Core community structure recovery and phase transition detection in temporally evolving networks

Wei Bao,George Michailidis

doi:10.1038/s41598-018-29964-9

Wei Bao, George Michailidis

Open Access

https://doi.org/10.1038/s41598-018-29964-9

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Abstract

Community detection in time series networks represents a timely and significant research topic due to its applications in a broad range of scientific fields, including biology, social sciences and engineering. In this work, we introduce methodology to address this problem, based on a decomposition of the network adjacency matrices into low-rank components that capture the community structure and sparse & dense noise perturbation components. It is further assumed that the low-rank structure exhibits sharp changes (phase transitions) at certain epochs that our methodology successfully detects and identifies. The latter is achieved by averaging the low-rank component over time windows, which in turn enables us to precisely select the correct rank and monitor its evolution over time and thus identify the phase transition epochs. The methodology is illustrated on both synthetic networks generated by various network formation models, as well as the Kuramoto model of coupled oscillators and on real data reflecting the US Senate’s voting record from 1979–2014. In the latter application, we identify that party polarization exhibited a sharp change and increased after 1993, a finding broadly concordant with the political science literature on the subject.

Highlights

There has been a lot of work across different scientific communities including computer science, applied physics, statistics and the social sciences in developing methods for the analysis of network data[1]
It is assumed that A(t) can be decomposed as follows: A(t) = L(t) + S(t) + E(t), where L(t) is a low-rank matrix, S(t) is a sparse one with most of its elements being zero and E(t) a dense matrix with E(t) F < ε for some small ε > 0 and where ⋅ F denotes the Frobenius norm. This model captures the presence of community structure in networks through the low-rank component, as well as possible small sparse and/or dense perturbations as explained in the previous section. It is compatible with the popular network formation model that gives rise to community structure, namely the Stochastic Block Model (SBM)[15]
The stochastic block model (SBM) assumes an undirected network on n nodes and that the nodes are partitioned into K blocks

Summary

Consider a sequence of T weighted adjacency matrices

{A(t)}Tt=1 that encaptulate the structure of a network comprising of n nodes and their corresponding edges. Once the network adjacency matrices have been decomposed, the task becomes on how to identify the phase transition epochs and determine the number of communities and their membership These issues are discussed and illustrated on synthetic data generated according to the following mechanism. If a window is within a certain stable period, following the argument in the section of model formulation, we would expect an enhanced modularity and the thresholded rank would be consistent with the number of communities in the network across a wide range of threshold values. Note here we identified the phase transition time position t = 30 when new nodes join the network, while if we look at the time dependent rank of L(t) in Fig. 1(c) we would fail to detect this since the two regimes share the same rank but different community structures. The specifics of the network formation mechanisms are described :

Recovery Rate

Lin stands for connections within communities and

Party D R M

Concluding Remarks

Additional Information