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Graph mining 2: Statistical approaches for graph mining

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Workshop "Advanced mathematics for network analysis"
organized by Institut des Systèmes Complexes de Toulouse
http://isc-t.fr/evenements/?event_id1=2
Luchon, France
May, 3rd 2016

Published in: Science
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Graph mining 2: Statistical approaches for graph mining

  1. 1. Graph mining 2 Statistical approaches for graph mining Nathalie Villa-Vialaneix nathalie.villa@toulouse.inra.fr http://www.nathalievilla.org Advanced mathematics for network analysis Luchon, May 3rd 2016 Nathalie Villa-Vialaneix | Graph mining 2 1/48
  2. 2. Talk map... Who am I? Statistician working in biostatistics at INRA Toulouse My research interests are: data mining, network inference and mining, machine learning Purpose of this talk: presenting a few statistical tools for graph mining (graph structure, important vertices) and clustering Nathalie Villa-Vialaneix | Graph mining 2 2/48
  3. 3. Background Unlike said so, G: undirected and connected graph; Nathalie Villa-Vialaneix | Graph mining 2 3/48
  4. 4. Background Unlike said so, G: undirected and connected graph; with vertices V = {x1, ..., xn}; with set of edges E; Nathalie Villa-Vialaneix | Graph mining 2 3/48
  5. 5. Background Unlike said so, G: undirected and connected graph; with vertices V = {x1, ..., xn}; with set of edges E; eventually with (positive and symmetric) weights on edges, wij (st wii = 0, no self loop) adjacency matrix A = (wij)i,j=1,...,n Nathalie Villa-Vialaneix | Graph mining 2 3/48
  6. 6. Examples are made with... the toy example “Les Misérables” (co-appearance network in Hugo’s novel) Myriel Napoleon MlleBaptistine MmeMagloire CountessDeLoGeborand Champtercier Cravatte Count OldMan Labarre Valjean Marguerite MmeDeR Isabeau Gervais Tholomyes Listolier Fameuil Blacheville Favourite Dahlia Zephine Fantine MmeThenardier Thenardier Cosette Javert Fauchelevent Bamatabois Perpetue Simplice Scaufflaire Woman1 Judge Champmathieu Brevet Chenildieu Cochepaille Pontmercy Boulatruelle Eponine Anzelma Woman2 MotherInnocent Gribier Jondrette MmeBurgon Gavroche Gillenormand Magnon MlleGillenormand MmePontmercy MlleVaubois LtGillenormand Marius BaronessT Mabeuf Enjolras Combeferre Prouvaire Feuilly Courfeyrac Bahorel Bossuet Joly Grantaire MotherPlutarch GueulemerBabet Claquesous Montparnasse Toussaint Child1Child2 Brujon MmeHucheloup Nathalie Villa-Vialaneix | Graph mining 2 4/48
  7. 7. Examples are made with... the toy example “Les Misérables” (co-appearance network in Hugo’s novel) software and especially the R package igraph Nathalie Villa-Vialaneix | Graph mining 2 4/48
  8. 8. Examples are made with... the toy example “Les Misérables” (co-appearance network in Hugo’s novel) software and especially the R package igraph the full script and the dataset is available on my website at: http://www.nathalievilla.org/teaching/toconet.html Nathalie Villa-Vialaneix | Graph mining 2 4/48
  9. 9. Basic description of the graph lesmis ## IGRAPH U--- 77 254 -- ## + attr: layout (g/n), id (v/n), label (v/c), value (e/n) ## + edges: ## [1] 1-- 2 1-- 3 1-- 4 3-- 4 1-- 5 1-- 6 1-- 7 1-- 8 1-- 9 1--10 ## [11] 11--12 4--12 3--12 1--12 12--13 12--14 12--15 12--16 17--18 17--19 ## [21] 18--19 17--20 18--20 19--20 17--21 18--21 19--21 20--21 17--22 18--22 ## [31] 19--22 20--22 21--22 17--23 18--23 19--23 20--23 21--23 22--23 17--24 ## [41] 18--24 19--24 20--24 21--24 22--24 23--24 13--24 12--24 24--25 12--25 ## [51] 25--26 24--26 12--26 25--27 12--27 17--27 26--27 12--28 24--28 26--28 ## [61] 25--28 27--28 12--29 28--29 24--30 28--30 12--30 24--31 31--32 12--32 ## [71] 24--32 28--32 12--33 12--34 28--34 12--35 30--35 12--36 35--36 30--36 ## + ... omitted several edges U--- means: Undirected, not Named (no name attribute for the vertices), not Weighted (no weight attribute for the edges) and not Bipartite Nathalie Villa-Vialaneix | Graph mining 2 5/48
  10. 10. System information ## R version 3.2.5 (2016-04-14) ## Platform: x86_64-pc-linux-gnu (64-bit) ## Running under: Ubuntu 14.04.4 LTS ## ## locale: ## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C ## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8 ## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8 ## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C ## [9] LC_ADDRESS=C LC_TELEPHONE=C ## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C ## ## attached base packages: ## [1] stats graphics grDevices utils datasets methods base ## ## other attached packages: ## [1] igraph_1.0.1 knitr_1.12.3 ## ## loaded via a namespace (and not attached): ## [1] magrittr_1.5 formatR_1.3 tools_3.2.5 stringi_1.0-1 ## [5] highr_0.5.1 stringr_1.0.0 evaluate_0.8.3 Nathalie Villa-Vialaneix | Graph mining 2 6/48
  11. 11. Outline Numerical characteristics Clustering Modularity optimization Spectral clustering Model based clustering Nathalie Villa-Vialaneix | Graph mining 2 7/48
