Alignment and Comparison of Directed Networks via Transition Couplings of Random Walks

TL;DR

NetOTC introduces a parameter-free, transport-based framework for comparing and aligning directed or undirected networks by coupling the full random walks of two graphs through stationary transition couplings that minimize the expected cost $\mathbb{E} c(\tilde{X}_0, \tilde{Y}_0)$. The method outputs a dissimilarity $\rho(G_1,G_2)$ and probabilistic vertex and edge alignments $\pi_v$, $\pi_e$, while preserving edges; it scales to networks of different sizes and has no Monte Carlo randomness. Theoretical results show edge preservation, metric properties on undirected common-vertex networks when $c$ is a metric, and a natural notion of network factors that relate extension and factor graphs via deterministic couplings. Empirically, NetOTC is competitive with or superior to existing OT-based methods in tasks like isomorphism recovery and SBM block alignment, and can detect exact or approximate factor relations, all without tunable parameters.

Abstract

We describe and study a transport based procedure called NetOTC (network optimal transition coupling) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may have distinct vertex sets of different sizes. Given two networks and a cost function relating their vertices, NetOTC finds a transition coupling of their associated random walks having minimum expected cost. The minimizing cost quantifies the difference between the networks, while the optimal transport plan itself provides alignments of both the vertices and the edges of the two networks. Coupling of the full random walks, rather than their marginal distributions, ensures that NetOTC captures local and global information about the networks, and preserves edges. NetOTC has no free parameters, and does not rely on randomization. We investigate a number of theoretical properties of NetOTC and present experiments establishing its empirical performance.

Figures (8)

• Figure 1: An example of two networks related by a factor map. Here $G_2$ is a factor of $G_1$ via the map that collapses vertices along vertical lines.
• Figure 2: An illustration of the relationship between factors and the NetOTC problem. Under the conditions described in Corollary \ref{['cor:factor']}, the NetOTC problem aligns vertices according to the factor map relating the two networks. In this example, $G_2$ is a factor of $G_1$. Figure \ref{['fig:factor_gotc']} illustrates the NetOTC vertex alignment, which is supported on pairs of the form $(u, f(u))$.
• Figure 3: An illustration of Theorem \ref{['thm:two_factor']}. If $G_1$ and $H_1$, and $G_2$ and $H_2$ are related by factor maps $f$ and $g$, respectively, the NetOTC of $H_1$ and $H_2$ can be naturally induced by the NetOTC of $G_1$ and $G_2$ using the maps $f$ and $g$.
• Figure 4: Three networks in which all vertices are located on the unit circle in $\mathbb{R}^2$. $G_1$ is an octagon network. $G_2$ is obtained by removing an edge $G_1$. In $G_3$, the vertices are uniformly distributed in the left semicircle.
• Figure 5: Vertex alignment of two isomorphic lollipop networks obtained by NetOTC, OT, and FGW. NetOTC correctly finds the isomorphism map, while other methods do not.

Theorems & Definitions (41)

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