Quantum link prediction in complex networks

João P. Moutinho, André Melo, Bruno Coutinho, István A. Kovács, and Yasser Omar

Phys. Rev. A 107, 032605 – Published 10 March 2023

Abstract

Predicting new links in physical, biological, social, or technological networks has a significant scientific and societal impact. Path-based link prediction methods utilize the explicit counting of even- and odd-length paths between nodes to quantify a score function and infer new or unobserved links. Here, we propose a quantum algorithm for path-based link prediction using a controlled continuous-time quantum walk to encode even and odd path-based prediction scores. Through classical simulations on a few real networks, we confirm that the quantum walk scoring function performs similarly to other path-based link predictors. In a brief complexity analysis we identify the potential of our approach in uncovering a quantum speedup for path-based link prediction.

Received 25 November 2022
Revised 12 February 2023
Accepted 14 February 2023

DOI:https://doi.org/10.1103/PhysRevA.107.032605

Physics Subject Headings (PhySH)

Quantum algorithms Quantum walks

Complex networks

Quantum Information, Science & Technology

Authors & Affiliations

João P. Moutinho^1,2,*, André Melo³, Bruno Coutinho², István A. Kovács^4,5,6, and Yasser Omar^1,7,8

¹Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
²Instituto de Telecomunicações, 1049-001 Lisboa, Portugal
³Kavli Institute of Nanoscience, Delft University of Technology, 2628 CD Delft, The Netherlands
⁴Department of Physics and Astronomy, Northwestern University, Evanston, 60208 Illinois, USA
⁵Northwestern Institute on Complex Systems, Northwestern University, Evanston, 60208 Illinois, USA
⁶Department of Network and Data Science, Central European University, 1051 Budapest, Hungary
⁷Portuguese Quantum Institute, 1049-001 Lisboa, Portugal
⁸Centro de Física e Engenharia de Materiais Avançados (CeFEMA), Physics of Information and Quantum Technologies Group, 1049-001 Lisboa, Portugal

^*Corresponding author: joao.p.moutinho@tecnico.ulisboa.pt

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Vol. 107, Iss. 3 — March 2023

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Figure 1
Classical path-based link prediction. Link prediction methods take as input in a complex network with a corresponding adjacency matrix $A$ . Each method then associates a prediction value $p_{i j}$ , or score, to every pair of nodes ${i, j}$ , such that a higher value $p_{i j}$ correlates to a higher probability of the link ${i, j}$ appearing. This requires assumptions about the organizing principles of each network. Predictions based on the triangle closure principle (TCP) rely on similarity between nodes, represented as a matrix $S$ , a common assumption about connections in social networks. This can be quantified in the simplest case as $P \sim S \sim A^{2}$ , counting paths of length 2 between pairs of nodes. As an alternative, proteins often connect to others that are similar to their neighbors, but not necessarily similar to themselves, quantified for example as $P \sim A S \sim A^{3}$ , counting paths of length 3 between nodes [19]. Most classical link prediction methods output all nonzero entries of matrix $P$ , organized in a ranked list of scores from highest to lowest, where the relevant top $l$ predictions are those where the precision is above a user-determined threshold $δ$ . The relevant predictions are considered as new inferred links, represented in yellow, while the rest are discarded.
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Figure 2
Quantum link prediction (QLP) circuit. The algorithm requires a total of $n = {log}_{2} (N)$ qubits to encode each of the $N$ nodes as a basis state and an extra ancilla qubit $q_{a}$ to perform controlled operations. The ancilla qubit is initialized to $| 0 〉$ and the remaining $n$ qubits are initialized to some basis state $| j 〉$ corresponding to a node in the network. The first Hadamard gate creates a superposition of $| 0 〉$ and $| 1 〉$ in $q_{a}$ . A controlled QW, represented here by two operators, applies $e^{- i A t}$ to the remaining $n$ qubits if $q_{a} = | 0 〉$ and $e^{+ i A t}$ if $q_{a} = | 1 〉$ . Given that $q_{a}$ is in a superposition of $| 0 〉$ and $| 1 〉$ , the controlled QW creates a superposition of the two evolutions entangled to the ancilla qubit. A second Hadamard gate applied to the ancilla qubit mixes the two subspaces together and creates an interference between the two quantum walks. Finally, all qubits are measured. The measurement of $q_{a}$ collapses the network to one of two possible cases, imposing either a sum or subtraction of the two conjugate evolutions, which encodes even powers of $A$ (even predictions) for $q_{a} = | 0 〉$ and odd powers of $A$ (odd predictions) for $q_{a} = | 1 〉$ [Eqs. (6) and (7)]. The measurement of the remaining $n$ qubits returns a bit string marking a node $i$ , which together with the initial node $j$ forms a sample of a link $(i, j)$ [Eq. (12)].
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Figure 3
QLP algorithm. Starting with an initial node $j$ , the QLP circuit samples a node $i$ corresponding to an even or odd prediction of a link $(i, j)$ according to the value measured in $q_{a}$ . Repeating this procedure for each node $j$ allows the larger values of the even and odd predictions scores $p_{i j}$ to be approximated.
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Figure 4
Computational cross-validation: Precision. Cumulative precision over the list of ranked scores for each network, averaged over a ten-fold cross validation procedure. The shaded regions correspond to the standard deviation. Details on the precision and score rank metrics are described in the text. The networks used correspond to the PPI networks HI-III-20, the most recent PPI mapping of the human interactome [20], Yeast-Bio, a PPI network of a yeast organism [35], Messel, a food web [36], Hamsterster [37] and Facebook [38], two online social networks, and Wiki-Vote, a vote network between users for adminship of Wikipedia [39]. For comparison, we implemented five classical link prediction methods: the L3 method [19], the LO method [28], the CH-L3 method [29], and two even power methods, RA-L2 (resource allocation) [40], and CH-L2 [29, 41]. The dataset parameters characterizing each network are shown in Table 2 and the values selected for the optimal parameters $t$ in the QLP method and $α$ in the LO method are shown in Table 3.
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Figure 5
Extra cross-validation results: Cumulative precision over the top-ranked predictions (the top $0.05 * N k_{av}$ scores) for each network, averaged over a ten-fold cross-validation procedure. The shaded regions correspond to the standard deviation. In each trial 10% of the links were randomly removed and the remainder were used as input to the link prediction methods. The networks used correspond to the PPI network Arabidopsis [45], Pombe [35], AS Routes [46], Citeseer [47], Cora [47], and P2P-Gnutella [46]. We note that P2P-Gnutella is a bipartite network and thus the TCP based methods QLP-Even, RA-L2 and CH-L2 have null precision. The dataset parameters characterizing each network are shown in Table 2, and the values selected for the optimal 610 parameters $t$ in the QLP method and $α$ in the LO method are shown in Table 3.
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Figure 6
Prediction of missing PPIs in the human interactome validated against experimental screens. Procedure described in Appendix pp4.
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