A Framework for Imperfectly Observed Networks

Abstract

Model a network as an edge-weighted graph, where the (unknown) weight w e of edge e indicates the frequency of observed interactions, and over time t we observe a Poisson(tw e ) number of interactions across edges e. How should we estimate some given statistic of the underlying network? This leads to wide-ranging and challenging problems, on which this article makes only partial progress.

Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

“…we clearly have a(S) ≤ κb(S); note this is where we use the monotonicity hypothesis (3). The result now follows from (6,8).…”

“…Setting κ = max{h(S) − h(S ′ ) : S → S ′ a possible transition} we clearly have a(S) ≤ κb(S); note this is where we use the monotonicity hypothesis (3). The result now follows from (6,8).…”

“…The process here arises in a broad "imperfectly observed networks" program described in [6]. We give two examples of the application of Lemma 1 to this process.…”

The incipient giant component in bond percolation on general finite weighted graphs

“…A technically more complicated application of that bound to first passage percolation on general weighted graphs, plus other simple applications, can be found in [2]. "Big picture" discussions of various random processes over finite edge-weighted graphs can be found in [1] and [3].…”