Fig 1.
(Color online) Convergence behavior of the LLP process on real-world networks.
(a) and (b) show the number of active nodes per step during the LLP process on some snapshots of AS-Internet and AS-Oregon datasets, respectively. The labels in the legend indicate the date of the snapshots.
Fig 2.
(Color online) Schematic illustrations of how community structure can be affected by network modifications.
The node to be added or removed is stroked by red, and the edge to be added or removed is represented by red dashed line. Communities are distinguished by different filled colors of the nodes.
Fig 3.
Schematic diagram of the scope of the warm-up step and the LLP process.
The warm-up step only propagates labels inside the target community (or communities), while the LLP process may involve some nodes outside of the target community (or communities). These two processes generally affect a small portion of the whole network, especially when the network has many small communities.
Fig 4.
(Color online) Incremental detection of communities on the LFR network.
The color code in panel (a) corresponds to our incremental detection of the different communities in the network over time. Each node (vertical axes) at every time step belongs to a community, which is distinguished by different colors. White indicates that the node does not exist at that snapshot. From left to right, as new edges are added, nodes tends to group together, hence, many colors disappear and the community structure becomes clear. Eventually, our algorithm could correctly detect the true community structure. In particular, the number of active nodes involved in each event is shown in panel (b). The parameters of the LFR network are: N = 1000, μ = 0.3, 〈k〉 = 20, kmax = 50, γ = 2, β = 1, minimum and maximum community sizes are 90 and 125 respectively.
Fig 5.
(Color online) Visualization of communities’ evolution for time-varying LFR network.
The color code in panel (a) corresponds to our adaptive detection of communities in the network over time. Each node (vertical axes) at every time step belongs to a community, which is distinguished by different colors. White indicates that the node does not exist at that snapshot. From left to right, embedded community events occur sequentially. Panel (b) visualizes the communities’ evolution by using Netgram with parameters ρ = 0.4 and ν = 0.1 [43]. The parameters of the LFR network are the same as those in Fig 4.
Fig 6.
(Color online) Results of the application of different methods on the three standard benchmarks (in columns).
The first row corresponds to the planted partition of each benchmark, while the three remaining rows are the partitions detected by different algorithms. In each plot, the vertical axis corresponds to the index of nodes in the network, while the horizontal axis represents the time. The color of each pair {node, time} indicates the community to which the node is belongs at that specific time.
Fig 7.
(Color online) Three different measures (NMI, NVI and Jaccard index) between the planted partitions and the partitions detected by different algorithms for the three standard benchmarks.
There is a column for each benchmark and a row for each measure.
Fig 8.
(Color online) Normalized mutual information (NMI) as a function of the mixing parameter μ in LFR networks.
Four network scenarios are shown, which correspond to two different network sizes (N = 1000,5000) and, for a given size, to two different ranges for the community sizes (C = [10, 50], [20, 100]). Each point on the curves corresponds to the average value of the NMI value over 100 network realizations.
Fig 9.
(Color online) Structural changes over time in the datasets of interest.
(a) The structural changes over 733 snapshots in the AS-Internet dataset, including the number of edges added (E+) and deleted (E−), as well as the number of nodes involved in changes (N+,N−). (b) The structural changes in the AS-Oregon dataset over 9 snapshots.
Fig 10.
(Color online) Results on the AS-Internet dataset.
Comparison of (a) modularity and (b) consuming time of ALPA at each snapshot with FacetNet, iLCD and Infomap on the AS-Internet dataset.
Fig 11.
(Color online) Results on the AS-Oregon dataset.
Comparison of modularity (a) and consuming time (b) of ALPA at each snapshot with iLCD and Infomap on the AS-Oregon dataset.