Limits and trade-offs of topological network robustness
LLimits and trade-offs of topological network robustness
Christopher Priester , ∗ , Sebastian Schmitt , Tiago P. Peixoto ∗ E-mail: [email protected]
Abstract
We investigate the trade-off between the robustness against random and targeted removal of nodesfrom a network. To this end we utilize the stochastic block model to study ensembles of infinitelylarge networks with arbitrary large-scale structures. We present results from numerical two-objectiveoptimization simulations for networks with various fixed mean degree and number of blocks. Theresults provide strong evidence that three different blocks are sufficient to realize the best trade-offbetween the two measures of robustness, i.e. to obtain the complete front of Pareto-optimal networks.For all values of the mean degree, a characteristic three block structure emerges over large parts ofthe Pareto-optimal front. This structure can be often characterized as a core-periphery structure,composed of a group of core nodes with high degree connected among themselves and to a periphery oflow-degree nodes, in addition to a third group of nodes which is disconnected from the periphery, andweakly connected to the core. Only at both extremes of the Pareto-optimal front, corresponding tomaximal robustness against random and targeted node removal, a two-block core-periphery structureor a one-block fully random network are found, respectively.
The theoretical investigation of complex networks has proven to be a valuable tool for the study of manyreal-world systems [1–6]. One important aspect is how the topological properties of networks are linkedto their function and robustness [7, 8]. Robustness is defined as the correct functioning in the presenceof disturbances, and it is a desired property of many empirical network systems. The robustness ofnetworks to topological disturbances is a very active field of research [8–11], since it is often assumedthat it is a necessary ingredient for higher-order forms of robustness associated with specific networkdynamics [12–16].One popular way to address topological robustness is by removing nodes from a given network andthen analyzing how connected the network remains as function of the number of nodes removed [7,17,18].In this way, the problem of robustness is mapped to the classical phenomenon of percolation, and theformation of a giant component in the remaining network after the node removals.Recent studies focused on the optimization of the topological robustness of networks, when a givenset of constraints are imposed [11, 19–25]. Most recent works have focused on optimization according todifferent robustness criteria, such as targeted attacks [11, 25, 26] and random failure [25, 26]. However,most real systems are subject to simultaneous types of perturbations, which individually require different,and thus competing strategies to mitigate failure. In order to properly access the inherent trade-offs insuch situations, one needs to combine multiple robustness criteria. A standard technique is to chose aweighted sum of the relevant criteria as the objective function to be minimized or maximized. However,such an approach can be ineffective if the goal is to map all possible trade-off values between theseobjectives. In addition, it also bears the difficulty to define properly scaled objective functions for eachcriterion, such that a weighted sum really reflects the relative importance the multiple criteria.In order to avoid such issues we use a multi-objective optimization approach [27–30], where a completeset of Pareto-optimal solutions is directly obtained. The two objectives we focus on are the topological1 a r X i v : . [ c s . N I] O c t obustness of networks against random and targeted removal of nodes. These two types of robustness areknown to be in a trade-off relation, where increasing the robustness with respect to one type of removalis likely to decrease the other [18, 25, 26]. In particular, it has been recently shown that in absence ofany constraints other than a fixed average degree, the optimization of robustness against random failureleads to a core-periphery structure, where most nodes are connected to a core group, possessing a highaverage degree, which is also internally connected [25]. Although being maximally robust against randomfailure, this core-periphery topology is minimally robust against targeted attacks, since the removal ofthe few core node immediately leads to the vanishing of the giant component. This robustness-fragilityduality is a common feature of real networks with heterogeneous structure; a