Over a century ago, Thiele (1889) introduced cumulants, a concept now fundamental to the field of statistics, which has justifiably percolated throughout the scientific community. In this work, we introduce *graph cumulants*, their generalization to networks. This principled hierarchy of network statistics provides a framework to systematically describe and compare networks, naturally including those with additional features, such as directed edges, node attributes, and edge weights. Moreover, through the lens of the maximum entropy principle, these statistics induce a natural hierarchical family of network models. These models are immune to the "degeneracy problem", providing a principled prescription for obtaining distributions that are clustered around the properties of the network they intend to model.

- Videos:

MLSS 2020 (9m21s)

ERGM visualization (0m11s) - Paper:

**LM Gunderson**and G Bravo-Hermsdorff

"Introducing graph cumulants: What is the variance of your social network?"

*arXiv preprint*(2020)

link - Code:

GitHub

The deletion of edges (sparsification) and the merging of adjacent vertices (coarsening) are two common methods for reducing a graph. We analytically unify these two operations using a single objective function based on the graph Laplacian pseudoinverse, providing a principled algorithm that simultaneously sparsifies and coarsens a graph while preserving its large-scale structure.

- Videos:

NeurIPS 2019 (4m14s)

playlist (1m53s, 0m23s, 0m19s, 0m15s) - Paper:

G Bravo-Hermsdorff and**LM Gunderson**

"A unifying framework for spectrum-preserving graph sparsification and coarsening"

*Advances in Neural Information Processing Systems*(2019)

link - Poster:

NeurIPS 2019 - Code:

GitHub

We constructed a gender-labeled temporal bipartite graph of academic collaborations in the Institute for Operations Research and the Management Sciences (INFORMS) spanning over six decades. Employing two metrics based on the graph Laplacian which are sensitive to global connectivity, we analyze the relevance of gender in these collaboration patterns.

- Paper:

G Bravo-Hermsdorff, V Felso, E Ray,**LM Gunderson**, ME Helander, J Maria, and Y Niv

"Gender and collaboration patterns in a temporal scientific authorship network"

*Applied Network Science*(2019)

link - Dataset:

GitHub

Helioseismology has greatly advanced our understanding of the internal structure of the Sun. Its first contribution was to constrain the 1D radial entropy profile, establishing the depth of the convection zone. Later, through the splitting of prograde and retrograde modes, it became possible to infer the 2D internal rotation profile. Arguably the most surprising result from these inversions is the sharp change to uniform rotation at the base of the convection zone, occurring in a thin layer known as the tachocline. This region is important for many models of the Sun, yet its existence is often forced in an ad hoc way, and there are still many open questions about its structure.

- Thesis:

Solar Equilibrium à la Grad-Shafranov - Paper:

**LM Gunderson**and A Bhattacharjee

"A model of solar equilibrium: the hydrodynamic limit"

*The Astrophysical Journal*(2019)

link

Stellarators and tokamaks are devices that confine hot ionized gases, known as plasmas, using a strong, suitably-shaped magnetic field. This magnetic field provies a Lorentz force that forces the particles to make tight helical paths along the magnetic field lines. To prevent so-called end losses, the magnetic field is bent into a topological torus. If the magnetic field can be so designed that sufficiently hot (i.e., fast) particles can be confined for sufficiently long times at sufficient densities, then the particles will fuse and release energy.

- Paper:

D Pfefferlé,**LM Gunderson**, SR Hudson, and L Noakes

"Non-planar elasticae as optimal curves for the magnetic axis of stellarators"

*Physics of Plasmas*(2018)

link - Paper:

SR Hudson, C Zhu, D Pfefferlé, and**LM Gunderson**

"Differentiating the shape of stellarator coils with respect to the plasma boundary"

*Physics Letters A*(2018)

link

Aerodynamic lensing has long been used to focus the particulates in an aerosol into a columated beam. These beams could also be used in laser-plasma applications, such as generation of extreme laser intensities through Raman or Brillouin backscattering. Here, we address some theoretical and experimental considerations when producing beams with nonlinear particulate densities.

- Paper:

DE Ruiz,**LM Gunderson**, MJ Hay, E Merino, EJ Valeo, SJ Zweben, and NJ Fisch

"Aerodynamic focusing of high-density aerosols"

*Journal of Aerosol Science*(2014)

link