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- 1. Community DetectionIlio Catallo, catallo@elet.polimi.itPolitecnico di Milano
- 2. Outline¡ Communities and Partitions ¡ What is a community? ¡ What is a partition?¡ Partitioning algorithms ¡ Kerninghan and Lin, 1970 ¡ Newman and Girvan, 2004 ¡ Bagrow and Bollt, 2008¡ Assess the quality of good partitions ¡ The impossibility theorem ¡ Quality functions
- 3. Communities and Partitions
- 4. 4What is a community?Intuition¡ Community: a set of tightly connected nodes¡ Examples: ¡ People with common interests ¡ Papers on the same topics ¡ Scholars working on the same field
- 5. 5What is a community?Local definitions (1/3)clique (complete subgraph) ¡ Too strict definition (what to do if just one link is missing?) ¡ Cliques are hard to find (exponential complexity in the graph size)
- 6. 6 What is a community? Local definitions (2/3) Strong community: subgraph V ⊆ G such that each vertex has more connection within the community than with the rest of the graph in out ki (V ) > ki (V ) 8i 2 VThe number of edges The number ofconnecting node i to connections towardother nodes belonging nodes in the rest of theto V graph
- 7. 7 What is a community? Local definitions (3/3) ¡ Strong communitiy definition is too strict ¡ Unrealistic in many real cases ¡ Weak communities: subgraph V ⊆ G such that the sum of all degrees within V in greater than the sum of all degrees toward the rest of the network ¡ A strong community is also weak, while the converse is not generally true P in P out i2V ki (V )> i2V ki (V ) number of edges connectingnumber of edges connecting nodes in V toward nodes in thenodes in V to other nodes rest of the graphbelonging to V
- 8. 8What is a community?Global definitions (1/2)¡ Idea: the graph has a community structure if it is different from the random graph¡ Random graph: graph such that each pair of vertices is connected with equal probability p, independently on the other pairs ¡ Any two vertices have the same probability to be adjacent ¡ No preferential linking involving
- 9. 9What is a community?Global definitions (2/2)¡ The graph of interest is compared with the null model¡ Null model: a graph which matches the original in some of its structural features, but which is otherwise a random graph ¡ Used as term of comparison to verify whether the graph of interest shows community structures
- 10. 10What is a community?Vertex-based definitions¡ Idea: communities are subgraphs of vertices similar to each other ¡ A measure of similarity needs to be defined¡ If it is possible to embed the vertices in an n- dimensional Euclidian space, possible (dis)similarity measures are: q PN 2 ¡ Euclidian distance dA,B = j (ak bk ) PN 2 ¡ Manhattan distance dA,B = j |(ak bk ) | A·B ¡ Cosine similarity dA,B = kAkkBk¡ With A = (a1, a2, …, aN) and B = (b1, b2, …, bN) vertex feature vectors
- 11. 11What is a community?Vertex-based definitions¡ If it is not possible to embed the vertices in Euclidian space the similarity must be inferred from the adjacency relationships¡ Dissimilarity measure based on structural equivalence: qP dij = k6=i,j (Aik Ajk )2¡ Structural equivalence: two vertices are structural equivalent if they have the same neighbors, even if they are not adjacent themselves ¡ if i and j are structural equivalent then dij = 0
- 12. 12What is a partition?¡ Partition: a division of a graph in clusters, such that each vertex belongs to one cluster¡ If the vertices can be shared among different communities the division is called cover
