Multislice community detection

Transcription

1 Multislice community detection P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, J.-P. Onnela Jukka-Pekka JP Onnela Harvard University NetSci2010, MIT; May 13, 2010

2 Outline (1) Background (2) Multislice formulation (3) Applications Preprint arxiv: Under embargo until 2PM Thursday afternoon 2

3 Background Community? A group of nodes that are relatively densely connected to each other but relatively sparsely connected to other nodes Communities are thought to have a strong bearing on functional units Introductory review: Communities in networks, M. A. Porter, J.-P. Onnela, P. J. Mucha, Notices of the American Mathematical Society 56, 1082 (2009) Comprehensive review: Community detection in graphs by Santo Fortunato, Physics Reports 486, 75 (2010) 3

4 Background Aim: Extend quality-function based community detection methods to deal with (1) Time-dependent networks: Nodes and ties may change in time (2) Multiscale networks: Structure simultaneously present at multiple scales (3) Multiplex networks: Multiple types of ties 4

5 Background Modularity maximization is possibly the most commonly used method Goal is to assign nodes to communities to maximize modularity Cannot guarantee optimal modularity without full enumeration of all partitions More intra-community edges than one would expect at random But what is random? Need to specify a null model Modularity and community structure in networks, M. E. J. Newman, PNAS 103, 8577 (2006) 5

6 Background Erdös-Rényi (Bernoulli) Newman-Girvan* Leicht-Newman* (directed) Barber* (bipartite) * with resolution parameters (Reichardt & Bornholdt, Lambiotte et al.) 6

7 Multislice formulation Easy part: Glue common individuals across slices (ordered vs. categorical) Difficult part: Null model? (cross-ties) ORDERED: neighbors CATEGORICAL: all to all 7

8 Multislice formulation Lambiotte, Delvenne and Barahona (arxiv: ) showed a way to derive modularity from normalized Laplacian dynamics Quality of a partition in terms of its stability, which is an autocovariance function of an ergodic Markov process on the network: R M (t) = [P (C, t) P (C, )] ṗ i = C A ij p j p i k j j Expansion of matrix exponential to first-order in time recovers Newman- Girvan modularity with resolution parameter 8

9 Multislice formulation Undirected network slices Undirected couplings Define multislice strength A ijs = A jis C jrs = C jsr κ js = k js + c js Density of random walkers in node-slice is: ṗ is = jr (A ijs δ sr + δ ij C jsr )p jr /κ jr p is within slice between slices Steady state probability distribution: p jr = κ jr /(2µ) 2µ = jr κ jr

10 Multislice formulation Null model: Probability of sampling node-slice is conditional on whether the multislice structure allows one to step from node-slice jr to node-slice is: ] κjr ρ is jr p jr = [ kis 2m s k jr κ jr δ sr + C jsr c jr c jr κ jr δ ij 2µ Subtracting this conditional joint probability from the linear in time approximation of the exponential describing the Laplacian dynamics gives Q multislice = 1 2µ ijsr {( A ijs γ s k is k js 2m s δ sr ) + δ ij C jsr } δ(c is,c jr ) Each slice has its own resolution parameter γ s Intra-slice couplings C jsr = {0, ω}

11 Application 1: Multiplexity Tastes, ties, and time data, which is a multiplex network of 1640 college students at an anonymous, northeastern American university Examine the following symmetrized ties from one wave of data: 1. Facebook friendships 2. Facebook picture friendships (upload & tag a photo) 3. Roommates (share dormitory room, creating clusters of 1-6 students) 4. Housing group (preference to be placed in same upper-class residence) Tie types are categorical, hence inter-slice coupling from all slices to all slices

12 Application 1: Multiplexity When omega = 0, each individual placed into four separate communities Increasing omega causes communities to merge across slices, especially if the patterns of connection are similar between slices (tie types) For intermediate omega, most individuals are placed in 1 or 2 communities, indicating their networks maintain group-level similarities across tie types Small minority maintain 4 separate assignments => different positions in slices ω #communities # communities per individual % % 40.5% 37.3% 8.2% % 49.1% 25.3% 5.7% % 48.3% 21.6% 3.9% % 47% 18.4% 2.8% % 42.4% 16.8% 1.5% %

13 Application 2: Time-dependence 100 Senators serving staggered six-year terms (2+2+2=6) Study Congresses 1-110, covering , with 1884 individual Senators Define weighted connections between each pair of Senators in terms of similarity of their voting dynamics (independently for each two-year Congress) Define adjacency matrices based on roll-call votes: A ij = (1/b ij ) k α ijk where α ijk equals unity if and only if i and j voted the same on bill k and b ij is the total number of bills on which both legislators voted Inter-slice coupling from each individual Senator to himself only when in consecutive Congresses Note that link strengths and nodes change from one slice to another

14 Application 2: Time-dependence Obtain 9 communities using inter-slice coupling omega = 0.5 Dark blue and red correspond to modern Democratic and Republican parties Vertical gray bars indicate Congresses in which three communities appeared Nominal party affiliations: Pro-Administration (PA) Anti-Administration (AA) Federalist (F) Democratic-Republican (DR) Whig (W) Anti-Jackson (AJ) Adams (A) Jackson (J) Democratic (D) Republican (R)

15 Application 2: Time-dependence Obtain 9 communities using inter-slice coupling omega = 0.5 Dark blue and red correspond to modern Democratic and Republican parties Vertical gray bars indicate Congresses in which three communities appeared Gray areas: 4th and 5th: First with political parties 10th and 11th: Vice President Aaron Burr's indictment for treason 14th and 15th: Changing structures in Democratic-Republican party 31st: Compromise of th: Beginning of the American Civil War 73rd and 74th: Landslide 1932 election amidst the Great Depression 85th to 88th: Brought the major American civil rights acts

16 Conclusion Many real-world networks are time-dependent, multiscale, or multiplex Multislice framework enables community detection in networks with any combination of the three properties