Giant Component in Random Multipartite Graphs with given Degree Sequences

We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with p parts generated according to a given degree sequence n[superscript d][subscript i](n),n≥1 which denotes the number of vertices in part i of the multipartite graph with degree given by the vector d in an n-node graph. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed Molloy and Reed (1995) to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the random processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimensional Galton-Watson processes along with Lyapunov function technique to overcome the challenges

The Critical Phase for Random Graphs with a Given Degree Sequence

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence

Random graphs containing arbitrary distributions of subgraphs

Traditional random graph models of networks generate networks that are locally tree-like, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly non-tree-like neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a new class of random graph models that incorporates general subgraphs, allowing for non-tree-like neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl

Generating random networks that consist of a single connected component with a given degree distribution

We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations. Moreover, they are found to be disassortative, namely high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for single-component networks with ternary, exponential and power-law degree distributions.Comment: 37 pages, 8 figure

Diffusion and Cascading Behavior in Random Networks

The spread of new ideas, behaviors or technologies has been extensively studied using epidemic models. Here we consider a model of diffusion where the individuals' behavior is the result of a strategic choice. We study a simple coordination game with binary choice and give a condition for a new action to become widespread in a random network. We also analyze the possible equilibria of this game and identify conditions for the coexistence of both strategies in large connected sets. Finally we look at how can firms use social networks to promote their goals with limited information. Our results differ strongly from the one derived with epidemic models and show that connectivity plays an ambiguous role: while it allows the diffusion to spread, when the network is highly connected, the diffusion is also limited by high-degree nodes which are very stable