The 2-core of a Non-homogeneous Hypergraph

Transcription

1 July 16, 2012

2 k-cores A Hypergraph G on vertex set V is a collection E of subsets of V. E is the set of hyperedges. For ordinary graphs, e = 2 for all e E. The k-core of a (hyper)graph is the maximal subgraph among subgraphs with minimum degree k.

3 Random hypergraph models A hypergraph is r-uniform if e = r for all e E. The random r-uniform hypergraph: m iid hyperedges, e i uniform from { e = r : e E} The Erdos-Renyi random graph: The indicators 1{e E} are iid for all edges e, equal to 1 with probability p. Expected number of edges = p n(n+1) 2 if V = n. Interested in properties when V = n and n.

4 Studying the k-core Interested in properties when V = n and n with density paramater c fixed. Theorem There exists a critical value c for having a large k-core: If c > c then the size of the k-core is Θ(n) with high probability. If c < c then the size of the k-core is o(n) with high probability.

5 My model Label the vertices as V = {1, 2,..., n}. A non-homogeneous hypergraph model: m iid hyperedges, each edge e i being distributed so that the indicators 1{j e i } are independent (but not identical!), and equal to 1 with probability 1 j+1. Mean hyperedge size log n. m = m(n) = cn. Expected degree of vertex j = cn j+1

6 Unique features Expected degree of vertex j = cn j+1 = c x where x = j+1 n is the rescaled position of vertex j. In the hypergraph, x is evenly spread across [0, 1]. Any subinterval of [0, 1] represents a positive fraction of the graph, and subintervals near 0 represent positive fractions of the graph with arbitrarily large degree!

7 Removal process The k-core of a hypergraph G is the maximal subgraph among subgraphs with minimum degree k. One can find the k-core of G by iterating a removal process: Layer 0 Let G 0 := G. Layer 1 G 1 := Remove from layer 0 all vertices whose degree is less than k (and remove any hyperedges containing them.)... Layer t G t := Remove from layer t 1 all vertices whose degree in G t 1 is less than k.

8 Main idea We can study the 2-core by using the removal process: The n + 1 random variables ( G t, d 1 (t),..., d n (t)) at time step t form a Markov chain. For n the chain remains close to a non-stochastic, limiting trajectory with high probability. The limiting trajectory can be characterized by a deterministic dynamical system.

9 Degree density Degree of vertex j in layer 0 = Poiss(λ j ) where λ j = m j+1 = c x degree k density := dµ k := n λk k! e λ dx, meaning that the integral of this density over x [0, 1] gives (asymptotically) the total number: # degree k vertices in layer dµ k(x). The limiting trajectory can in fact be modeled by tracking just G t, µ 0 (t) and µ 1 (t).

10 Main results Theorem Fluctuations of the Markov Chain about the limiting trajectory are o(n) after t = O(log n) many steps.. Theorem The critical value is c = e γ for the 2-core: If c > c then the size of the 2-core is Θ(n) with high probability. If c < c then the size of the 2-core is o(n) with high probability.

11 Open questions What can we say near the critical region? k-core versus 2-core Generalize to weakly dependent vertex distributions.

12 Thank you for your attention!