Modularity is designed to measure the strength of division of a network into clusters (known also as communities). Networks with high modularity have dense connections between the vertices within clusters but sparse connections between vertices of different clusters. As a result, modularity is often used in optimization methods for detecting community structure in networks, and so it is an important graph parameter from a practical point of view. Unfortunately, many existing non-spatial models of complex networks do not generate graphs with high modularity; on the other hand, spatial models naturally create clusters. We investigate this phenomenon by considering a few examples from both sub-classes. We prove precise theoretical results for the classical model of random d-regular graphs as well as the preferential attachment model, and contrast these results with the ones for the spatial preferential attachment (SPA) model that is a model for complex networks in which vertices are embedded in a metric space, and each vertex has a sphere of influence whose size increases if the vertex gains an in-link, and otherwise decreases with time. The results obtained in this paper can be used for developing statistical tests for models selection and to measure statistical significance of clusters observed in complex networks.

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6, 23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs for any integer k ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform 'quasi-random' hypergraphs.

Chvátal, Rödl, Szemerédi and Trotter [V. Chvátal, V. Rödl, E. Szemerédi, W.T. Trotter Jr., The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B 34 (1983) 239-243] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform 'pseudo-random' hypergraphs.

Abstract. We study the two most common types of percolation processes on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p, and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that, in fact, these coincide. As a special case, our results apply to power-law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.

We consider a model for complex networks that was introduced by Krioukov et al.  In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an  edge if they are within a certain hyperbolic distance.  The $N$ points are distributed according to a quasi-uniform distribution, which is a distorted version of  the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power-law and $\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph.  We show that (a) for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component  of linear order,  and (c) when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$. A corrigendum was added to this paper 29 Dec 2018.

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r 2 is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals.We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2 < β < 3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability.More specifically, we determine explicitly a critical function a c (n) such that a c (n) = o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n) ≪ a c (n), then the process does not evolve at all, with high probability as n grows, whereas if a(n) ≫ a c (n), then there is a constant ε > 0 such that, with high probability, the final set of infected vertices has size at least εn. This behaviour is in sharp contrast with the case where the underlying graph is a G(n, p) random graph with p = d/n. Recent results of Janson, Luczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then there is lack of evolution of the process.It turns out that when the maximum degree is o(n 1/(β−1) ), then a c (n) depends also on r. But when the maximum degree is Θ(n 1/(β−1) ), then a c (n) = n β−2 β−1 .

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The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least r≥2 infected neighbours becomes infected and remains so forever. Assume that initially a(t) vertices are randomly infected, where t is the total number of vertices of the graph. Suppose also that r