Comparing linear modularization criteria using the relational notation

Transcription

1 Comparing linear modularization criteria using the relational notation Université Pierre et Marie Curie Laboratoire de Statistique Théorique et Appliquée April 30th, /35

2 Table of contents 1 Introduction and objective Definitions The definition of community 2 Relational approach 3 Contributions 4 Applications 5 Conclusions 2/35

3 1 Introduction and objective Definitions The definition of community 2 Relational approach Mathematical Relational modelling Properties verified by linear modularization criteria 3 Contributions Linear criteria Comparison of linear criteria 4 Applications The generalized Louvain algorithm Examples: real large graphs 5 Conclusions 3/35

4 Description of the problem Nowadays, we can find networks everywhere: biology, sociology, computer programming, marketing, etc). cyber-marketing, cyber-security. It is difficult to analyse a network directly because of its big size. Therefore, we need to decompose it in clusters or modules modularize it. Different modularization criteria have been formulated in different contexts in the last few years and we need to compare them. Objective: Compare the partitions found by different linear criteria We will provide a unified notation of different linear modularization criteria to understand the properties of the clusters found by their optimization. Moreover, this notation allows to easily identify the criteria having a resolution limit. 4/35

5 Definitions Definitions and notations A graph G(V, E) is a set of objects V, called nodes, linked by edges E. 5/35

6 Definitions Definitions and notations A graph G(V, E) is a set of objects V, called nodes, linked by edges E. N = V is the number of nodes and M = E is the number of edges. 5/35

7 Definitions Definitions and notations A graph G(V, E) is a set of objects V, called nodes, linked by edges E. N = V is the number of nodes and M = E is the number of edges. A graph is completely described by a N N matrix called the Adjacency Matrix A defined as follows { 1 if there is an edge between nodes i and i, a ii = 0 otherwise. Example: given a graph with N = 6 and M = 7. its adjacency matrix is: A = (1) 5/35

8 Definitions Definitions and notations A graph G(V, E) is a set of objects V, called nodes, linked by edges E. N = V is the number of nodes and M = E is the number of edges. A graph is completely described by a N N matrix called the Adjacency Matrix A defined as follows { 1 if there is an edge between nodes i and i, a ii = 0 otherwise. Example: given a graph with N = 6 and M = 7. its adjacency matrix is: A = (1) The degree d i of node i is the number of edges incident to i. d i = i a ii = a i. = a.i The average degree of the graph is d av = 2M N. The Density of the graph is δ = 2M N 2. 5/35

9 Definitions Definitions and notations A graph G(V, E) is a set of objects V, called nodes, linked by edges E. N = V is the number of nodes and M = E is the number of edges. A graph is completely described by a N N matrix called the Adjacency Matrix A defined as follows { 1 if there is an edge between nodes i and i, a ii = 0 otherwise. Example: given a graph with N = 6 and M = 7. its adjacency matrix is: A = (1) The degree d i of node i is the number of edges incident to i. d i = i a ii = a i. = a.i The average degree of the graph is d av = 2M N. The Density of the graph is δ = 2M N 2. Just in case the graph is weighted we will denote the adjacency matrix W. 5/35

10 The definition of community What is a community? Each criterion is based on its own definition of community. Definitions of community: Densely connected groups of nodes with few connections between groups whose density is higher than the density of the whole graph. Set of nodes where the connections are not randomly distributed. many others... The words community, module and cluster will be treated as synonyms 6/35

11 Table of contents 1 Introduction and objective 2 Relational approach Mathematical Relational modelling Properties verified by linear modularization criteria 3 Contributions 4 Applications 5 Conclusions 7/35

12 Mathematical Relational modelling Mathematical Relational modelling Let X be a square matrix of order N defining an equivalence relation on V as follows: { 1 if i and i are in the same cluster i, i V V x ii = (2) 0 otherwise 8/35

13 Mathematical Relational modelling Mathematical Relational modelling Let X be a square matrix of order N defining an equivalence relation on V as follows: { 1 if i and i are in the same cluster i, i V V x ii = (2) 0 otherwise We present a modularization criterion as a function F to optimize: max X subject to the constraints of an equivalence relation: or min F (A, X). (3) X x ii {0, 1} Binarity (4) x ii = 1 i Reflexivity x ii x i i = 0 (i, i ) Symmetry x ii + x i i x ii 1 (i, i, i ) Transitivity Finding the exact solution of this problem turns impractical for large graphs, therefore we will use heuristics ad-hoc. 8/35

14 Properties verified by linear modularization criteria Properties verified by linear modularization criteria A criterion is linear if it can be written in the general form: N N F (X) = φ(a ii )x ii + K (5) i=1 i =1 9/35

