A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Many studies have attempted to determine whether an observed network exhibits a so-called “small-world structure.” Such determinations have often relied on a conceptual definition of small worldliness proposed by Watts and Strogatz in their seminal 1998 paper, but recently several quantitative indices of network small worldliness have emerged. This paper reviews and compares three such indices—the small-world quotient (Q), a small-world metric (ω), and the small-world index(SWI)—in the canonical Watts–Strogatz re-wiring model and in four real-world networks. These analyses suggest that researchers should avoid Q, and identify considerations that should guide the choice between ω and SWI.

‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2n and an upper bound of order log3n.

A scale-free tree with the parameter β is very close to a star if β is just a bit larger than −1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from −1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.

In a recent paper, Kleinberg (2000) considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power, r-α, of the distance, r, between them. Kleinberg showed that greedy routeing is efficient if α = d and that there is no efficient decentralised routeing algorithm if α ≠ d. The results were extended to a continuum model by Franceschetti and Meester (2003). In our work, we extend the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.