A QUBO Formulation for Eigencentrality Prosper D. Akrobotu 1,2 , Tamsin E. James 2 , Christian F. A. Negre 3 , Susan M. Mniszewski 2* 1 Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, USA 2 Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, USA 3 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA These authors contributed equally to this work. * smm@lanl.gov Abstract The efficient calculation of the centrality or “hierarchy” of nodes in a network has gained great relevance in recent years due to the generation of large amounts of data. The eigenvector centrality is quickly becoming a good metric for centrality due to both its simplicity and fidelity. In this work we lay the foundations for the calculation of eigenvector centrality using quantum computational paradigms such as quantum annealing and gate-based quantum computing. The problem is reformulated as a quadratic unconstrained binary optimization (QUBO) that can be solved on both quantum architectures. The results focus on correctly identifying a given number of the most important nodes in numerous networks given by our QUBO formulation of eigenvector centrality on both the D-Wave and IBM quantum computers. Introduction There are several centrality measures used to identify the most influential node or nodes within a network, each having their own benefits dependent on the data at hand or the results desired. For example, degree centrality [1], which is based purely on the number of connections a node has, could be used for identifying the most popular person within a group of people on a social media platform (number of followers). Closeness centrality [2] is dependent on the length of the paths from one node to all other nodes in a network, prioritizing nodes that are “closer” to all other nodes as more central. This has been used for predicting enzyme catalytic residues from topological descriptions of protein structures [3]. Betweenness centrality [4], is based on the number of times a node appears when two other nodes are connected by their shortest path. This measure is often used in biological networks, for example identifying a specific protein that is important for information flow within a network, which could be used in drug discovery [5]. Katz centrality [6], measures the importance of a node through its immediate connections, and also the connections of other nodes through the immediate neighbors. Katz centrality has also been used within a biological setting, such as identifying disease genes [7]. PageRank centrality [8] is a variant of eigenvector centrality designed to rank web pages by importance based on links between pages or October 19, 2021 1/18 arXiv:2105.00172v3 [quant-ph] 16 Oct 2021