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On the asymptotics of constrained exponential random graphs
"... Abstract. The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but in many situations partial information of the graph is already known beforehand. A natural question to ask is what would be a typical random graph drawn from an exponential m ..."
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Abstract. The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but in many situations partial information of the graph is already known beforehand. A natural question to ask is what would be a typical random graph drawn from an exponential model subject to certain constraints? In particular, will there be a similar phase transition phenomenon as that which occurs in the unconstrained exponential model? We present some general results for the constrained model and then apply them to get concrete answers in the edgetriangle model. 1.
Differential calculus on the space of countable labelled graphs
, 2014
"... Abstract. The study of very large graphs is becoming increasingly prominent in modernday mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable graphs, and the completed graph spac ..."
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Abstract. The study of very large graphs is becoming increasingly prominent in modernday mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable graphs, and the completed graph space G (V) is identified with the 2adic integers as well as the Cantor set. The goal of this paper is to develop a model for differentiation on graph space in the spirit of the NewtonLeibnitz calculus. To this end, we first study the space of all finite labelled graphs and their limiting objects, and establish analogues of leftconvergence, homomorphism densities, a Counting Lemma, and a large family of topologically equivalent metrics on labelled graph space. We then establish results akin to the First and Second Derivative Tests for realvalued functions on countable graphs, and completely classify the permutation automorphisms of graph space that preserve its topological and differential structures. 1.
An Lp theory of sparse graph convergence II: LD convergence, quotients, and right convergence, in preparation
"... Abstract. We extend the Lp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy c ..."
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Abstract. We extend the Lp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence, and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs, and sparse versions of Wrandom graphs. Contents
arXiv: arXiv:0000.0000 On the Propagation of LowRate Measurement Error to Subgraph Counts in Large, Sparse Networks ∗
"... Abstract: Our work in this paper is motivated by an elementary but also fundamental and highly practical observation – that uncertainty in constructing a network graph Ĝ, as an approximation (or estimate) of some true graph G, manifests as errors in the status of (non)edges that must necessarily pr ..."
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Abstract: Our work in this paper is motivated by an elementary but also fundamental and highly practical observation – that uncertainty in constructing a network graph Ĝ, as an approximation (or estimate) of some true graph G, manifests as errors in the status of (non)edges that must necessarily propagate to any summaries η(G) we seek. Mimicking the common practice of using plugin estimates η(Ĝ) as proxies for η(G), our goal is to characterize the distribution of the discrepencyD = η(Ĝ)−η(G), in the specific case where η(·) is a subgraph count. In the empirically relevant setting of large, sparse graphs with lowrate measurement errors, we demonstrate under an independent and unbiased error model and for the specific case of counting edges that a Poissonlike regime maintains. Specifically, we show that the appropriate limiting distribution is a Skellam distribution, rather than a normal distribution. Next, because dependent errors typically can be expected when counting subgraphs in practice, either at the level of the edges themselves or due to overlap among subgraphs, we develop a parallel formalism for using the Skellam distribution in such cases. In particular, using Stein’s method, we present a series of results leading to the quantification of the accuracy with which the difference of two sums of dependent Bernoulli random variables may be approximated by a Skellam. This formulation is general and likely of some independent interest. We then illustrate the use of these results in our original context of subgraph counts, where we examine (i) the case of counting edges, under a simple dependent error model, and (ii) the case of counting chains of length 2 under an independent error model. We finish with a discussion of various open problems raised by our work.