Maria Deijfen

Generating Simple Random Graphs with Prescribed Degree Distribution

Let F be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution F are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erdös-Rényi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution F and also using different order of operations.

Random intersection graphs with tunable degree distribution and clustering

A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this article a model is developed in which each vertex is given a random weight and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree, and—in the power-law case—tail exponent.

A preferential attachment model with random initial degrees

In this paper, a random graph process {G(t)}t≥1 is studied and its degree sequence is analyzed. Let {Wt}t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with Wt edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to di(t-1)+δ, where di(t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τW,τP}, where τW is the power-law exponent of the initial degrees {Wt}t≥1 and τP the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

ASYMPTOTIC SHAPE IN A CONTINUUM GROWTH MODEL

A continuum growth model is introduced. The state at time t, S t , is a subset of ℝ d and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in S t after an exponentially distributed time with expected value |S t |-1 and the location of the outburst is uniformly distributed over S t . The main result is that, if the distribution of the radii of the outburst balls has bounded support, then S t grows linearly and S t /t has a nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.

Scale-free percolation

Abstract We formulate and study a model for inhomogeneous long-range percolation on Zd. Each vertex x?Zd is assigned a non-negative weight Wx, where (Wx)x?Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters a,?>0, the edges are independent and the probability that there is an edge between x and y is given by pxy=1-exp{-?WxWy/|x-y|a}. The parameter ? is the percolation parameter, while a describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of Wx is regularly varying with exponent t-1, then the tail of the degree distribution is regularly varying with exponent ?=a(t-1)/d. The parameter ? turns out to be crucial for the behavior of the model. Conditions on the weight distribution and ? are formulated for the existence of a critical value ?c?(0,8) such that the graph contains an infinite component when ?>?c and no infinite component when ?0, les aretes sont independantes et la probabilite qu’il existe un lien entre x et y est pxy=1-exp{-?WxWy/|x-y|a}. Le parametre ? est le parametre de percolation tandis que a caracterise la portee des interactions. Nous etudierons la distribution des degres dans le graphe resultant et l’existence eventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites eloignes. Nous montrons d’abord que la queue de la distribution des degres est liee a la queue de la distribution des poids. Quand la queue de la distribution de Wx est a variation reguliere d’indice t-1, alors la queue de la distribution des degres est a variation reguliere d’indice ?=a(t-1)/d. Le parametre ? s’avere crucial pour decrire le modele. Des conditions sur la distribution des poids et de ? sont formulees pour l’existence d’une valeur critique ?c?(0,8) telle que le graphe contienne une composante infinie quand ?>?c et aucune composante infinie quand ?

Epidemics and vaccination on weighted graphs.

A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge (u,v) in the graph is equipped with two weights W((u,v)) and W((v,u)) that represent the (subjective) strength of the connection and determine the probability that u infects v in case u is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d. weights and for weights that are functions of the degrees. For i.i.d. weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.

A Weighted Configuration Model and Inhomogeneous Epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight (reflecting how closely ‘connected’ the corresponding vertices are). It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that R0 can be substantially over- or underestimated if the correlation between degree and weight is not taken into account.

The Initial Configuration is Irrelevant for the Possibility of Mutual Unbounded Growth in the Two-Type Richardson Model

The two-type Richardson model describes the growth of two competing infections on

The two-type Richardson model with unbounded initial configurations

\mathbb{Z}^d

Growing networks with preferential deletion and addition of edges

. At time 0 two disjoint finite sets