Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 09, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Johannes Lengler

Weighted Percolation on Geometric Inhomogeneous Random Graphs

Geometric Inhomogeneous Random Graphs (GIRGs) are a random graph model that is supposed to capture some properties of social networks, including a power law degree distributions and community structures. These networks are ultra-small worlds, i.e., the average distance in the giant component is O(log log n). If we assign an i.i.d. cost W to each edge (think of a delay for communicating along this edge), then the average cost distance (i.e., the minimal total cost of a connecting path) drops to O(1) unless the distribution of W is doubly exponentially decaying at zero (Komjathy and Lodewijk, 2018).

However, in social networks communication along high-degree vertices is often slower, so together with Komjathy and Lapinskas we studied the scenario where large-degree vertices are penalised by a factor (deg v)^\mu, i.e., an edge u-v has cost (deg u * deg v)^\mu * W, where again all W are drawn i.i.d. In this case we get a much richer behaviour, depending on \mu, the distribution of W, and the model parameters: The number of vertices in cost-distance C may be \Omega(n) ("infinite regime"), or it may grow doubly exponentially in C, or it may grow exponentially in C, or it may grow polynomially in C.

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