Predicting Graph Structure via Adapted Flux Balance Analysis

TL;DR

The paper tackles forecasting dynamic graphs with growing vertex sets by merging ARIMA-based degree/time-series predictions with a constraint-based edge allocation inspired by Flux Balance Analysis. It introduces a hypothetical augmented graph $\mathcal{G}_{T}^{H}$ and uses its incidence matrix $\mathbf{S}$ to formulate a binary edge-assembly optimisation, with degree bounds $f(\mathbf{d})$ derived from predictive distributions. Empirical results on synthetic Preferential Attachment graphs and four real networks (UCI, HePH, Facebook, Bitcoin) show substantial reductions in vertex and edge prediction errors compared to using the last observed graph, with larger gains for longer horizons. The work demonstrates the viability of a constraint-based framework for growing-graph prediction and sets the stage for extensions to weighted/directed graphs and spectral similarity methods for evaluation. $\widehat{\mathcal{G}}_{T+h}$, $\mathcal{G}_{T}^{H}$, $\mathbf{S}$, $f(\mathbf{d})$, $\xi_{ij}$, $\hat n_{T+h}$, and $\hat d_{i,T+h}$ are central quantities enabling the approach.

Abstract

Many dynamic processes such as telecommunication and transport networks can be described through discrete time series of graphs. Modelling the dynamics of such time series enables prediction of graph structure at future time steps, which can be used in applications such as detection of anomalies. Existing approaches for graph prediction have limitations such as assuming that the vertices do not to change between consecutive graphs. To address this, we propose to exploit time series prediction methods in combination with an adapted form of flux balance analysis (FBA), a linear programming method originating from biochemistry. FBA is adapted to incorporate various constraints applicable to the scenario of growing graphs. Empirical evaluations on synthetic datasets (constructed via Preferential Attachment model) and real datasets (UCI Message, HePH, Facebook, Bitcoin) demonstrate the efficacy of the proposed approach.

Figures (1)

• Figure 1: A conceptual illustration of the prediction graph distribution in terms of $\gamma$ and $u$. The parameter $\gamma$ gives various $\widehat{n}_{T+h}$ values. The two graphs on bottom left each have 4 vertices, corresponding to a single $\gamma$ (or $\widehat{n}_{T+h}$) value, but have differing number of edges corresponding to various $u$ values. Similarly, the two graphs on the bottom right have 7 vertices corresponding to a higher value of $\gamma$ with the rightmost graph having more edges due to a higher $u$.