State Aggregations in Markov Chains and Block Models of Networks

Mauro Faccin, Michael T. Schaub, and Jean-Charles Delvenne

Phys. Rev. Lett. 127, 078301 – Published 12 August 2021

Abstract

We consider state-aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by $T$ time steps is maximized. We show that for $T = 1$ this recovers the maximum-likelihood estimator of the degree-corrected stochastic block model as a particular case, which enables us to explain certain features of the likelihood landscape of this generative network model from a dynamical lens. We further highlight how we can uncover coherent, long-range dynamical modules for which considering a timescale $T ≫ 1$ is essential. We demonstrate our results using synthetic flows and real-world ocean currents, where we are able to recover the fundamental features of the surface currents of the oceans.

Received 4 May 2020
Revised 17 June 2021
Accepted 15 July 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.078301

Physics Subject Headings (PhySH)

Clustering Community structure

Geophysical networks

Block models Metropolis algorithm

Interdisciplinary PhysicsNetworks

Authors & Affiliations

Mauro Faccin ¹, Michael T. Schaub ^2,3, and Jean-Charles Delvenne^1,4

¹ICTEAM, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
²Department of Engineering Science, University of Oxford, Oxford OX1 2JD, United Kingdom
³Department of Computer Science, RWTH Aachen University, 52074 Aachen, Germany
⁴CORE, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

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Vol. 127, Iss. 7 — 13 August 2021

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Images

Figure 1
Transition graph of a Markov chain with two alternative aggregations (inset schematic not of original size): an “assortative” split into two almost regular cyclic structures and a “disassortative,” almost bipartite split. The black edges exemplify the linking pattern. Two additional edges (red) break the bipartite symmetry and one joins the two cycles (green). The autoinformation results are shown for $N = 120 + 240$ nodes with average degree $⟨ k ⟩ = 10.02$ . At short and long timescales, autoinformation is optimized in the bipartite or the assortative partition, respectively.
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Figure 2
Markov chains with natural timescales: a $k$ -nearest-neighbor cycle with $N$ nodes (green), two cycles of $N / 2$ nodes connected by a single edge (orange), and a bipartite graph with a single link breaking the symmetry (blue); see insets for schematics. The plots correspond to graphs with 360 nodes with average degree $⟨ k ⟩ \approx 36$ . The color of each line encodes the corresponding graph topology. At short timescales, the maximization of the regularized autoinformation ( $β = 0.1$ ) tends to overfit the structure of these graphs with dense diagonal blocks (similar results hold for many community detection methods; see text). When increasing $T$ , the algorithm finds the solution with the expected number of classes.
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Figure 3
State aggregations for Markov chains with hierarchical timescales on a stochastic block model. Left: We plot the difference in the regularized autoinformation between a fine-state aggregation $h_{f}$ into the four planted aggregation classes, and a coarse two-class state aggregation $h_{c}$ for a hierarchical state-transition graph of a Markov chain (inset). The plot shows that at longer timescales (green shade) $I_{β, T} (h_{c}) > I_{β, T} (h_{f})$ and hence the coarser aggregation is preferred over the fine aggregation, which is preferred at shorter timescales (gray shade). Right: Difference in the regularized autoinformation between the two-class split $h_{c}$ , describing either a core-periphery (orange) or an assortative (violet) aggregation, and the underlying planted aggregation into four classes ( $h_{f}$ ). The four-class partition has a higher autoinformation than the two-class split at short timescales (gray shade). The assortative partition has highest autoinformation for middle range timescales (orange shade), and the core-periphery partition is preferred at longer timescales (blue shade). All graphs consist of 400 nodes and expected average degree $⟨ k ⟩ = 15$ , with $β = 0.05$ .
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Figure 4
State aggregation of ocean currents. The above maps compare two partitions induced by aggregating the states of the ocean currents according to the regularized autoinformation for short timescales (top) or longer timescales (bottom) with $β = 0.5$ . At shorter timescales, a higher number of classes is found. At longer timescales, the aggregation classes reveal well-known features of global ocean dynamics such as the Antarctic Circumpolar Current, subtropical gyres, and, in general, a marked separation of the polar, midlatitude, and equatorial regions. The quiver plot overlay displays the average drifter’s velocity. Each time step $t$ corresponds to 16 days.
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