribution to approximate samples from a discrete distribution such that the back-propagation algorithm can be also applied.
Suppose we will reconstruct a network with size , and the adjacency matrix is . Where, s are real values with gumbel-softmax function with the components of and the temperature parameter , which can be written by:
where s and s are i.i.d. random numbers following the gumbel distribution. This calculation use a continuous function with random noise to simulate a discontinuous sampling process. And the temperature parameter adjust the sharpness of the output. When ,
will take 1 with probabilityand 0 with probability .
s are all trainable parameters, which can be adjusted according to the back propagation algorithm. Thanks to the features of Gumbel-softmax trick, the gradient information can be back propagated through the whole computation graph although the process of sampling a random number is non-differentiable.
2.3 Dynamics Learner
Learning with graph-structured data is a hot trend in deep learning research areas. It focuses on the effective representation learning of nodes in a graph. Recently, Graph Networks (GNs) battaglia2018relational have been widely investigated and have achieved state-of-the-art performance in node classification, link prediction, et al. In general, a GN takes the graph structure and node features as its input to learn a representation of each node. We input the adjacency matrix constructed by the generator into the GN directly, and the errors are calculated and back forwarded. The whole dynamics learner can be presented as a function:
is the state vector of allnodes at time step , and is the adjacency matrix constructed by the network generator. If we want to make multiple prediction, we have:
where represents the states of the system from time to . is the transitional function, whose structure can be visualized by Figure 2
In figure 2, the symbol represents the Kronecker product while the time operator is replaced with the operator to make two elements a pair. Therefore, if is a vector, then is an matrix, and the element at the th row and th column is , where is the th element in . The symbol “*” represents the element-wised product, and the symbol represents the concatenation operator.
We have considered three representative dynamical systems on networks to validate our model: Coupled Map Lattice model standing for discrete dynamics, Kuramoto model representing for continuous dynamics and Boolean Network dynamics representing for boolean dynamics. Next, we will briefly discuss these three model and show our results.
3.1 Coupled Map Lattices
A coupled map lattice is a dynamical system with discrete time and continuous state variables defined on a chain with a periodic boundary condition. In the past 30 years, studies in coupled map lattices have improved our understanding of spatiotemporal chaos systems. In our work, we consider coupled systems on a complex network:
where denotes a node and we choose the following logistic map:
With adequate parameters of and , the model exhibits abundant quantitative universality classes such as spatial bifurcation and frozen chaos. We simulate the coupled map lattices model with node number ranging from 10 to 100 on different complex networks in the different region of chaotic behaviour.
And we input all these generated data into our framework. Figure 3 presents our results obtained on a random 4-regular network with the number of nodes
. It is interesting to see that after 5 epochs, our model has revealed the network topology accurately.
Table 1 lists three numerical experimental results on different networks with various parameters of the dynamics.
3.2 Kuramoto model
The Kuramoto model is a nonlinear system of phase-coupled oscillators. The following differential equation is the model we use in the paper. We simulate 1D trajectories by solving this equation with a fourth-order Runge-Kutta integrator with step size 0.01.
The nodes which are phase-coupled oscillators here with undirected edges. if two oscillators have a connection, or . is intrinsic frequency which is sampled uniformly from , and is initial phase which is sampled uniformly from . Then we subsample the simulated by a fator of 10 and create the trectories by concatenating and .
We simulate Kuramoto dynamics on a sample small network which is shown in Figure 4. We generate 50k training examples to train and test our method, among which 10k are tested. The following figure 5 shows a sample of the test data. The left part is the trajectory learned by our method, and the right part is the results generated by the simulation. The mean square errors of the 20 steps prediction according to only one step input is 1.20e-2. And we compare our result with two baseline methods as shown in Table 2. This means that our method is capable of capturing the graph dynamics better than the state-of-the-art algorithms.
3.3 Boolean Network
In this model, every variable in a Boolean Network has a possible value of 0 or 1 and a boolean function is assigned to the node. The function takes the states of its neighbours as inputs and return a binary value that determines the state of the current node.
We simulate the Boolean Network model with node numbers varying from 5 to 50 on different complex networks. And we set adjust dynamics for each model to change their behaviours from low randomness to chaos.
shows the training process of our model with 20 nodes, the number of the incorrect entries that our network can recover is shown in the left sub-figure, and the loss function which is the cross entropy is shown in the right sub-figure. Both curves drop to low values closed to zero. Results in Boolean Networks are summarized in Table3
In this work, we present Gumbel Graph Network, a model-free deep learning framework for dynamics learning and network reconstruction from the observed time series data. Our method requires no prior knowledge about underlying dynamics and has shown the state-of-the-art performance in three typical dynamical systems on complex networks. Further studies considered larger networks and more detailed experiments are ongoing and will be finished soon.