Finding remarkably dense sequences of contacts in link streams

Abstract

A link stream is a set of quadruplets (b, e, u, v) meaning that a link exists between u and v from time b to time e. Link streams model many real-world situations like contacts between individuals, connections between devices, and others. Much work is currently devoted to the generalization of classical graph and network concepts to link streams. We argue that the density is a valuable notion for understanding and characterizing links streams. We propose a method to capture specific groups of links that are structurally and temporally densely connected and show that they are meaningful for the description of link streams. To find such groups, we use classical graph community detection algorithms, and we assess obtained groups. We apply our method to several real-world contact traces (captured by sensors) and demonstrate the relevance of the obtained structures.

Appendices

Appendix 1: Rollernet data set

See Figs. 12, 13, 14, 15 and 16.

Fig. 12

Number of active links (left axis) and average number of active links per node (right axis) as a function of time on the Rollernet data set

Fig. 13

Inverse cumulative distributions of the number of links, nodes and duration (a) and density (b) for the candidates found by the Louvain method on the Rollernet data set

Fig. 14

Inverse cumulative distribution of the number of groups captured per node for the Rollernet data set

Fig. 15

Inverse cumulative distribution of scores for each aspect for the data set Rollernet. a Start time: $p_{t}.$ b Duration: $p_{\delta }.$ c Node set:$p_{node}.$ d Correlation between start time and duration aspects

Fig. 16

Inverse cumulative distributions of the number of links, nodes and duration in a and density in b for the candidates captured by our method on the Rollernet data set

Appendix 2: Baboon data set

See Figs. 17, 18, 19, 20 and 21.

Fig. 17

Number of active links (left axis) and average number of active links per node (right axis) as a function of time on the Baboon data set

Fig. 18

Inverse cumulative distributions of the number of links, nodes and duration (a) and density (b) for the candidates found by the Louvain method on the Baboon data set

Fig. 19

Inverse cumulative distribution of the number of groups captured per node for the Baboon data set

Fig. 20

Inverse cumulative distribution of scores for each aspect for the data set baboon. a Start time: $p_{t}.$ b Duration: $p_{\delta }.$ c Node set:$p_{node}.$ d Correlation between start time and duration aspects

Fig. 21

Inverse cumulative distributions of the number of links, nodes and duration in (a) and density in (b) for the candidates captured by our method on the Baboon data set

Appendix 3: Reality mining data set

See Figs. 22, 23, 24, 25 and 26.

Fig. 22

Number of active links (left axis) and average number of active links per node (right axis) as a function of time on the Reality Mining data set

Fig. 23

Inverse cumulative distributions of the number of links, nodes and duration (a) and density (b) for the candidates found by the Louvain method on the Reality Mining data set

Fig. 24

Inverse cumulative distribution of the number of groups captured per node for the Reality Mining data set

Fig. 25

Inverse cumulative distribution of scores for each aspect for the data set Reality Mining. a Start time: $p_{t}.$ b Duration: $p_{\delta }.$ c Node set:$p_{node}.$ d Correlation between start time and duration aspects

Fig. 26

Inverse cumulative distributions of the number of links, nodes and duration in (a) and density in (b) for the candidates captured by our method on the Reality Mining data set