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Dynamical systems in which local interactions among agents give rise to complex emerging phenomena are ubiquitous in nature and society.
This work explores the problem of inferring the unknown interaction structure (represented as a graph) of such a system from measurements of its constituent agents or individual components (represented as nodes).
We consider a setting where the underlying dynamical model is unknown and where different measurements (i.e., snapshots) may be independent (e.g., may stem from different experiments).
We propose GINA (Graph Inference Network Architecture), a graph neural network (GNN) to simultaneously learn the latent interaction graph and, conditioned on the interaction graph, the prediction of a node's observable state based on adjacent vertices.
GINA is based on the hypothesis that the ground truth interaction graph---among all other potential graphs---allows to predict the state of a node, given the states of its neighbors, with the highest accuracy.
We test this hypothesis and demonstrate GINA's effectiveness on a wide range of interaction graphs and dynamical processes.

In the recent COVID-19 pandemic, mathematical modeling constitutes an important tool to evaluate the prospective effectiveness of non-pharmaceutical interventions (NPIs) and to guide policy-making.
Most research is, however, centered around characterizing the epidemic based on point estimates like the average infectiousness or the average number of contacts.
In this work, we use stochastic simulations to investigate the consequences of a population's heterogeneity regarding connectivity and individual viral load levels.
Therefore, we translate a COVID-19 ODE model to a stochastic multi-agent system. We use contact networks to model complex interaction structures and a probabilistic infection rate to model individual viral load variation.
We observe a large dependency of the dispersion and dynamical evolution on the population's heterogeneity that is not adequately captured by point estimates, for instance, used in ODE models.
In particular, models that assume the same clinical and transmission parameters may lead to different conclusions, depending on different types of heterogeneity in the population.
For instance, the existence of hubs in the contact network leads to an initial increase of dispersion and the effective reproduction number, but to a lower herd immunity threshold (HIT) compared to homogeneous populations or a population where the heterogeneity stems solely from individual infectivity variations.

Human mobility is the fuel of global pandemics.
In this simulation study, we analyze how mobility restrictions mitigate epidemic processes and how this mitigation is influenced by the epidemic's degree of dispersion.
We find that (even imperfect) mobility restrictions are generally efficient in mitigating epidemic spreading. Notably, the effectiveness strongly depends on the dispersion of the offspring distribution associated with the epidemic.
We also find that mobility restrictions are useful even when the pathogen is already prevalent in the whole population. However, also a delayed implementation is more efficient in the presence of overdispersion. Conclusively, this means that implementing green zones is easier for epidemics with overdispersed transmission dynamics (e.g., COVID-19).
To study these relationships at an appropriate level of abstraction, we propose a spatial branching process model combining the flexibility of stochastic branching processes with an agent-based approach allowing a conceptualization of locality, saturation, and interaction structure.

We study continuous-time multi-agent models, where agents interact according to a network topology. At any point in time, each agent occupies a specific local node state. Agents change their state at random through interactions with neighboring agents. The time until a transition happens can follow an arbitrary probability density. Stochastic (Monte-Carlo) simulations are often the preferred—sometimes the only feasible—approach to study the complex emerging dynamical patterns of such systems. However, each simulation run comes with high computational costs mostly due to updating the instantaneous rates of interconnected agents after each transition. This work proposes a stochastic rejection-based, event-driven simulation algorithm that scales extremely well with the size and connectivity of the underlying contact network and produces statistically correct samples. We demonstrate the effectiveness of our method on different information spreading models.

We address the problem of reducing the spread of an epidemic over a contact network by vaccinating a limited number of nodes that represent individuals or agents.
We propose a Simulation based vaccine allocation method (Simba), a combination of (i) numerous repetitions of an efficient Monte-Carlo simulation , (ii) a PageRank-type influence analysis on an empirical transmission graph which is learned from the simulations, and (iii) discrete stochastic optimization.
Our method scales very well with the size of the network and is suitable for networks with millions of nodes. Moreover, in contrast to most approaches that are model-agnostic approaches and solely perform graph-analysis on the contact graph, the stochastic simulations explicitly take the exact diffusion dynamics of the epidemic into account. Thereby, we make our vaccination strategy sensitive to the specific clinical and transmission parameters of the epidemic.

In the recent COVID-19 pandemic, computer simulations are used to predict the evolution of the virus propagation and to evaluate the prospective effectiveness of non-pharmaceutical interventions. As such, the corresponding mathematical models and their simulations are central tools to guide political decision-making. Typically, ODE-based models are considered, in which fractions of infected and healthy individuals change deterministically and continuously over time.
In this work, we translate an ODE-based COVID-19 spreading model from literature to a stochastic multi-agent system and use a contact network to mimic complex interaction structures. We observe a large dependency of the epidemic's dynamics on the structure of the underlying contact graph, which is not adequately captured by existing ODE-models. For instance, existence of super-spreaders leads to a higher infection peak but a lower death toll compared to interaction structures without super-spreaders. Overall, we observe that the interaction structure has a crucial impact on the spreading dynamics, which exceeds the effects of other parameters such as the basic reproduction number R0.
We conclude that deterministic models fitted to COVID-19 outbreak data have limited predictive power or may even lead to wrong conclusions while stochastic models taking interaction structure into account offer different and probably more realistic epidemiological insights.

