Longitudinal Modularity, a Modularity for Link Streams

TL;DR

This article introduces the first adaptation of the well-known Modularity quality function to link streams, and its relation to existing static and dynamic definitions of Modularity.

Abstract

Temporal networks are commonly used to model real-life phenomena. When these phenomena represent interactions and are captured at a fine-grained temporal resolution, they are modeled as link streams. Community detection is an essential network analysis task. Although many methods exist for static networks, and some methods have been developed for temporal networks represented as sequences of snapshots, few works can handle link streams. This article introduces the first adaptation of the well-known Modularity quality function to link streams. Unlike existing methods, it is independent of the time scale of analysis. After introducing the quality function, and its relation to existing static and dynamic definitions of Modularity, we show experimentally its relevance for dynamic community evaluation.

Figures (7)

• Figure 1: Illustration of a link stream with a community structure depicted in blue and red. The figure includes representations of temporal community notations. The total interactions between nodes $b$ and $c$ are quantified as $L_{bc} = 4$. Within community $B$, these interactions total $L_{bc \in B}=3$. Additionally, the degree of node $a$ is represented as $k_a=8$.
• Figure 2: CSC Examples. The figure is a 5 nodes link stream with 2 communities in red and blue. Nodes $a$ and $b$ leave the red community to form the two-nodes community in blue and then rejoin the red community again, so $\eta_a = \eta_b = 2$. Node $e$ temporarily leaves the red community, so $\eta_e = 1$. Since $c$ and $d$ never change communities, $\eta_c = \eta_d = 0$. Finally, $\rho=0.08$.
• Figure 3: Illustration of the independence of L-Modularity from the time granularity. All 3 examples are composed of the same interactions occurring at the same time, but aggregated at different time scales. The L-Modularity score is the same for the blue/red communities in all 3 cases, since 1)the number of interactions inside each community, 2)the relative duration of communities, 3)the global nodes degree, and 4)the number of affiliation discontinuity are all identical in all 3 cases.
• Figure 4: Illustration of the smoothness incentive property. The same link stream is partitioned in different ways, each color representing a community. In Fig. (b), the three nodes on the top change communities, although there is not change in the network structure. A quality function respecting the smoothness incentive property must strictly favor partition (a) over (b)
• Figure 5: Topochrone disconnections. A quality function respecting the Topochrone disconnection property should strictly favor the partition in Fig. a) over the one in b).

Theorems & Definitions (6)

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