famous example of whichis the Internet [31].In order to investigate this multi-objective optimization scenario, we follow Ref. [25] and focus onlarge-scale topological features, as parametrized by a stochastic block model [32,33]. This parametrizationallows for arbitrary large-scale mixing patterns, such as assortativity, dissortativity, community structure,core-peripheries, etc., as well as arbitrary local degree distributions. This model is also convenient,since it allows the exact computation of the percolation properties of the system in the limit of largenetworks [25, 34].By analyzing the Pareto-optimal fronts according to the two robustness criteria, we observe that aminimal number of three blocks is sufficient to obtain the optimal fronts, and that in most cases the besttrade-off is realized by a hybrid structure composed of a core-periphery and a third “secluded” group,which is strongly connected internally and marginally connected to the core nodes. The two-block core-periphery of Ref. [25] and the fully random network are recovered at the two extremes of the Paretofronts, for maximum robustness against random and targeted node removal, respectively.This paper is organized as follows. In Sec. 2.1, we define the stochastic block model and in Sec. 2.2our robustness criteria. In Sec. 2.3, the evolutionary multi-objective optimization algorithm is describedbriefly. Sec. 3 presents the results of the optimization for several parameter choices, including the Pareto-optimal fronts and the resulting topologies. In Sec. 4, we finalize with an overall discussion. The stochastic block model defines an ensemble of random networks, in which nodes belong to differentgroups (also called “blocks”), and the probability of an edge existing between nodes is a function of theblock membership of each node. Each block holds a fraction n r of the N nodes of the whole network,where r ∈ [1 , B ] enumerates these blocks and B is the total number of blocks, such that (cid:80) Br =1 n r = 1.Following Ref. [33], each of the B blocks is characterized by an independent degree distribution p rk , whichspecifies the fraction of nodes with degree k in block r .The connections between the blocks are described with a matrix e , where the elements e rs specify thenumber of half-edges per node in block r connecting to nodes in block s . For simplicity of notation, thediagonal elements e rr encode twice the number of edges per node within block r .In the framework of the stochastic block model, the network structure becomes locally tree-like whenthe number of nodes N n r inside each block is sufficiently large. Since the probability of an edge existingbetween any two nodes of groups r and s scales as is Ee rs /N n r N n s ∼ O (1 /N ), with n r ∼ /B and e rs ∼ /B , the probability of an edge existing between any two chosen neighbours will become vanishinglysmall as N → ∞ . Therefore, since local substructures such as triangles are not generated by the model,predictions based on block model calculations can only be accurate for (large) tree-like networks withoutthese local substructures. However, global and meso-scale properties such as community structure [33],assortativity [35], bipartite, core-periphery structures [25], or any other arbitrary mixing pattern are wellcaptured. 2ach block of the network can in principle have an arbitrary degree distribution p rk . However, in thiswork we restrict the degree distribution of each block to be a modified Poisson degree distribution, p MP k = (1 − δ k, ) κ k ( e κ − k ! , (1)where δ ij is the Kronecker delta function. Thereby nodes with zero degree ( k = 0) are explicitly ex-cluded, since they can never belong to the giant component. In contrast to a regular Poisson distribution p P k = κ k e − κ /k !, where the mean degree is directly given by κ , the mean degree of the modified Poissondistribution is given by (cid:104) k (cid:105) MP = κ/ (1 − e − κ ), which is always bigger than κ . In particular, the meandegree cannot be less than one, (cid:104) k (cid:105) MP ≥ S i ze o f t h e g i a n t c o m p o n e n t h k i p k k h k i = 1 . Regular Poisson distributionModi(cid:28)ed Poisson distribution
Figure 1:
Giant connected component of a Erd˝os-R´enyi random network with a Poisson(dashed green) and modified Poisson (solid red) degree distribution as function the averagemean degree of the network.
The inset shows the regular and modified Poisson distribution for amean degree of (cid:104) k (cid:105) = 1 . (cid:104) k (cid:105) CP = 1. In the case of the modified Poisson distri-bution, the transition is shifted to (cid:104) k (cid:105) CMP = e/ ( e − ≈ .
