Expanders in Models of Social Networks

Abstract: A common model for social networks are Geometric Inhomogeneous Random Graphs (GIRGs), in which vertices draw a random position in some latent geometric space, and the probability of two vertices forming an edge depends on their geometric distance. The geometry may be modelled in two ways: either two points are defined as close if they are similar in all dimensions, or they are defined as close if they are similar in some dimensions. The first option is mathematically more natural since it can be described by metrics. However, the second option is arguably the better model for social networks if the different dimensions represent features like profession, kinship, or interests. In such cases, nodes already form bonds if they align in some, but not all dimensions. For the first option, it is known that the resulting networks are poor expanders. We study the second option in the form of Minimum-Component Distance GIRGs, and find that those behave the opposite way for dimension $d\ge 2$, and that they have strong expanding properties. More precisely, for a suitable constant $C>0$, the subgraph induced by vertices of (expected) degree at least $(\log n)^C$ forms an expander. Moreover, we study how the expansion factor of the resulting subgraph depends on the choice of $C$, and show that this expansion factor is $\omega(1)$ except for sets that already take up a constant fraction of the vertices. This has far-reaching consequences, since many algorithms and mixing processes are fast on expander graphs.