- 13. 13How many partitions wemay have in a graph?¡ Stirling number of second kind: the number of possible partitions in k clusters of a graph with n vertices ⇢ 1 k = n, k = 1 S(n, k) = kS(n 1, k) + S(n 1, k 1) otherwise¡ Nth Bell number: the total number of possible partitions n X Bn = S(n, k) k=1¡ The nth Bell number is huge, even for relatively small graphs
- 14. Partitioning algorithms
- 15. 15Kernighan and Lin, 1970:Basic concepts (1/2)¡ Given: ¡ A graph G = (N,A) of n vertices of weights wi > 0 ¡ p a positive number s.t. wi ≤ p ¡ C = (cij) the weighted adjacency matrix (cost matrix)¡ A k-way partition 𝚪 of G is a set of non-empty, pairwise disjoint set 𝜐1, …, 𝜐k such that: k [ i =G i=1 The sum of weights of¡ A partition is admissible if: vertices in 𝜐i is less or X equal to p wj p 8i = 1, . . . , k j2 i
- 16. 16Kernighan and Lin, 1970:Basic concepts (2/2)¡ The cost T of a partition 𝚪 is the summation of cij over all i and j such that i and j are in different clusters 5 b cb2 a 1 2 f cf 4 e c 4 3 T ( ) = cb2 + cf 4
- 17. 17Kernighan and Lin, 1970:2-way uniform partitioning prob.¡ 2-way uniform partitioning problem: finding a minimal cost partition of a given graph of 2n vertices (of equal weights) into two subsets of n vertices 5 b cb2 a 1 2 f cf 4 e c 4 3¡ The Kernighan and Lin algorithm is a heuristic for solving the 2-way uniform partitioning problem
- 18. 18Kernighan and Lin, 1970:Basic principle (1/2)¡ Basic principle: starting with any arbitrary partition 𝛤 = {A, B} of N try to decrease the initial cost T by a series of interchanges of elements of A and B¡ When no further improvement is possible, the resulting partition 𝛤’ is locally minimum with respect to the algorithm
- 19. 19Kernighan and Lin, 1970:Basic principle (2/2)¡ Given: ¡ 𝛤* = {A*, B*} is a minimum cost 2-way uniform partition ¡ 𝛤 = {A, B} is a arbitrary 2-way uniform partition¡ There are subsets X⊂A, Y⊂B with |X| = |Y| such that interchanging X and Y produces A* and B* X Y A B A⇤ = A X +Y B⇤ = B Y +X Y X A⇤ B⇤
- 20. 20Kernighan and Lin, 1970:Internal and external cost¡ Let’s define for each a∈A : X ¡ External cost: Ea = cay y2B X ¡ Internal cost: Ia = cax x2A ¡ Cost difference: D a = Ea Ia¡ Similarly, define Eb, Ib, Db for each b∈B
- 21. 21Kernighan and Lin, 1970:Cost reduction¡ Lemma 1: Consider any a∈A, b∈B. If a and b are interchanged, the reduction in cost (i.e., the gain) is g=T T 0 = Da + Db 2cab¡ Lemma 2: Consider any a∈A, b∈B. If a and b are interchanged, the variations in the cost difference for all the other nodes are 0 Dx = Dx + 2cxa 2cxb x ⇥ A {a} 0 Dy = Dy + 2cyb 2cya y ⇥ B {b}
- 22. 22Kernighan and Lin, 1970:The algorithm1. Compute the D values for all elements of N2. A1 A, B1 B; X1 = ;, Y1 = ;; i 13. While i < n Lemma 1 (a) arg maxai 2A,bi 2B gi = Dai + Dbi 2cai bi (b) Xi+1 Xi [ {ai }, Yi+1 Yi [ {bi }; Lemma 2 (c) Ai+1 Ai {ai }, Bi+1 Bi {bi } (d) Recalculate the D values for the elements of Ai+1 , Bi+1 (e) i i+1 Pk4. Choose k to maximize G = i gi k = 1, . . . , n5. If G > 0 then swap Xk , Yk and go back to 1; if G = 0 exit
- 23. 23Newman and Girvan, 2004:Betweenness (1/2)¡ All paths from any two vertices in different communities pass along the few inter-community edges¡ Betweenness: a measure j that favors edges that lie i between communities and disfavors those that lie inside communities Bij ≫ 0
- 24. 24Newman and Girvan, 2004:Betweenness (2/2)¡ Different implementation of betweenness: ¡ Shortest-path betweenness: find the shortest path between all pairs of vertices and count how many run along each edge ¡ Random-walk betweenness: expected number of times that a random walk between a particular pair of vertices will pass down a particular edge and sum over all vertex pairs ¡ Current-flow betweenness: absolute value of current along the edge summed over all source/sink pairs
