Abstract
Robustness and smallworld effect are two crucial structural features of complex networks and have attracted increasing attention. However, little is known about the relation between them. Here we demonstrate that, there is a conflicting relation between robustness and smallworld effect for a given degree sequence. We suggest that the robustnessoriented optimization will weaken the smallworld effect and vice versa. Then, we propose a multiobjective tradeoff optimization model and develop a heuristic algorithm to obtain the optimal tradeoff topology for robustness and smallworld effect. We show that the optimal network topology exhibits a pronounced coreperiphery structure and investigate the structural properties of the optimized networks in detail.
Introduction
Many realworld complex systems can be modeled as networks. Examples include the Internet, metabolic networks, electric power grids, supply chains, urban road networks, and the world trade web among many others. In the last decades, an emergent science of networks has attempted to explain, predict, and control these networked systems. The study of complex networks has become an important area of multidisciplinary research involving physics, mathematics, biology, social sciences, informatics, and other theoretical and applied sciences^{1,2,3,4,5,6}.
The function and behavior of networked systems can be largely influenced by their structural features. As one of the fundamental structural features, network robustness (also known as resilience, tolerance, survivability, invulnerability), i.e., the ability of a network to maintain its connectivity when a fraction of nodes (links) is damaged, has received growing attention in many fields^{7,8,9}. In ecology, network robustness is an important attribute of ecosystems, and can give insight into the reaction to disturbances such as the extinction of species^{10}. For biologists, network robustness can help the study of diseases and mutations, and how to recover from some mutations^{11}. In economics, network robustness principles can help our understanding of the stability and risks of banking systems^{12}. And in engineering, network robustness can help to evaluate the resilience of infrastructure networks such as the Internet or power grids^{13}. Many studies have been devoted to designing robust networks^{14,15,16,17,18,19,20,21} motivated by applications in communication networks^{22}, power grid^{23}, transportation network^{24}, sensor networks^{25}, logistic network^{26} and so on. However, many real networks are the results of complex and extended processes; thus, designing networks from scratch is practically impossible. Therefore, the study of improving existing networks is of great interest. Recently, some methods for enhancing network robustness by modifying the topology have been proposed, e.g., adding links^{27,28}, deleting^{29}, or rewiring links^{30,31,32,33,34}. However, despite recent research focusing on the enhancement of network robustness, little has been done on the joint optimization of network robustness and other desirable structural features. This is the motivation for our study.
In this paper, along with the network robustness, we focus on another important and wellknown structural feature: smallworld effect, the fact that most pairs of nodes are connected by a relatively short path through the network^{2}. The existence of the smallworld effect had been speculated upon before the Milgram’s experiment and the more rigorously in the mathematical work of Pool and Kochen^{35}. Nowadays, the smallworld effect has been studied and verified directly in a large number of different networks. The smallworld effect has obvious implications for the diffusion processes taking place on networks. For example, if one considers the spread of information across a network, the smallworld effect implies that the spread will be fast on most realworld networks. If it takes only six steps for a rumor to spread from any person to any other, for instance, then the rumor will spread much faster than if it takes a hundred steps, or a million. Moreover, smalleffect effect is also related to the synchronization of oscillator networks^{36}. The smallworld effect has been used across the many diverse applications^{37}. The smallworld effect is useful in analysis of manmade networks such as transportation networks and communications networks. It is used to help determine how costefficient a particular network construction is. Beyond human constructed networks, the smallworld effect is a useful metric when talking about physical biological networks. For example, it is used in neuroscience to discuss information transfer across neural networks, where the physical space and resource constraints are a major factor^{38}. There are some previous works addressing the relation between robustness and smallworld effect. For example, Netotea and Pongor studied the evolution of robust yet smallworld network topologies and how the selection for robustness or smallworld effect influences network topology^{39}. Brede et al. studied networks that optimize a tradeoff between smallworld effect and resilience^{40}. However, these works did not preserve node degrees. For the practical purposes, changing the degree of a node can be more expensive than changing the connection in networked system. For example, the adjustment of airline is more easier than increasing the capacity of airports. Therefore, conserving the degrees is a reasonable constraint that we will use.
