S S symmetry Article Reconstructing Damaged Complex Networks Based on Neural Networks Ye Hoon Lee 1 and Insoo Sohn 2,* 1 Department of Electronic and IT Media Engineering, Seoul National University of Science and Technology, Seoul 01811, Korea; [email protected] 2 Division of Electronics and Electrical Engineering, Dongguk University, Seoul 04620, Korea * Correspondence: [email protected]; Tel.: +82-2-2260-8604 Received: 2 October 2017; Accepted: 9 December 2017; Published: 9 December 2017 Abstract: Despite recent progress in the study of complex systems, reconstruction of damaged networks due to random and targeted attack has not been addressed before. In this paper, we formulate the network reconstruction problem as an identification of network structure based on much reduced link information. Furthermore, a novel method based on multilayer perceptron neural network is proposed as a solution to the problem of network reconstruction. Based on simulation results, it was demonstrated that the proposed scheme achieves very high reconstruction accuracy in small-world network model and a robust performance in scale-free network model. Keywords: network reconstruction; neural networks; small world networks; scale free networks 1. Introduction Complex networks have received growing interest from various disciplines to model and study the network topology and interaction between nodes within a modeled network [1–3]. One of the important problems that are actively studied in the area of network science is the robustness of a network under random failure of nodes and intentional attack on the network. For example, it has been found that scale-free networks are more robust compared to random networks against random removal of nodes, but are more sensitive to targeted attacks [4–6]. Furthermore, many approaches have been proposed to optimize conventional networks against random failure and intentional attack compared to the conventional complex networks [7–10]. However, these approaches have been mainly concentrated on designing network topology based on various optimization techniques to minimize the damage to the network. So far, no work has been reported on techniques to repair and recover the network topology after the network has been damaged. Numerous solutions have been proposed on reconstruction of spreading networks based on spreading data [11–13], but the problem on the reconstruction of damaged networks due to random and targeted attacks has not been addressed before. Artificial neural networks (NNs) have been applied to solve various problems in complex systems due to powerful generalization abilities of NNs [14]. Some of the applications where NNs have been successfully used are radar waveform recognition [15], image recognition [16,17], indoor localization [18,19], and peak-to-average power reduction [20,21]. In this paper, we propose a novel network reconstruction method based on the NN technique. To the best of our knowledge, this work is the first attempt to recover the network topology after the network has been damaged and also the first attempt to apply NN technique for complex network optimization. We formulate the network reconstruction problem as an identification of a network structure based on a much reduced amount of link information contained in the adjacency matrix of the damaged network. The problem is especially challenging due to (1) very large number of possible network configurations 2 on the order of 2N , where N is the number of nodes and (2) very small number of node interaction Symmetry 2017, 9, 310; doi:10.3390/sym9120310 www.mdpi.com/journal/symmetry Symmetry 2017, 9, 310 2 of 11 Symmetryespecially2017 challenging, 9, 310 due to (1) very large number of possible network configurations on the 2order of 11 2 of 2N , where N is the number of nodes and (2) very small number of node interaction information informationdue to node due removals. to node We removals. simplify We the simplify problem the by problem the following by the assumptions following assumptions (1) average (1) number average of numberreal connections of real connections of a network of is a networkmuch smaller is much than smaller all thethan possible all the link possible configurations link configurations and (2) link andinformation (2) link information of M undamaged of M undamaged networks are networks available. are available.Based on these Based assumptions, on these assumptions, we chose wethe chosemultiple the-layer multiple-layer perceptron perceptron neural network neural network (MLPNN), (MLPNN), which whichis one isof one the of frequently the frequently used used NN NNtechniques, techniques, as the as basis the basis of our of method. our method. We evaluate We evaluate the performance the performance of