  12. 12. Sketch of this section Issue at stake: a graph is given Nathalie Villa-Vialaneix | Graph mining 2 8/48
  13. 13. Sketch of this section Issue at stake: a graph is given numerical characteristics describing the graph, the nodes, are a standard approach to describe it Nathalie Villa-Vialaneix | Graph mining 2 8/48
  14. 14. Sketch of this section Issue at stake: a graph is given numerical characteristics describing the graph, the nodes, are a standard approach to describe it how to know that the observed value are unexpected according to a so-called “null model”? Nathalie Villa-Vialaneix | Graph mining 2 8/48
  15. 15. Standard (global) characteristics density: |E| n(n−1)/2 graph.density number of triangles: triangles (see also motifs) transitivity: number of triangles divided by the number of triplets with at least two edges transitivity diameter: length of the longest shortest paths between two nodes diameter radius: minimal length, over all vertices in the graph, of the longest shortest path linking this vertex to another vertex radius girth: length of the shortest circle in the graph girth cohesion: minimum number of vertices to remove to disconnect the graph Nathalie Villa-Vialaneix | Graph mining 2 9/48
  16. 16. Standard (global) characteristics for “Les misérables” graph.density(lesmis); triangles(lesmis); length(triangles(lesmis))/3 ## [1] 0.08680793 ## + 1401/77 vertices: ## [1] 12 1 3 12 1 4 12 3 4 12 24 32 12 24 13 12 24 25 12 24 30 12 25 ## [24] 71 12 25 70 12 25 69 12 25 27 12 26 24 12 26 25 12 26 27 12 26 72 12 ## [47] 26 71 12 26 70 12 26 69 12 27 73 12 27 52 12 27 50 12 27 44 12 28 73 ## [70] 12 28 24 12 28 25 12 28 26 12 28 27 12 28 29 12 28 30 12 28 32 12 28 ## [93] 34 12 28 44 12 28 72 12 28 59 12 28 69 12 28 70 12 28 71 12 29 45 12 ## [116] 30 39 12 30 38 12 30 37 12 30 35 12 30 36 12 35 39 12 35 38 12 35 36 ## [139] 12 35 37 12 36 39 12 36 38 12 36 37 12 37 39 12 37 38 12 38 39 12 49 ## [162] 26 12 49 28 12 49 56 12 49 59 12 49 65 12 49 69 12 49 70 12 49 72 12 ## [185] 50 52 12 56 26 12 56 27 12 56 65 12 56 50 12 56 52 12 56 59 12 59 71 ## [208] 12 59 65 12 69 72 12 69 71 12 69 70 12 70 72 12 70 71 12 71 72 49 26 ## + ... omitted several vertices ## [1] 467 transitivity(lesmis); diameter(lesmis); radius(lesmis); girth(lesmis) ## [1] 0.4989316 ## [1] 5 ## [1] 3 ## $girth ## [1] 3 ## ## $circle ## + 3/77 vertices: ## [1] 3 1 4 Nathalie Villa-Vialaneix | Graph mining 2 10/48
  17. 17. Comparison with random graphs... Erdos-Renyi model with the same number of nodes and the same number of edges than the original graph (uniform probability to observe an edge between two given nodes) Nathalie Villa-Vialaneix | Graph mining 2 11/48
  18. 18. Comparison with random graphs... Erdos-Renyi model with the same number of nodes and the same number of edges than the original graph (uniform probability to observe an edge between two given nodes) Method: compare the observed values with those of a large number of randomly generated random graphs (with no loop, only connected graphs are kept) sample_gnm(vcount(lesmis), ecount(lesmis)) Nathalie Villa-Vialaneix | Graph mining 2 11/48
  19. 19. Results of the comparison with random graphs... For B = 500 graphs (only connected graphs are kept), we have: ## density triangles transitivity diameter ## Min. :0.08681 Min. :31.00 Min. :0.05834 Min. :4.000 ## 1st Qu.:0.08681 1st Qu.:43.00 1st Qu.:0.07907 1st Qu.:4.000 ## Median :0.08681 Median :47.00 Median :0.08701 Median :5.000 ## Mean :0.08681 Mean :47.55 Mean :0.08660 Mean :4.627 ## 3rd Qu.:0.08681 3rd Qu.:52.00 3rd Qu.:0.09415 3rd Qu.:5.000 ## Max. :0.08681 Max. :67.00 Max. :0.11793 Max. :6.000 ## radius girth cohesion ## Min. :3.000 Min. :3 Min. :1.000 ## 1st Qu.:3.000 1st Qu.:3 1st Qu.:1.000 ## Median :3.000 Median :3 Median :2.000 ## Mean :3.004 Mean :3 Mean :1.599 ## 3rd Qu.:3.000 3rd Qu.:3 3rd Qu.:2.000 ## Max. :4.000 Max. :3 Max. :3.000 compared to: 0.0868079, 467, 0.4989316, 5, 3, 3, 1 ⇒ all values are standard except for: the number of triangles and the transitivity which are larger: local connectivity is strongest than expected in Erdos-Renyi random graphs the cohesion which is in the lowest values of what is expected in Erdos-Renyi random graphs: this again indicates a strongest local connectivity Nathalie Villa-Vialaneix | Graph mining 2 12/48
  20. 20. Standard (local) characteristics ... for the vertex xi: degree: {xj : (xi, xj) ∈ E, j i} degree (or strength for the weighted version, j i wij) betweenness (or centrality): number of shortest paths between any pair of vertices in the graph which pass through xi betweenness eccentricity: maximal length of all the shortest paths going from xi to any other vertex in the graph eccentricity closeness (or closeness centrality): 1 j i d(xi,xj) in which d(xi, xj) is the length of the shortest path between xi and xj closeness ...and their distributions among all vertices. Nathalie Villa-Vialaneix | Graph mining 2 13/48
