Quantifying community evolution in temporal networks

TL;DR

This work tackles the problem of comparing evolving community partitions in temporal networks where the node set changes over time. It introduces Union-AMI (UAMI) and Intersection-AMI (IAMI), two Normalised Mutual Information extensions that handle node additions/deletions by operating on the union or intersection of node sets and by incorporating null-model corrections. The framework combines a snapshot-based temporal-network model with significance testing and detailed normalization to provide robust, interpretable similarity measures across time. Empirical validation on synthetic data and five real temporal networks demonstrates that UAMI and IAMI capture complementary aspects of community evolution and offer a practical tool for window selection and dynamic-network analysis.

Abstract

When we detect communities in temporal networks it is important to ask questions about how they change in time. Normalised Mutual Information (NMI) has been used to measure the similarity of communities when the nodes on a network do not change. We propose two extensions namely Union-Normalised Mutual Information (UNMI) and Intersection-Normalised Mutual Information (INMI). UNMI and INMI evaluate the similarity of community structure under the condition of node variation. Experiments show that these methods are effective in dealing with temporal networks with the changes in the set of nodes, and can capture the dynamic evolution of community structure in both synthetic and real temporal networks. This study not only provides a new similarity measurement method for network analysis but also helps to deepen the understanding of community change in complex temporal networks.

Figures (7)

• Figure 1: The diagram shows the principles of union extension and intersection extension respectively. In the two networks, the nodes labelled 6 and 9 appear in $G_1$ but are removed in $G_2$. The nodes labelled 1 and 4 are new in $G_2$. (a) The union extension compares the union of two node sets and adds a community labelled with $m^{(1)}$ in the partition of $G_1$, including nodes 6, and 9. A new community labelled $m^{(2)}$ is added to the partition of $G_2$, including nodes 1 and 4. (B) The intersection extension compares the intersection of two node sets, ignoring the added/removed nodes. In the example, nodes 1, 4, 6 and 9 are excluded when comparing the similarity.
• Figure 2: Example dynamic community structures with $n=400$ nodes and their pair-to-pair UAMI, IAMI measurement, where $p_{\mathrm{rot}} \in [0,1]$ is a parameter which, when high, means nodes leave/arrive in the network quickly (it increases from left to right), and $p_{\mathrm{move}} \in [0,1]$ is a parameter which, when high, means nodes move to new communities quickly (it increases from top to bottom). The first graph for each set of experiments represents the community assignments of nodes in different snapshots. We use four colours to represent the four communities specified in this experiment ( ). The two subsequent diagrams show the similarity measured by UAMI and IAMI in community structure between each pair of network slices. (0 1).
• Figure 3: (a) shows edge and node counts. (b) shows the proportion of nodes appears only in the $1^{\text{st}}$ window, the $2^{\text{nd}}$ window, and appears in both windows. (c) shows the similarity in community measures in the email-EU-core network.
• Figure 4: (a) shows edge and node counts. (b) shows the proportion of nodes appears only in the $1^{\text{st}}$ window, the $2^{\text{nd}}$ window, and appears in both windows. (c) shows the similarity in community measures in the Math Overflow network.
• Figure 5: (a) shows edge and node counts. (b) shows the proportion of nodes appears only in the $1^{\text{st}}$ window, the $2^{\text{nd}}$ window, and appears in both windows. (c) shows the similarity in community measures in the arXiv HEP-TH network.