Typical distances in a geometric model for complex networks

Abstract

Modularity is designed to measure the strength of division of a network into clusters (known also as communities). Networks with high modularity have dense connections between the vertices within clusters but sparse connections between vertices of different clusters. As a result, modularity is often used in optimization methods for detecting community structure in networks, and so it is an important graph parameter from a practical point of view. Unfortunately, many existing non-spatial models of complex networks do not generate graphs with high modularity; on the other hand, spatial models naturally create clusters. We investigate this phenomenon by considering a few examples from both sub-classes. We prove precise theoretical results for the classical model of random d-regular graphs as well as the preferential attachment model, and contrast these results with the ones for the spatial preferential attachment (SPA) model that is a model for complex networks in which vertices are embedded in a metric space, and each vertex has a sphere of influence whose size increases if the vertex gains an in-link, and otherwise decreases with time. The results obtained in this paper can be used for developing statistical tests for models selection and to measure statistical significance of clusters observed in complex networks.

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

“…where 1−P(A c kn ) → 0 by Proposition 3.6. Next we show that the joint distribution of these two variables tend to (Y (1) , Y (2) ) as n → ∞. The model EGIRG W,L (1) is translation invariant, thus, marginally,…”

“…see Figure 2, thus these sets with index 1, 2 are disjoint on E (1) n . Next we determine k n such that E n,kn holds whp.…”

“…Set (X , ν) := ([−1/2, 1/2] d , Leb) for some integer d ≥ 1. We assume that (a) there exists an event L n measurable wrt to the σ-algebra generated by the weights (W (n) i ) i∈[n] that satisfies lim n→∞ P(L n ) = 1, (b) there exist intervals I ∆ (n) ⊆ R + , I w (n) ⊆ [1, ∞) and a sequence (n) ∈ R + , such that I ∆ (n) → (0, ∞), I w (n) → [1, ∞), (n) → 0 as n → ∞, (c) there exists a function h : 1], such that whenever the (fixed) triplet (∆, w u , w v ) ∈ R d × R + × R + satisfies that ∆ ∈ I ∆ (n), w u , w v ∈ I w (n), on the event L n , g n in (2.1) satisfies for ν-almost every x ∈ X , all u, v ∈ [n] that | g n (x, x + ∆/n 1/d , w u , w v , (W…”

“…Claim 7.1. Let E 7.1 (n) be the event that both uniformly chosen vertices v 1 n , v 2 n ∈ C max of BGIRG W,L (n) are also in C 1 ∞ , the infinite component of EGIRG W,L (1). Then E 7.1 (n) holds with probability 1 − e 7.1 (n), for some e 7.1 (n) ↓ 0.…”

On the Diameter of Hyperbolic Random Graphs

“…, v p . Similarly, the probability that sector p + 1 contains exactly one node v p+1 in L 3 is again e −Θ (1) . From here, we expose a path to the inner band B I as follows.…”