  21. 21. Standard (local) characteristics for “Les misérables” summary(degree(lesmis)) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 1.000 2.000 6.000 6.597 10.000 36.000 summary(betweenness(lesmis)) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.00 0.00 0.00 62.36 22.92 1624.00 summary(eccentricity(lesmis)) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 3.00 4.00 4.00 4.13 5.00 5.00 summary(closeness(lesmis)) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.003378 0.004484 0.005181 0.005123 0.005435 0.008475 Nathalie Villa-Vialaneix | Graph mining 2 14/48
  22. 22. Comparison with random graphs... Erdos-Renyi model with the same number of nodes and the same number of edges than the original graph (uniform probability to observe an edge between two given nodes) Nathalie Villa-Vialaneix | Graph mining 2 15/48
  23. 23. Comparison with random graphs... Erdos-Renyi model with the same number of nodes and the same number of edges than the original graph (uniform probability to observe an edge between two given nodes) Method: compare the observed values (average betweenness and degree) with those of a large number of randomly generated random graphs (with no loop, only connected graphs are kept) sample_gnm(vcount(lesmis), ecount(lesmis)) Nathalie Villa-Vialaneix | Graph mining 2 15/48
  24. 24. Results of the comparison with random graphs... For B = 500 graphs (only connected graphs are kept), we have: ## degree betweenness eccentricity closeness ## Min. :6.597 Min. :54.64 Min. :3.597 Min. :0.005249 ## 1st Qu.:6.597 1st Qu.:55.93 1st Qu.:3.779 1st Qu.:0.005322 ## Median :6.597 Median :56.32 Median :3.857 Median :0.005340 ## Mean :6.597 Mean :56.36 Mean :3.863 Mean :0.005340 ## 3rd Qu.:6.597 3rd Qu.:56.71 3rd Qu.:3.909 3rd Qu.:0.005361 ## Max. :6.597 Max. :58.79 Max. :4.688 Max. :0.005430 compared to: 6.597, 62.364, 4.13, 0.00512 ⇒ the observed average betweenness is higher and the observed average closeness is smaller for all the randomly generated graphs: this seems to indicate that, in average, shortest paths in the graphs are longer than expected for graphs with uniform distribution of the edges. Nathalie Villa-Vialaneix | Graph mining 2 16/48
  25. 25. Degree distribution for “Les misérables” + + + + + + + + + + +++ ++++ + 0 1 2 3 −4.0−3.5−3.0−2.5−2.0−1.5 log(k) log(P(k)) Estimation of power law fit (left: α = 1.49) with fit_power_law(degree(lesmis) + 1, implementation = "R.mle") Nathalie Villa-Vialaneix | Graph mining 2 17/48
  26. 26. Comparison with random graphs... Scale free model with a parameter for the power law identical to the one previously estimated and the same number of nodes. Barabási and Albert model is used with a number of edges added at each step which is chosen so that the final number of edges resembles that of the original graph (3 edges, which gives 225 edges in the final graph, compared to 254) P(degree = k) = k−α Nathalie Villa-Vialaneix | Graph mining 2 18/48
  27. 27. Comparison with random graphs... Scale free model with a parameter for the power law identical to the one previously estimated and the same number of nodes. Barabási and Albert model is used with a number of edges added at each step which is chosen so that the final number of edges resembles that of the original graph (3 edges, which gives 225 edges in the final graph, compared to 254) P(degree = k) = k−α Method: compare the observed values with those of a large number of randomly generated random graphs sample_pa(vcount(lesmis), m = 3, power = ..., directed = FALSE) Nathalie Villa-Vialaneix | Graph mining 2 18/48
  28. 28. Results of the comparison with random graphs... For B = 500 graphs, we have: ## density triangles transitivity diameter ## Min. :0.0769 Min. : 72 Min. :0.1075 Min. :3.000 ## 1st Qu.:0.0769 1st Qu.:102 1st Qu.:0.1250 1st Qu.:4.000 ## Median :0.0769 Median :112 Median :0.1307 Median :4.000 ## Mean :0.0769 Mean :113 Mean :0.1303 Mean :3.988 ## 3rd Qu.:0.0769 3rd Qu.:124 3rd Qu.:0.1359 3rd Qu.:4.000 ## Max. :0.0769 Max. :153 Max. :0.1530 Max. :5.000 ## radius girth cohesion degree betweenness ## Min. :2.000 Min. :3 Min. :3 Min. :5.844 Min. :41.86 ## 1st Qu.:2.000 1st Qu.:3 1st Qu.:3 1st Qu.:5.844 1st Qu.:47.88 ## Median :2.000 Median :3 Median :3 Median :5.844 Median :49.55 ## Mean :2.314 Mean :3 Mean :3 Mean :5.844 Mean :49.35 ## 3rd Qu.:3.000 3rd Qu.:3 3rd Qu.:3 3rd Qu.:5.844 3rd Qu.:50.97 ## Max. :3.000 Max. :3 Max. :3 Max. :5.844 Max. :55.73 ## eccentricity closeness ## Min. :2.935 Min. :0.005407 ## 1st Qu.:3.130 1st Qu.:0.005695 ## Median :3.221 Median :0.005788 ## Mean :3.234 Mean :0.005805 ## 3rd Qu.:3.325 3rd Qu.:0.005901 ## Max. :3.662 Max. :0.006334 compared to: 0.087, 467, 0.499, 5, 3, 3, 1, 6.597, 62.364, 4.13, 0.00512 ⇒ the number of triangles, the transitivity, the radius, the average degree, the average betweenness and the eccentricity are larger than in power law graphs with power 1.495, whereas the cohesion and the closeness are smaller. Nathalie Villa-Vialaneix | Graph mining 2 19/48
