The Trouble with Community Detection

Transcription

1 HV 3 The Trouble with Community Detection Aaron Computer Science Dept. & BioFrontiers Institute University of Colorado, Boulder External Faculty, Santa Fe Institute 2016 Aaron Clauset 1 June 2016

2 D community detection large-scale structure = community structure vertices with same pattern of inter-community connections community interaction matrix 9 s D t 600 YQKDAPNY DPNY PNY 13

3 lies, damned lies, and community detection

4 lies, damned lies, and community detection many networks include metadata on their nodes: social networks food webs Internet protein interactions age, sex, ethnicity or race, etc. feeding mode, species body mass, etc. data capacity, physical location, etc. molecular weight, association with cancer, etc. metadata x is often used to evaluate the accuracy of community detection algs. if community detection method finds a partition that correlates with then we say that is good A A P x

5 ct and ation, we ally relevant d vice versa. We rates (Fig. 4). As e the highest loss rate ost conserved across all f p lost (R) for ultra-peripheral ative, but hardly surprising. The n-hub connectors (role R3) and wever, yields a surprising finding. The ial hubs class have many within-module lies, damned lies, and community detection re 3 Cartographic representation of the metabolic network of E. coli. Each circle presents a module and is coloured according to the KEGG pathway classification of the etabolites it contains. Certain important nodes are depicted as triangles (non-hub ctors), hexagons (connector hubs) and squares (provincial hubs). Interactions modules and nodes are depicted using lines, with thickness proportional to the Nature Publishing Group representation number of actual links. Inset: metabolic network of E. coli, which contains 473 metabolites and 574 links. This representation was obtained using the program Pajek. Each node is coloured according to the main colour of its module, as obtained from the cartographic NATURE VOL FEBRUARY !"#$%$&'()')*+,$

6 lies, damned lies, and community detection several recent studies claim these are the exception real networks either do not contain structural communities or communities exist but they do not correlate with metadata groups "classic" data sets online data sets Hric et al. (2014) maximum NMI between any partition layer of the metadata partitions and any layer returned by the community detection method [1] see Leskovec et al. (2009), and Yang & Leskovec (2012), and Hric, Darst & Fortunato (2014)

7 but wait! the idea x P 2 {P} use metadata to help select a partition that correlates with, from among the exponential number of plausible partitions a metadata-aware degree-corrected stochastic block model x metadata degree prior on community assignments edge probability block parameters x = {x u } d = {d u } P (s, x) = Y i p uv = d u d v su,s v st s i,x i [1] technical details in Newman & Clauset (2015) arxiv:

8 but wait! the idea x P 2 {P} use metadata to help select a partition that correlates with, from among the exponential number of plausible partitions a metadata-aware degree-corrected stochastic block model model likelihood (with adjacency matrix A) x P (A,, x) = X s network metadata P (A, s)p (s, x) = X s Y u<v p A uv uv (1 p uv ) 1 A uv Y u s u,x u inference performed using EM algorithm to choose and [1] technical details in Newman & Clauset (2015) arxiv:

9 networks with planted structure does this method recover known structure in synthetic data? we use the planted partition test: 2 groups, equal sizes, constant mean degree assign metadata x u = ground truth with prob. = {0.5,...,0.9} c in n c out n c out n c in n

10 networks with planted structure let mean degree c =8 when =0.5 metadata is irrelevant and we recover regular SBM behavior Fraction of correctly assigned nodes weaker undetectable stronger easy to detect [1] n = c in -c out

11 networks with planted structure let mean degree when for =0.5 > 0.5 metadata + SBM c =8 beats any algorithm without metadata, and beats metadata alone. metadata is irrelevant and we recover regular SBM behavior Fraction of correctly assigned nodes weaker undetectable stronger easy to detect [1] n = c in -c out

12 real-world networks

13 real-world networks 1. high school social network: 795 students in a medium-sized American high school and its feeder middle school 2. marine food web: predator-prey interactions among 488 species in Weddell Sea in Antarctica 3. Malaria gene recombinations: recombination events among 297 var genes 4. Facebook friendships: online friendships among 15,126 Harvard students and alumni 5. Internet graph: peering relations among 46,676 Autonomous Systems

14 real-world networks 1. high school social network: 795 students in a medium-sized American high school and its feeder middle school x = {grade 7-12, ethnicity, gender} method finds a good partition between high-school and middle-school NMI. = without metadata: NMI 2 [0.105, 0.384] [1] Add Health network data, designed by Udry, Bearman & Harris