Here we investigate the relation between robustness and smallworld effect while keeping the degrees of each node fixed in optimization. We focus on the complex networks with nontrivial topological features that do not occur in simple networks such as lattices or random graphs. To realize the tradeoff between robustness and smallworld effect, we build a multiobjective optimization model and develop a heuristic algorithm. We report on successful results from applying the method to scalefree network for discussing how robustness and smallworld effect influence on network topology.
Results
Conflicting relation between robustness and smallworld effect
In this paper, we use a spectral measure of robustness—natural connectivity . It measures the robustness of a network by quantifying the redundancy of paths between each pairs of nodes, and has been proved to sensitively exhibit the variation of the robustness^{41,42}. Besides, a measure E defined as the reciprocal harmonic average of shortest distances is adapted to describe the extent of smallworld effect^{43}. To analyze the relation between robustness and smallworld effect, we first propose a singleobjective optimization model to optimize robustness and smallworld effect separately and then investigate the change of one property when optimizing another one. We consider scenarios where changing the degree of a node is significantly more expensive than rewiring a link, thus we impose the constraint that each node degree d_{i} is fixed in optimization process. Besides, the optimized network should be connected. We use the degreepreserving greedy optimization algorithm^{44,45} shown in Fig. 1 to search the optimal robust network or the optimal network with smallworld effect. Starting from an initial network, we adopt the rewiring method at each time step. Only if the objective is improved and the generated network is still a connected simple network, the rewiring is accepted. We repeat this procedure for many iterations to obtain the optimized network. We apply the singleobjective optimization method to model networks. Figure 2 shows the change of the robustness and the smallworld effect E versus iterations t in scalefree networks and ER random graphs, respectively. We find that optimizing will reduce E obviously while increasing E also leads to a great loss of robustness . It means that robustness and smallworld effect are in a conflicting relation.
Since the degree of each node is fixed in optimization, the degree correlation is a significant network property that deserve investigation. In a work by Milo et al.^{46}, they presented an approach for comparing network local structure based on the significance profile (SP). To obtain the SP of a network, the statistical significance for each subgraph is calculated by comparing the network to an ensemble of randomized network with the same degree sequence. Similarly, this method can also be used to analyze the degree correlations in networks. Specifically, when studying the degree correlations, the statistical significance is described by the Z score:
where m(d_{i}, d_{j}) is the number of links between the nodes with degree d_{i} and nodes with degree d_{j} in the network, and 〈m_{r}(d_{i}, d_{j})〉 and σ_{r}(d_{i}, d_{j}) are the mean and standard deviation of the values of m(d_{i}, d_{j}) in a randomized network sets, where the networks are randomized from the specific network by executing the degreepreserving rewiring algorithm^{44} enough times. Therefore, according to the equation, Z score reflects the density of connections between nodes with different degrees. The higher the value of Z(d_{i}, d_{j}) score is, the more connections between d_{i}degree nodes and d_{j}degree nodes the network has. Figure 3 shows the correlation profiles of the optimized network and the Eoptimized network based on scalefree network and ER random network. The Z score value is normalized to a range of [−1, 1] and indicated by the color in the correlation profiles. The randomized network is generated by rewiring the original networks in Fig. 3 for 10^{4} times while keeping node degrees fixed. We generate 10^{3} randomized networks to ensure the stability of results. The results in Fig. 3(a) and (c) show that the optimization for results in a more assortative topology compared to the randomized network with the same degree sequence. Nodes in optimized network prefer to attach to those nodes with similar degree. On the contrary, the Eoptimized network is inclined to be a disassortative network, in which highdegree nodes prefer to connect lowdegree nodes rather than connect the nodes with similar degree. Therefore, the Eoptimized network shows a high degree of difference with optimized network, which implies that robustness and smallworld effect are in a conflicting relation.