  29. 29. Limits of the previous approaches Until now, we have compared the real graph to graphs randomly generated according to a given random model but: this approach only gives information about global characteristics of the observed graph; none of the distributions of the current characteristics is preserved during the process, especially not the degree distribution which is central for controlling local/global connectivity, counts of specific patterns... Nathalie Villa-Vialaneix | Graph mining 2 20/48
  30. 30. A null model closer to the real graph... Sketch of statistical tests on graphs 1. sample at random within the set of graphs with the same degree distribution than the observed graph (B times) 2. compute a numerical statistics for each of these randomly generated graphs 3. comparing the observed value of the statistics and its distribution over the random graphs, a p-value can be derived (for B large enough) Nathalie Villa-Vialaneix | Graph mining 2 21/48
  31. 31. A null model closer to the real graph... Sketch of statistical tests on graphs 1. sample at random within the set of graphs with the same degree distribution than the observed graph (B times) 2. compute a numerical statistics for each of these randomly generated graphs 3. comparing the observed value of the statistics and its distribution over the random graphs, a p-value can be derived (for B large enough) Two main approaches to sample at random with fixed degrees: configuration model [Bender and Canfield, 1978] permutation approach [Rao et al., 1996, Roberts Jr., 2000] Nathalie Villa-Vialaneix | Graph mining 2 21/48
  32. 32. Sampling at random within the set of graphs with a given degree distribution Aim: all graphs can exhaustively be sampled all graphs have the same probability to be sampled ⇒ MCMC approach Nathalie Villa-Vialaneix | Graph mining 2 22/48
  33. 33. Sampling at random within the set of graphs with a given degree distribution Aim: all graphs can exhaustively be sampled all graphs have the same probability to be sampled ⇒ MCMC approach Method: 1: Start from the observed graph G 2: for t = 1 → T do 3: Select uniformly at random two edges e1 = (x1 i , x1 j ) and e2 = (x2 i , x2 j ) ∈ E 4: E ← E {e1 , e2 } ∪ {e1 s , e2 s } with e1 s = (x1 i , x2 j ) and e2 s = (x2 i , x1 j ) 5: if G = (V, E ) is simple and connected then 6: G ← G 7: end if 8: end for 9: return G Nathalie Villa-Vialaneix | Graph mining 2 22/48
  34. 34. In practice... This method is used in [Milo et al., 2004] with T = 100. It can be performed using rewire(lesmis, keeping_degseq(n = 100)) Number of triangles Frequency 200 300 400 020406080100120 transitivity Frequency 0.25 0.35 0.45 020406080100 Nathalie Villa-Vialaneix | Graph mining 2 23/48
  35. 35. In practice... for the vertex characteristics Find a(n empirical) p-value for all vertices which indicates if its betweenness is higher or lower than expected with respect to its degree: ratio of random graphs for which the observed betweenness is higher (resp. lower) than 95% of the betweennesses for the corresponding vertex in random graphs. Myriel Valjean Listolier Fameuil Blacheville Favourite Dahlia Zephine Fantine Judge Champmathieu Brevet Chenildieu Cochepaille LtGillenormand Marius Combeferre Prouvaire FeuillyCourfeyrac BahorelJoly Grantaire GueulemerBabet Claquesous MontparnasseBrujon MmeHucheloup Nathalie Villa-Vialaneix | Graph mining 2 24/48
  36. 36. More on random graphs generation Sometimes, one wants to compare the observed graph with a more sophisticated (constrained) null model (taking into account some additional information on edges or nodes for instance): This can be achieved using the same principle and throwing away the random graphs which do not satisfy the constrains. Nathalie Villa-Vialaneix | Graph mining 2 25/48
  37. 37. More on random graphs generation Sometimes, one wants to compare the observed graph with a more sophisticated (constrained) null model (taking into account some additional information on edges or nodes for instance): This can be achieved using the same principle and throwing away the random graphs which do not satisfy the constrains. Warning: The more sophisticated the model is, the more costly the simulation would be. For instance, only removing graphs with multiple edges and graphs which are not connected leads to throw away 47 simulations over 500. Nathalie Villa-Vialaneix | Graph mining 2 25/48
  38. 38. More on random graphs generation Sometimes, one wants to compare the observed graph with a more sophisticated (constrained) null model (taking into account some additional information on edges or nodes for instance): This can be achieved using the same principle and throwing away the random graphs which do not satisfy the constrains. Warning: The more sophisticated the model is, the more costly the simulation would be. For instance, only removing graphs with multiple edges and graphs which are not connected leads to throw away 47 simulations over 500. Possible solution: [Tabourier and Cointet, 2011] use multiple edge switching to improve the simulations such simulations. Nathalie Villa-Vialaneix | Graph mining 2 25/48