58, as can be seen in Fig. 1. This is a directconsequence of the fact that no nodes with degree k = 0 are allowed in the later case. For a modifiedPoisson network to have low mean degree, (cid:104) k (cid:105) MP (cid:39)
1, a large fraction of nodes needs to have degree one.In order to achieve this, many of the k − (cid:104) k (cid:105) MP (cid:39) k = 1 diminishes, a macroscopic giant component can form. In this case, the nonexistence of discon-nected k = 0 nodes results in larger connected components in general and leads to the stronger increaseof the size of the giant components as can be seen in Fig. 1.Apart from the degree distribution of each block, p rk , more parameters need to be specified in orderto define a realization of a block model ensemble. These are the total number of blocks B , the relativesize of each block n r , the mean degree of each block (cid:104) k (cid:105) r , as well as the edges connecting the blocks givenby e rs . These parameters are, however, not completely independent as the relative sizes n r of all blocksmust add up to one, (cid:80) r n r = 1, and the sum of all the edges incident to one block is related to its meandegree, (cid:104) k (cid:105) r = (cid:80) s e rs /n r . Since we will always consider networks with a given total mean degree (cid:104) k (cid:105) ,the following constraint will need to be fulfilled, (cid:104) k (cid:105) = (cid:80) r n r (cid:104) k (cid:105) r . Failure in networks is modeled by removing a finite fraction q of nodes from the network. We will considertwo different strategies for selecting which nodes are removed. The first is random removal where thenodes to be removed are selected purely randomly. The second is targeted removal where nodes withhigher degree are more likely to be removed.Both types of failures are inspired by real-world technical networks. Random removal is consideredto model fatigue of parts or other random influences. Targeted removal is inspired by the fact thathighly loaded nodes are more likely to fail or, in the context of critical infrastructure, malicious damageis preferably brought to important nodes.In the context of block models, where we only model representative nodes in an statistical ensemble,we employ a slight variation of the targeted removal which was also used in Ref. [25]. The targetedcriterion is only applied to the selection of blocks where the fraction of nodes to be removed from block r is proportional to e (cid:104) k (cid:105) r and thus increases with the mean degree of the block. However, within eachblock no further targeted removal of nodes is performed and nodes are removed at random. In case of allblocks having the same mean degree targeted removal is identical to random node removal.As a measure of robustness of a network we use the size of the macroscopic component S ( q ) aftera finite fraction q of nodes has been removed. Instead of focusing on the robustness when removing asingle fraction q , all possible values 0 ≤ q ≤ R = 2 (cid:90) S ( q ) d q , (2)where the factor of 2 serves to adjust the range of R to be [0 , R = 0 is achievedby networks without a macroscopic component, even when no nodes are removed at all. The oppositelimiting case of R = 1 requires a fully connected network where S ( q ) = 1 − q .Following Ref. [25] using the generating function formalism [36] the size of the macroscopic componentis calculated using u r , which is the probability that a node in block r is not connected to the macroscopiccomponent via one of its neighbors. These probabilities for all blocks have to fulfill a system of B u r = (cid:88) s m rs (cid:20) φ s κ s (cid:0) g (cid:48) ,s ( u s ) − (cid:1)(cid:21) , (3)where m rs ≡ e rs /n r κ r is the fraction of edges in block r leading to block s , n r and (cid:104) k (cid:105) r are the relativenumber of nodes and mean degree of block r , respectively, and g ,r ( z ) = (cid:80) k p rk z k is the generatingfunction of the degree distribution of block r and g (cid:48) ,r ( z ) = ∂∂z g ,r ( z ) is its derivative. φ r ∈ [0 ,
1] is thefraction of nodes not removed from block r . The φ r have to be chosen in accordance with the noderemoval strategy, for example, φ r = φ for random removal or φ r ∝ e −(cid:104) k (cid:105) r for targeted removal. Sincethe total fraction of removed nodes is given by q , the φ r need to satisfy the relation q = 1 − (cid:80) s φ s n s .Due to this requirement, the φ r for targeted removal need to be determined by numerically solving0 = 1 − q − (cid:80) r n r exp( −(cid:104) k (cid:105) r (1 − x ) /x ) for x and using the solution x ∗ to get φ r = exp( −(cid:104) k (cid:105) r (1 − x ∗ ) /x ∗ ).The solutions of these equations for all u r allows for the calculation of the size of the giant connectedcomponent S ( q ), S ( q ) = (cid:88) s n s φ s [1 − g ,s ( u s )] . (4)At this point, a few remarks about the interpretation of the value of S ( q ) should be made. Since weare parametrizing the system with intensive quantities ( e rs , n r , u r , etc.) which specify fractions of nodesand edges in infinitely large systems, we cannot differentiate between the existence of single or multiplemacroscopic components for a given value of S ( q ). In other words, if two macroscopic components areconnected by a single edge (or more generally, any intensive number of edges) the probability of edgesbetween them vanishes in the infinite size limit. Thus, this situation cannot be distinguished from twotruly disconnected macroscopic components where no edges exists between the two components. Forthe purposes of this work, we consider this issue to be unimportant, and we focus on the existence ofmacroscopic components in the more abstract sense as given by the value of S ( q ) directly.For each node removal strategy, Eqs. (3) have to be solved for all q in order to calculate the robustness R of a specific block model ensemble. In our case, this leads to two different measures of robustness, R Random and R Targeted , for random and targeted node removal, respectively.
