# Dynamic embedding of motifs into networks

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We study various structural information of a large network $G$ by randomly embedding a small motif $F$ of choice. We propose two randomized algorithms to effectively sample such a random embedding by a Markov chain Monte Carlo method. Time averages of various functionals of these chains give structural information on $G$ via conditional homomorphism densities and density profiles of its filtration. We show such observables are stable with respect to various notions of network distance. Our efficient sampling algorithm and stability inequalities allow us to use our techniques for hypothesis testing on and hierarchical clustering of large networks. We demonstrate this by analyzing both synthetic and real world network data. Join with Facundo Memoli and David Sivakoff.