Predicting Graph Structure via Adapted Flux Balance Analysis

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.

💡 Research Summary

The paper tackles the problem of predicting future graph structures in dynamic networks where both vertices and edges can appear over time—a scenario commonly encountered in telecommunications, transportation, and social media but largely ignored by existing graph‑prediction methods that assume a fixed vertex set. The authors propose a novel framework that couples classical time‑series forecasting with an adapted version of Flux Balance Analysis (FBA), a linear programming technique originally devised for reconstructing metabolic networks.

The workflow consists of two main stages. First, the authors extract two families of time series from the observed graph sequence {G₁,…,G_T}: (i) the total number of vertices n_t, and (ii) for each vertex i that has already appeared, its degree d_{i,t} from its birth time t₀ onward. Both series are modeled with automatically‑parameterized ARIMA models, yielding forecasts of the total vertex count b_n_{T+h} and the future degrees b_d_{i,T+h} of existing vertices at the target horizon h. For vertices that will be newly introduced, the authors compute the average degree of all historically “t‑new” vertices and use this mean as a proxy upper bound for the degree of any new node.

Second, the predicted degree information is fed into a constrained edge‑allocation problem. Starting from the last observed graph G_T, the method constructs a hypothetical super‑graph G_H_T by (a) adding the predicted number of new vertices (b_n_new = max(b_n_{T+h} – n_T, 0)) and (b) attaching each new vertex to at most k existing vertices (k = 10 in the experiments) to keep the problem size manageable. The incidence matrix S of G_H_T serves as the constraint matrix in an FBA‑style linear program. Decision variables u correspond to binary edge selections b_e_{ij,T+h} ∈ {0,1}. The objective maximizes a weighted sum of selected edges: existing edges receive weight 1, while potential new edges receive a small weight α (α = 10⁻³), encouraging the solution to preserve known structure while allowing limited growth. Two sets of constraints are imposed: (1) for each vertex i, the sum of incident edges must not exceed the predicted degree upper bound f(d)i, and (2) the total number of edges must not exceed the ARIMA‑predicted total edge count f(|b_E{T+h}|). The latter is implemented by appending a row of ones to S, thereby coupling all vertex constraints into a global edge‑budget constraint.

Solving this mixed‑integer linear program yields a concrete graph b_G_{T+h} that satisfies the degree forecasts and the global edge budget while being as close as possible to the previous topology. Because the forecasts themselves are stochastic (they come from ARIMA predictive distributions), the authors can generate a family of possible graphs by varying the quantiles used for b_n_{T+h} (γ) and for the degree bounds (u). This defines a prediction distribution 𝔊(γ, u) over graphs.

The empirical evaluation comprises synthetic data generated by the preferential‑attachment model and four real‑world dynamic networks: UCI Message, HePH, Facebook, and Bitcoin transaction graphs. For each dataset, the authors predict graphs at horizons h = 1,…,5 and report two simple yet interpretable error metrics: vertex error |b_n – n| / n and edge error |b_m – m| / m. Across all experiments, the method achieves vertex and edge errors typically below 5 %, with synthetic graphs reaching over 95 % accuracy in terms of matching the true counts. Compared to baseline link‑prediction or node‑attribute‑forecasting approaches, the proposed framework delivers substantially lower count errors, demonstrating its ability to handle growing vertex sets.

The paper also discusses limitations. ARIMA assumes linear, stationary dynamics, which may be insufficient for abrupt, non‑linear growth phases. The model deliberately ignores edges between two simultaneously added new vertices (case iii), which could be relevant in some social‑network scenarios. The MILP size grows with the number of potential edges, potentially leading to high computational cost for very large graphs. Finally, the evaluation focuses on count‑based errors and does not assess higher‑order topological fidelity (e.g., clustering coefficient, community structure).

In conclusion, the authors show that adapting FBA to graph prediction—by treating degree constraints as “mass‑balance” equations and using a simple linear objective—provides a flexible, interpretable, and computationally tractable way to forecast the evolution of growing networks. Future work is suggested in several directions: integrating non‑linear time‑series models (e.g., LSTM, Prophet), incorporating case iii edges, adding global topological constraints (modularity, average path length), and exploring distributed optimization techniques to scale the approach to massive networks. This work opens a promising interdisciplinary bridge between systems biology optimization methods and dynamic network analysis.