arXiv subjects

Riccardo Michielan

Publications and source records attributed to Riccardo Michielan.

9 recordsLinked to original sources

arXiv2026

Efficient generation of expected-degree graphs via edge-arrivals

We study the efficient generation of random graphs with a prescribed expected degree sequence, focusing on rank-1 inhomogeneous models in which vertices are assigned weights and edges are drawn independently with probabilities proportional to the product of endpoint weights. We adopt a temporal viewpoint, adding edges to the graph one at a time up to a fixed time horizon, and allowing for self-loops or duplicate edges in the first stage. Then, the simple projection of the resulting multigraph recovers exactly the simple Norros--Reittu random graph, whose expected degrees match the prescribed targets under mild conditions. Building on this representation, we develop an exact generator based on \textit{edge-arrivals} for expected-degree random graphs with running time $O(n+m)$, where $m$ is the number of generated edges, and hence proportional to the output size. This removes the typical vertex sorting used by widely-used fast generator algorithms based on \textit{edge-skipping} for rank-1 expected-degree models, which leads to a total running time of $O(n \log n + m)$. In addition, our algorithm is simpler than those in the literature, easy to implement, and very flexible, thus opening up to extensions to directed and temporal random graphs, generalization to higher-order structures, and improvements through parallelization.

cs.DS↗

arXiv2026

Planted clique recovery in random geometric graphs

We investigate the problem of identifying planted cliques in random geometric graphs, focusing on two distinct algorithmic approaches: the first based on vertex degrees (VD) and the other on common neighbors (CN). We analyze the performance of these methods under varying regimes of key parameters, namely the average degree of the graph and the size of the planted clique. We demonstrate that exact recovery is achieved with high probability as the graph size increases, in a specific set of parameters. Notably, our results reveal that the CN-algorithm significantly outperforms the VD-algorithm. In particular, in the connectivity regime, tiny planted cliques (even edges) are correctly identified by the CN-algorithm, yielding a significant impact on anomaly detection. Finally, our results are confirmed by a series of numerical experiments, showing that the devised algorithms are effective in practice.

math.PR↗

arXiv2026

Large deviations for subgraphs in inhomogeneous random graphs

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the heavy-tailed nature observed in many contexts. While various graph functionals have been studied in this setting, inhomogeneity makes their analysis significantly more challenging. The goal of this paper is to investigate the large deviations of subgraph counts in inhomogeneous random graphs. Rare events concerning these functionals translate into quantifying the probability that extremely large hubs appear in the graph. This can be achieved by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs. When the expected number of subgraphs is sublinear in the graph size, polynomially large deviations are possible, and in this case, we can derive sharp results on clique counts.

math.PR↗

arXiv2025

Homophily Within and Across Groups

Homophily -- the tendency of individuals to interact with similar others -- shapes how networks form and function. Yet existing approaches typically collapse homophily to a single scale, either one parameter for the whole network or one per community, thereby detaching it from other structural features. Here, we introduce a maximum-entropy random graph model that moves beyond these limits, capturing homophily across all social scales in the network, with parameters for each group size. The framework decomposes homophily into within- and across-group contributions, recovering the stochastic block model as a special case. As an exponential-family model, it fits empirical data and enables inference of group-level variation of homophily that aggregate metrics miss. The group-dependence of homophily substantially impacts network percolation thresholds, altering predictions for epidemic spread, information diffusion, and the effectiveness of interventions. Ignoring such heterogeneity risks systematically misjudging connectivity and dynamics in complex systems.

physics.soc-ph↗

arXiv2024

Optimal subgraphs in geometric scale-free random graphs

Geometric scale-free random graphs are popular models for networks that exhibit as heavy-tailed degree distributions, small-worldness and high clustering. In these models, vertices have weights that cause the heavy-tailed degrees and are embedded in a metric space so that close-by groups of vertices tend to cluster. The interplay between the vertex weights and positions heavily affects the local structure of the random graph, in particular the occurrence of subgraph patterns, but the dependencies in these structures and weights make them difficult to analyze. In this paper we investigate subgraph counts using a \textit{divide et impera} strategy: first counting the number of subgraphs in specific classes of vertices; then computing which class yields maximum contribution. Interestingly, the scaling behavior of induced and general subgraphs in such geometric heavy-tailed random graphs is closely related to the solution of a mixed-integer linear program which also shows that subgraphs appear predominantly on vertices with some prescribed degrees and inter-distances. Finally, we derive precise asymptotics for trees and Hamiltonian subgraphs.

math.PR↗

arXiv2024

Detecting hyperbolic geometry in networks: why triangles are not enough

In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.

physics.soc-ph↗

arXiv2023

Geometric random intersection graphs with general connection probabilities

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points $(v,u)$ with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$, where $g$ is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random intersection graph on $\mathbb{R}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function $g$. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether $g$ has bounded or unbounded support.

math.PR↗

arXiv2023

Localized geometry detection in scale-free random graphs

We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as an hypothesis testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k = o(n)$ vertices are given spatial locations and connect between each other following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test, which is based on counting normalized triangles, to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n \to \infty$, and identifies large-degree vertices of the community with high probability.

math.ST↗

arXiv2022

Cliques in geometric inhomogeneous random graph

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model containing geometry. The interplay between scale-freeness and geometry ensures that connections are likely to form between either high-degree vertices, or between close by vertices. At the same time it is rare for a vertex to have a high degree, and most vertices are not close to one another. This trade-off makes cliques more likely to appear between specific vertices. In this paper, we formalize this trade-off and prove that there exists a typical type of clique in terms of the degrees and the positions of the vertices that span the clique. Moreover, we show that the asymptotic number of cliques as well as the typical clique type undergoes a phase transition, in which only k and the degree-exponent tau are involved. Interestingly, this phase transition shows that for small values of tau, the underlying geometry of the model is irrelevant: the number of cliques scales the same as in a non-geometric network

math.PR↗