15 Properties verified by linear modularization criteria Properties verified by linear modularization criteria A criterion is linear if it can be written in the general form: N N F (X) = φ(a ii )x ii + K (5) i=1 i =1 Besides that, the criterion has the property of General balance if it can be written in the form: N N N N F (X) = φ(a ii )x ii + φ(a ii ) x ii (6) i=1 i =1 i=1 i =1 where K is any constant depending only on the original data; φ(a ii ) 0 i, i and φ(a ii ) 0 i, i are non negative functions verifying: N N i=1 i =1 φii > 0 and N N φ i=1 i =1 ii > 0;. N N N N The quantities φ(a ii )x ii and φ(a ii ) x ii are called positive (+) i=1 i =1 and negative (-) agreements respectively. i=1 i =1 9/35

16 Properties verified by linear modularization criteria The impact of the property of General balance Let κ denote the number of clusters obtained after optimization of the criterion. Given a graph 10/35

17 Properties verified by linear modularization criteria The impact of the property of General balance Let κ denote the number of clusters obtained after optimization of the criterion. If (-) agreements missing ( φ ii = 0 i, i ) Given a graph then all nodes are clustered together, κ = 1 10/35

18 Properties verified by linear modularization criteria The impact of the property of General balance Let κ denote the number of clusters obtained after optimization of the criterion. If (-) agreements If (+) agreements missing ( φ ii = 0 i, i ) missing (φ ii = 0 i, i ) Given a graph then all nodes are clustered together, κ = 1 then all nodes are separated, κ = N 10/35

19 Properties verified by linear modularization criteria Contribution: Different levels of general balance for linear criteria Property of Local balance A balanced linear criterion whose functions φ ii and φ ii satisfy φ ii + φ ii = K L (i, i ) where K L is a constant depending only upon the pair (i, i ) has the property of local balance. Therefore K L must not depend on global properties of the graph. 11/35

20 Properties verified by linear modularization criteria Contribution: Different levels of general balance for linear criteria Property of Local balance A balanced linear criterion whose functions φ ii and φ ii satisfy φ ii + φ ii = K L (i, i ) where K L is a constant depending only upon the pair (i, i ) has the property of local balance. Therefore K L must not depend on global properties of the graph. Property of Global balance A balanced linear criterion whose functions φ ii and φ ii satisfy N N ( ) φii + φ ii = KG i=1 i =1 where K G is a constant depending on global properties of the graph has the property of global balance. 11/35

21 Properties verified by linear modularization criteria Contribution: Different levels of general balance for linear criteria Null Model A global balanced linear criterion whose functions φ ii and φ ii satisfy the following conditions: N N N N φ ii = φ ii i=1 i =1 i=1 i =1 φ ii + φ ii = K G (i, i ) where K G is a constant depending on global properties of the graph is a criterion based on a null model. 12/35

22 Properties verified by linear modularization criteria Different levels of general balance for linear criteria Figure: Venn Diagram of criteria verifying the property of general balance. 13/35

23 Properties verified by linear modularization criteria Resolution limit [Fortunato and Barthelemy (2006)] A null model has a resolution limit. Figure: A null model may fail to identify modules smaller than a scale depending on the global network, even cliques (complete graphs with m nodes k m ) connected by single edges. 14/35

24 Table of contents 1 Introduction and objective 2 Relational approach 3 Contributions Linear criteria Comparison of linear criteria 4 Applications 5 Conclusions 15/35

25 Linear criteria Some linear criteria in relational notation Criterion Zahn-Condorcet (1785, 1964) Owsiński - Zadrożny (1986) Newman-Girvan (2004) Relational notation N N F ZC (X) = (a ii x ii + ā ii x ii ) F ZOZ (X) = i=1 i =1 N i=1 i =1 with 0 < α < 1 F NG (X) = 1 N 2M N ((1 α)a ii x ii + αā ii x ii ) i=1 i =1 N (a ii a i.a.i 2M ) x ii Table: Relational notation of linear modularity functions. 16/35

26 Linear criteria Some linear criteria in relational notation (continuation) Criterion Deviation to Uniformity (2013) Deviation to Indetermination (2013) The Balanced Modularity (2013) Relational notation F UNIF (X) = 1 N N 2M F DI (X) = 1 2M F BM (X) = N N i=1 i =1 N i=1 i =1 i=1 i =1 where P ii = ai.a.i 2M and P ii = ( a ii 2M ) N 2 x ii ( a ii a i. N a.i N + 2M ) N 2 x ii N ( (aii P ii ) x ii + (ā ii P ii ) x ii ) ( ā ii (N ai.)(n a.i ) N 2 2M ) Table: Relational notation of linear modularity functions. 17/35