To verify the inferences, we also investigated the trial on a real network, Zachary’s karate club, which is a social network in a karate club at a US university in the 1970s^{47}. It has 34 nodes and 78 links (see Fig. 4(a)). Figure 4(b) shows the value of and E in the original network, the optimized network and Eoptimized network. The results shows similar conclusions, i.e., optimizing will result in a decrease of E, while maximizing E will decrease of network. Several primary topological properties are expressed through radar charts in Fig. 5. We can clearly see the different topologies of optimized and Eoptimized networks. The Eoptimized network (Fig. 5(b)) exhibits roughly a multihub, local starlike structure in which highdegree nodes (hubs) are interconnected through lowdegree nodes, and lowdegree nodes rarely attach to each other. In the multihub starlike structure, information on any node can be quickly transmitted to other nodes through the hubs they connected. Moreover, lower D, lower σ_{d} shown in radar chart illustrates that each pair of nodes have approximately the same short length of distance. The multihubs starlike network has a disassortative structure and a lower cluster coefficient. On the contrary, the optimized network (Fig. 5(c)) has a corechain with a core comprised of highdegree nodes and other lowdegree nodes that are interconnected to form a long chain. The highly interconnected hubs guarantee the connectivity of network after the removals of some nodes, even the highdegree nodes, whereas the structure with a long chain composed of lowdegree nodes will leads to a high cost of communication between nodes. The corechain topology displays a high degree of assortativity and clustering. That multihub starlike represent the extreme case of disassortativity and the corechain topology represent networks optimized for assortativity was found in ref. 48. The results in radar charts show that the robust networks and the efficient networks are completely different in our network metrics.
Tradeoff between robustness and smallworld effect
To consider both the robustness and smallworld effect of network in optimization, SMSMOEA is utilized to obtain the Paretooptimal front, namely the best possible set of nondominating solutions. In Fig. 6, we show the Paretooptimal front of and E optimized from the initial solutions, which is generated from an initial scalefree network with nodes N = 100, links M = 179 and powerlaw exponent γ = 3 by executing the mutation operator for 10^{3} times. Here we state the parameter used for SMSEMOA: The population size is 50, the crossover probability is P_{c} = 0.9, and the mutation probability is P_{m} = 0.05. Besides, we iterate the optimization 20,000 times. As expected, the index E of networks in Paretooptimal front decreases monotonically with their robustness in Fig. 6. Hence, the results are welldistributed and have a good convergence, which vertify the effectiveness of SMSMOEA when optimizing the network topology.
We analyze the change in network structure as the optimization orientation for robustness and smallworld effect are shifted. Figure 7 gives the correlation profiles and the exhibition of three selected networks in Paretooptimal solutions. Figure 7(a) and (c) show the topology of the network with high E and the network with high respectively, while the intermediate network with both high and E is shown in Fig. 7(b). In Fig. 8, the topological properties of optimized networks in Paretooptimal front is depicted. Here we can clearly observe the transition from smallworld network structure towards robust network structure.
As described above, the network with low and high E roughly display a multihubs starlike structure, in which most of lowdegree nodes are directly connected to highdegree nodes. Therefore, the network has low assortativity coefficient r, low clustering coefficient C, low network diameter D and low standard deviation of distance distribution σ_{d} (see Fig. 8). Naturally, this structure is very fragile for the removal of hubs. With increasing robustness , the index E of the optimal networks decreases monotonously, and the degree assortativity of network is enhanced. As shown in Fig. 7, links between highdegree nodes and lowdegree nodes obviously decreases, and links among highdegree nodes and links among lowdegree nodes increases. The multihubs starlike structure is gradually changed into a coreperiphery structure. In this structure, the interconnected highdegree nodes form the core, while the periphery is composed of interconnected lowdegree nodes. Increasing , reduce the number of links between the core and the periphery. The coreperiphery structure gradually evolves into a corechain structure. The link density of the core increases, and the periphery expands, and then evolves into long chain or ring substructures (see Fig. 7(c)). The increasing variation of clustering coefficient C, network diameter D and standard deviation of distance distribution σ_{d} validates the transition process of network topology. In addition, we find that as the increase of robustness, the spectral gap SG increases. It indicates that the optimization for robustness will improve the communicability and the expansion of network.