In order to consider both robustness measures, R Targeted and R Random , at once, we utilize a multi-objective [27–30] evolutionary optimization [37, 38] algorithm. Unlike in the optimization of a singleobjective, where it is always possible to state if a certain solution A is better, worse or equally goodcompared to a solution B , this is not necessarily possible in multi-objective optimization. If a solution A performs better than a different solution B in one objective, but worse in a second objective, no statementis possible which of the solutions is better. Only if solution A is better than B in at least one objectiveand not worse in any objective it can be considered generally better and it is then said that A dominates B . Sets in which no solution dominates any other solution are called non-dominating . In general, a multi-objective optimization will not result in a single best solution but in a set of non-dominating solutionswhich ideally is close to the best possible set of non-dominating solutions, the Pareto-optimal front. Thesenon-dominating sets are very useful to study the trade-off relation between the robustness R Targeted and R Random and their relation to the structure of optimal networks.The algorithm we use here is the so called
S-metric selection evolutionary multi-objective optimizationalgorithm (SMS-EMOA) [39]. It is a population based evolutionary stochastic search algorithm whichdoes not utilize any gradient information and is well suited for non-convex and noisy optimization prob-lems. The algorithm does not optimize the objectives directly but instead maximizes the hypervolume inobjective-space dominated by the population and bound by a reference point. In the present case of twoobjectives, the hypervolume is given by the area under the Pareto-curves as, for example, shown in Fig. 7.5t each iteration the solution whose removal leads to the lowest decrease in the dominated hypervolumeis removed from the population and a new solution is generated by recombination and mutation (for moredetails see [40]).Repeating the steps of removing the least contributing solution and generating a new solution notonly shifts the solution set closer to the Pareto-optimal front but also leads to a broad distribution alongthe front, two desired properties of an optimal set of solutions. For each optimization run, we fix the number of blocks B and the mean degree of the completenetwork, (cid:104) k (cid:105) . Each block has a modified Poisson distribution as its degree distribution (cf. Eq. (1)), butthe average mean degree of each block can vary. Therefore, the free variables subject to optimization,i.e. the search parameters, are the relative size of each block, n r , the mean degree of each block, κ r ,and the entries in the matrix containing edges within and between the blocks, e rs . With the sum rulesand constraints stated at the end of Section 2.1, this results in B ( B + 1) + B − In Fig. 7 we show robustness values obtained from different optimizations for several numbers of blocks( B = 2 , , , (cid:104) k (cid:105) = 2 .
5. The Pareto-optimal fronts for optimizationswith B = 3, 4, and 5 blocks match exactly. Only the B = 2 result deviates and yields lower robustnessover large parts of the Pareto-optimal front.The network structures corresponding to the Pareto-optimal solutions for B = 3, 4, and 5 blocks arealso identical (not shown) . The same behavior was found for other values of the mean degree (cid:104) k (cid:105) , wherethe results for B ≥ B = 2.This leads to the conclusion that three blocks are sufficient to describe networks which are maximallyrobust against random and targeted node removal. At both extremes of the Pareto-optimal fronts allcurves coincide, which means that for optimizing only with respect to one objective (i.e. single-objectiveoptimization), B ≤ B = 2 core-periphery and a B = 1 fully random structures were found as optimum for random and targeted noderemoval, respectively. This is also consistent with the findings of Valente et al. [26] who showed two-and three-peak degree distributions to be optimal when minimizing percolation thresholds of networkssubject to random and targeted removal of nodes.The Pareto-optimal fronts of optimized block model networks with three blocks ( B = 3) and a meandegree (cid:104) k (cid:105) between 1 . . (cid:104) k (cid:105) (cid:46) .
