SOCIAL NETWORKS AND SIMPLICIAL COMPLEXES IN HONOR OF

SOCIAL NETWORKS AND SIMPLICIAL COMPLEXES (IN HONOR OF ED WEGMAN) DANIELE STRUPPA CHAPMAN UNIVERSITY

STARTING POINT • Many (social) phenomena cannot be efficiently described by graphs (networks) • Example: papers coauthorship B B A, B: Paper 1 A, B, C: Paper 1 B, C: Paper 2 A, C: Paper 3 A C

HYPERGRAPHS • In a graph, an edge only join two vertices, while in a hypergraph an edge (or hyperedge) can join any number of vertices. • Hypergraphs are examples of incidence structures. • An important example of hypergraphs is given by abstract simplicial complexes, which are hypergraphs that contain all subsets of every edge.

SIMPLICIAL COMPLEXES • A set composed of points, line segments, triangles, tetrahedra, and their n-dimensional analogs. • From a purely combinatorial point of view they are referred to as simplicial complexes.

SOME QUESTIONS ON NETWORKS: CAN THEY BE EXTENDED TO HYPERGRAPHS AND SIMPLICIAL COMPLEXES? • Centrality measures (Identification of thought leaders) • Random networks (Evolution of social systems) • Applications to a wide variety of settings (from social sciences to natural sciences)

POLYNOMIAL ALGEBRA FOR SIMPLICIAL COMPLEXES Node Variable Face Monomial of Nodes in the Face

C n = simplicial complex on n nodes R = Q[x 1, …. , xn]

EXAMPLE u z y v x w (x, y, z, u, v, w, uv, xy, xz, yz, xyz)

FACE RING Stanley-Reisner ideal I is the ideal generated by faces not in the complex. We associate to this ideal the face ring of the simplicial complex defined by F I : = Q [x 1, ……, xn]/ I

EXAMPLE z u y v x w I = {xu, xv, xw, yu, yv, yw, zu, zv, zw, uw, vw} But what about xyw? Minimality

FACET RING Facet ideal L is the ideal generated by maximal faces (facets), i. e. faces not contained in faces of higher dimension, and the facet ring is the quotient defined by F L : = Q [x 1, ……, xn]/ L

EXAMPLE z u y v x w J = {xyz, uv, w}

KEY IDEA • Quotients of rings of polynomials can be described in terms of algebraic invariants, which give important information on the associated simplicial complexes • The theory of Grobner bases allows explicit calculation of these invariants • These invariants, in turn, may shed light on the social structure described by the simplicial complex

THE CASE OF THE FACE RING OF A SIMPLICIAL COMPLEX C • Connectedness and high-dimensional holes are topological properties of the complex C that have important meaning in the social context • These objects can be calculated through the dimension of the homology groups of the Stanley-Reisner ideal associated to the dual of C • The Hilbert series of C, H(z) = h(n)z/(1 -z)d and its coefficients can offer additional understanding on its structure

THE SYZYGIES OF THE FACET IDEAL OF A COMPLEX C • If L is the facet ideal generated by r monomials m 1, …mr, we can compute its syzygies, i. e. the submodule of Rr generated by (s 1, …sr) such that m 1 s 1+…mrsr=0, and then the minimal graded free resolution of L. that can be obtained by constructing the syzygies of the syzygies, etc. • These algebraic objects are a good descriptor of the connectedness of the faces of the complex C. • For example, if a syzygy between two facets appears as a minimal generator of the ideal of syzygies, then the two facets are either connected by a common face, or are totally separated, but if the syzygy is not a minimal generator, then the facets are connected by at least one face, but not directly connected.

EXAMPLE: MINIMAL QUADRATIC SYZYGIES z u y x w v

EXAMPLE: NON MINIMAL SYZYGIES z u y x w v

EXAMPLE: LINEAR SYZYGIES z u y x w v

FINAL CONSIDERATIONS • What we just said about syzygies can be made precise, both to describe local connectivity properties, as well as to define global connectivity properties of a complex • This can allow definitions of active and inactive teams within a complex, and can justify the use of these instruments to detect key components of the complex (e. g. faces that allow or accelerate diffusion across the complex) • Finally, one can use more sophisticated techniques, such as the primary decomposition of the ideals, to detect potential leadership subsets in the complex.

REFERENCES • K. Kees, L. Sparks, M. A. Mannucci, and DCS, Social Groups, Social Media, and Higher Dimensional Social Structures: Towards a Simplicial Model of Social Aggregation for Computational Communication Research, Communication Quarterly, 2013, pp. 35 -58. • K. Kee, L. Sparks, M. A. Mannucci, and DCS, Information diffusion, Facebook clusters, and the simplicial model of social aggregation: a computational simulation of simplicial diffusers for community health interventions, Health Communication 2016, pp. 385399. • A. Damiano, M. A. Mannucci, L. Sparks and DCS, Combinatorial Invariants of Social Groups: A Computational Commutative Algebra Approach, in preparation.