  39. 39. Outline Numerical characteristics Clustering Modularity optimization Spectral clustering Model based clustering Nathalie Villa-Vialaneix | Graph mining 2 26/48
  40. 40. Sketch of this section Issue at stake: short overview of different types of methods for vertex clustering only simple clustering (although some methods for overlapping clustering, clustering according to vertex/edge attributes, clustering of bipartite graphs... also exist) statistical relevance and comparison of clustering results Nathalie Villa-Vialaneix | Graph mining 2 27/48
  41. 41. A short overview of vertex clustering Purpose: Find communities or modules (i.e., groups of vertices) st vertices inside the community are strongly connected whereas vertices between two communities are slightly connected. Nathalie Villa-Vialaneix | Graph mining 2 28/48
  42. 42. A short overview of vertex clustering Purpose: Find communities or modules (i.e., groups of vertices) st vertices inside the community are strongly connected whereas vertices between two communities are slightly connected. Some approaches to perform such task: optimizing a given criterion (e.g., modularity maximization) spectral clustering model based clustering ... (see [Fortunato and Barthélémy, 2007, Schaeffer, 2007, Brohée and van Helden, 2006]) Nathalie Villa-Vialaneix | Graph mining 2 28/48
  43. 43. Clustering based on criterion optimization “Cut” criteria: Given a number of clusters, K, find the partition of V, C1, . . . , CK such that it solves the mincut problem, i.e., it minimizes cut(A1, . . . , AK ) = 1 2 K k=1 xi∈Ak , xj Ak wij Nathalie Villa-Vialaneix | Graph mining 2 29/48
  44. 44. Clustering based on criterion optimization “Cut” criteria: Given a number of clusters, K, find the partition of V, C1, . . . , CK such that it solves the mincut problem, i.e., it minimizes cut(A1, . . . , AK ) = 1 2 K k=1 xi∈Ak , xj Ak wij Problem: The mincut problem often separates individual vertices from the rest of the graph. Nathalie Villa-Vialaneix | Graph mining 2 29/48
  45. 45. Clustering based on criterion optimization “Cut” criteria: Given a number of clusters, K, find the partition of V, C1, . . . , CK such that it solves the “RatioCut” problem, i.e., it minimizes RatioCut(A1, . . . , AK ) = 1 2 K k=1 xi∈Ak , xj Ak wij |Ak | (forces larger communities than the mincut problem). Nathalie Villa-Vialaneix | Graph mining 2 29/48
  46. 46. Clustering based on criterion optimization “Cut” criteria: Given a number of clusters, K, find the partition of V, C1, . . . , CK such that it solves the “NCut” problem, i.e., it minimizes NCut(A1, . . . , AK ) = 1 2 K k=1 xi∈Ak , xj Ak wij Vol(Ak ) in which Vol(Ak ) = xi, xj∈Ak wij (also forces larger communities than the mincut problem). Nathalie Villa-Vialaneix | Graph mining 2 29/48
  47. 47. Clustering based on criterion optimization “Cut” criteria “Modularity” criterion [Newman and Girvan, 2004]: Given a number of clusters, K, find the partition of V, C1, . . . , CK which maximizes Q(A1, . . . , Ak ) = 1 2m K k=1 xi, xj∈Ck (wij − Pij) with Pij: weight of a “null model” (graph with the same degree distribution but no preferential attachment): Pij = didj 2m with di = 1 2 j i wij. Nathalie Villa-Vialaneix | Graph mining 2 29/48
  48. 48. Advantages and drawbacks mincut is not adapted to vertex clustering in practice (clusters with isolated vertices) the other three methods are NP hard to solve... Nathalie Villa-Vialaneix | Graph mining 2 30/48
  49. 49. Advantages and drawbacks mincut is not adapted to vertex clustering in practice (clusters with isolated vertices) the other three methods are NP hard to solve... the modularity takes into account asymmetry in degree distribution by correcting the importance of a vertex by its degree: it is often more adapted to real life graphs [Fortunato and Barthélémy, 2007] showed that modularity has a small resolution issue. [Bickel and Chen, 2009] gave conditions for consistency of the clusters obtained by modularity optimization in Stochastic Block Models (SBM). Nathalie Villa-Vialaneix | Graph mining 2 30/48
  50. 50. Advantages and drawbacks mincut is not adapted to vertex clustering in practice (clusters with isolated vertices) the other three methods are NP hard to solve... the modularity takes into account asymmetry in degree distribution by correcting the importance of a vertex by its degree: it is often more adapted to real life graphs [Fortunato and Barthélémy, 2007] showed that modularity has a small resolution issue. [Bickel and Chen, 2009] gave conditions for consistency of the clusters obtained by modularity optimization in Stochastic Block Models (SBM). Remark: Relaxation of RatioCut problem and NCut problem gives spectral clustering. Modularity optimization is often solved by approximation methods. Nathalie Villa-Vialaneix | Graph mining 2 30/48
  51. 51. A short description of approximation methods for modularity optimization simple greedy algorithms ([Newman, 2004] and [Clauset et al., 2004] for a fast version): hierarchical clustering which merges pairs of vertices with the highest contribution to modularity cluster_fast_greedy Nathalie Villa-Vialaneix | Graph mining 2 31/48