the of proposed the proposed method method based basedon simulations on simulations in two in twoclassical classical complex complex networks networks (1) (1) small small-world-world network network and and (2) scale-freescale-free networks,networks, generatedgenerated by by Watts Watts and and Strogatz Strogatz model model and and Barab Barabásiási and and Albert Albert model, model, respectively. respectively. TheThe restrest of the the paper paper is is organized organized as asfollows. follows. Section Section 2 describes2 describes the small the small-world-world network network model modeland scale and-free scale-free network, network, followed followed by the by thenetwork network damage damage model. model. In In Section Section 3,3, wewe proposepropose the the reconstructionreconstruction methodmethod basedbased onon NNNN technique.technique. InIn SectionSection4 ,4, we we present present the the numerical numerical results results of of the the proposedproposed method, method, and and conclusions conclusions are are given given Section Section5 .5. 2.2. ModelModel 2.1.2.1. Small-WorldSmall-World NetworkNetwork Small-worldSmall-world networknetwork isis anan importantimportant networknetwork modelmodel withwith lowlow averageaverage pathpath lengthlength andand highhigh clusteringclustering coefficientcoefficient [22[22––2424].]. Small-worldSmall-world networksnetworks havehave homogeneoushomogeneous networknetwork topologytopology withwith aa degreedegree distribution distribution approximated approximated by by the the Poisson Poisson distribution. distribution. A small-world A small-worl networkd network is created is created based onbased a regular on a regular lattice such lattice a ring such of aN ringnodes, of N where nodes, each where node each is connected node is connected to J nearest to nodes. J nearest Next, nodes. the linksNext, are the randomly links are rewired randomly to one rewired of N nodes to one in theof N network nodes within the probability network withp. The probability rewiring process p. The isrewiring repeated process for all is the repeated nodes nfor= 1all... theN nodes. By controlling n = 1 … N. theBy controlling rewiring probability the rewiringp, the probability network p will, the interpolatenetwork will between interpolate a regular between lattice a regular (p = 0) tolattice a random (p = 0) network to a random (p = 1).network Figure (1p shows= 1). Figure the detailed 1 shows algorithmthe detailed for algorithm small-world for networksmall-world construction network inconstruction pseudocode in format.pseudocode format. 1: Initialization: 2: Set the total number of nodes N 3: Set the rewiring probability p 4: Set the node degree K 5: end initialization 6: 7: for each node n (1, N) 8: Connect to nearest K nodes. 9: end for 10: 11: for each node n (1, N) 12: for each link k (1, K) 13: Generate a random number r (0, 1) 14: if r < p 15: Select a node randomly m (1, N) 16: Rewire to node m 17: end if 18: end for 19: end for FigureFigure1. 1.Small-world Small-world networknetwork algorithm.algorithm. 2.2.2.2. Scale-FreeScale-Free NetworkNetwork Scale-freeScale-free networks, networks, such such as asBarab Barabáásisi and and AlbertAlbert (BA)(BA) network,network, areare evolvingevolving networksnetworks thatthat havehave degreedegree distributiondistribution followingfollowing power-lawpower-lawmodel model[ 4[4––66].]. Scale-freeScale-free networksnetworks consistconsist ofof aa smallsmall numbernumber ofof nodesnodes withwith veryvery high degree of of connections connections and and rest rest of of the the nodes nodes with with low low degree degree connections. connections. A Ascale scale-free-free network network starts starts the the evolution evolution process process with a small number ofof mm00 nodes. Next, aa newnew nodenode isis introduced to the network and attaches to m existing nodes with high degree k. The process consisting Symmetry 2017, 9, 310 3 of 11 Symmetry 2017, 9, 310 3 of 11 introduced to the network and attaches to m existing nodes with high degree k. The process consisting of new node introduction and preferential attachment is repeated until a network with N = t + m0 of new node introduction and preferential attachment is repeated until a network with N = t + m0 nodesnodes hashas been been constructed. constructed. Figure Figure2 shows 2 theshows detailed the algorithmdetailed foralgorithm scale-free for network scale-free construction network inconstruction pseudocode in format. pseudocode format. 1: Initialization: 2: Set the initial number of nodes m0 3: Set the total number of nodes N 4: Set the node degree K 5: end initialization 6: 7: for each node n (1, m0) 8: Connect to nearest K nodes. 9: end for 10: 11: for each new node 12: Calculate the