Ueli Peter

Universality of random graphs and rainbow embedding

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rodl and Rucincki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij. Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors).

Robust hamiltonicity of random directed graphs

Abstract In his seminal paper from 1952 Dirac showed that the complete graph on n ≥ 3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊ n / 2 ⌋ edges touching each vertex. In 1960 Ghouila–Houri obtained an analogue statement for digraphs by showing that every directed graph on n ≥ 3 vertices with minimum in- and out-degree at least n / 2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in- and out-going) edges touching each vertex. The theorems of Dirac and Ghouila–Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = ω ( log ⁡ n / n ) with respect to Hamiltonicity is 1 / 2 ± o ( 1 ) . For random directed graphs, Hefetz, Steger and Sudakov (2014) proved an analogue statement, but only for edge probability p = ω ( log ⁡ n / n ) . In this paper we significantly improve their result to p = ω ( log 8 ⁡ n / n ) , which is optimal up to the polylogarithmic factor.

Internal DLA: Efficient Simulation of a Physical Growth Model

The internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position.

Nongrowing Preferential Attachment Random Graphs

Abstract We consider an edge rewiring process that is widely used to model the dynamics of scale-free weblike networks. This process uses preferential attachment and operates on sparse multigraphs with n vertices and m edges. We prove that its mixing time is optimal and develop a framework that simplifies the calculation of graph properties in the steady state. The applicability of this framework is demonstrated by calculating the degree distribution, the number of self-loops, and the threshold for the appearance of the giant component.

Clustering Signature in Complex Social Networks

An important aspect in social computing is the structure of social networks,which build the underlying substrate for the exchange of information. With the growing importance of microblogging networks like Twitter a new class ofdirected social networks appeared. Recently, Ahnert and Fink showed that some classes of directed networks are cleanly separated in the space of the clustering signature. In this paper, we study the structural dynamics which defines the clustering signature in scale free equilibrium networks. Moreover, we also study the hierarchical features of the network topology which lead to a deeper understanding of the clustering signature.