5, the two types of robustness are in strong trade-offrelation: Increasing the robustness against targeted removal strongly decreases the robustness againstrandom removal (and vice-versa).Pareto-optimal solutions of networks with (cid:104) k (cid:105) (cid:46) R Targeted = 0.But for (cid:104) k (cid:105) >
2, networks with the maximal value of R Random shift to have a finite robustness againsttargeted removal, R Targeted >
0. In this case, networks with lower robustness against targeted removalare not accessible via the multi-objective optimization, since they are not Pareto-optimal (i.e. they are For completeness, we state the parameters used for the SMS-EMOA: A population size of 50 is used, the crossoverprobability is p c = 1, the crossover distribution parameter is η m = 20, the mutation probability is p m = 1, and the mutationdistribution parameter is η c = 15. Two structures with different B values are considered identical when their structural entropy is the same. See Sec. 3.2for more details. .50.550.60.650.70.75 0.1 0.2 0.3 0.4 0.5 0.6 R R a nd o m R Targeted h k i = 2 . B = 2 B = 3 B = 4 B = 5 Figure 2:
Pareto-optimal fonts of robustness against targeted and random removal of nodesfor mean degree (cid:104) k (cid:105) = 2 . and various number of blocks. dominated by the solutions with maximal R Random , see Refs. [27–30]). However, they can be found byperforming an optimization with the value of R Targeted fixed, and such results are shown as the smallergray symbols in Fig. 8. The Pareto optimal front together with these additional solutions form the wholetrade-off curve for each (cid:104) k (cid:105) .With increasing mean degree, the trade-off curves become very flat, indicating that a slight sacrificeon the robustness with respect to random removal yields a great enhancement in the robustness againsttargeted removal. Additionally, the curves increasingly approach the diagonal where R Random = R Targeted ,which means that there are solutions which are equally good in both measures.In general R Random is always greater or equal to R Targeted , and for (cid:104) k (cid:105) (cid:38) .
5, the Pareto-optimalfronts extend to the diagonal. In random networks, nodes with high degree are important for the sizeof the giant component since they naturally are more likely to connect different components. Due tothis, a removal mechanism targeting high degree nodes is able to degrade the giant component easily byremoving a relatively small amount of high degree nodes. Therefore, making the degree distribution of a7 R R a nd o m R Targeted h k i = 1 . h k i = 2 . h k i = 2 . h k i = 3 . h k i = 3 . Figure 3:
Pareto-optimal fronts of R Random versus R Targeted for optimal block model networkswith B = 3 and for various mean degrees (colored symbols). For (cid:104) k (cid:105) >
2, the smaller graysymbols to the left of each Pareto front indicate solutions which maximize R Random for fixed R Targeted but which are not Pareto-optimal (see main text).network narrow should increase the robustness against targeted removal since there are less high-degreenodes. In a block model with several blocks a narrow degree distribution implies that all blocks have thesame mean degree (cid:104) k (cid:105) r = (cid:104) k (cid:105) . Since, in this work, targeted removal only differentiates between blocksbut not between nodes inside the block, targeted and random removal are identical if all blocks have thesame mean degree. As a consequence, the robustness values are then equal, R Random = R Targeted .In contrast, for (cid:104) k (cid:105) (cid:46) .
0, the Pareto-optimal fronts do not extend to the diagonal, which is aconsequence of the percolation properties of fully random B = 1 networks (cf. Section 2.1). For lowmean degrees, the giant connected component of a fully random network is very small even without noderemoval ( q = 0). Due to the steep increase of the giant component with increasing mean degree (cf.Fig. 1), it is beneficial to have two blocks with differing mean degree, one higher and the other lowerthan the total average mean degree (cid:104) k (cid:105) . The block having a mean degree greater than (cid:104) k (cid:105) , also has asubstantial larger giant component, while the giant component of the other block is still small (or evenzero). Therefore, the argument presented above for (cid:104) k (cid:105) (cid:38) .
5, where a finite giant component at q = 0always exists, is not effective for (cid:104) k (cid:105) (cid:46)
2. It is always beneficial to have (at least) two blocks in order tohave increase the size of the giant component for q = 0.8 .2 Network structures In our approach, the number of blocks B is set a priori and kept fixed during a single optimizationprocedure. However, networks with different values of B could have equivalent topologies. This canhappen if one or more blocks have a vanishing size n r and mean degree (cid:104) k (cid:105) r , or when two or more blockscan be merged together without altering the ensemble of generated networks.For a clearer visualization and analysis of block model structures, we reduce the number of blocksby removing insignificant blocks and by merging multiple blocks into one if they are equivalent. For twoblocks to be equivalent, we require that the entropy of the merged and the original network ensemblesdiffer by a very small amount. The entropy of the stochastic block model ensemble is simply the logarithmof the total number of networks which can be generated given a specific parametrization, i.e. choices of n r and e rs . The entropy is a signature of the ensemble, and determines how random it is. If the entropyremains the same after two blocks are merged into one, this means that these two groups correspondsimply to an arbitrary subdivision of a larger group, and they do not in fact constrain the topology inany way. The entropy of block model networks is calculated as described in Ref. [41]. We emphasize that,since the topologies in this case are in fact equivalent, the effect of the merging process on the robustnessvalues was found to be negligible.We now consider the Pareto fronts separately for different values of the mean degree. (cid:104) k (cid:105) = 2 . (cid:104) k (cid:105) = 2 . R Targeted → n periphery ≈
1) and which has the lowest mean degree possible in this kind of structure (cid:104) k (cid:105) periphery ≈ (cid:104) k (cid:105) / .
