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In Euclidean Geometric Inhomogeneous Random Graphs (GIRG), a model for real-networks recently presented by Bringmann, Keusch, and Lengler, each vertex is equipped with a weight and a position. The weights are drawn from a power-law distribution (with a fixed exponent 2 < β < 3), while the positions are drawn uniformly from the d-dimensional torus Td. Then, the edge probability of two vertices is roughly polynomial in the product of their weights, and the inverse of their Euclidean distance. The Euclidean GIRG model has a wide range of interesting properties, such as a small diameter, small average distance, small separators, and a large clustering coefficient. In this thesis, we study a variant of the GIRG model that uses minimum component distance (MCD) instead of Euclidean distance, but is otherwise defined in the same way as its Euclidean counterpart. The main result of the thesis is that the MCD model (for d > 1) has no small (edge) separators, setting it apart from Euclidean GIRGs.

Size of Edge Separators in Geometric Inhomogeneous Random Graphs Depends on Underlying Geometry