  52. 52. A short description of approximation methods for modularity optimization simple greedy algorithms ([Newman, 2004] and [Clauset et al., 2004] for a fast version): hierarchical clustering which merges pairs of vertices with the highest contribution to modularity cluster_fast_greedy multi-level greedy algorithms ([Blondel et al., 2008], also known as “Louvain algorithm” and [Noack and Rotta, 2009] for an improved version): hierarchical approach in which vertices are sometimes re-assigned to a different community in a greedy way cluster_louvain Nathalie Villa-Vialaneix | Graph mining 2 31/48
  53. 53. A short description of approximation methods for modularity optimization simple greedy algorithms ([Newman, 2004] and [Clauset et al., 2004] for a fast version): hierarchical clustering which merges pairs of vertices with the highest contribution to modularity cluster_fast_greedy multi-level greedy algorithms ([Blondel et al., 2008], also known as “Louvain algorithm” and [Noack and Rotta, 2009] for an improved version): hierarchical approach in which vertices are sometimes re-assigned to a different community in a greedy way cluster_louvain simulated annealing ([Reichardt and Bornholdt, 2006] uses a spin-glass model which, in some cases, is equivalent to modularity maximization) cluster_spinglass(..., gamma = 1, update.rule = "config") Nathalie Villa-Vialaneix | Graph mining 2 31/48
  54. 54. A short description of approximation methods for modularity optimization simple greedy algorithms ([Newman, 2004] and [Clauset et al., 2004] for a fast version): hierarchical clustering which merges pairs of vertices with the highest contribution to modularity cluster_fast_greedy multi-level greedy algorithms ([Blondel et al., 2008], also known as “Louvain algorithm” and [Noack and Rotta, 2009] for an improved version): hierarchical approach in which vertices are sometimes re-assigned to a different community in a greedy way cluster_louvain simulated annealing ([Reichardt and Bornholdt, 2006] uses a spin-glass model which, in some cases, is equivalent to modularity maximization) cluster_spinglass(..., gamma = 1, update.rule = "config") ...to be compared (when usable) with the exact optimization cluster_optimal. Nathalie Villa-Vialaneix | Graph mining 2 31/48
  55. 55. Examples res_time <- cbind( system.time(res_hierarchical <- cluster_fast_greedy(lesmis)), system.time(res_multilevel <- cluster_louvain(lesmis)), system.time(res_annealing <- cluster_spinglass(lesmis)), system.time(res_exact <- cluster_optimal(lesmis)) )[3, ] ## hierarchical multilevel annealing exact ## 0.002 0.002 1.907 21.656 Nathalie Villa-Vialaneix | Graph mining 2 32/48
  56. 56. Computational time (greedy approaches) Difference (computational time) between the first two approaches (100 evaluations): library(microbenchmark) res_micro <- microbenchmark(cluster_fast_greedy(lesmis), cluster_louvain(lesmis)) cluster_fast_greedy(lesmis) cluster_louvain(lesmis) 1000 Time [microseconds] Nathalie Villa-Vialaneix | Graph mining 2 33/48
  57. 57. Accuracy of the clustering hierarchical − 0.5006 − 5 multilevel − 0.5556 − 6 simulated annealing − 0.5596 − 7 exact − 0.56 − 6 Nathalie Villa-Vialaneix | Graph mining 2 34/48
  58. 58. Assessing the relevance of a clustering Given a graph, the modularity optimization will always return a clustering: how to know that this clustering is meaningful? (i.e., that its modularity is large) Nathalie Villa-Vialaneix | Graph mining 2 35/48
  59. 59. Assessing the relevance of a clustering Given a graph, the modularity optimization will always return a clustering: how to know that this clustering is meaningful? (i.e., that its modularity is large) Similarly as previously, compare the maximum modularity to the maximum modularity over a large number of randomly generated graphs (with same degree sequence). Modularity Frequency 0.30 0.35 0.40 0.45 0.50 0.55 020406080 Nathalie Villa-Vialaneix | Graph mining 2 35/48
  60. 60. Relation between RatioCut and Laplacian [von Luxburg, 2007] shows that minimizing RatioCut(A1, A2) = 1 2 2 k=1 xi∈Ak , xj Ak wij |Ak | is equivalent to the following constrained problem: min A1, ,A2 v Lv st v ⊥ 1n and v = √ n for v the vector of Rn obtained from the partition by: vi = (|A2|)/|A1| if vi ∈ A1 − |A1|/(|A2|) otherwise. and L is the Laplacian of the graph, n × n-matrix with entries: Lij = −wij if i j di = j i wij otherwise . Nathalie Villa-Vialaneix | Graph mining 2 36/48
  61. 61. ... and more remarks this is a discrete (since v can only have two values) and NP-hard problem; Nathalie Villa-Vialaneix | Graph mining 2 37/48
  62. 62. ... and more remarks this is a discrete (since v can only have two values) and NP-hard problem; the same relation holds between NCut problem and normalized Laplacian D−1/2 LD−1/2 is which D = Diag(d1, . . . , dn); Nathalie Villa-Vialaneix | Graph mining 2 37/48
