## Section: Research Program

### Network Diffusion Processes

Social, biological, and technological networks can serve as conduits for the spread of ideas, trends, diseases, or viruses. In social networks, rumors, trends and behaviors, or the adoption of new products, spread from person to person. In biological networks, diseases spread through contact between individuals, and mutations spread from an individual to its offsprings. In technological networks, such as the Internet and the power grid, viruses and worms spread from computer to computer, and power failures often lead to cascading failures. The common theme in all the examples above is that the rumor, disease, or failure starts out with a single or a few individual nodes, and propagates through the network, from node to node, to reach a potentially much larger number of nodes.

These types of *network diffusion processes* have long been a topic of study in various disciplines, including sociology, biology, physics, mathematics, and more recently, computer science.
A main goal has been to devise mathematical models for these processes, describing how the state of an individual node can change as a function of the state of its neighbors in the network, and then analyse the role of the network structure in the outcome of the process.
Based on our previous work, we would like to study to what extent one can affect the outcome of the diffusion process by controlling a small, possibly carefully selected fraction of the network.

For example, we plan to explore how we may increase the spread or speed of diffusion by choosing an appropriate set of seed nodes (a standard goal in viral marketing by word-of-mouth), or achieve the opposite effect either by choosing a small set of nodes to remove (a goal in immunization against diseases), or by seeding a competing diffusion (e.g., to limit the spread of misinformation in a social network).

Our goal is to provide a framework for a systematic and rigorous study of these problems. We will consider several standard diffusion models and extensions of them, including models from mathematical sociology, mathematical epidemiology, and interacting particle systems. We will consider existing and new variants of spread maximization/limitation problems, and will provide (approximation) algorithms or show negative (inapproximability) results. In case of negative results, we will investigate general conditions that make the problem tractable. We will consider both general network topologies and specific network models, and will relate the efficiency of solutions to structural properties of the topology. Finally, we will use these insights to engineer new network diffusion processes for efficient data dissemination.

#### Spread maximization

Our goal is in particular to study spread maximization in a broader class of diffusion processes than the basic independent cascade (IC) and linear threshold (LT) models of influence [64], [65], [66] that have been studied in this context so far. This includes the *randomized rumor spreading (RS)* model for information dissemination [57], *biased* versions of the *voter model* [61] modelling influence, and the (graph-based) *Moran processes* [68] modelling the spread of mutations.
We would like to consider several natural versions of the spread maximization problem, and the relationships between them.
For these problems we will use the greedy algorithm and the submodularity-based analytical framework of [64], and will also explore new approaches.

#### Immunization optimization

Conversely we would also like to explore immunization optimization problems. Existing works on these types of problem assume a *perfect-contagion* model, i.e., once a node gets infected, it deterministically infects all its non-immunized neighbors.
We plan to consider various diffusion processes, including the
standard *susceptible–infected* (SI), *susceptible–infected–recovered* (SIR) and *susceptible–infected–susceptible* (SIS) epidemic models, and explore the extent to which results and techniques for the perfect-contagion model carry over to these probabilistic models.
We will also investigate whether techniques for spread maximization could be applied to immunization problems.

Some immunization problems are known to be hard to approximate in general graphs, even for the perfect-contagion model, e.g., the fixed-budget version of the fire-fighter problem cannot be approximated to any ${n}^{1-\u03f5}$ factor [46]. This strand of work will consider restricted graph families, such as trees or graphs of small treewidth, for such problems. In addition, for some immunization problems, there is a large gap between the best known approximation algorithm and the best known inaproximability result, and we would like to make progress in reducing these gaps.