Directed hybrid random networks mixing preferential attachment with uniform attachment mechanisms

Annals of the Institute of Statistical Mathematics - Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with...     

A preferential attachment model with Poisson growth for scale-free networks

We propose a scale-free network model with a tunable power-law exponent. The Poisson growth model, as we call it, is an offshoot of the celebrated model of Barabási and Albert where a network is generated iteratively from a small seed network; at each step a node is added together with a number of incident edges preferentially attached to nodes already in the network. A key feature of our model is that the number of edges added at each step is a random variable with Poisson distribution, and, unlike the Barabási–Albert model where this quantity is fixed, it can generate any network. Our model is motivated by an application in Bayesian inference implemented as Markov chain Monte Carlo to estimate a network; for this purpose, we also give a formula for the probability of a network under our model.     

Local clustering coefficients in preferential attachment models

The local clustering coefficients of preferential attachment models are analyzed. Previously, a general approach to preferential attachment was proposed (the PA-class was introduced); it was shown that the degree distribution in all models of the PA-class obeys a power law. The global clustering coefficient was also analyzed, and a lower bound for the mean local clustering coefficient was found. In the paper, new results are obtained by analyzing the local clustering coefficients of models of the PA-class. Namely, the behavior of the mean value C ₂(n, d) of local clustering over vertices of degree d is studied.     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

General results on preferential attachment and clustering coefficient

This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.     

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 {W _t } _t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W _t 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 d _i (t-1)+δ, where d _i (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{τ_W,τ_P}, where τ_W is the power-law exponent of the initial degrees {W _t } _t≥1 and τ_P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.     

Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G₀ as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r−1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN.In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N₀(F,r,n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1−o(1) as n→∞) avoids creating a copy of F for as long as N=o(N₀), and prove that any online strategy will a.a.s. create such a copy once N=ω(N₀).     

Growth of preferential attachment random graphs via continuous-time branching processes

Some growth asymptotics of a version of ‘preferential attachment’ random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.     

Asymptotic dependence of in- and out-degrees in a preferential attachment model with reciprocity

Extremes - Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges....     

Multigraph limit of the dense configuration model and the preferential attachment graph

The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn models.     

Geodesic subgraphs

Define a geodesic subgraph of a graph to be a subgraph H with the property that any geodesic of two points of H is in H. The trivial geodesic subgraphs are the complete graphs K_n' n ≧ 0, and G itself. We characterize all (finite, simple, connected) graphs with only the trivial geodesic subgraphs, and give an algorithm for their construction. We do this also for triangle-free graphs.     

Bipartite subgraphs

It is shown that there exists a positivec so that for any large integerm, any graph with 2m ²edges contains a bipartite subgraph with at least edges. This is tight up to the constantc and settles a problem of Erdös. It is also proved that any triangle-free graph withe>1 edges contains a bipartite subgraph with at least e/2+c e ^4/5 edges for some absolute positive constantc. This is tight up to the constantc.Research supported in part by a USA Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

Distance-residual subgraphs

For a connected finite graph G and a subset V₀ of its vertex set, a distance-residual subgraph is a subgraph induced on the set of vertices at the maximal distance from V₀. Some properties and examples of distance-residual subgraphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are presented. The relations between the distance-residual subgraphs of product graphs and their factors are explored.     

Degree constrained subgraphs

In this paper, we present new structural results about the existence of a subgraph where the degrees of the vertices are pre-specified. Further, we use these results to prove a 16-edge-weighting version of a conjecture by Karoński, ?uczak and Thomason, an asymptotic 2-edge-weighting version of the same conjecture, and a version of Louigi's Conjecture.     

Maximum k-colorable subgraphs

A lower bound is established on the number of edges in a maximum k-colorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.     

Maximal chordal subgraphs

An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations.