Exchangeable Random Networks

We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that carachterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. These results are used to compare to other models of simple and complex networks. A particular case of directed networks with power-law out--degree is studied in more detail, as an example of the flexibility of the model in applications.Comment: to appear on "Internet Mathematics

Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent \tau>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent \tau exceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for \tau>3, distances are of the order \log{t}. For \tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when \tau\in (2,3), and of order \log{t} when \tau>3

Percolation in a hierarchical random graph

We study asymptotic percolation as $N\to \infty$ in an infinite random graph ${\cal G}_N$ embedded in the hierarchical group of order $N$, with connection probabilities depending on an ultrametric distance between vertices. ${\cal G}_N$ is structured as a cascade of finite random subgraphs of (approximate) Erd\"os-Renyi type. We give a criterion for percolation, and show that percolation takes place along giant components of giant components at the previous level in the cascade of subgraphs for all consecutive hierarchical distances. The proof involves a hierarchy of random graphs with vertices having an internal structure and random connection probabilities.Comment: 19 pages and 1 figur

Distances in random graphs with finite variance degrees

In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with \prob(D_j\leq x)=F(x). We assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_{\nu}N$, when the base of the logarithm equals \nu=\expec[D_j(D_j -1)]/\expec[D_j]>1. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.Comment: 40 pages, 2 figure

Distances in random graphs with finite mean and infinite variance degrees

In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent $\tau\in (2,3)$. The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. When $\tau\in (2,3)$, this graph distance grows like $2\frac{\log\log N}{|\log(\tau-2)|}$. In different papers, the cases $\tau>3$ and $\tau\in (1,2)$ have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.Comment: 52 pages, 4 figure

Combinative Cumulative Knowledge Processes

We analyze Cumulative Knowledge Processes, introduced by Ben-Eliezer, Mikulincer, Mossel, and Sudan (ITCS 2023), in the setting of "directed acyclic graphs", i.e., when new units of knowledge may be derived by combining multiple previous units of knowledge. The main considerations in this model are the role of errors (when new units may be erroneous) and local checking (where a few antecedent units of knowledge are checked when a new unit of knowledge is discovered). The aforementioned work defined this model but only analyzed an idealized and simplified "tree-like" setting, i.e., a setting where new units of knowledge only depended directly on one previously generated unit of knowledge. The main goal of our work is to understand when the general process is safe, i.e., when the effect of errors remains under control. We provide some necessary and some sufficient conditions for safety. As in the earlier work, we demonstrate that the frequency of checking as well as the depth of the checks play a crucial role in determining safety. A key new parameter in the current work is the $\textit{combination factor}$ which is the distribution of the number of units $M$ of old knowledge that a new unit of knowledge depends on. Our results indicate that a large combination factor can compensate for a small depth of checking. The dependency of the safety on the combination factor is far from trivial. Indeed some of our main results are stated in terms of $\mathbb{E}\{1/M\}$ while others depend on $\mathbb{E}\{M\}$.Comment: 28 pages, 8 figure

An evolving network model of credit risk contagion in the financial market

This paper introduces an evolving network model of credit risk contagion containing the average fitness of credit risk contagion, the risk aversion sentiments, and the ability of resist risk of credit risk holders. We discuss the effects of the aforementioned factors on credit risk contagion in the financial market through a series of theoretical analysis and numerical simulations. We find that, on one hand, the infected path distribution of the network gradually increases with the increase in the average fitness of credit risk contagion and the risk aversion sentiments of nodes, but gradually decreases with the increase in the ability to resist risk of nodes. On the other hand, the average fitness of credit risk contagion and the risk aversion sentiments of nodes increase the average clustering coefficient of nodes, whereas the ability to resist risk of nodes decreases this coefficient. Moreover, network size also decreases the average clustering coefficient. First published online: 29 Feb 201

The phase transition in inhomogeneous random graphs

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that under one very weak assumption (that the expected number of edges is `what it should be'), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is `stable': for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random Structures and Algorithm