Many researchers have considered multi-agent systems over single-layer networks as models for studying diffusion phenomena. Since real-world networks involve connections between agents with different semantics (e.g., family member, friend, colleague), the study of multi-agent systems over multilayer networks has assumed importance. Our focus is on one class of multi-agent system models over multilayer networks, namely multilayer synchronous dynamical systems (MSyDSs). We study several fundamental problems for this model. We establish properties of the phase spaces of MSyDSs and bring out interesting differences between single-layer and multilayer dynamical systems. We show that, in general, the problem of determining whether two given MSyDSs are inequivalent is NP-complete. This hardness result holds even when the only difference between the two systems is the local function at just one node in one layer. We also present efficient algorithms for the equivalence problem for restricted versions of MSyDSs (e.g., systems where each local function is a bounded-threshold function, systems where the number of layers is fixed and each local function is symmetric). In addition, we investigate the expressive power of MSyDSs based on the number of layers. In particular, we examine conditions under which a system with k >= 2 layers has an equivalent system with k-1 or fewer layers.

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.

Discrete dynamical systems are commonly used to model the spread of contagions on real-world networks. Under the PAC framework, existing research has studied the problem of learning the behavior of a system, assuming that the underlying network is known. In this work, we focus on a more challenging setting: to learn both the behavior and the underlying topology of a black-box system. We show that, in general, this learning problem is computationally intractable. On the positive side, we present efficient learning methods under the PAC model when the underlying graph of the dynamical system belongs to some classes. Further, we examine a relaxed setting where the topology of an unknown system is partially observed. For this case, we develop an efficient PAC learner to infer the system and establish the sample complexity. Lastly, we present a formal analysis of the expressive power of the hypothesis class of dynamical systems where both the topology and behavior are unknown, using the well-known formalism of the Natarajan dimension. Our results provide a theoretical foundation for learning both the behavior and topology of discrete dynamical systems.

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial time algorithm for approximating a solution to this problem to within the factor n^1-\epsilon for any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics.

Using a graphical discrete dynamical system to model a networked social system, the problem of inferring the behavior of the system can be formulated as the problem of learning the local functions of the dynamical system. We investigate the problem assuming an active form of interaction with the system through queries. We consider two classes of local functions (namely, symmetric and threshold functions) and two interaction modes, namely batch mode (where all the queries must be submitted together) and adaptive mode (where the set of queries submitted at a stage may rely on the answers received to previous queries). We develop complexity results and efficient heuristics that produce query sets under both query modes. We demonstrate the performance of our heuristics through experiments on over 20 well-known networks.