@MISC{Fronczak12exponentialrandom,

author = {Agata Fronczak},

title = {Exponential Random Graph Models },

year = {2012}

}

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 realizations) 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) 2 of a particular graph G is proportional to eH(G) , where H(G) = ∑ iθixi(G). In the Hamiltonian, {xi} is the set of graph observables upon which the relevant constraints act, and {θi} 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 Aij = 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. Aij = Aji) and has 0s on the diagonal (i.e. Aii = 0). Accordingly, for a simple directed graph the symmetry condition may not be fulfilled, i.e. it can be that Aij = Aji. 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.

exponential random graph model synonym describes tendency adjacency matrix possible graph real-world network aij aji vertex pair ensemble parameter network realization assigned probability distribution probability distribution graph vertex element aij symmetry condition glossary graph nearest neighbor graph observables exponential random graph model clustering coefficient so-called simple graph specified set relevant constraint act maximum entropy random network measurable property simple directed graph p1 model multiple edge particular realization p-star model random graph average probability network data exponential family particular type statistical weight particular graph real network undirected graph model network observed network logit model

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