Poster Abstract on how birth-death processes can reproduce SIS-type epidemic dynamics on (random) complex networks.

Stochastic processes can model many emerging phenomena on networks, like the spread of computer viruses, rumors, or infectious diseases. Understanding the dynamics of such stochastic spreading processes is therefore of fundamental interest. In this work we consider the wide-spread compartment model where each node is in one of several states (or compartments). Nodes change their state randomly after an exponentially distributed waiting time and according to a given set of rules. For networks of realistic size, even the generation of only a single stochastic trajectory of a spreading process is computationally very expensive.

Stochastic processes on complex networks, where each node is in one of several compartments, and neighboring nodes interact with each other, can be used to describe a variety of real-world spreading phenomena. However, computational analysis of such processes is hindered by the enormous size of their underlying state space.
In this work, we demonstrate that lumping can be used to reduce any epidemic model to a Markov Population Model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions, we obtain coarse-grained Markov models with a natural MPM representation that approximate the original systems. This makes it possible to transfer the rich pool of approximation techniques developed for MPMs to the computational analysis of complex networks’ dynamics.
We present numerical examples to investigate the relationship between the accuracy of the MPMs, the size of the lumped state space, and the type of counting abstraction.

Stochastic models in which agents interact with their neighborhood according to a network topology are a powerful modeling framework to study the emergence of complex dynamic patterns in real-world systems. Stochastic simulations are often the preferred-sometimes the only feasible-way to investigate such systems. Previous research focused primarily on Markovian models where the random time until an interaction happens follows an exponential distribution. In this work, we study a general framework to model systems where each agent is in one of several states. Agents can change their state at random, influenced by their complete neighborhood, while the time to the next event can follow an arbitrary probability distribution. Classically, these simulations are hindered by high computational costs of updating the rates of interconnected agents and sampling the random residence times from arbitrary distributions. We propose a rejection-based, event-driven simulation algorithm to overcome these limitations. Our method over-approximates the instantaneous rates corresponding to inter-event times while rejection events counterbalance these over-approximations. We demonstrate the effectiveness of our approach on models of epidemic and information spreading.

Stochastic processes on complex networks, where each node is in one of several compartments and neighboring nodes interact with each other, can be used to describe a variety of real-world spreading phenomena. However, computational analysis of such processes is hindered by the enormous size of their underlying state space.
In this work, we demonstrate that lumping can be used to reduce any epidemic model to a Markov population model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions, we obtain coarse-grained Markov models with a natural MPM representation that approximate the original systems. This makes it possible to transfer the rich pool of approximation techniques developed for MPMs to the computational analysis of complex networks' dynamics.
We present numerical examples to investigate the relationship between, the accuracy of the MPMs, the size of the lumped state space, and the type of counting abstraction.

Stochastic processes can model many emerging phenomena on networks, like the spread of computer viruses, rumors, or infectious diseases. Understanding the dynamics of such stochastic spreading processes is therefore of fundamental interest. In this work we consider the wide-spread compartment model where each node is in one of several states (or compartments). Nodes change their state randomly after an exponentially distributed waiting time and according to a given set of rules. For networks of realistic size, even the generation of only a single stochastic trajectory of a spreading process is computationally very expensive. Here, we propose a novel simulation approach, which combines the advantages of event-based simulation and rejection sampling. Our method outperforms state-of-the-art methods in terms of absolute run-time and scales significantly better, while being statistically equivalent.

Complex networks play an important role in human society and in nature. Stochastic multistate processes provide a powerful framework to model a variety of emerging phenomena such as the dynamics of an epidemic or the spreading of information on complex networks. In recent years, mean-field type approximations gained widespread attention as a tool to analyze and understand complex network dynamics. They reduce the model's complexity by assuming that all nodes with a similar local structure behave identically. Among these methods the approximate master equation (AME) provides the most accurate description of complex networks' dynamics by considering the whole neighborhood of a node. The size of a typical network though renders the numerical solution of multistate AME infeasible. Here, we propose an efficient approach for the numerical solution of the AME that exploits similarities between the differential equations of structurally similar groups of nodes. We cluster a large number of similar equations together and solve only a single lumped equation per cluster. Our method allows the application of the AME to real-world networks, while preserving its accuracy in computing estimates of global network properties, such as the fraction of nodes in a state at a given time.