The results also show that the network with both high robustness and smallworld effect exhibits a coreperiphery structure, which is quite similar to the results by Netotea and Pongor^{39}. It is obviously a compromised structure between the multihubs starlike structure and the corechain structure. Optimization for robustness will make the core denser, and expand the chains of the periphery, while optimization for smallworld effect make the core sparser, and fragment the periphery so that each lowdegree node connects to one or more hubs.
Discussion
In this paper, we have emphasized that both robustness and smallworld effect are of great importance for designing or optimizing the network topology. Generally, smallworld effect can be interpreted by the reciprocal of average shortest path lengths, while robustness is positively related to the redundancy of alternative routes in network which can be precisely measured by natural connectivity.
We have verified that robustness and smallworld effect are in a conflicting relation in optimization while conserving node degree by means of large amount of optimization experiments. The efficient network shows a multihubs starlike structure, which is proved to be fragile for the removals of highdegree nodes. Conversely, the robust network has a corechain topology in which highdegree with a cluster of highdegree nodes comprise the core and other lowdegree nodes that are interconnected to form long chain substructures. Obviously, such long chain substructure has a serious problem with the communication among nodes.
To optimize both robustness and smallworld effect, we proposed a multiobjective evolutionary algorithm including crossover, mutation and reduce operators. We have demonstrated that our algorithm can optimize the network topology for both robustness and smallworld effect. We found the networks that realize the tradeoff between robustness and smallworld effect exhibit a coreperiphery structure where highdegree nodes comprise the core and lowdegree nodes form the periphery. Optimizing robustness will strengthen the link density of core and expand the periphery, while optimization for smallworld effect will weaken the core and fragment the periphery. We also investigate several primary topological properties in optimized networks for robustness and smallworld effect to discuss their relation.
As a final point, we note that constraints such as geography or rewiring limitations should be considered in practical applications. When such constraints are imposed, the multiobjective optimization results could potentially be quite different. Moreover, the tradeoff between robustness and smallworld effect in directed or weighted network are also interesting future directions.
Methods
Robustness, smallworld effect and topology metrics of network
In this paper, we employ a rather new robustness measure—natural connectivity —which has been received growing attention^{41,42,49,50,51,52,53,54}. Natural connectivity, derived from the Estrada index^{42,52}, has an intuitive physical meaning and a simple mathematical formulation. Physically, it characterizes the redundancy of alternative paths by quantifying the weighted number of closed walks of all lengths and can also be interpreted as the Helmholtz free energy of a network^{41,42}. Mathematically, the natural connectivity can be derived from the graph spectrum as an average eigenvalue and increases strictly monotonically with the addition of edges. It is defined as:
where N is the number of nodes in a network G and λ_{i} is the ith element of the set {λ_{1}, λ_{2}, …, λ_{N}}, which is called the spectrum of G. It is shown that the natural connectivity has a strong discrimination in measuring the robustness of complex networks and lower computation complexity. The natural connectivity of a network increases in the optimization process, the robustness of the network is improved. Similarly, the network become more fragile as the natural connectivity decreases.
To measure the extent of smallworld effect, as an alternative to the average path length L of a network, we employ the reciprocal harmonic average of shortest distances, also known as efficiency^{43},
where l_{ij} is the geodesic distance from node v_{i} to node v_{j}. We do not use the average path length L or other index related to distances considering allwalks (e.g. Estrada community index^{52}), because the reciprocal harmonic average of shortest distances has a simple definition and desirable mathematical properties. For example, it is normalized to a range of [0,1] and is valid for disconnected networks. We remark that it can be related to the Harary index H(G)^{55}, which is a measure of the compactness of the molecule, as follows
where . The index E is a summary statistic that combines aspects of the sizes of connected components and the average distances within these. Although it is a quite an arbitrary combination of these more fundamental properties (component size and distance), many authors have argued that it is a powerful predictor for how well the both natural and manmade networks function. It is clear, the smallworld effect decreases with both increasing fragmentation (especially splitting large components into similar sizes) and increasing distances. The increase of index E indicates that the information exchanges more fast among nodes in the network. Conversely, the decrease of E means the loss of smallworld effect.
