Exponential random graph models

Synonyms p* models, p-star models, p1 models, exponential family of random graphs, maximum entropy random networks, logit models, Markov graphs Glossary • Graph and network: the terms are used interchangeably in this essay. • Real-world network: (real network, observed network) means network data the researcher has collected and is interested in modelling. • Ensemble of graphs: means the set of all possible graphs (network realiza-tions) that the (real-world) network may reasonably be expected to become, with the assigned probability distribution, which specifies how likely it is that the network will be found in a particular realization. In other words, ensemble of graphs is defined by ascribing a statistical weight to every graph in the given set. • Graph observable: measurable property of a graph. • Network Hamiltonian: is a particular type of objective (fitness) function, H(G). The exponential random graph model defines a probability distribution over a specified set of possible graphs, G = {G}, such that the probability P (G) the Hamiltonian, {x i } is the set of graph observables upon which the relevant constraints act, and {θ i } is a set of ensemble parameters which we can vary so as to match the properties of the model network to the real-world network under investigation. • Adjacency matrix: is a matrix with rows and columns labelled by graph vertices i and j, with elements A ij = 1 or 0 according to whether the vertices, i and j, are connected/adjacent or not. In the case of an undirected graph with no self-loops or multiple edges (the so-called simple graph), the adjacency matrix is symmetric (i.e. A ij = A ji) and has 0s on the diagonal (i.e. A ii = 0). Accordingly, for a simple directed graph the symmetry condition may not be fulfilled, i.e. it can be that A ij = A ji. • Reciprocity: describes tendency of vertex pairs to form mutual directed connections between each other. • Clustering: describes tendency of nodes to cluster together. Clustering is measured by the clustering coefficient which calculates the average probability that two neighbors of a vertex are themselves nearest neighbors. A graph consists of a set of objects or individuals, called nodes (points, vertices), connected by links (edges). The idea of a graph is a powerful simplification for different phenomena-a way of specifying pairwise relations among a collection of items or agents. Graph models are introduced in …