Simplicial Complexes as Complex Networks Statistical Mechanics of Simplicial Complexes

1. 1  Simplicial Complexes as Complex Networks (Statistical Mechanics of Simplicial Complexes) Slobodan Maletic, Zoran Mihailovic and Milan Rajkovic Instute of Nuclear Sciences Vinca, Belgrade

2. 2 Why need for simplicial complexes? Use of simple or directed graphs to represent complex networks does not provide an adequate description Examples: In a collaboration network represented as simple graph we only know whether 3 or more authors linked together in the network were couthors of the same paper or not.

3. 3 Social networks – necessary to consider coordinated action of more than 2 agents, such as a buyer, a seller and a broker. Network theory doesn’t have flexibility to represent higher order aggregations, where several agents interact as a group, rather than as a collection of pairs. Not only agents taking part in the actions are important but time, place etc. Protein complex networks – need information about proteins, regulation, localization, turnover etc. Reaction and metabolic networks

4. 4 Definition of a Simplex Def: A simplicial complex K on a finite set V = {v0, v1, …, vn} of vertices is a nonempty subset of the power set of V with the property that K is closed under the formation of subsets, i.e. if s 3 K and t 2 s, then t 2 K. The dimension of a simplex s is equal to one less than the number of vertices defining it. The dimension of K is the max of dim of all simplices in K.

5. 5 Example:

6. 6 Simplical Complex: finite set of simplices

7. 7 Simplices and relations Any relation l between the elements of a set X and the elements of a set Y is associated with two simplicial complexes K(l) and the conjugate K(l-1). A simplex of K(l) is a finite set of elements of X related to a common element of Y, a simplex of K(l-1) is a finite set of elements of Y related to a common element of X. Dowker, Homology groups of relations (Annals of Math., 1952)

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10. 10 Why need for simplicial complexes (cont.)? To detect qualitative features of the network structure S.C’s are combinatorial versions of topological spaces that can be analyzed with combinatorial, topological, or algebraic methods

11. 11 Simplicial complex may be viewed as: a combinatorial object (consider numerical invariants); Q-analysis of R. Atkin a combinatorial model of a topological space (consider algebraic topological measures, such as homotopy and homology groups). an algebraic model of the complex, its so called Stanley-Reisner ring (the quotient of a polynomial ring on variables corresponding to vertices, divided by the ideal generated by the non-faces of the complex).

12. 12 Statistical mechanics of S.C.? Can we gauge topological objects with statistical mechanical tools? Are there power laws connected to the measures of the combinatorial, topological or algebraic aspect of the simplicial complex obtained from scale-free networks?

13. 13 Simplicial complexes from digraphs 1 Neighborhood complex N(G) of graph G. Its vertices are the vertices of G. For each vertex v of G there is a simplex containing the vertex v, together with all vertices w corresponding to directed edges v ?w. By including all faces of those simplices, the neighborhood simplex is obtained.

14. 14 Simplicial complexes from digraphs 2 The complex C(G) has complete subgraphs of G as simplices. C(G) has as vertices again the vertices of G. The maximal simplices are given by collections of vertices that make up maximal (un)directed complete subgraphs, or cliques of G. In general, any property of G that is monotone (preserved under deletion of vertices or eges) may be used; e.g. simplices consist of all subgraphs of G without (dir or undirected) cycles.

15. 15 Combinatorial aspect: Q-analysisInvariants and useful quantities Dimension of the complex K Vector valued quantities: f-vector (second structure vector) (f0,….fD) fi number of i-dim simplices in K

16. 16 q-nearness and q-connectedness Two simplices K1 and K2 are q-near in the simplicial family iff they share at least q+1 vertices. Two simplices are q-connected by a chain of complexes of length r iff there is sequence of r pair-wise q-near simplices. Equivalence relation ? generates the partition of the simplicial family into q-connected components (q-chains). Enumeration of all q-connected components for each q is the essence of Q-analysis (R. Atkin).

17. 17 Structure vectors

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20. 20 Third structure vector

21. 21 Obstruction vector Q*={QN-1,QN-1-1,…,Q0-1} Measures structural limitations in simplicial connectivity at q. Gives some sense of the “gaps” in the complex.

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23. 23 Eccentricity

24. 24 Eccentricity

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26. 26 Vertex significance

27. 27 Vertex significance ?(vi) - weight of vertex vi equal to the number of simplices which are formed by vertex vi. Any simplex sq in simplicial complex is formed by exactly q+1 different vertices, calculate sum weight ? (sq ) of its vertices and define vertex significance

28. 28 Cohesiveness

29. 29 Cohesiveness

30. 30 Clustering coefficient (1)

31. 31 Clustering coefficient (2)

32. 32 Algebraic model of the complex x1,…,xn - vertices of K; k a field, k=R. Consider polynomial ring k[x1,…,xn]=k[x], containing all polynomials in x1,…,xn (addition and multiplication of polynomials as ring operations). Each simplex {xi1,…,xir} of K corresponds to a unique (square-free) monomial in k[x]. Let I ? k[x] be the ideal generated by all square-free monomials that correspond to non-faces in K, i.e. collection of vertices that do not represent simplices in K.

33. 33 Algebraic model of the complex (cont.) The Stanley-Reisner ring of K is the quotient ring RK= k[x]/I Algebraic model of K and many combinatorial properties of K are contained in RK. Betti numbers – measure of the relationships among monomials (among non-faces of K).

34. 34 Algebraic model of the complex (cont.) Betti vector B={ b0=1,b1,…,br } rmax=N (no. of vertices) simplest homological invariants The number of disjointed components that make up the simplicial complex at each level (dimension). What kind of information does it convey to …. ….social analyst?

35. 35 Algebraic model of the complex (cont.) Assume b3=1 ? somewhere in the complex a 3-dim subcomplex is missing (i.e. a 3-dim object assembled from simplices of dim at most 3), though all of its faces are already there. Conclusion: the complex is weak in triadic relations. Depending on the statistics governing the complex, the group may complete by “filling the whole” at some later stage.

36. 36 Complex as a topological spaceSingular homology Singular homolgy groups of a topological space measure the existence of holes of various dimensions in the space HK ={ H0 ,…,Hd }; dmax=dim K Example: K is a hollow tetrahedron, with a one-cycle graph attached to one if its vertices ? HK ={ 1 ,1,1 }.

37. 37 Random network2000 nodes; p=0.005

38. 38 random network

39. 39 random network

40. 40 Protein-protein interaction network in yeast S. cerevisiae (2361 nodes)

41. 41 Protein-protein interaction network in yeast S. cerevisiae

42. 42 Protein-protein interaction network in yeast S. cerevisiae

43. 43 Protein-protein interaction network in yeast S. cerevisiae

44. 44 US Power grid: 4941 nodes

45. 45 US Power grid: 4941 nodes

46. 46 US Power grid: 4941 nodes

47. 47 Computation geometry collaboration network

48. 48 Computation geometry collaboration network

49. 49 Conclusion: Versatile method (combinatorial, topoloical and algebraic aspects) Distance measures Time series of graphs – complexes Measures for time series of complexes Computational methods for high q

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