Using higher-order Markov models to reveal flow-based communities in networks.

Vsevolod Salnikov,Michael T Schaub,Renaud Lambiotte

doi:10.1038/srep23194

Vsevolod Salnikov, Michael T Schaub + Show 1 more

Open Access

https://doi.org/10.1038/srep23194

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Abstract

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges. Markov processes have been the prevailing paradigm to model such a network-based dynamics, for instance in the form of random walks or other types of diffusions. Despite the success of this modelling perspective for numerous applications, it represents an over-simplification of several real-world systems. Importantly, simple Markov models lack memory in their dynamics, an assumption often not realistic in practice. Here, we explore possibilities to enrich the system description by means of second-order Markov models, exploiting empirical pathway information. We focus on the problem of community detection and show that standard network algorithms can be generalized in order to extract novel temporal information about the system under investigation. We also apply our methodology to temporal networks, where we can uncover communities shaped by the temporal correlations in the system. Finally, we discuss relations of the framework of second order Markov processes and the recently proposed formalism of using non-backtracking matrices for community detection.

Highlights

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges
Dynamics on complex networks, such as the diffusion of information in social networks, are commonly modelled as Markov processes. An advantage of this approach is that for every network with positive edge-weights we can define a corresponding Markov process by interpreting the network as the state space of a random walker, and assigning the state-transition probabilities according to the link weights
While simple Markov models have been very successful in modelling dynamics of complex systems and found many applications, they have one obvious disadvantage

Summary

Aij kiout π j

Similar adjusted adjacency (or Laplacian) matrices have been proposed as means for community detection in the literature based on different reasoning[24,25]. If there are multiple possible transitions, the walker takes each edge with a probability proportional to its weight This process is repeated multiple times for the observed interval [1, T] in order to generate sufficiently many trajectories. The slowest time-scale of the diffusion may not be correlated with moving from one planted (node) community to the other, but local obstacles become more important making the second eigenvector a bad predictor of the bi-partition This is in particular interesting, as for most memory dynamics reported[6] the return flow appears to be significantly large, which would imply that the dynamical constraints on the flow are much more pronounced on a local level than one may expect from the perspective of an aggregated network. This observation appears to be aligned with the fact that many real-world systems tend to be composed of overlapping communities[21]

Conclusions

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