27 Linear criteria Existing linear criteria: Deviation to a particular structure Principe: A random graph does not have community structure. Uniformity structure Independence structure Multiplicative model: Indetermination structure Additive model: Geometric mean Arithmetic mean 18/35

28 Linear criteria Properties of these linear criteria The 6 criteria have the property of General balance. Criterion Zahn-Condorcet Owsiński-Zadrożny Newman-Girvan Deviation to Uniformity Deviation to Indetermination Balanced modularity Global balance Local Null Balance model X X X X X X Table: Balance Property for Linear criteria 19/35

29 Comparison of linear criteria The number of clusters The number of clusters found by the generalized Louvain algorithm for two real networks. Network Jazz Internet N = 198 N = 69,949 M = 2,742 M = 351,380 Criterion κ κ Zahn-Condorcet 38 40,123 Owsiński-Zadrożny 6 α = 2% 456 α < 1% Deviation to Uniformity Newman-Girvan 4 46 Deviation to Indetermination 6 39 Balanced Modularity 5 41 How to explain these differences? 20/35

30 Comparison of linear criteria First approach: the deviation form notation Criterion Notation F (X) = Zahn-Condorcet F ZC (X) = Owsiński-Zadrożny F OZ (X) = Deviation to uniformity F UNIF (X) = Newman-Girvan F NG (X) = Deviation to indetermination F DI (X) = N N i=1 i =1 N i=1 i =1 N N i=1 i =1 ( a ii 1 2 N (φ ii φ ii )x ii ) x ii N (a ii α)x ii N i=1 i =1 N N ( i=1 i =1 N N i=1 i =1 ( a ii 2M N 2 ) x ii a ii ai.a.i 2M ( a ii ) x ii ( ai. N + a.i N 2M N 2 )) x ii 21/35

31 Comparison of linear criteria First approach: the deviation form notation The (+) agreements term is the same for the 5 criteria: φ ZC ii = φoz ii = φng ii = φdi ii = φunif ii = a ii (i, i ). So we have to analyse the (-) agreements term. We distinguish 2 cases: 22/35

32 Comparison of linear criteria First approach: the deviation form notation The (+) agreements term is the same for the 5 criteria: φ ZC ii = φoz ii = φng ii = φdi ii = φunif ii = a ii (i, i ). So we have to analyse the (-) agreements term. We distinguish 2 cases: Case 1: φ is a constant not depending on each pair of nodes. Thas is the case of Zahn-Condorcet, Owsiński-Zadrożny and Deviation to Uniformity criteria. 22/35

33 Comparison of linear criteria First approach: The degree distribution Case 2: φ depends on the degree of each pair of nodes as for Newman-Girvan and Deviation to Indetermination criteria. 23/35

34 Comparison of linear criteria First approach: The degree distribution Case 2: φ depends on the degree of each pair of nodes as for Newman-Girvan and Deviation to Indetermination criteria. Many real networks fall into the class of scale-free networks, meaning that their degree distribution follows a power-law which distinguish them from random networks [Barabasi and Albert (1999)]. In a scale-free network a few nodes called hubs have many connexions whereas most nodes have few connexions. 23/35

35 Comparison of linear criteria Comparison between Newman-Girvan, Deviation to Indetermination and the Balanced Modularity Maximizing the Balanced Modularity turns out to maximize the following expressions depending upon the Newman-Girvan criterion and the Deviation to Indetermination respectively. N N ( ) (ai. d av )(a.i d av ) F BM = 2F NG + x ii. 2M(1 δ) i=1 i =1 ( F BM = 2F DI ) N δ i=1 i =1 N ( (ai. d av )(a.i d av ) N 2 (1 δ) The Balanced Modularity behaves as a regulator between the Newman-Girvan criterion and the Deviation to Indetermination. ) x ii. 24/35

36 Comparison of linear criteria Second approach: Impact of merging two clusters Now let us suppose we want to merge two clusters C 1 and C 2 in the network of sizes n 1 and n 2 respectively. Let us suppose as well they are connected by l edges and they have average degree d 1 av et d 2 av respectively. 25/35

37 Comparison of linear criteria Impact of merging two clusters What is the contribution of merging two clusters to the value of each criterion? 26/35

38 Comparison of linear criteria Impact of merging two clusters What is the contribution of merging two clusters to the value of each criterion? The contribution C of merging two clusters will be: C = n 1 i C 1 n 2 i C 2 (φ ii φ ii ) (7) 26/35

