We find scaling limits for the sizes of the largest components at criticality for the rank-1 inhomogeneous random graphs with power-law degrees with exponent τ . We investigate the case where τ ∈ (3, 4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n −(τ −2)/(τ −1) , converge to hitting times of a 'thinned' Lévy process. This process is intimately connected to the general multiplicative coalescents studied in [1] and [3]. In particular, we use the results in [3] to show that, when interpreting the location λ inside the critical window as time, the limiting process is a multiplicative process with diffusion constant 0 and the entrance boundary describing the size of relative components in the λ → −∞ regime proportional to i. A crucial ingredient is the identification of the scaling of the largest connected components in the barely subcritical regime.Our results should be contrasted to the case where the degree exponent τ satisfies τ > 4, so that the third moment is finite. There, instead, we see that the sizes of the components rescaled by n −2/3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in [1] for the Erdős-Rényi random graph and extended to the present setting in [6,26]. The limit again is a multiplicative coalescent, the only difference with the limit for τ ∈ (3, 4) being the initial state, corresponding to the barely subcritical regime.

Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both enforce as well as inhibit diffusion processes. Our models further show that it is the mesoscopic set of communities that matters. The exact internal structures of communities barely influence the behavior of percolation processes across networks. This insensitivity is likely due to the relative denseness of the communities.

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent τ ∈ (2,3), so that the degrees have infinite variance. The natural cutoff h c characterizes the largest degrees in the hidden variable models, and a structural cutoff h s introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient C scales with the network size N , as a function of h s and h c . For scale-free networks with exponent 2 < τ < 3 and the default choices h s ∼ N 1/2 and h c ∼ N 1/(τ −1) this gives C ∼ N 2−τ ln N for the universality class at hand. We characterize the extremely slow decay of C when τ ≈ 2 and show that for τ = 2.1, say, clustering starts to vanish only for networks as large as N = 10 9 .

Real-world networks often have power-law degrees and scale-free properties, such as ultrasmall distances and ultrafast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms ofc(k), the probability that two neighbors of a degree-k node are neighbors themselves. We investigate how the clustering spectrum k →c(k) scales with k in the hidden-variable model and show thatc(k) follows a universal curve that consists of three k ranges wherec(k) remains flat, starts declining, and eventually settles on a power-lawc(k) ∼ k −α with α depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks.

This paper deals with the large deviations behavior of a stochastic process called thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs [3]. The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

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Most random graph models are locally tree-like-do not contain short cycles-rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of HCM, we observe two previously undiscovered power-law relations: between the number of edges inside a community and the community sizes, and between the number of edges going out of a community and the community sizes. We also relate the power-law exponent τ of the degree distribution with the power-law exponent of the community-size distribution γ . In the case of extremely dense communities (e.g., complete graphs), this relation takes the simple form τ = γ − 1.

We investigate the final size distribution of the SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of Martin-Löf (1998) for the hitting time of a Brownian motion with parabolic drift, we derive asymptotic expressions for the final size distribution that capture the effect of the initial number of infectives and the closeness of the reproduction number to zero. These asymptotics shed light on the bimodularity of the limiting density of the final size observed in Martin-Löf (1998). We also discuss the connection to the largest component in the Erdős-Rényi random graph, and, using this connection, find an integral expression of the Laplace transform of the normalized Brownian excursion area in terms of Airy functions.