Oliver Riordan

The Diameter of a Scale-Free Random Graph

We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabási, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately logn. We show that while this holds for m=1, for m≥2 the diameter is asymptotically log n/log logn.

Explosive Percolation Is Continuous

A mathematical proof shows that in many models of the growth of network connectivity, phase transitions are continuous. “Explosive percolation” is said to occur in an evolving network when a macroscopic connected component emerges in a number of steps that is much smaller than the system size. Recent predictions based on simulations suggested that certain Achlioptas processes (much-studied local modifications of the classical mean-field growth model of Erdős and Rényi) exhibit this phenomenon, undergoing a phase transition that is discontinuous in the scaling limit. We show that, in fact, all Achlioptas processes have continuous phase transitions, although related models in which the number of nodes sampled may grow with the network size can indeed exhibit explosive percolation.

Nonrepetitive colorings of graphs

A sequence a = a1a2. . . . an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance, the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thues sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and Δ(G). We show, by an application of the Lovasz Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ cΔ(G)2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n - 3, and π(T) ≤ 4(Δ(T - 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.

Sparse graphs: Metrics and random models

Recently, Bollobas, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]2 → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real-world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ. The analogous questions for dense graphs, with Θ(n2) edges, are answered by recent results of Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1]. Possible generalizations of these results to graphs with o(n2) but ω(n) edges are discussed in a companion article [Bollobas and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa.

The consequences of Zipf's law for syntax and symbolic reference

Although many species possess rudimentary communication systems, humans seem to be unique with regard to making use of syntax and symbolic reference. Recent approaches to the evolution of language formalize why syntax is selectively advantageous compared with isolated signal communication systems, but do not explain how signals naturally combine. Even more recent work has shown that if a communication system maximizes communicative efficiency while minimizing the cost of communication, or if a communication system constrains ambiguity in a non-trivial way while a certain entropy is maximized, signal frequencies will be distributed according to Zipfs law. Here we show that such communication principles give rise not only to signals that have many traits in common with the linking words in real human languages, but also to a rudimentary sort of syntax and symbolic reference.

Degree distribution of the FKP network model

Recently, Fabrikant, Koutsoupias and Papadimitriou [7] introduced a natural and beautifully simple model of network growth involving a trade-off between geometric and network objectives, with relative strength characterized by a single parameter which scales as a power of the number of nodes. In addition to giving experimental results, they proved a power-law lower bound on part of the degree sequence, for a wide range of scalings of the parameter. Here we prove that, despite the FKP results, the overall degree distribution is very far from satisfying a power law. First, we establish that for almost all scalings of the parameter, either all but a vanishingly small fraction of the nodes have degree 1, or there is exponential decay of node degrees. In the former case, a power law can hold for only a vanishingly small fraction of the nodes. Furthermore, we show that in this case there is a large number of nodes with almost maximum degree. So a power law fails to hold even approximately at either end of the degree range. Thus the power laws found in [7] are very different from those given by other internet models or found experimentally [8].

Sparse random graphs with clustering

In 2007, we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this article is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with (conditionally) independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.

Percolation on dense graph sequences

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G n ). Let λn be the largest eigenvalue of the adjacency matrix of G n , and let G n (p n ) be the random subgraph of G n obtained by keeping each edge independently with probability p n . We show that the appearance of a giant component in G n (p n ) has a sharp threshold at p n = 1/λ n . In fact, we prove much more: if (G n ) converges to an irreducible limit, then the density of the largest component of G n (c/n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovasz, S6s and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.

A Short Proof of the Harris–Kesten Theorem

We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice Z 2 is equal to 1/2. The upper bound was proved by Harris, who showed in 1960 that percolation does not occur at p = 1/2. The other, more difficult, bound was proved by Kesten, who showed in 1980 that percolation does occur for any p > 1/2.

Contraction-Deletion Invariants for Graphs

We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(G/e)+T(G?e) for e?E(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc.43, 26?40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z. Extending results of Tutte (1974, J. Combin. Theory Ser. B16, 168?174) and Brylawski (1981, J. Combin. Theory Ser. B30, 233?246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this.