Kaleidoscopic reorganization of network communities across different scales

TL;DR

This paper shows that community structure in real networks reorganizes across scales in ways that cannot be captured by a single resolution parameter. By analyzing the modularity function $Q(\mathcal{G};\gamma)$ and its change under merging, it demonstrates that increasing or decreasing scale can trigger simultaneous splitting and merging of communities, leading to non-monotonic changes in the number of communities. Empirical evidence from real networks (e.g., cond-mat collaboration and OpenFlights) plus a simple stochastic block model reveal a characteristic dip and reorganization pattern driven by core-periphery interactions. The findings suggest a paradigm shift in network science: scale-dependent reorganization should be accounted for to faithfully characterize mesoscale structure, with implications for how we detect and interpret communities in large, heterogeneous networks.

Abstract

The notion of structural heterogeneity is pervasive in real networks, and their community organization is no exception. Still, a vast majority of community detection methods assume neatly hierarchically organized communities of a characteristic scale for a given hierarchical level. In this work, we demonstrate that the reality of scale-dependent community reorganization is convoluted with simultaneous processes of community splitting and merging, challenging the conventional understanding of community-scale adjustment. We provide a mathematical argument concerning the modularity function, the results from real-network analysis, and a simple network model for a comprehensive understanding of the nontrivial community reorganization process. The reorganization is characterized by a local drop in the number of communities as the resolution parameter varies. This study suggests a need for a paradigm shift in the study of network communities, which emphasizes the importance of considering scale-dependent reorganization to better understand the genuine structural organization of networks.

Figures (5)

• Figure 1: A schematic diagram of community reorganization with the fixed number (two) of communities. The left panel has a lower value of $\gamma$ than the right panel, i.e., $\gamma_1 < \gamma_2$. The set of nodes enclosed within the dashed circles indicates that they belong to the same cohesive cluster, and a set with the same background color corresponds to each community for each $\gamma$ value. The edges represented by black and red lines indicate intra- and inter-community connections, respectively.
• Figure 2: (a) The number $n_c$ of communities as a function of the resolution parameter $\gamma$. (b) The sizes of the communities $C_s$ with the first (the black-filled squares) and the second (the red-filled circles) largest sizes as a function of the resolution parameter $\gamma$. For (a) and (b), the results are from the averaged values over $100$ independent community detection results applied to the cond-mat network, and the blue vertical lines indicate $\gamma=0.08$ and $0.1$. The error bars represent the standard deviation, as do all of the other figures in this paper. (c) and (d) show the CGN of communities at $\gamma=0.08$ and $\gamma=0.1$. Each node here indicates a community in the original network, and its size is proportional to the number of nodes composing communities. The edges represent inter-community connections. The communities with red color have two or more neighbors, and the communities with blue color have only one neighbor. The community size is proportional to $(\textrm{the number of nodes in the community} + 5)$ for ideal visibility.
• Figure 3: The Sankey diagram describes the membership change across communities for different $\gamma$ values. This diagram visually represents the community membership of nodes in a network for given resolution parameters $\gamma$. The flows illustrate nodes' community membership changes as the resolution parameter value varies. The communities are sorted descendingly by the number of nodes from bottom to top, and the three largest communities and their members (and the flows from them) at $\gamma = 0.3$ are colored purple, blue, and red, respectively. The nodes are sorted descendingly by the size of communities to which they belong (from bottom to top) at the one-step larger $\gamma$ value.
• Figure 4: (a) The mean number $n_c$ of communities as a function of the resolution parameter $\gamma$, averaged over $100$ community detection results in the case of OpenFlights. We color-code each airport's community membership on the world map, for (b) $\gamma = 0.11$ (seven communities) and (c) $\gamma = 0.15$ (four communities), where the two gamma values are indicated with the blue vertical lines in the panel (a).
• Figure 5: (a) The mean number of communities as a function of the resolution parameter in the SBM-based model network, averaging over $100$ independent community detection results from each of $100$ independent network realizations for each $\gamma$ value. (b)--(e) we show the community division (marked with the green boundary) of the same coarse-grained network composed of elementary building-block communities generated by the SBM at given resolution parameters (b) $\gamma = 0$, (c) $\gamma = 0.16$, (d) $\gamma = 0.36$, and (e) $\gamma = 0.51$, respectively.