Theoretical Analysis of Geographic Routing in Social Networks

We introduce a formal model for geographic social networks, and introduce the notion of rank-based friendship, in which the probability that a person v is a friend of a person u is inversely proportional to the number of people w who live closer to u than v does. We then prove our main theorem, showing that rank-based friendship is a sufficient explanation of the navigability of any geographic social network that adheres to it. 1 A Model of Population Networks There are two key features that we wish to incorporate into our social-network model: geography and population density. We will first describe a very general abstract model; in later sections we examine a concrete grid-based instantiation of it. Definition 1.1 (Population network) A population network is a 5-tuple 〈L, d, P, loc, E〉 where • L is a finite set of locations ( , s, t, x, y, z, . . .); • d : L× L −→ R+ is an arbitrary distance function on the locations; • P is a finite ordered set of people (u, v, w, . . .); • loc : P −→ L is the location function, which maps people to the location in which they live; and • E ⊆ P × P is the set of friendships between people in the network. The ordering on people is required only to break ties when comparing distances between two people. Let P(L) denote the power set of L. Let pop : L −→ Z+ denote the population of each point on L, where pop( ) := |{u ∈ P : loc(u) = }|. We overload notation, and let pop : P(L) −→ Z+ denote the population of a subset of the locations, so that pop(L′) := ∑ ∈L′ pop( ). We will write n := pop(L) = |P | to denote the total population, and m := |L| to denote the total number of locations in the network. ∗IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120. ravi,jnovak,[email protected]. †MIT Computer Science and Artificial Intelligence Laboratory (CSAIL), 32 Vassar Street, Cambridge, MA 02139. [email protected]. The majority of this work was performed while the author was visiting the IBM Almaden Research Center. ‡Verity, Inc., 892 Ross Drive, Sunnyvale, CA 94089. [email protected]