25. The core block contains only very few ( n core ∼ − ) but very high-degreenodes ( (cid:104) k (cid:105) core ∼ ). Almost all of the edges are between the core and the periphery.The core is central for forming the giant component, but takes up only a very small fraction of thenetwork. Therefore, random removal will almost always affect periphery nodes and the giant componentwill shrink approximately linearly with the number of removed nodes, which is as slow as possible.On the other hand, the core-periphery structure is maximally fragile with respect to targeted removal,since removing the core completely removes the giant component.With increasing robustness against targeted removal, a third block emerges for R Targeted (cid:38) .
07 inaddition to the core and periphery block. This new block, which we will call the secluded block is firstof medium size ( n secluded ≈ .
16) and has a mean degree of (cid:104) k (cid:105) secluded ≈
4. In contrast to the core andthe periphery block, it has a substantial amount of edges internally, i.e. edges between nodes within thisblock (green square in the Hinton plots). The secluded block is only lightly connected to the core blockand no edges exist between secluded and periphery block.Increasing R Targeted further, the mean degree of the secluded block slightly decreases, while it growsin size. The number of nodes in the periphery continuously decreases and around R Targeted ≈ .
24 thesecluded block is larger than the periphery. For very high R Targeted (cid:38) .
52 the secluded block dominatesand the core periphery structure vanishes. The result is a single block network with a modified Poissondegree distribution, as it was already mentioned in the discussion at the end of Section 3.1 in connectionwith Fig. 8.Considering the complete Pareto-optimal set of solutions, the dominant structure is a three-blockstructure with a small but very high degree core, a large but low degree periphery and an additionalsecluded block which has a medium mean degree. Connections only exist between the core and theperiphery, and between the core and the secluded block. The structure is best qualified as a modified9 .540.600.660.72 R R a nd o m (cid:1) k (cid:0) r (cid:1) k (cid:0) r Targeted -4 -2 n r Figure 4:
Parameters of the optimized networks as a function of R Targeted obtained from athree-block optimization with (cid:104) k (cid:105) = 2 . . The upper row shows the elements of the edge matrix e rs ,where the areas of the squares is proportional to the logarithm of the element. The positions for whichthese Hinton plots are shown are marked with dashed lines in the other panels. The second row showsthe trade-off curve already displayed in Fig. 7, while the third and fourth display the mean degree ofthe blocks, on a logarithmic and a linear scale, respectively. The last row shows the relative sizes of theblocks. The coloring of the blocks and their index is determined by their mean degree, where the blockwith the highest mean degree is shown in blue and always has index r = 0, followed by green with r = 1(second highest) and red ( r = 2, lowest degree).core-periphery with a regular Erd˝os-R´enyi network attached to the core. The relative size of the secludedErd˝os-R´enyi block compared to the core-periphery structure grows with an increased robustness againsttargeted node removal. (cid:104) k (cid:105) = 2The structures of Pareto-optimal networks with a low mean degree of (cid:104) k (cid:105) = 2 are shown in Figure 10.The resulting structures are overall quite similar to the previously discussed case with (cid:104) k (cid:105) = 2 .
