Efficient PAC Learnability of Dynamical Systems Over Multilayer Networks

TL;DR

Efficient PAC Learnability of Dynamical Systems Over Multilayer Networks studies how to learn the dynamics of cascade processes on multilayer networks when only a subset of vertex interaction functions are unknown. It develops an efficient PAC learner with a sample complexity that scales as $m_{\mathcal{H}}(\delta,\epsilon) \le \frac{1}{\epsilon}\, \sigma k \log(\frac{\sigma k}{\delta})$, and extends to the PMAC setting for reduced data needs. The work also delivers a tight characterization of model complexity via the Natarajan dimension, showing exact results for single-layer systems ($\text{Ndim}(\mathcal{H})=\sigma$), bounds for multi-layer systems ($\sigma \le \text{Ndim}(\mathcal{H}) \le k\sigma$), and asymptotic equivalence to $k\sigma$ for almost all graphs. The combination of an efficient learning algorithm, rigorous sample-complexity and complexity bounds, and experimental validation provides a solid theoretical foundation for learning multilayer dynamical systems and guides future research on more complex network structures and noisy data.

Abstract

Networked dynamical systems are widely used as formal models of real-world cascading phenomena, such as the spread of diseases and information. Prior research has addressed the problem of learning the behavior of an unknown dynamical system when the underlying network has a single layer. In this work, we study the learnability of dynamical systems over multilayer networks, which are more realistic and challenging. First, we present an efficient PAC learning algorithm with provable guarantees to show that the learner only requires a small number of training examples to infer an unknown system. We further provide a tight analysis of the Natarajan dimension which measures the model complexity. Asymptotically, our bound on the Nararajan dimension is tight for almost all multilayer graphs. The techniques and insights from our work provide the theoretical foundations for future investigations of learning problems for multilayer dynamical systems.

Figures (8)

• Figure 1: A 2-layer threshold system with OR master functions. Threshold values of vertices $v_1$ to $v_4$ in layer 1 are $(2, 3, 3, 2)$, and in layer 2 are $(3, 3, 2, 1)$. State-1 vertices are in blue. The configuration $\mathcal{C} = (1, 1, 1, 0)$, and its successor is $\mathcal{C}' = (1, 0, 0, 1)$.
• Figure 2: (a): $\ell$ vs $|\mathcal{T}|$ and (b): $\ell$ vs $\sigma$, over different distributions specified by $p$. The underlying network is Multi-Gnp (Table \ref{['tab:networks']}). The stdev for all data points is less than $0.09$.
• Figure 3: (a): $\ell$ vs $|\mathcal{T}|$, over different real-world networks (Table \ref{['tab:networks']}), and (b): $\ell$ vs $k$ over different values of $\sigma$, where the underlying network is Gnp. The stdev for all data points is less than $0.08$.
• Figure 4: An alternative interpretation of shattering: associated configurations and $2^{|\mathcal{R}|}$ mappings for a set $\mathcal{R}$.
• Figure 5: An example of a contested vertex $v$ for a configuration $\mathcal{C}$. In particular, $\mathcal{C}^A$ and $\mathcal{C}^B$ are the two associated configurations of $\mathcal{C}$. The state of $v$ is highlighted in blue.

Theorems & Definitions (27)

• Theorem 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Definition 4.1: Landmark Vertices
• Definition 4.2: Canonical Set
• Definition 4.3: Contested Vertices
• Lemma 4.4
• Theorem 4.5
• Lemma 4.6