  63. 63. ... and more remarks this is a discrete (since v can only have two values) and NP-hard problem; the same relation holds between NCut problem and normalized Laplacian D−1/2 LD−1/2 is which D = Diag(d1, . . . , dn); a generalization of these results exist for K > 2. Nathalie Villa-Vialaneix | Graph mining 2 37/48
  64. 64. Some properties of the Laplacian Relations with the graph structure: 1 2 3 4 5 has a null space spanned by the vectors   1 1 1 0 0   and   0 0 0 1 1   . Nathalie Villa-Vialaneix | Graph mining 2 38/48
  65. 65. Some properties of the Laplacian Relations with the graph structure: the vector 1n spans the null space for connected graphs. Nathalie Villa-Vialaneix | Graph mining 2 38/48
  66. 66. Some properties of the Laplacian Relations with the graph structure: Random walk point of view: If we consider a random walk on the graph with probability to jump from one node to the other equal to wij di then NCut(A1, A2) is interpreted as the probability to go from A1 to A2 or from A2 to A1. Nathalie Villa-Vialaneix | Graph mining 2 38/48
  67. 67. Some properties of the Laplacian Relations with the graph structure: Random walk point of view: If we consider a random walk on the graph with probability to jump from one node to the other equal to wij di then the average time to go from one node to another (commute time) is given by L+ [Fouss et al., 2007]. Nathalie Villa-Vialaneix | Graph mining 2 38/48
  68. 68. Spectral clustering: relaxing the constrains K has to be given. Solving minA1, ,A2 Tr(U LU) for a K × n matrix U st U U = 1: 1. Compute the first K eigenvectors of L, u1 , . . . , uK and write U = (u1 , . . . , uK ) (a n × K matrix). Nathalie Villa-Vialaneix | Graph mining 2 39/48
  69. 69. Spectral clustering: relaxing the constrains K has to be given. Solving minA1, ,A2 Tr(U LU) for a K × n matrix U st U U = 1: 1. Compute the first K eigenvectors of L, u1 , . . . , uK and write U = (u1 , . . . , uK ) (a n × K matrix). 2. For i = 1, . . . , n, denote ui ∈ RK the i-th row of U. Cluster the points (ui)i=1,...,n using a clustering algorithm (e.g., k-means). Nathalie Villa-Vialaneix | Graph mining 2 39/48
  70. 70. Spectral clustering: relaxing the constrains K has to be given. Solving minA1, ,A2 Tr(U LU) for a K × n matrix U st U U = 1: 1. Compute the first K eigenvectors of L, u1 , . . . , uK and write U = (u1 , . . . , uK ) (a n × K matrix). 2. For i = 1, . . . , n, denote ui ∈ RK the i-th row of U. Cluster the points (ui)i=1,...,n using a clustering algorithm (e.g., k-means). embed_laplacian_matrix(..., no = ..., which = "sa", scaled = ...) et kmeans(..., centers = ..., nstart = 10) Nathalie Villa-Vialaneix | Graph mining 2 39/48
  71. 71. Spectral clustering in practice res_time_spec <- system.time({ spec_embed <- embed_laplacian_matrix(lesmis, no = 6, which = "sa", scaled = FALSE) res_spectral <- kmeans(spec_embed$X[ ,-1], centers = 6, nstart = 1) })[3] res_time_spec ## elapsed ## 0.017 Time is between the greedy approaches for modularity optimization and simulated annealing for modularity optimization. Nathalie Villa-Vialaneix | Graph mining 2 40/48
  72. 72. Accuracy of the clustering spectral clustering − 0.4461 − 6 exact − 0.56 − 6 Modularity is smaller (as expected) and clusters tend to be more unbalanced. An empirical comparison between the performance of spectral clustering and modularity optimization is provided in [Bickel and Chen, 2009]. [Lei and Rinaldo, 2015] gives conditions for the consistency of spectral clustering in stochastic block models. Nathalie Villa-Vialaneix | Graph mining 2 41/48
  73. 73. A mixture model for networks [Snijders and Nowicki, 1997]: The observed network G is supposed to be the realization of some random graph model in which vertices are organized in groups. description of the model vertices xi belong to an unknow class in {C1, ..., CK } (K is given) ⇒ latent (unobserved) variables Zi ∼ M(1, α = (α1, . . . , αK )) in which αk is the probability that xi belongs to Ck Nathalie Villa-Vialaneix | Graph mining 2 42/48
  74. 74. A mixture model for networks [Snijders and Nowicki, 1997]: The observed network G is supposed to be the realization of some random graph model in which vertices are organized in groups. description of the model vertices xi belong to an unknow class in {C1, ..., CK } (K is given) ⇒ latent (unobserved) variables Zi ∼ M(1, α = (α1, . . . , αK )) in which αk is the probability that xi belongs to Ck given the class membership, the probabilities to have an edge between xi and xj are all independant and obtained by: wij = 1|Zik Zik = 1 ∼ L(., πkk ) for a given distribution L Nathalie Villa-Vialaneix | Graph mining 2 42/48
  75. 75. A mixture model for networks [Snijders and Nowicki, 1997]: The observed network G is supposed to be the realization of some random graph model in which vertices are organized in groups. description of the model vertices xi belong to an unknow class in {C1, ..., CK } (K is given) ⇒ latent (unobserved) variables Zi ∼ M(1, α = (α1, . . . , αK )) in which αk is the probability that xi belongs to Ck given the class membership, the probabilities to have an edge between xi and xj are all independant and obtained by: typically, the Bernouilli distribution with probability πkk with πkk = p1 if k = k p0 if k k for p1 > p0. Nathalie Villa-Vialaneix | Graph mining 2 42/48