- 25. 25Newman and Girvan, 2004:Basic principle¡ Algorithm based on a divisive approach¡ Basic principle: removes links with the highest betweenness
- 26. 26Newman and Girvan, 2004:Algorithm1. Calculate betweennes scores for all edges in the network2. Find the edge with the highest score and remove it from the network3. Recalculate betweennes for all remaining edges4. Repeat from step 2
- 27. 27Newman and Girvan, 2004:Dendrogram¡ The output of the algorithms is called dendrogram¡ Cutting the diagram horizontally at some height displays a possible partition of the graph FIG. 2: A hierarchical tree or dendrogram illustrating the type of output generated by the algorithms described here. The circles at the bottom of the ﬁgure represent the indi- FIG. 3 vidual vertices of the network. As we move up the tree the at disc vertices join together to form larger and larger communities, vertice as indicated by the lines, until we reach the top, where all are even w joined together in a single community. Alternatively, we the munity
- 28. 28Bagrow and Bollt, 2008:L-shell¡ L-shell: given a starting vertex i, the l-shell is the set of all the i’s neighbors within a shortest path distance i d≤l¡ Example: 1-shell from starting vertex i
- 29. 29Bagrow and Bollt, 2008:Emerging degree (1/2) 1¡ Emerging degree kj(i) of K0 = 6 internal vertex j: the number 0 of edges that connect j to 1 vertices external to the l- 2 shell 3¡ Total emerging degree Kjl: 4 the total number of emerging edges from that l- shell k1 (0) = 1 k2 (0) = 2¡ Leading edge Sil: the set of all vertices exactly l steps k3 (0) = 1 away from vertex i k4 (0) = 2
- 30. 30Bagrow and Bollt, 2008:Emerging degree (2/2) 1¡ Change in the total K0 = 6 emerging degree: for a shell 0 at depth l starting from 1 vertex i is 2 l l Ki 3 Ki = l 1 4 Ki k1 (0) = 1 k2 (0) = 2 k3 (0) = 1 k4 (0) = 2
- 31. 31Bagrow and Bollt, 2008:Basic principle¡ Basic principle: expanding an l-shell outward from some starting vertex i and comparing the change in total emerging to some thresholdα l Ki < ↵¡ There are many interconnections within a community ¡ The total emerging degree tends to increase¡ The edges connecting the community to the rest of the graph are less in number ¡ The total emerging degree tends to decrease sharply
- 32. 32Bagrow and Bollt, 2008:Algorithm1. Select starting vertex i; l 02. CM = ; 03. Compute Ki l4. While Ki < ↵ (a) l l+1 l l (b) Compute Si ; CM CM [ Si l l (c) Compute Ki and Ki
- 33. 33Bagrow and Bollt, 2008:αas “Social acceptance”¡ The performance of the algorithm is strictly dependent on the value of α¡ αcan be thought as a measure of social acceptance ¡ α≪1 indicates people who are more welcoming of their neighbors (the l-shell will spread to much of the network) ¡ α≫1 indicates hermit-like people who are unwilling to accept even their immediate neighbors into their communities (the l-shell will stop growing immediately)
- 34. Assess thequality of good partitions
- 35. 35Expected properties of agood partition (1/3)¡ Problem: How to say that the partition my algorithm found is good?¡ Given: ¡ A set N of n ≥ 2 points ¡ A distance function d: N x N → ℝ ¡ A partitioning function f that takes a distance function d on N and returns a partition 𝚪 on N
- 36. 36Expected properties of agood partition (2/3)¡ A partition is “good” if it satisfies a set of basic properties: ¡ Scale invariance: for any distance function d and any α> 0, we have f(d) = f(α⋅d) ¡ Richness: every partition of N must be a possible output of f(d) ¡ Consistency: if we produce a d’ by reducing distances within the clusters and enlarging distance between the clusters, the same same partition 𝚪 should arise from d’
