[Paper Review] Predicting Dynamics on Networks Hardly Depends on the Topology

Bastian Prasse, Piet Van Mieghem|arXiv (Cornell University)|May 29, 2020

Opinion Dynamics and Social Influence58 references4 citations

TL;DR

This paper proposes a prediction method for network dynamics that bypasses network reconstruction by leveraging temporal dynamics directly, demonstrating that accurate prediction is possible even when the inferred network topology bears no resemblance to the true network. The key insight is that dynamics prediction depends minimally on topology, relying instead on low-dimensional dynamical patterns captured via polynomial approximations of interaction functions.

ABSTRACT

Processes on networks consist of two interdependent parts: the network topology, consisting of the links between nodes, and the dynamics, specified by some governing equations. This work considers the prediction of the future dynamics on an unknown network, based on past observations of the dynamics. For a general class of governing equations, we propose a prediction algorithm which infers the network as an intermediate step. Inferring the network is impossible in practice, due to a dramatically ill-conditioned linear system. Surprisingly, a highly accurate prediction of the dynamics is possible nonetheless: Even though the inferred network has no topological similarity with the true network, both networks result in practically the same future dynamics.

Motivation & Objective

• To address the challenge of predicting future dynamics on an unknown network using only past observations of nodal states.
• To investigate whether accurate prediction of network dynamics requires precise reconstruction of the underlying network topology.
• To develop a method that predicts dynamics without relying on topological structure, focusing instead on dynamical patterns in observed data.
• To demonstrate that network reconstruction is ill-conditioned and often infeasible, yet prediction remains highly accurate.

Proposed method

• The method models network dynamics using a general class of continuous-time equations involving self-dynamics and pairwise interactions governed by a function $ g(x_i, x_j) $, with dynamics evolving via a system of ODEs.
• It employs polynomial approximation of the interaction function $ g $, truncating the series expansion at a finite degree $ q $, which allows the transformation of the dynamics into a linear regression problem.
• The algorithm constructs a design matrix $ F_{q,i} $ from observed nodal states over time, where each row corresponds to a time point and each column to a monomial term of degree up to $ q $.
• The method uses truncated singular value decomposition (SVD) to solve the ill-conditioned linear system for estimating interaction weights, effectively regularizing the inference process.
• Prediction is performed directly from the fitted coefficients without reconstructing the network, relying on the low-rank structure of the state evolution.
• Theoretical analysis shows that the rank of the design matrix is bounded by a function of the number of dynamical modes $ m $ and the polynomial degree $ q $, explaining the ill-conditioning and enabling stable prediction.

Experimental results

Research questions

• RQ1 Can future network dynamics be accurately predicted without reconstructing the underlying network topology?
• RQ2 To what extent does the accuracy of dynamics prediction depend on the fidelity of network reconstruction?
• RQ3 Why is network reconstruction ill-conditioned, and what structural properties of the data cause this instability?
• RQ4 Can low-dimensional dynamical patterns in nodal states enable accurate prediction despite topological ambiguity?
• RQ5 What is the role of the interaction function's polynomial structure in enabling stable prediction without topology?

Key findings

• Despite the network reconstruction problem being dramatically ill-conditioned, the proposed method achieves highly accurate prediction of future dynamics.
• The inferred network bears no topological similarity to the true network, yet both produce nearly identical future dynamics.
• The rank of the design matrix $ F_{q,i} $ is bounded by $ inom{eta + m - 1}{m - 1} $ for each degree $ eta $, where $ m $ is the number of dynamical modes, explaining the ill-conditioning.
• For the SIS epidemic model with $ m = 10 $ dynamical modes and polynomial degree $ q = 2 $, the maximum rank of $ F_i $ is bounded by 66, indicating a low-dimensional subspace of dynamics.
• Prediction accuracy remains high even when the inferred adjacency matrix $ ilde{A} $ is topologically dissimilar to the true $ A $, confirming that dynamics depend weakly on topology.
• The method achieves accurate prediction by focusing on the intrinsic low-dimensional manifold of nodal state evolution, bypassing the need for structural inference.