24 research outputs found
Multi-group Binary Choice with Social Interaction and a Random Communication Structure -- a Random Graph Approach
We construct and analyze a random graph model for discrete choice with social
interaction and several groups of equal size. We concentrate on the case of two
groups of equal sizes and we allow the interaction strength within a group to
differ from the interaction strength between the two groups. Given that the
resulting graph is sufficiently dense we show that, with probability one, the
average decision in each of the two groups is the same as in the fully
connected model. In particular, we show that there is a phase transition: If
the interaction among a group and between the groups is strong enough the
average decision per group will either be positive or negative and the decision
of the two groups will be correlated. We also compute the free energy per
particle in our model
Fluctuation Results for General Block Spin Ising Models
Knopfel H, Lowe M, Schubert K, Sinulis A. Fluctuation Results for General Block Spin Ising Models. JOURNAL OF STATISTICAL PHYSICS. 2020;178(5):1175-1200.We study a block spin mean-field Ising model, i.e. a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices, and where pair interactions are given according to their blocks. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein's method we establish a rate of convergence in the Central Limit Theorem for the block magnetization vector in the high temperature regime
Detection of an Arbitrary Number of Communities in a Block Spin Ising Model
We study the problem of community detection in a general version of the block
spin Ising model featuring M groups, a model inspired by the Curie-Weiss model
of ferromagnetism in statistical mechanics. We solve the general problem of
identifying any number of groups with any possible coupling constants. Up to
now, the problem was only solved for the specific situation with two groups of
identical size and identical interactions. Our results can be applied to the
most realistic situations, in which there are many groups of different sizes
and different interactions. In addition, we give an explicit algorithm that
permits the reconstruction of the structure of the model from a sample of
observations based on the comparison of empirical correlations of the spin
variables, thus unveiling easy applications of the model to real-world voting
data and communities in biology.Comment: 31 page
Partial recovery bounds for clustering with the relaxed means
We investigate the clustering performances of the relaxed means in the
setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM).
After identifying the appropriate signal-to-noise ratio (SNR), we prove that
the misclassification error decay exponentially fast with respect to this SNR.
These partial recovery bounds for the relaxed means improve upon results
currently known in the sGMM setting. In the SBM setting, applying the relaxed
means SDP allows to handle general connection probabilities whereas other
SDPs investigated in the literature are restricted to the assortative case
(where within group probabilities are larger than between group probabilities).
Again, this partial recovery bound complements the state-of-the-art results.
All together, these results put forward the versatility of the relaxed
means.Comment: 39 page