In this chapter, we develop spectral techniques. We highlight some applications to Markov chain mixing and network analysis. The main tools are the spectral theorem and the variational characterization of eigenvalues, which we review together with some related results. We also give a brief introduction to spectral graph theory and detail an application to community recovery. Then we apply the spectral theorem to reversible Markov chains. In particular we define the spectral gap and establish its close relationship to the mixing time. We also show in that the spectral gap can be bounded using certain isoperimetric properties of the underlying network. We prove Cheeger’s inequality, which quantifies this relationship, and introduce expander graphs, an important family of graphs with good “expansion.” Applications to mixing times are also discussed. One specific technique is the “canonical paths method,” which bounds the spectral graph by formalizing a notion of congestion in the network.

Resilience has been studied in many fields and contexts. While there are different definitions and perspectives, it refers to a process or outcome of adapting to and recovering from a disruption and ideally bouncing forward to an improved state of functioning. In this chapter, the focus is on community resilience using a systems perspective, including the elements that support resilience in adverse circumstances. The global experience with the COVID-19 pandemic is discussed as an example of how social actions can support community resilience, as well as two examples of citizen engagement focused on preparedness for and recovery from disaster. Essential elements of any community preparedness, response and recovery activities include recognition of the importance of addressing inequities and social justice, while managing complexity and tailoring strategies to the unique cultures within each community.

This chapter provides a survey of the common techniques for determining the sharp statistical and computational limits in high-dimensional statistical problems with planted structures, using community detection and submatrix detection problems as illustrative examples. We discuss tools including the first- and second-moment methods for analyzing the maximum-likelihood estimator, information-theoretic methods for proving impossibility results using mutual information and rate-distortion theory, and methods originating from statistical physics such as the interpolation method. To investigate computational limits, we describe a common recipe to construct a randomized polynomial-time reduction scheme that approximately maps instances of the planted clique problem to the problem of interest in total variation distance.