Properties of Stochastic Kronecker Graph

TL;DR

This paper analyzes the properties of stochastic Kronecker graphs, providing bounds and conditions for key features like isolated vertices, edges, self-loops, and triangles, and identifying phase transitions for their appearance.

Contribution

It offers new bounds and conditions for various graph properties and reveals phase transitions in the stochastic Kronecker graph model.

Findings

Bounds on expected number of isolated vertices, edges, self-loops, and triangles.

Conditions for generating graphs with specific properties.

Identification of phase transitions for edge and self-loop appearance.

Abstract

The stochastic Kronecker Graph model can generate large random graph that closely resembles many real world networks. For example, the output graph has a heavy-tailed degree distribution, has a (low) diameter that effectively remains constant over time and obeys the so-called densification power law [1]. Aside from this list of very important graph properties, one may ask for some additional information about the output graph: What will be the expected number of isolated vertices? How many edges, self loops are there in the graph? What will be the expected number of triangles in a random realization? Here we try to answer the above questions. In the first phase, we bound the expected values of the aforementioned features from above. Next we establish the sufficient conditions to generate stochastic Kronecker graph with a wide range of interesting properties. Finally we show two phase transitions for the appearance of edges and self loops in stochastic Kronecker graph.