  76. 76. Basic principle for using SBM 1. assignments of vertices to groups; 2. parameter estimation ((αk )k and (πkk )k,k ); 3. estimation of the number of groups. Nathalie Villa-Vialaneix | Graph mining 2 43/48
  77. 77. Basic principle for using SBM 1. assignments of vertices to groups; 2. parameter estimation ((αk )k and (πkk )k,k ); 3. estimation of the number of groups. Estimation is made by Bayesian or frequentist approaches and Variational EM (see e.g., [Daudin et al., 2008] for the more computationally efficient frequentist approach). Number of nodes can be chosen using ICL [Biernacki et al., 2000]. Nathalie Villa-Vialaneix | Graph mining 2 43/48
  78. 78. Basic principle for using SBM 1. assignments of vertices to groups; 2. parameter estimation ((αk )k and (πkk )k,k ); 3. estimation of the number of groups. Estimation is made by Bayesian or frequentist approaches and Variational EM (see e.g., [Daudin et al., 2008] for the more computationally efficient frequentist approach). Number of nodes can be chosen using ICL [Biernacki et al., 2000]. All this is implemented in the package blockmodels [Léger, 2016]. BM_bernoulli("SBM_sym", as_adjacency_matrix(..., sparse = FALSE)) BM_bernoulli$estimate() Nathalie Villa-Vialaneix | Graph mining 2 43/48
  79. 79. SBM in practice library(blockmodels) res_time_sbm <- system.time({ res_sbm <- BM_bernoulli("SBM_sym", as_adjacency_matrix(lesmis, sparse = FALSE)) res_sbm$estimate() })[3] res_time_sbm ## elapsed ## 1.821 opt_K <- which.max(res_sbm$ICL) opt_K ## [1] 6 sbm_clust <- apply(res_sbm$memberships[[opt_K]]$Z, 1, which.max) Nathalie Villa-Vialaneix | Graph mining 2 44/48
  80. 80. Accuracy of the clustering SBM clustering − 0.4556 − 6 exact − 0.56 − 6 Modularity is smaller (as expected) but groups can be interpreted by being sets of vertices with similar connecting patterns. Nathalie Villa-Vialaneix | Graph mining 2 45/48
  81. 81. Comparing clustering Various metrics ((di)similarities) exist to compare clustering, among which: Rand Index [Rand, 1971] compare(..., method = "rand"): number of agreements between the two clusterings n Normalized Mutual Information [Danon et al., 2005] compare(..., method = "nmi") K1 k=1 K2 k =1 nkk n log   nkk n n1 k n2 k   in which Kj is the number of clusters in clustering j, n j k is the number of vertices classified into cluster k for clustering j and nkk is the number of vertices classified into cluster k for clustering 1 and cluster k for clustering 2. The similarity is normalized so that it is between 0 and 1 (1 is for a perfect match). Nathalie Villa-Vialaneix | Graph mining 2 46/48
  82. 82. How do clusterings relate? Method: 1. compute a dissimilarity based on Rand index or NMI (1 − value) 2. perform clustering (of the results of vertex clustering) using hierarchical clustering hclust Nathalie Villa-Vialaneix | Graph mining 2 47/48
  83. 83. How do clusterings relate? sbm spectral hierarchical multilevel annealing exact 0.00.10.20.3 Rand index hclust (*, "complete") as.dist(compare_rand) Height sbm spectral hierarchical multilevel annealing exact 0.00.20.40.6 NMI hclust (*, "complete") as.dist(compare_nmi) Height Nathalie Villa-Vialaneix | Graph mining 2 47/48
  84. 84. Any question? Nathalie Villa-Vialaneix | Graph mining 2 48/48
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  86. 86. In Proceedings of the National Academy of Sciences, volume 104, pages 36–41. doi:10.1073/pnas.0605965104; URL: http://www.pnas.org/content/104/1/36.abstract. Fouss, F., Pirotte, A., Renders, J., and Saerens, M. (2007). Random-walk computation of similarities between nodes of a graph, with application to collaborative recommendation. IEEE Transactions on Knowledge and Data Engineering, 19(3):355–369. Léger, J. (2016). Blockmodels: a R-package for estimating in LBM and SBM, with many pdf, with or without covariates. Preprint arXiv 1602.07587v1. Submitted for publication. Lei, J. and Rinaldo, A. (2015). Consistency of spectral clustering in stochastic block models. The Annals of Statistics, 43(1):215–237. Milo, R., Kashtan, N., Itzkovitz, S., Newman, M., and Alon, U. (2004). On the uniform generation of random graphs with prescribed degree sequences. eprint arXiv: cond-mat/0312028v2. Newman, M. (2004). Fast algorithm for detecting community structure in networks. Physical Review E, 69:066133. Newman, M. and Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review, E, 69:026113. Noack, A. and Rotta, R. (2009). Multi-level algorithms for modularity clustering. In SEA 2009: Proceedings of the 8th International Symposium on Experimental Algorithms, pages 257–268, Berlin, Heidelberg. Springer-Verlag. Rand, W. (1971). Nathalie Villa-Vialaneix | Graph mining 2 48/48
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