To describe the structure of our optimized networks, we outline a list of standard network topology metrics: the assortativity coefficient r, clustering coefficient C, network diameter D, standard deviation of distance distribution σ_{d} and spectral gap SG. The assortativity coefficient r is a measure of assortative mixing by degree in networks, while the clustering coefficient of a node is the fraction of connected node pairs in the node’s neighborhood. The network diameter D (the maximal distance among node pairs), average shortest path length L and standard deviation of distance distribution σ_{d} are utilized to measure the characteristics of distance distribution. Spectral gap (SG), the difference between the two largest eigenvalues of the adjacency matrix of a network, is also an important metric for network features. For instance, the good expansion (GE) properties in complex networks is shown to be related to the network spectral gap^{56}.
Multiobjective evolutionary optimization algorithm
To solve the multiobjectives optimization model, we utilize an applicable optimization algorithm, Smetric selection evolutionary multiobjective optimization algorithm (SMSEMOA)^{57}. SMSMOEA is proved to be suited for optimization with two and three objectives. It is better than the conventional algorithms such as SPEA2 in convergence and distribution uniformity. Here we describe the three primary parts of SMSEMOA for network robustness and smallworld effect: crossover operator, mutation operator and reduce operator.
Crossover operator
The crossover operator fuses the genetic information from a pair of chromosomes and generate a new chromosome at each iteration. When we swap the local structure information from two selected networks, the main constraint we should consider is to keep each node degree fixed after crossover operation. We use an improved version of the crossover operator proposed in ref. 58. Figure 9 shows the process of crossover operator on node i. If the two networks are unconnected networks, the crossover operator is invalid, and we start the process again. The process of crossover operator on G_{r1} and G_{r2} is to employ the crossover operator on each node in networks according to the crossover probability p_{c}. When the crossover operator is finished, one of the new generated networks is randomly selected as the new solution.
Mutation operator
Mutation operator aims to search new solutions in a local area, which can accelerate the convergence of the algorithm. The neighborhood of the current network is the set of networks that are obtained by rewiring the current network while keeping degrees fixed. At each iteration, the rewiring process shown in Fig. 1 is executed as the mutation operation.
Reduce operator
When a new network is added to the population, the inferior solution in the population should be removed. Unlike the conventional algorithm, SMSEMOA does not maximize the objectives individually, but maximizes the hypervolume spanned by them in the space of objectives. The hypervolume is the area under the Paretocurves and bounded by a reference point^{59}. During the reduce operation, the contributing hypervolumes of each solutions is be calculated, and the inferior solution with lowest contributing hypervolume is eliminated from current population. In the optimization model, hypervolume of solution p_{i} can be calculated by:
where is the robustness of solution p_{i} and E(p_{i}) is the smallworld effect of solution p_{i}. P = {p_{1}, p_{2}, …, p_{v}} is the current population which is arranged by the order of the value of robustness from low to high.
At each iteration, a new solution is generated from the current population by executing the mutation and crossover operations sequentially. Then the solution which has the lowest contribution to the hypervolume is removed. By repeatedly executing the steps above within a large enough number of iterations, a best possible set of nondominating solutions, called Paretooptimal front, is obtained.
Additional Information
How to cite this article: Peng, G.S. et al. Tradeoffs between robustness and smallworld effect in complex networks. Sci. Rep. 6, 37317; doi: 10.1038/srep37317 (2016).
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Acknowledgements
J.W. was supported by the National Natural Science Foundation of China under Grant No. 71371185, and the Program for New Century Excellent Talents in University under Grant No. NCET120141. P.H. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B01007774). We thank Raissa D’Souza, Gene Stanley, Vito Latora, Keith Burghardt, YuanSheng Lin, HaoChen Wu, Andrew Smith and XinCheng Lei for useful discussions and comments.
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J.W. designed research; G.S.P., J.W. and S.Y.T. performed research; G.S.P. conducted the experiments; J.W., G.S.P. and P.H. analysed the results; J.W., G.S.P., P.H. and S.Y.T. wrote the paper.
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Peng, GS., Tan, SY., Wu, J. et al. Tradeoffs between robustness and smallworld effect in complex networks. Sci Rep 6, 37317 (2016). https://doi.org/10.1038/srep37317
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