Graphons and sampled networks A graphon W : [0 , 1] 2 [0 , 1] is a - - PowerPoint PPT Presentation

Graphons and sampled networks

A graphon W : [0, 1]2 → [0, 1] is a measurable function. W (u, v) measures the level of interaction between two infinitessimal agents with labels u and v. We can define a sampled network of size n from graphon W as follows:

I Sample n labels {u(n)

i

}n

i=1 uniformly from [0, 1]; attach u(n) i

to agent i

I Connect agents i and j with Bernoulli probability W (u(n)

i

, u(n)

j

).

I Obtain adjacency matrix A(n) ∈ {0, 1}n×n. I E.g., Erd˝

• s-R´

enyi graphon: W (u, v) = 0.4

Contagion in graphons

Initial seed set: C0 ⊆ [0, 1] (measurable). Threshold function: τ : [0, 1] → [0, 1] (piecewise continuous). Set of infected labels in period t: Ct Label u is exposed if R 1

0 W (u, v)1Ct−1(v)dv

R 1

0 W (u, v)dv.

> τ(u). Label u is added to Ct if either u ∈ Ct−1 or if u is exposed. In a sampled network: standard linear threshold contagion (Granovetter’78/Morris’00/Kempe-Kleinberg-Tardos’03)

Main theoretical result

Given a graphon W , threshold function τ, and initial seed set C0, can we predict the terminal set C∞ of (non-)infected agents in the sampled network?

Theorem (informal)

In a large enough sampled network, the terminal set of agents (not) infected in a sampled network can be reconstructed with high probability from the terminal set of labels (not) infected in the graphon.

Stochastic Block Model C0 = [0, 0.1], τ(u) = 0.16