15 real-world networks 1. high school social network: 795 students in a medium-sized American high school and its feeder middle school x = {grade 7-12, ethnicity, gender} method finds a good partition between blacks and whites (with others scattered among) NMI = without metadata: NMI 2 [0.120, 0.239] [1] Add Health network data, designed by Udry, Bearman & Harris

16 real-world networks 1. high school social network: 795 students in a medium-sized American high school and its feeder middle school x = {grade 7-12, ethnicity, gender} method finds no good partition between males/ females. instead, chooses a mixture of grade/ethnicity partitions NMI = without metadata: NMI 2 [0.000, 0.010] [1] Add Health network data, designed by Udry, Bearman & Harris

17 PNY DDPNY conclusions PNY 13

18 PNY DDPNY conclusions PNY 13 the trouble with community detection 1. many good local optima 2. naive optimization inconsistent against node metadata

19 PNY DDPNY conclusions PNY 13 the trouble with community detection 1. many good local optima 2. naive optimization inconsistent against node metadata two birds with one stone a metadata-aware stochastic block model choose a partition that correlates with edges, and metadata (if possible) learns metadata associations with communities more accurate than any algorithm that uses only edges or metadata highly scalable EM + Belief Propagation algorithm P (A,, x)

20 acknowledgements Funding support: Structure and inference in annotated networks M. E. J. Newman 1, 2 and Aaron Clauset 2, 3 1 Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA 2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM Department of Computer Science and BioFrontiers Institute, University of Colorado, Boulder, CO 80309, USA Mark Newman (Michigan) to appear, Nature Communications (arxiv: ) PHYSICAL REVIEW E 81, Benjamin H. Good (Harvard) Yva de Montjoye (MIT) Performance of modularity maximization in practical contexts Benjamin H. Good, 1,2, * Yves-Alexandre de Montjoye, 3,2, and Aaron Clauset 2, 1 Department of Physics, Swarthmore College, Swarthmore, Pennsylvania 19081, USA 2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA 3 Department of Applied Mathematics, Université Catholique de Louvain, 4 Avenue Georges Lemaitre, B-1348 Louvain-la-Neuve, Belgium Received 1 October 2009; published 15 April 2010 [1] a C implementation of the metadata SBM is available by request

21 PNY DDPNY PNY 13 fin community detection

22 real-world networks ut metadata 2. marine food web: predator-prey interactions among 488 species in Weddell Sea in Antarctica x = {species body mass, feeding mode, oceanic zone} partition recovers known correlation between body mass, trophic level, and ecosystem role: 8 Probability of community membership Mean body mass (g) [1] here, we re using a continuous FIG. metadata S4: Learned model priors, as a function of body mass, for the [2] Brose et al. (2005) three-community division of the Weddell Sea network shown Detritivore Carnivore Omnivore Herbivore Primary producer

23 real-world networks 3. Malaria gene recombinations: recombination events among 297 var genes x = {Cys-PoLV labels for HVR6 region} with metadata, partition discovers correlation with Cys labels (which are associated with severe disease) HVR6 without metadata with metadata NMI 2 [0.077, 0.675] NMI = [1] Larremore, Clauset & Buckee (2013)

24 real-world networks 3. Malaria gene recombinations: recombination events among 297 var genes x = {Cys-PoLV labels for HVR6 region} on adjacent region of gene, we find Cys-PoLV labels correlate with recombinant structure here, too HVR5 without metadata with metadata [1] Larremore, Clauset & Buckee (2013)

25 real-world networks 4. Facebook friendships: online friendships among 15,126 Harvard students and alumni (in Sept. 2005) x = {graduation year, dormitory} method finds a good partition between alumni, recent graduates, upperclassmen, sophomores, and freshmen NMI. = without metadata: NMI 2 [0.573, 0.641] Prior probability of membership None Year [1] Traud, Mucha & Porter (20012)

26 real-world networks 4. Facebook friendships: online friendships among 15,126 Harvard students and alumni (in Sept. 2005) x = {graduation year, dormitory} method finds a good partition among the dorms NMI. = without metadata: NMI 2 [0.074, 0.224] Prior probability of membership Dorm [1] Traud, Mucha & Porter (20012)

27 real-world networks 5. Internet graph: 262,953 peering relations among 46,676 Autonomous Systems x = {country location of AS} method finds a good partition along the lines of the 173 countries NMI. = without metadata: NMI 2 [0.398, 0.626] [1] here, we re using a continuous metadata model