39 Comparison of linear criteria Impact of merging two clusters What is the contribution of merging two clusters to the value of each criterion? The contribution C of merging two clusters will be: C = n 1 i C 1 n 2 i C 2 (φ ii φ ii ) (7) The objective is to compare function φ(.) to function φ(.) If C > 0 the criterion merges the two clusters, it is a gain. If C < 0 the criterion separates the two clusters, it is a cost. 26/35

40 Comparison of linear criteria The Contribution of merging two clusters Contribution of merging two clusters for linear criteria. n 1 n 2 Criterion: F C F = (φ ii φ ii ) i C 1 i C ( 2 Zahn-Condorcet C ZC = l n ) 1n 2 2 Owsiński-Zadrożny C OZ = (l n 1 n 2 α) 0 < α < 1 Deviation to Uniformity C UNIF = ( (l n 1 n 2 δ) d Newman-Girvan C NG = l n 1 n avd 1 2 ) av 2 2M Deviation to Indetermination C DI = ( ( d 1 l n 1 n av 2 N + d av 2 N 2M N 2 )) 27/35

41 Comparison of linear criteria Summary by criterion Criterion Zahn-Condorcet Owsiński-Zadrożny Deviation to Uniformity Characteristics of the clustering The density of edges of each cluster is at least equal to 50%. No resolution limit. For real networks the optimal partition contains many small clusters or single nodes. It gives the choice to define the minimum required within-cluster density, α. For α = 0.5 the Owsiński-Zadrożny criterion the Zahn-Condorcet criterion. No resolution limit. A particular case of Owsiński-Zadrożny criterion with α = δ. The density of within cluster edges of each cluster is at least δ. It has a resolution limit. 28/35

42 Comparison of linear criteria Summary by criterion Criterion Newman-Girvan Deviation to Indetermination Balanced modularity Characteristics of the clustering It has a resolution limit. The contribution depends on the degree distribution of the clusters. The optimal partition has no single nodes. It has a resolution limit. The contribution depends on the degree distribution of the clusters. It has a resolution limit. The contribution depends on the degree distribution of the clusters. Depending upon δ and d av this criterion behaves like a regulator between the NG criterion and the DI criterion. 29/35

43 Table of contents 1 Introduction and objective 2 Relational approach 3 Contributions 4 Applications The generalized Louvain algorithm Examples: real large graphs 5 Conclusions 30/35

44 Examples: real large graphs Modularizing large graphs with the generalized Louvain algorithm Number of clusters found by the generalized Louvain algorithm (see [Campigotto et al. (2014)]) Web nd.edu Amazon Youtube WebBase01 N 325k 334k 1M 118M M 1M 925k 3M 1B δ Criterion κ κ κ κ ZC 201, , ,849 71,806,729 UNIF ,584 2,777,580 NG ,567 2,759,248 DI ,985 2,770,647 BM ,410 2,736,808 Ref: Zahn-Condorcet (ZC), Deviation to Uniformity (UNIF), Newman-Girvan (NG), Deviation to Indetermination(DI) and Balanced Modularity (BM) 31/35

45 Examples: real large graphs Running times obtained for each criterion N M NG ZC DI UNIF BM karate arxiv 9k 48k internet 70k 351k webndu 325k 2M webuk M 783M webbase M 1B 430 1, Table: Running times (in seconds) obtained for each criterion 32/35

46 Table of contents 1 Introduction and objective 2 Relational approach 3 Contributions 4 Applications 5 Conclusions 33/35

47 Conclusions We presented six different modularization criteria in Relational notation. 34/35

48 Conclusions We presented six different modularization criteria in Relational notation. We described and clearly defined the property of balance by making the link between this property and the resolution limit property. 34/35

49 Conclusions We presented six different modularization criteria in Relational notation. We described and clearly defined the property of balance by making the link between this property and the resolution limit property. The generic Louvain algorithm allowed us to modularize real large graphs and we could compare the number of clusters found by the different criteria. 34/35

50 Conclusions We presented six different modularization criteria in Relational notation. We described and clearly defined the property of balance by making the link between this property and the resolution limit property. The generic Louvain algorithm allowed us to modularize real large graphs and we could compare the number of clusters found by the different criteria. We characterized the partitions found by six linear modularization criteria. We saw that two criteria who have a local definition are based on a the density of within-cluster edges (Zahn-Condorcet and Owsiński-Zadrożny), whereas others are based on a null model (Newman-Girvan, Deviation to Uniformity, Deviation to Indetermination and the Balanced Modularity). These criteria have a resolution limit. 34/35

51 Thanks for your attention! 35/35