5. For R Targeted → R Targeted is increased. 10igure 5:
Parameters of the networks along the trade-off curve for the five block optimizationwith (cid:104) k (cid:105) = 2 . . The panels are the same as in Fig.9However, a striking difference to the situation for (cid:104) k (cid:105) = 2 . (cid:104) k (cid:105) periphery (cid:38)
1, which implies thatthe majority of nodes has exactly degree one. Therefore, most edges within the periphery produce anisolated pair of two nodes not connected to any macroscopic component (cf. discussion of the modifiedPoisson distribution in Section 2.1).At first glance, this seems contradictory as the giant component is already reduced without any noderemoval ( q = 0). However, this is beneficial for the overall robustness as it allows for the rest of the nodesto have a higher mean degree putting it further above the percolation threshold. As can be seen fromFig. 1, this is especially effective for increasing the macroscopic component of the secluded block as itsmean degree of (cid:104) k (cid:105) secluded ≈ . q after the optimization. However, this would make the analysis significantlymore complicated, and would only affect the outcome of very sparse networks.Interestingly, this holds for the two-block core-periphery structure as well for R Targeted = 0. A pure11ore-periphery structure is especially expected for (cid:104) k (cid:105) = 2, since then the extreme topology of a star canbe realized (the Hinton of which plot is not shown in Fig. 10). However, a very slight increase in therobustness against targeted removal to R Targeted ≈ . n core ≈ × − to n core ≈ × − while its mean degree is reducedfrom (cid:104) k (cid:105) core ≈ × to (cid:104) k (cid:105) core ≈ . × . With this structural change, a little robustness againstrandom removal is lost, but the a finite number of edges is realized within the core which provides a finiterobustness against targeted removal.As expected from the discussion at the end of Section 3.1, the structure with a maximal R Targeted consists of two blocks, one with a mean degree (cid:104) k (cid:105) > (cid:104) k (cid:105) and another with (cid:104) k (cid:105) < (cid:104) k (cid:105) . Both blocksin fact form largely independent components since there are very few connections between them. Thisis a situation similar to the “onion-like” structure found in Ref. [11] when optimizing against targetednode removal while preserving a heterogeneous degree sequence. There, the nodes with higher degree arekept isolated from the rest of the network, hence effectively functioning simply as “bait” for the targetedremoval, whereas the rest of the system remains intact.It is very remarkable that for (cid:104) k (cid:105) = 2 the mean degree of the three blocks stay constant over thecomplete range of the Pareto-optimal front (apart from the far extremes). The optimal trade-off betweenthe two robustness measures can be achieved by only changing the connection matrix and the relativesizes of the blocks. (cid:104) k (cid:105) = 3 . (cid:104) k (cid:105) = 3 . R Targeted (cid:38) .
48, displays the same three-block structures as for (cid:104) k (cid:105) = 2 . R Targeted where R Targeted = R Random (cf. Fig. 9).For the part of the trade-off curve to the left of the maximum of R Random (i.e. for R Targeted (cid:46) . n (cid:38) .
9, and has the lowest mean degree of (cid:104) k (cid:105) ≈
2. Asecond block is very small with 10 − < n < − and has a high degree 10 < (cid:104) k (cid:105) < and thereforestrongly resembles the core block. The third block is of intermediate size and degree, 10 − < n < − and 10 < (cid:104) k (cid:105) < , respectively. The midsized and the small block are only connected via the largestblock since there are no direct edges between them.For very low robustness against targeted removal, R Targeted (cid:38)
0, most of the edges are between thecore and the largest block with (cid:104) k (cid:105) = 2. With increasing robustness against targeted removal the numberof edges between core and the (cid:104) k (cid:105) = 2 block decreases while more edges emerge between the (cid:104) k (cid:105) = 2 blockand the midsized block. At around R Targeted ≈ .
19 the same number of edges exist from the (cid:104) k (cid:105) = 2block to both of the other two blocks. For higher R Targeted more edges exist between the (cid:104) k (cid:105) = 2 blockand the midsized block.This structural evolution can be understood by noting that the largest part of the network always hasa mean degree very close to two and acts a connecting layer between the core and the midsized block.For low R Targeted , very few edges are between the block with (cid:104) k (cid:105) = 2 and the midsized block, so that aconnecting path between two different nodes of the midsized block is very likely to traverse one of thefew core nodes. Therefore, removing the core quickly fragments the network into small components. Onthe other hand, increasing the number of edges between the (cid:104) k (cid:105) = 2 and the midsized block, a connectingpath between nodes within the midsized block is more likely to involve no nodes from the core. Thecore becomes increasingly unimportant and therefore the robustness with respect to targeted removalincreases. 12igure 6: Parameters of the networks along the trade-off curve for the five block optimizationwith (cid:104) k (cid:105) = 3 . . The panels are the same as in Fig.9.
