In online social networks, it is crucial for a service consumer to find the most trustworthy path to a target service provider from numerous social trust paths between them. The selection of the most trustworthy path (namely, optimal social trust path (OSTP)) with multiple endtoend quality of trust (QoT) constraints has been proved to be NPComplete. Heuristic algorithms with polynomial and pseudopolynomialtime complexities are often used to deal with this challenging problem. However, existing solutions cannot guarantee the efficiency of searching; that is, they can hardly avoid obtaining partial optimal solutions during searching process. Quantum annealing uses delocalization and tunneling to avoid falling into local minima without sacrificing execution time. It has been proved to be a promising way to many optimization problems in recently published literature. In this paper, for the first time, QA based OSTP algorithm (QA_OSTP) is applied to the selection of the most trustworthy path. The experiment results show that QA based algorithm has better performance than its heuristic opponents.
Online social networks (OSNs) [
Korkmaz and Krunz [
Yu et al. [
Until now, there are only a few works that are proposed to address the problem in online social networks where some significant influence factors including trust, recommendation roles, and social relationships are taken into account.
Liu et al. [
In statistical mechanics, a physical process called simulated annealing (SA) is often performed in order to relax the system to the state with the minimum energy. In the basic form of SA, it first generates an initial solution as the current feasible solution using Metropolis algorithm [
It is worth mentioning that introducing thermal fluctuation is not the only way to perform annealing; QA depends on quantum fluctuation instead [
The structure of the present paper is as follows. In Section
In order to give prominence to the main problem of OSTP in online social networks, the
In order to solve OSTP in large scale online social networks using quantum annealing, we first need to map the problem onto a highly constrained Ising model [
Before searching the OSTP, three QoT parameters, that is, the values of trust, the social intimacy degree between participants, and the role impact factor of participants mentioned in [
Since the transverse Ising spin glass (TISG) model is the simplest model in which quantum effects in a random system can and have been studied extensively and systematically [
The OSNs studied here are
Online social networks.
In a TISG model, phase space is spanned by all the set of spin variables
In TISG model, the total Hamiltonian in (
A TISG model consists of a set of spins, each of which can only be in one of two states. Each of these spin variable usually takes on the value of either
We use Figure
Possible social trust path and corresponding state vector in Figure

Social trust path  State vector  Utility 


1—2—4 

0.744 

1—3—4 

0.384 

Invalid path 

Invalid 

Invalid path 

Invalid 
An example of OSNs (
In Figure
Note that the realization of quantum annealing requires introducing an artificial and adjustable quantum kinetic operator that can provide the quantum fluctuations to escape the local minima. Moreover, the quantum annealing process is required to be slow enough to approximate the adiabatic evolution. Reasonably, the choice of
Initially the strength of the transverse field
With
Then an important issue arises, that is, how slowly we should decrease
In this part, we will describe the components of our algorithm in detail. Notations that are used in QA_OSTP are shown in Table
Notations used in QA_OSTP.

Social graph of online social networks with 

The number of replicas 

Temperature parameter 

Optimal initial temperature 

An initial value of the transverse field 

Fixed number of nearest neighbors of participant 

Number of nodes in online social networks 

Total Monte Carlo time 
Max_{steps}  Maximum number of Monte Carlo steps 

Number of iterations 

A social trust path configuration 

A series of 

Neighbor of configuration 
Flow chart of QA_OSTP (
The convergence conditions for the implementation of QA_OSTP with the quantum Monte Carlo evolution are investigated in this part.
In our method, the
A Monte Carlo step is characterized by the transition probability from configuration
To derive the convergence conditions for the implementation of QA_OSTP with the quantum Monte Carlo evolution, we should prove that inhomogeneous Markov chain associated with QA_OSTP is strongly ergodic under appropriate conditions [
For a causal system, the transition matrix
Directly following the definition of the transition probability and the property of the acceptance function in (
The inhomogeneous Markov chain generated by (
In order to prove strong ergodicity, we refer to the conditions for strong ergodicity [
Since the Markov chain is proved to be weakly ergodic in [
In our experiments, if no otherwise specified, all the related parameters are set following reference [
For QA_OSTP, we implement a similar GFMC that was used in [
In our experiments, the three QoT parameters are randomly generated. The endtoend QoT constraints specified by a source participant are set as
Both algorithms were implemented in Matlab 7.0 and run on a PC with a 3 GHz Intel processor and 3 GB of RAM with Windows 7.
Figure
Comparison of path utilities of networks.
Network ID with WID = 1
Network ID with WID = 2
Network ID with WID = 3
Network ID with WID = 4
Figure
Comparison of execution time of algorithms.
Network ID with WID = 1
Network ID with WID = 2
Network ID with WID = 3
Network ID with WID = 4
Based on the above experiments conducted with different scales and parameters, we can observe that QA_OSTP is a promising algorithm and it performs better than its heuristic opponent in both the quality of the selected social trust path and the execution time.
A novel quantum annealing based OSTP algorithm, that is, QA_OSTP for selection of the most trustworthy path to service provider in online social networks, was proposed. To the best our knowledge, this is the first application of quantum annealing to the challenging NPComplete OSTP problem in online social networks. Since quantum mechanics work with wave functions that can sample different regions of phase space equally well and quantum systems can tunnel through classically impenetrable potential barriers between energy valleys, a process that might prove more effective than waiting for them to be overcome thermally as in SA, QA_OSTP is able to outperform its heuristic opponents and even find configuration of comparable quality to the best algorithm. Thus, QA_OSTP is proven to be a very promising tool for solving the OSTP in online social networks.
As for the future work, understanding how quantum mechanics can quantitatively improve the quality of solution of OSTP is still an important open issue. Moreover, since GFMC simulations using quantumclassical mapping with the aid of a SuzukiTrotter transformation only simulate the equilibrium behavior at finite temperature, we plan to devise another effective and alternative scheme to solve the infinite time Schrödinger equation with stochastic processes.
This work is supported partially by National Natural Science Foundation of China (NSFC) under Grant nos. 61002016 and 61101111, Zhejiang Provincial Natural Science Foundation of China under Grant no. LY13F010016, and Qianjiang Talent Project of Zhejiang Province under Grant no. QJD1302014.