Coarse-graining Directed Networks with Ergodic Sets Preserving Diffusive Dynamics

TL;DR

This work addresses preserving diffusive, random-walk dynamics when coarse-graining directed networks, where non-ergodicity undermines standard methods. It defines generalized ergodic sets (forward and backward) and a two-step ESCA algorithm: first, compress forward/backward ergodic sets while preserving flows; second, compress the transient core via a long-time mixing matrix $B$ and singular value decomposition $B = M_{bw} C M_{fw}$ to reveal dominant interaction modes. The approach yields substantial compression (e.g., $C_1\approx 0.01$, $C_2\approx 0.5$ on real networks) while preserving long-time diffusion, and analyses of cores show retained structural non-randomness; this provides a dynamics-aware alternative to purely topological coarse-graining. Overall, ESCA offers a fast, dynamics-preserving method applicable across synthetic and real directed networks, enabling efficient analysis of sources, sinks, and core mixing in complex systems.

Abstract

In this paper, we introduce ergodic sets, subsets of nodes of the networks that are dynamically disjoint from the rest of the network (i.e. that can never be reached or left following to the network dynamics). We connect their definition to purely structural considerations of the network and study some of their basic properties. We study numerically the presence of such structures in a number of synthetic network models and in classes of networks from a variety of real-world applications, and we use them to present a compression algorithm that preserve the random walk diffusive dynamics of the original network.

Figures (6)

• Figure 1: A representation of a generic network division. Top: the classical bow-tie structure broder2000graph. Bottom: the division in ergodic sets. BW: backward ergodic set, FW: forward ergodic set.
• Figure 2: Example of \ref{['alg:coarseFW']} applied to a forward ergodic set. The dashed edges to nodes of the forward ergodic sets are replaced by the plain edges to the ergodic sets. The latter are weighted such as to preserve the flows of random walkers.
• Figure 3: Example of \ref{['alg:coarseBtW']} applied to a backward ergodic set. Dashed edges from nodes in backward ergodic sets are replaced by plain edges from the ergodic sets, with weights chosen to preserve flows of walkers.
• Figure 4: Fraction of nodes of an Erdős-Rényi random graph belonging to any ergodic set (above) and the largest ergodic set (below), for different connection probabilities $p$ and network sizes $N$.
• Figure 5: Compression factor $C_1$ for the first step of the ESCA. The blue data is the compression factor $C_1$, the red data is the total compression factor $C_2$.

Theorems & Definitions (5)

• Definition 1
• Proposition 1
• Definition 2
• Definition 3
• Definition 4