In this paper we investigated the trade-offs between topological robustness of networks against randomand targeted node removals. We used the stochastic block model to parametrize arbitrary mixing patterns,and a multi-objective optimization algorithm to obtain the Pareto-optimal fronts. It was found that inorder to achieve a Pareto-optimal combination of robustness against random and targeted removal, anetwork composed of at most three different blocks is sufficient. In many cases the networks along thePareto-optimal fronts are composed of a hybrid topology, comprised of a core-periphery structure, inaddition to a secluded group, which is only sparsely connected to the core of the network, and not at allwith the periphery.At the edges of the Pareto-fronts, where one of the two robustness criteria is maximized, one or twoblocks suffice to obtain optimal networks: A two-block structure is maximally robust against randomfailure, and a fully random network with one block is sufficient in the case of targeted removal. Thisreproduces the results of Ref. [25], and is also consistent with the earlier findings of Valente et al. [26]who found two- or three-peak degree distributions to be optimal when minimizing percolation thresholds,with networks which are otherwise fully random. 13or low mean degrees of the overall network, the optimal robustness values are generally lower thanfor higher mean degrees and a significant trade-off exists between robustness against random and targetedremoval. With increasing mean degree the strong trade-off diminishes and it is increasingly possible toobtain a high robustness with respect to both criteria. This implies that a network optimized againstone type of failure does not necessarily lose much of its robustness when it is subsequently optimizedagainst the other type of failure or attack. Hence this shows that increasing the overall connectivity of thenetwork not only has the expected trivial effect of increasing each robustness criterion individually, but toa large extent also allows for them to be fulfilled simultaneously. This suggests that the simple strategy ofincreasing the total number of edges in the network, if combined with the optimal large-scale structurespresent in the Pareto-optimal front, can be much more beneficial than could be expected otherwise.A comparison of the large-scale structures we find with the ones observed in empirical systems [ ? ,31,42]is a natural and important extension of this work, and one we intend to pursue in the future. The mostappropriate approach is to search for precisely the same type of model we are using in the analysis,which can be done by inferring the parameters of the stochastic block model itself from empirical data,which is a very active field of research [32,33,43,44]. In fact, core-periphery structures have already beendetected, such as the topology of the internet at the autonomous systems level presented recently in [ ? ].However, to our knowledge, an empirical verification of the specific structures we have found has not yetbeen made.In this work we have considered maximally robust networks that are obtained when very few con-straints are imposed. This gives us fundamental limits on the multi-objective optimization against randomfailures and targeted attacks. However, in empirical systems, exogenous constraints are almost alwayspresent, such as geographical confinement, and restrictions due to functional performance. In previousstudies [11,45], the optimization against targeted node removal was considered when the degree sequenceof an empirical network is preserved. It was found that an assortative structure emerges as the optimumin this case, where nodes become connected with other nodes with similar degree. This has been obtainedas well by imposing similar constraints with the block model approach in Ref [25]. However, it is stillunknown how the multi-objective optimization would behave when these constraints (and other morerealistically motivated ones) are simultaneously imposed. We leave these questions for future work.14 upporting Information R R a nd o m R Targeted h k i = 2 . B = 2 B = 3 B = 4 B = 5 Figure 7:
Pareto-optimal fonts of robustness against targeted and random removal of nodesfor mean degree (cid:104) k (cid:105) = 2 . and various number of blocks. For three, four and five blocks the curvesmatch exactly which implies that no more than three blocks are necessary to achieve the best robustnessvalues. Since at the and of the curves all of them match only two or even one block is enough to achievebest robustness.
Acknowledgments
The authors acknowledge fruitful discussions with Barbara Drossel. C.P. and S.S. additionally acknowl-edge the support and discussions with colleagues at the Honda Research Institute Europe GmbH. T.P.P.was funded by the University of Bremen, zentrale Forschunsf¨orderung Linie 04. C.P. was funded by theHonda Research Institute Europe GmbH. 15 R R a nd o m R Targeted h k i = 1 . h k i = 2 . h k i = 2 . h k i = 3 . h k i = 3 . Figure 8:
Pareto-optimal fronts of R Random versus R Targeted for optimal block model networkswith B = 3 and for various mean degrees (colored symbols). For (cid:104) k (cid:105) >
2, the smaller graysymbols to the left of each Pareto front indicate solutions which maximize R Random for fixed R Targeted but which are not Pareto-optimal (see main text).
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