Sampling random graph homomorphisms and applications to network data analysis

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $\mathcal{G}$. When $\mathcal{G}$ is the complete graph with $q$ nodes, this becomes the well-known problem of sampling uniform $q$-colorings of $F$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us real-, function-, and network-valued computable observables, including well-known ones such as homomorphism density and average clustering coefficient. One of the main observable we propose is called the conditional homomorphism density profile, which reveals hierarchical structure of the network. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We also provide various examples and simulations demonstrating our framework through synthetic and real-world networks. For instance, we apply our framework to analyze Word Adjacency Networks of a 45 novels data set and propose an authorship attribution scheme using motif sampling and conditional homomorphism density profiles.

Classifying Clustering Schemes

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

Persistent Clustering and a Theorem of J. Kleinberg

We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorem analogous to one of J. Kleinberg, in which one obtains an existence and uniqueness theorem instead of a non-existence result. We explore further properties of this unique scheme, stability and convergence are established.