Considering the difficulty of interpreting generative model output, there is significant current research focused on determining meaningful evaluation metrics. Several recent approaches utilize "precision" and "recall," borrowed from the classification domain, to individually quantify the output fidelity (realism) and output diversity (representation of the real data variation), respectively. With the increase in metric proposals, there is a need for a unifying perspective, allowing for easier comparison and clearer explanation of their benefits and drawbacks. To this end, we unify a class of kth-nearest-neighbors (kNN)-based metrics under an information-theoretic lens using approaches from kNN density estimation. Additionally, we propose a tri-dimensional metric composed of Precision Cross-Entropy (PCE), Recall Cross-Entropy (RCE), and Recall Entropy (RE), which separately measure fidelity and two distinct aspects of diversity, inter- and intra-class. Our domain-agnostic metric, derived from the information-theoretic concepts of entropy and cross-entropy, can be dissected for both sample- and mode-level analysis. Our detailed experimental results demonstrate the sensitivity of our metric components to their respective qualities and reveal undesirable behaviors of other metrics.

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.

Optimal allocation of agricultural water in the event of droughts is an important global problem. In addressing this problem, many aspects, including the welfare of farmers, the economy, and the environment, must be considered. Under this backdrop, our work focuses on several resource-matching problems accounting for agents with multi-crop portfolios, geographic constraints, and fairness. First, we address a matching problem where the goal is to maximize a welfare function in two-sided markets where buyers' requirements and sellers' supplies are represented by value functions that assign prices (or costs) to specified volumes of water. For the setting where the value functions satisfy certain monotonicity properties, we present an efficient algorithm that maximizes a social welfare function. When there are minimum water requirement constraints, we present a randomized algorithm which ensures that the constraints are satisfied in expectation. For a single seller--multiple buyers setting with fairness constraints, we design an efficient algorithm that maximizes the minimum level of satisfaction of any buyer. We also present computational complexity results that highlight the limits on the generalizability of our results. We evaluate the algorithms developed in our work with experiments on both real-world and synthetic data sets with respect to drought severity, value functions, and seniority of agents.

Applications such as employees sharing office spaces over a workweek can be modeled as problems where agents are matched to resources over multiple rounds. Agents' requirements limit the set of compatible resources and the rounds in which they want to be matched. Viewing such an application as a multi-round matching problem on a bipartite compatibility graph between agents and resources, we show that a solution (i.e., a set of matchings, with one matching per round) can be found efficiently if one exists. To cope with situations where a solution does not exist, we consider two extensions. In the first extension, a benefit function is defined for each agent and the objective is to find a multi-round matching to maximize the total benefit. For a general class of benefit functions satisfying certain properties (including diminishing returns), we show that this multi-round matching problem is efficiently solvable. This class includes utilitarian and Rawlsian welfare functions. For another benefit function, we show that the maximization problem is NP-hard. In the second extension, the objective is to generate advice to each agent (i.e., a subset of requirements to be relaxed) subject to a budget constraint so that the agent can be matched. We show that this budget-constrained advice generation problem is NP-hard. For this problem, we develop an integer linear programming formulation as well as a heuristic based on local search. We experimentally evaluate our algorithms on synthetic networks and apply them to two real-world situations: shared office spaces and matching courses to classrooms.

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.

The decision to take vaccinations and other protective interventions for avoiding an infection is a natural game-theoretic setting. Most of the work on vaccination games has focused on decisions at the start of an epidemic. However, a lot of people defer their vaccination decisions, in practice. For example, in the case of the seasonal flu, vaccination rates gradually increase, as the epidemic rate increases. This motivates the study of temporal vaccination games, in which vaccination decisions can be made more than once. An important issue in the context of temporal decisions is that of resource limitations, which may arise due to production and distribution constraints. While there has been some work on temporal vaccination games, resource constraints have not been considered. In this paper, we study temporal vaccination games for epidemics in the SI (susceptible-infectious) model, with resource constraints in the form of a repeated game in complex social networks, with budgets on the number of vaccines that can be taken at any time. We find that the resource constraints and the vaccination and infection costs have a significant impact on the structure of Nash equilibria (NE). In general, the budget constraints can cause NE to become very inefficient, and finding efficient NE as well as the social optimum are NP-hard problems. We develop algorithms for finding NE and approximating the social optimum. We evaluate our results using simulations on different kinds of networks.

The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent $\beta>2$, we show that the PoA is bounded by $O(T^{2(\beta-1)})$, where $T=\gamma/\alpha$ is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations.

Simple diffusion processes on networks have been used to model, analyze and predict diverse phenomena such as spread of diseases, information and memes. More often than not, the underlying network data is noisy and sampled. This prompts the following natural question: how sensitive are the diffusion dynamics and subsequent conclusions to uncertainty in the network structure? In this paper, we consider two popular diffusion models: Independent cascades (IC) model and Linear threshold (LT) model. We study how the expected number of vertices that are influenced/infected, given some initial conditions, are affected by network perturbation. By rigorous analysis under the assumption of a reasonable perturbation model we establish the following main results. (1) For the IC model, we characterize the susceptibility to network perturbation in terms of the critical probability for phase transition of the network. We find the expected number of infections is quite stable, unless the the transmission probability is close to the critical probability. (2) We show that the standard LT model with uniform edge weights is relatively stable under network perturbations. (3) Empirically, the transient behavior, i.e., the time series of the number of infections, in both models appears to be more sensitive to network perturbations. We also study these questions using extensive simulations on diverse real world networks, and find that our theoretical predictions for both models match the empirical observations quite closely.