- 37. 37Expected properties of agood partition (3/3)¡ The impossibility theorem: for each n ≥ 2, there’s no partitioning function f that satisfies Scale- Invariance, Richness and Consistency at the same time
- 38. 38Quality functions¡ Problem: In practical situations, the communities are not know ahead of time. ¡ How to asses the quality of the partition the algorithm found?¡ It may be convenient to have a quantitative criterion to assess the goodness of a graph partition¡ Quality function: a function that assigns a number to each partition of a graph ¡ Partitions can be ranked
- 39. 39Modularity:Trace as a metric (1/2)¡ Given a partition 𝛤 of G = (V,E), the fraction of edges that fall within the same community isP Aij (ci , cj ) ij 1 X P = Aij (ci , cj ) ij Aij 2m ij red green blue¡ Where: red 5 0 2 ¡ A is the adjacency matrix green 0 9 2 x(1/27) ¡ 𝛿(ci, cj) equals 1 iff ci = cj, 0 otherwise blue 2 2 11 matrix e
- 40. 40Modularity:Trace as a metric (2/2)¡ The trace Tr(e) gives the fraction of edges in the network that connect vertices in the same community¡ A good division in communities should have a high value of trace¡ Problem: the trace on its own it is not a good indicator of the quality of the division ¡ Example: placing all vertices in a single community would give maximal Tr(e) = 1
- 41. 41Modularity:Founding principle¡ Solution: random graph is not expected to have a cluster structure¡ The possible existence of clusters is revealed by the comparison between: ¡ The actual density of edges in a subgraph ¡ The density one would expect in the subgraph if the vertices of the graph were attached randomly (null model)
- 42. 42Quality functions:Modularity function¡ The modularity is the number of edges falling within groups minus the expected value of the same quantity in the case of a randomized network 1 X Q= (Aij Pij ) (ci , cj ) 2m ij¡ Pij is the expected number of edges between vertices i and j in the null model
- 43. 43Quality functions:Modularity’s null model (1/2) ¡ Modularity’s null model: the random graph has to keep the same degree distribution of the original graph ¡ A vertex can be attached to any other vertex ¡ It’s simple to compute Pij
- 44. 44Quality functions:Modularity’s null model (2/2)¡ What is the expected number of edges between i and j in the null model?¡ Given: (i) = ki (j) = kj ¡ Total number of edges m ¡ Degree of i (i) = ki ¡ Degree of j (j) = kj ¡ The number of possible edges kikj out of 2m¡ Expected number: ✓ ◆ ki kj 1 X ki kj Pij = Q= Aij (ci , cj ) 2m ij 2m 2m
- 45. 45Quality functions:Modularity function¡ Modularity, ¡ It can be negative ¡ It equals to 0 if there’s no community division (i.e., the whole graph is a single cluster) ¡ It is size-dependent: graphs of different size cannot be compared
- 46. 46Bibliography¡ F. Radicchi, C. Castellano, F. Cecconi, V. Loreto, D. Parisi - Defining and identifying communities in networks, Proc. Natl. Acad. Sci. USA, 2004¡ P. Erdős , A Rényi, On the evolution of random graphs, publication of the mathematical institute of the Hungarian Academy of Sciences, 1960¡ R.S. Burt, Positions in networks, Social Forces, 1976¡ Wikipedia contributors, Stirling numbers of the second kind, Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 1 Aug. 2012. Web. 19 Sep. 201¡ B.W. Kernighan, S. Lin, An Efficient Heuristic Procedure for Partitioning Graphs, Bell System Tech Journal No. 49, 1970¡ M.E. Newman, M. Girvan, Finding and evaluating community structure in networks, Physical Review E, Vol. 69, No. 2.,11 Aug 2003
- 47. 47Bibliography¡ J.P. Bagrow, E.M. Bollt, Local method for detecting communities, Physical Review E, 2005¡ J. Kleinberg. An Impossibility Theorem for Clustering. Advances in Neural Information Processing Systems (NIPS) 15, 2002

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