Directed network comparison using motifs

TL;DR

This work tackles the challenge of comparing directed networks by introducing a motif-based framework that accounts for directionality and higher-order substructure. The core idea is to compute a node-wise motif distribution across 2–4 node directed motifs, assemble an $N\times35$ motif distribution matrix, and quantify dissimilarity between networks using Jensen-Shannon divergence combined with a global-local dispersion metric (DNND) into the final $D_m$ score: $D_m(G_1,G_2)=\varphi \sqrt{\frac{\zeta(\mu^{G_1},\mu^{G_2})}{\ln 2}}+(1-\varphi)\left|\sqrt{DNND(G_1)}-\sqrt{DNND(G_2)}\right|$. The method is validated on six real directed networks against null models based on extended $dk$-series and against edge-perturbed versions, showing superior discrimination and robustness across domains. Key contributions include the node-level motif distribution approach, the $DNND$ dispersion measure, and the integrated dissimilarity $D_m$ that balances global motif-frequency differences with local connectivity heterogeneity. The approach offers a principled way to capture higher-order, directional patterns in networks, with potential applications in biology, sociology, and transportation analytics; future work may scale to larger motifs and broader network-task analyses.

Abstract

Analyzing and characterizing the differences between networks is a fundamental and challenging problem in network science. Previously, most network comparison methods that rely on topological properties have been restricted to measuring differences between two undirected networks. However, many networks, such as biological networks, social networks, and transportation networks, exhibit inherent directionality and higher-order attributes that should not be ignored when comparing networks. Therefore, we propose a motif-based directed network comparison method that captures local, global, and higher-order differences between two directed networks. Specifically, we first construct a motif distribution vector for each node, which captures the information of a node's involvement in different directed motifs. Then, the dissimilarity between two directed networks is defined on the basis of a matrix which is composed of the motif distribution vector of every node and Jensen-Shannon divergence. The performance of our method is evaluated via the comparison of six real directed networks with their null models as well as their perturbed networks based on edge perturbation. Our method is superior to the state-of-the-art baselines and is robust with different parameter settings.

Figures (5)

• Figure 1: Motifs formed by 2 to 4 nodes in directed networks. All the motifs are labeled from $m_1$ to $m_{35}$.
• Figure 2: Toy examples of three $dk$-series null models: (a)$Dk1.0$; (b)$Dk2.0$; (c)$Dk2.5$. The blue dashed lines indicate the newly connected edges. In (a), (b), and (c), the left panel shows the original network and the right panel shows the rewired network.
• Figure 3: Comparison between real directed networks and their null models via motif-based directed network comparison method. The null models are $Dk1.0$, $Dk2.0$ and $Dk2.5$. Smaller values in the heatmap indicate a higher similarity, and vice versa.
• Figure 4: Similarity between a real directed network and perturbed network generated by randomly adding or deleting edges, where positive values of $f$ indicate we randomly add $f$ fraction of edges, and vice versa. We show results for networks: (a) Mac; (b) Elegans; (c) Physicians; (d) Email; (e) US airport; (f) Chess. The parameter $\varphi$ of $D_m$ is set to $0.5$. Each point in the figure is averaged over 100 realizations.
• Figure 5: Parameter analysis for motif-based directed network comparison. We compare the real network with its perturbed network via edge addition or deletion. Different curves show we choose different values of $\varphi$, which is the only parameter in our method, $\varphi \in \left \{0.1, 0.3, 0.5, 0.7, 0.9\right \}$. Positive values of $f$ indicate the random edge addition, and vice versa. We show results for networks: (a) Mac; (b) Elegans; (c) Physicians; (d) Email; (e) US airport; (f) Chess. All results are averaged over 100 realizations.