Mixing time of vertex-weighted exponential random graphs

Abstract

We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in particular, the vector equation has a unique solution. Lastly, the behavior of vertex-weighted exponential random graphs counting triangles is studied in detail and the solution to the vector equation is shown to approach the zero vector as the weight diverges to negative infinity for sufficiently large networks.

Joint Estimation in Potts Model

“…Sampling from ERGMs plays a central role in both parameter estimation and hypothesis testing, and Glauber dynamics provide a standard and widely used approach for this purpose. The mixing properties of Glauber dynamics in ERGMs have been studied in several key works, including Bhamidi et al (2011) andDeMuse et al (2019), which demonstrate interesting phase transition properties in the mixing rate. In fact, even in the specialized Curie-Weiss Ising model, mixing rates can be either polynomial or exponential depending on the parameter regime.…”

Joint Estimation in Potts Model

“…Sampling from ERGMs plays a central role in both parameter estimation and hypothesis testing, and Glauber dynamics provide a standard and widely used approach for this purpose. The mixing properties of Glauber dynamics in ERGMs have been studied in several key works, including Bhamidi et al (2011) andDeMuse et al (2019), which demonstrate interesting phase transition properties in the mixing rate. In fact, even in the specialized Curie-Weiss Ising model, mixing rates can be either polynomial or exponential depending on the parameter regime.…”

Dynamic Hidden-Variable Network Models

“…We also note some additional works that are lessdirectly related to ours. Network-rewiring and MCMC algorithms are widely used to sample static networks [156][157][158][159][160]; in stationarity, these can be viewed as temporal networks satisfying the Equilibrium Property, with a level of persistence tunable via the number of iterations between adjacent snapshots. Adaptive network models (for instance, SIS-dynamics [161] alongside contactswitching [23,24]), have dynamic node-properties that evolve with time and guide network evolution, a commonality with THVMs.…”

Logarithmic Sobolev inequalities for finite spin systems and applications

“…Vertex-weighted random graph model. A second model of random graphsthe vertex-weighted exponential random graph model -was recently introduced in[DEY17]. The parameter-space is three-dimensional, i. e. β = (β 1 , β 2 , p) ∈ R 2 ×(0, 1), and the model is the spin system on Y = {0, 1} n defined via the Hamiltonian…”

Joint Estimation in Potts Model

“…Sampling from ERGMs plays a central role in both parameter estimation and hypothesis testing, and Glauber dynamics provide a standard and widely used approach for this purpose. The mixing properties of Glauber dynamics in ERGMs have been studied in several key works, including Bhamidi et al (2011) andDeMuse et al (2019), which demonstrate interesting phase transition properties in the mixing rate. In fact, even in the specialized Curie-Weiss Ising model, mixing rates can be either polynomial or exponential depending on the parameter regime.…”

Dynamic Hidden-Variable Network Models

“…We also note some additional works that are lessdirectly related to ours. Network-rewiring and MCMC algorithms are widely used to sample static networks [156][157][158][159][160]; in stationarity, these can be viewed as temporal networks satisfying the Equilibrium Property, with a level of persistence tunable via the number of iterations between adjacent snapshots. Adaptive network models (for instance, SIS-dynamics [161] alongside contactswitching [23,24]), have dynamic node-properties that evolve with time and guide network evolution, a commonality with THVMs.…”

Logarithmic Sobolev inequalities for finite spin systems and applications

“…Vertex-weighted random graph model. A second model of random graphsthe vertex-weighted exponential random graph model -was recently introduced in[DEY17]. The parameter-space is three-dimensional, i. e. β = (β 1 , β 2 , p) ∈ R 2 ×(0, 1), and the model is the spin system on Y = {0, 1} n defined via the Hamiltonian…”

Joint Estimation in Potts Model

“…Sampling from ERGMs plays a central role in both parameter estimation and hypothesis testing, and Glauber dynamics provide a standard and widely used approach for this purpose. The mixing properties of Glauber dynamics in ERGMs have been studied in several key works, including Bhamidi et al (2011) andDeMuse et al (2019), which demonstrate interesting phase transition properties in the mixing rate. In fact, even in the specialized Curie-Weiss Ising model, mixing rates can be either polynomial or exponential depending on the parameter regime.…”

Dynamic Hidden-Variable Network Models

“…We also note some additional works that are lessdirectly related to ours. Network-rewiring and MCMC algorithms are widely used to sample static networks [156][157][158][159][160]; in stationarity, these can be viewed as temporal networks satisfying the Equilibrium Property, with a level of persistence tunable via the number of iterations between adjacent snapshots. Adaptive network models (for instance, SIS-dynamics [161] alongside contactswitching [23,24]), have dynamic node-properties that evolve with time and guide network evolution, a commonality with THVMs.…”

Logarithmic Sobolev inequalities for finite spin systems and applications

“…Vertex-weighted random graph model. A second model of random graphsthe vertex-weighted exponential random graph model -was recently introduced in[DEY17]. The parameter-space is three-dimensional, i. e. β = (β 1 , β 2 , p) ∈ R 2 ×(0, 1), and the model is the spin system on Y = {0, 1} n defined via the Hamiltonian…”