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26
Structure and dynamics of molecular networks: A novel paradigm of drug discovery  A . . .
 PHARMACOLOGY THERAPEUTICS
, 2013
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An approximate version of Sidorenko’s conjecture
 Geom. Funct. Anal
"... A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent ana ..."
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Cited by 23 (7 self)
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A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs. 1
Subgraph Frequencies: Mapping the Empirical and Extremal Geography of Large Graph Collections
"... A growing set of online applications are generating data that can be viewed as very large collections of small, dense social graphs — these range from sets of social groups, events, or collaboration projects to the vast collection of graph neighborhoods in large social networks. A natural question ..."
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Cited by 18 (1 self)
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A growing set of online applications are generating data that can be viewed as very large collections of small, dense social graphs — these range from sets of social groups, events, or collaboration projects to the vast collection of graph neighborhoods in large social networks. A natural question is how to usefully define a domainindependent ‘coordinate system ’ for such a collection of graphs, so that the set of possible structures can be compactly represented and understood within a common space. In this work, we draw on the theory of graph homomorphisms to formulate and analyze such a representation, based on computing the frequencies of small induced subgraphs within each graph. We find that the space of subgraph frequencies is governed both by its combinatorial properties — based on extremal results that constrain all graphs — as well as by its empirical properties — manifested in the way that real social graphs appear to lie near a simple onedimensional curve through this space. We develop flexible frameworks for studying each of these aspects. For capturing empirical properties, we characterize a simple stochastic generative model, a singleparameter extension of ErdősRényi random graphs, whose stationary distribution over subgraphs closely tracks the onedimensional concentration of the real social graph families. For the extremal properties, we develop a tractable linear program for bounding the feasible space of subgraph frequencies by harnessing a toolkit of known extremal graph theory. Together, these two complementary frameworks shed light on a fundamental question pertaining to social graphs: what properties of social graphs are ‘social ’ properties and what properties are ‘graph ’ properties? We conclude with a brief demonstration of how the coordinate system we examine can also be used to perform classification tasks, distinguishing between structures arising from different types of social graphs.
Emergent structures in large networks
"... Abstract. Weconsider alarge class ofexponential randomgraphmodelsandprovetheexistence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs. 1 ..."
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Cited by 8 (6 self)
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Abstract. Weconsider alarge class ofexponential randomgraphmodelsandprovetheexistence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs. 1
An Analytic Approach to Stability
, 2010
"... The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n2) edges. Here we show h ..."
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Cited by 7 (0 self)
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The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdős–Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg. 1
Asymptotic quantization of exponential random graphs
, 2013
"... Abstract. We describe the asymptotic properties of the edgetriangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from ..."
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Cited by 7 (2 self)
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Abstract. We describe the asymptotic properties of the edgetriangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model. 1.
A Tensor Approach to Learning Mixed Membership Community Models
"... Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with nonoverlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed ..."
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Cited by 6 (0 self)
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Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with nonoverlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced by Airoldi et al. (2008). This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning these models via a tensor spectral decomposition method. Our estimator is based on loworder moment tensor of the observed network, consisting of 3star counts. Our learning method is fast and is based on simple linear algebraic operations, e.g., singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters and present a careful finite sample analysis of our learning method. As an important special case, our results match the best known scaling requirements for the (homogeneous) stochastic block model.
On the phase transition curve in a directed exponential random graph model
, 2014
"... Abstract. We consider a family of directed exponential random graph models parametrized by edges and outward stars. Essentially all of the statistical content of such models is given by the free energy density, which is an appropriately scaled version of the probability normalization. We derive prec ..."
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Cited by 5 (3 self)
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Abstract. We consider a family of directed exponential random graph models parametrized by edges and outward stars. Essentially all of the statistical content of such models is given by the free energy density, which is an appropriately scaled version of the probability normalization. We derive precise asymptotics for the free energy density of finite graphs. We use this to rederive a formula for the limiting free energy density first obtained by Chatterjee and Diaconis [3]. The limit is analytic everywhere except along a phase transition curve first identified by Radin and Yin [18]. Building on their results, we carefully study the model along the phase transition curve. In particular, we give precise scaling laws for the variance and covariance of edge and outward star densities, and we obtain an exact formula for the limiting edge probabilities, both on and off the phase transition curve. 1.
ASYMPTOTIC STRUCTURE AND SINGULARITIES IN CONSTRAINED DIRECTED GRAPHS
, 1405
"... Abstract. We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward pstars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of d ..."
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Cited by 3 (2 self)
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Abstract. We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward pstars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun [24]. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis [3]: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large graphs have either uniform or bipodal structure. When edge density (resp. pstar density) is fixed and pstar density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and pstar density are fixed, we find only bipodal structures and no phase transition. 1.