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PHASE TRANSITIONS IN A COMPLEX NETWORK
"... Abstract. We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle densit ..."
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Cited by 11 (5 self)
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Abstract. We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density. 1.
Singularities in the entropy of asymptotically large simple graphs
"... Abstract. We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. 1. ..."
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Cited by 9 (4 self)
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Abstract. We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. 1.
The asymptotics of large constrained graphs
 J. Phys. A: Math. Theor
, 2014
"... We show, through local estimates and simulation, that if one constrains simple graphs by their densities of edges and τ of triangles, then asymptotically (in the number of vertices) for over 95 % of the possible range of those densities there is a welldefined typical graph, and it has a very simpl ..."
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Cited by 6 (2 self)
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We show, through local estimates and simulation, that if one constrains simple graphs by their densities of edges and τ of triangles, then asymptotically (in the number of vertices) for over 95 % of the possible range of those densities there is a welldefined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets V1 and V2 of fixed relative size c and 1 − c, and there are welldefined probabilities of edges, gjk, between vj ∈ Vj, and vk ∈ Vk. Furthermore the four parameters c, g11, g22 and g12 are smooth functions of (, τ) except at two smooth ‘phase transition ’ curves. 1
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.
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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Cited by 3 (1 self)
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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.
A MEAN FIELD ANALYSIS OF THE FLUID/SOLID PHASE TRANSITION
"... Abstract. We study the fluid/solid phase transition via a mean field model. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite volume limit. We then determine the optimizing particle distributions for small energy density and a r ..."
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Abstract. We study the fluid/solid phase transition via a mean field model. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite volume limit. We then determine the optimizing particle distributions for small energy density and a range of particle density, though we can only prove they are local, not global, minima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous structures at low energy density, and a phase transition between these structured phases and the disordered phase at high energy density. 1.
ON THE LOWER TAIL VARIATIONAL PROBLEM FOR RANDOM GRAPHS
"... Abstract. We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1 − δ)EXH) for fixed 0 < δ < 1. Thanks to the re ..."
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Abstract. We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1 − δ)EXH) for fixed 0 < δ < 1. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the socalled “replica symmetric ” phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH> 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for nonbipartite H and δ close to 1. 1. Background We consider large deviations of subgraph counts in Erdős–Rényi random graphs. Fix a graph H, and let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). For a fixed δ> 0, the problem is to estimate the probabilities (upper tail) P(XH ≥ (1 + δ)EXH) and