Benchmark model to assess community structure in evolving networks

Detecting the time evolution of the community structure of networks is crucial to identify major changes in the internal organization of many complex systems, which may undergo important endogenous or exogenous events. This analysis can be done in two ways: considering each snapshot as an independent community detection problem or taking into account the whole evolution of the network. In the first case, one can apply static methods on the temporal snapshots, which correspond to configurations of the system in short time windows, and match afterwards the communities across layers. Alternatively, one can develop dedicated dynamic procedures, so that multiple snapshots are simultaneously taken into account while detecting communities, which allows us to keep memory of the flow. To check how well a method of any kind could capture the evolution of communities, suitable benchmarks are needed. Here we propose a model for generating simple dynamic benchmark graphs, based on stochastic block models. In them, the time evolution consists of a periodic oscillation of the system’s structure between configurations with built-in community structure. We also propose the extension of quality comparison indices to the dynamic scenario.

💡 Research Summary

The paper addresses a fundamental need in the study of temporal networks: the availability of realistic benchmark graphs that embed known community dynamics, enabling a rigorous assessment of both static and dynamic community‑detection algorithms. The authors build upon the classic stochastic block model (SBM) and introduce three families of synthetic dynamic networks that exhibit periodic structural changes.

The first family, the “grow‑shrink” benchmark, consists of two equally sized communities whose node membership oscillates over time. A fraction f of nodes migrates from one community to the other following a triangular wave of period τ. At each time step the community sizes are updated analytically, and all edges incident to moved nodes are removed and re‑drawn according to the fixed intra‑community probability p_in and inter‑community probability p_out. This construction guarantees that every snapshot remains a genuine SBM realization while the community composition evolves smoothly.

The second family, the “merge‑split” benchmark, models the gradual merging of two communities into a single dense block and their subsequent separation. Starting from two blocks of size n with internal link density p_in and cross‑block density p_out, the number of inter‑block edges is interpolated between a binomially sampled “unmerged” count m_um and a “merged” count m_m using the same triangular waveform x(t). At any time t the active inter‑block edge count is m*(t) =