Abstract. Several network models have been proposed to explain the link structure observed in online social networks. This paper addresses the problem of choosing the model that best fits a given real-world network. We implement a model-selection method based on unsupervised learning. An alternating decision tree is trained using synthetic graphs generated according to each of the models under consideration. We use a broad array of features, with the aim of representing different structural aspects of the network. Features include the frequency counts of small subgraphs (graphlets) as well as features capturing the degree distribution and small-world property. Our method correctly classifies synthetic graphs, and is robust under perturbations of the graphs. We show that the graphlet counts alone are sufficient in separating the training data, indicating that graphlet counts are a good way of capturing network structure. We tested our approach on four Facebook graphs from various American universities. The models that best fit these data are those that are based on the principle of preferential attachment.

Abstract. Consider a random graph process where vertices are chosen from the interval [0,1], and edges are chosen independently at random, but so that, for a given vertex x, the probability that there is an edge to a vertex y decreases as the distance between x and y increases. We call this a random graph with a linear embedding.We define a new graph parameter Γ * , which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph G, Γ * (G) = 0 if and only if G is a unit interval graph, and thus a deterministic example of a graph with a linear embedding.We show that the behaviour of Γ * is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on [0, 1] 2 . We define an operator Γ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence {Gn} converges to a function w, then {Γ * (Gn)} converges as well. Moreover, there exists a function w * arbitrarily close to w under the box distance, so that limn→∞ Γ * (Gn) is arbitrarily close to Γ(w * ).

Abstract. In this paper, we explore the mathematical properties of a distance function between graphs based on the maximum size of a common subgraph. The notion of distance between graphs has proven useful in many areas involving graph based structures such as chemistry, biology and pattern recognition. Graph distance has been used as a way of determining similarity between graphs. The distance function studied here forms a metric on isomorphism classes of graphs. We show that this metric induces the discrete topology. We also show that the distance between two graphs in the Erdös-Renyi probability space G(n, p) almost always is near the maximum attainable value. Finally, we define a notion of continuity of graph parameters and relate it to a property of graphs that can be easily verified. We also determine whether the normalized versions of some common graph parameters are continous in this framework.