Sensitivity of Diffusion Dynamics to Network Uncertainty

Abstract

Simple diffusion processes on networks have been used to model, analyze and predict diverse phenomena such as spread of diseases, information and memes. More often than not, the underlying network data is noisy and sampled. This prompts the following natural question: how sensitive are the diffusion dynamics and subsequent conclusions to uncertainty in the network structure?In this paper, we consider two popular diffusion models: Independent cascade (IC) model and Linear threshold (LT) model. We study how the expected number of vertices that are influenced/infected, for particular initial conditions, are affected by network perturbations. Through rigorous analysis under the assumption of a reasonable perturbation model we establish the following main results. (1) For the IC model, we characterize the sensitivity to network perturbation in terms of the critical probability for phase transition of the network. We find that the expected number of infections is quite stable, unless the transmission probability is close to the critical probability. (2) We show that the standard LT model with uniform edge weights is relatively stable under network perturbations. (3) We study these sensitivity questions using extensive simulations on diverse real world networks and find that our theoretical predictions for both models match the observations quite closely. (4) Experimentally, the transient behavior, i.e., the time series of the number of infections, in both models appears to be more sensitive to network perturbations.

Stability of influence maximization

“…When |σ(S) − σ (S)| is small compared to σ(A * 0 ) for all sets S, a user can have confidence that his optimization result will provide decent performance guarantees even if his input was perturbed. The converse 1 The example reveals a close connection between the stability of an IC instance and the question whether a uniform activation probability p lies close to the edge percolation threshold of the underlying graph. Characterizing the percolation threshold of families of graphs has been a notoriously hard problem.…”

“…[1].Goyal et al assume that for each edge (u, v), the value of p u,v is perturbed with uniformly random noise from a known interval. Adiga et al assume that each edge (u, v) that was observed to be present is actually absent with some probability , while each edge that was not observed is actually present with probability ; in other words, each edge's presence is independently flipped with probability .The standard Independent Cascade Model subsumes both models straightforwardly.…”

Robust Influence Maximization

“…However, when the algorithm does get to exceed the seed set target k by a factor of ln |Σ| (times a constant), much better bicriteria approximation guarantees can be obtained. 2 Specifically, we show that a modification of an algorithm of Krause et al [23] uses O(k ln |Σ|) seeds and finds a seed set whose influence is within a factor (1 − 1/e) of optimal.…”

“…A visual inspection of the nodes chosen by different algorithms reveals how the robust algorithm "hedges its bets" across models, while the non-robust heuristic tends to cluster selected nodes in one part of the network. 2 A bicriteria algorithm gets to pick more nodes than the optimal solution, but is only judged against the optimum solution with the original bound k on the number of nodes.…”

Scalable Vaccine Distribution in Large Graphs given Uncertain Data

“…Another related line of work studies the effect of sampling on measured structural properties [11,23,4] or network construction [25,29]. Correcting for the effects of missing data in cascades in general has not seen much attention-the exceptions are Sadikov et al [38] (who try to correct metrics like cascade size for sampling), and Adiga et al [1] (who study the effect of more general noise in the network structure on metrics like expected footprint in the IC and LT models). Here we study a specific algorithmic task (immunization) under uncertainty in observed infections.…”