We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

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Random Graphs and Complex Networks

On generalise l'estimateur bien connu de Hill de l'indice d'une fonction de reparatition avec queue de variation reguliere a une estimation de l'indice d'une loi de valeurs extremes. On demontre la convergence et la normalite asymptotique. On utilise l'estimateur pour certaines estimations comme celle d'une quantile elevee et d'un point d'extremite

A moment estimator for the index of an extreme-value distribution

BACKGROUND The increasing availability of digital data on scholarly inputs and outputs—from research funding, productivity, and collaboration to paper citations and scientist mobility—offers unprecedented opportunities to explore the structure and evolution of science. The science of science (SciSci) offers a quantitative understanding of the interactions among scientific agents across diverse geographic and temporal scales: It provides insights into the conditions underlying creativity and the genesis of scientific discovery, with the ultimate goal of developing tools and policies that have the potential to accelerate science. In the past decade, SciSci has benefited from an influx of natural, computational, and social scientists who together have developed big data–based capabilities for empirical analysis and generative modeling that capture the unfolding of science, its institutions, and its workforce. The value proposition of SciSci is that with a deeper understanding of the factors that drive successful science, we can more effectively address environmental, societal, and technological problems. ADVANCES Science can be described as a complex, self-organizing, and evolving network of scholars, projects, papers, and ideas. This representation has unveiled patterns characterizing the emergence of new scientific fields through the study of collaboration networks and the path of impactful discoveries through the study of citation networks. Microscopic models have traced the dynamics of citation accumulation, allowing us to predict the future impact of individual papers. SciSci has revealed choices and trade-offs that scientists face as they advance both their own careers and the scientific horizon. For example, measurements indicate that scholars are risk-averse, preferring to study topics related to their current expertise, which constrains the potential of future discoveries. Those willing to break this pattern engage in riskier careers but become more likely to make major breakthroughs. Overall, the highest-impact science is grounded in conventional combinations of prior work but features unusual combinations. Last, as the locus of research is shifting into teams, SciSci is increasingly focused on the impact of team research, finding that small teams tend to disrupt science and technology with new ideas drawing on older and less prevalent ones. In contrast, large teams tend to develop recent, popular ideas, obtaining high, but often short-lived, impact. OUTLOOK SciSci offers a deep quantitative understanding of the relational structure between scientists, institutions, and ideas because it facilitates the identification of fundamental mechanisms responsible for scientific discovery. These interdisciplinary data-driven efforts complement contributions from related fields such as scientometrics and the economics and sociology of science. Although SciSci seeks long-standing universal laws and mechanisms that apply across various fields of science, a fundamental challenge going forward is accounting for undeniable differences in culture, habits, and preferences between different fields and countries. This variation makes some cross-domain insights difficult to appreciate and associated science policies difficult to implement. The differences among the questions, data, and skills specific to each discipline suggest that further insights can be gained from domain-specific SciSci studies, which model and identify opportunities adapted to the needs of individual research fields.

Science of science

Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world networks. These subgraph counts, however, crucially depend on the assumed degree distribution. Fitting a degree distribution to network data is challenging, in particular for scale-free networks with power-law degrees. In this paper we develop robust subgraph counts that do not depend on the entire degree distribution, but only on the mean and mean absolute deviation (MAD), summary statistics that are easy to obtain for most real-world networks. By solving an optimization problem, we provide tight (the sharpest possible) bounds for the subgraph counts, for all possible subgraphs, and for all networks with degree distributions that share the same mean and MAD. We identify the extremal random graph that attains the tight bounds as the graph with a specific three-point degree distribution. We leverage the bounds to obtain robust scaling laws for how the numbers of subgraphs grow as function of the network size. The scaling laws indicate that sparse power-law networks are not the most extreme networks in terms of subgraph counts, but dense power-law networks are. The robust bounds are also shown to hold for several real-world data sets.

Robust subgraph counting with distribution-free random graph analysis

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.

Localized geometry detection in scale-free random graphs

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

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Limit theorems for assortativity and clustering in null models for scale-free networks

Quarantining and contact tracing are popular ad hoc practices for mitigating epidemic outbreaks. However, few mathematical theories are currently available to asses the role of a network in the effectiveness of these practices. In this paper, we study how the final size of an epidemic is influenced by the procedure that combines contact tracing and quarantining on a network null model: the configuration model. 
Namely, we suppose that infected vertices may self-quarantine and trace their infector with a given success probability. 
A traced infector is, in turn, less likely to infect others. We show that the effectiveness of such tracing process strongly depends on the network structure. In contrast to previous findings, the tracing 
procedure is not necessarily more effective on networks with heterogeneous degrees. We also show that network clustering influences the effectiveness of the tracing process in a non-trivial way: depending on the infectiousness parameter, contact tracing on clustered networks may either be more, or less efficient than on network without clustering.

Contact tracing in configuration models

We study the problem of finding a copy of a specific induced subgraph on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that finds a copy of any connected graph $H$ on a fixed number of $k$ vertices as an induced subgraph in a random graph with $n$ vertices. By exploiting the scale-free graph structure, the algorithm runs in $O(n k)$ time for small values of $k$. As a corollary, this shows that the induced subgraph isomorphism problem can be solved in time $O(nk)$ for the inhomogeneous random graph. We test our algorithm on several real-world data sets.

Clara Stegehuis

Papers

Robust subgraph counting with distribution-free random graph analysis

Localized geometry detection in scale-free random graphs

Limit theorems for assortativity and clustering in null models for scale-free networks

Contact tracing in configuration models

Finding induced subgraphs in scale-free inhomogeneous random graphs