Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks
Kaj-Kolja Kleineberg, Lubos Buzna, Fragkiskos Papadopoulos, Marián Boguñá, M. ?ngeles Serrano
GGeometric correlations mitigate the extreme vulnerability of multiplex networksagainst targeted attacks
Kaj-Kolja Kleineberg, ∗ Lubos Buzna, Fragkiskos Papadopoulos, Mari´an Bogu˜n´a,
4, 5 and M. ´Angeles Serrano
4, 5, 6 Computational Social Science, ETH Zurich, Clausiusstrasse 50, CH-8092 Zurich, Switzerland University of Zilina, Univerzitna 8215/1, SK-01026 Zilina, Slovakia Department of Electrical Engineering, Computer Engineering and Informatics,Cyprus University of Technology, 33 Saripolou Street, 3036 Limassol, Cyprus Departament de F´ısica de la Mat`eria Condensada,Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain ICREA, Passeig Llu´ıs Companys 23, E-08010 Barcelona, Spain (Dated: July 16, 2018)We show that real multiplex networks are unexpectedly robust against targeted attacks on highdegree nodes, and that hidden interlayer geometric correlations predict this robustness. Withoutgeometric correlations, multiplexes exhibit an abrupt breakdown of mutual connectivity, even withinterlayer degree correlations. With geometric correlations, we instead observe a multistep cascadingprocess leading into a continuous transition, which apparently becomes fully continuous in thethermodynamic limit. Our results are important for the design of efficient protection strategies andof robust interacting networks in many domains.
Networks are ubiquitous in many domains of scienceand engineering, ranging from ecology to economics, andoften form critical infrastructures, like the Internet andfinancial systems. Nowadays, these systems are increas-ingly interdependent [1] and form so-called multiplex ormultilayer networks [2, 3]. This interdependency impliesthat, if a node fails in one network layer, its counterpartsin the other layers also fail simultaneously. This processcan continue back and forth between the layers, whichmakes them especially vulnerable to failures. In particu-lar, an abrupt transition can arise in mutual percolationwhen nodes are removed at random [3–5]. Interestingly,interlayer degree correlations [6–9] mitigate this vulner-ability to random node removals and the transition be-comes continuous [10, 11].In real systems, failures may not always be randombut, instead, the result of targeted attacks. Multi-plexes are extremely vulnerable to them on high-degreenodes [12–14], and exhibit a discontinuous phase tran-sition even in the presence of interlayer degree correla-tions [13]. Although it is highly important for many realsystems, it is not well understood how this vulnerabil-ity can be mitigated. Previous results point to negativeinterlayer degree correlations as a mitigation factor [13],but real systems tend to show positive instead of nega-tive interlayer degree correlations [6]. Are there otherstructural features that render multiplex networks ro-bust against targeted attacks? And most importantly,are these properties present in real multiplexes?Here, we show that interlayer hidden geometric cor-relations [15] mitigate the vulnerability of multiplexesto targeted attacks. The removal of the highest degreenodes triggers multiple cascades which do not destroythe system completely, but eventually lead into a contin-uous percolation transition. Strikingly, we show that the strength of these geometric correlations in real systemsis a good predictor of their robustness.More specifically, we consider targeted attacks in two-layer multiplexes, where nodes are removed in decreas-ing order of their degrees among both layers. We rankall nodes i according to K i = max( k (1) i , k (2) i ), where k ( j ) i denotes the degree of node i in layer j = 1 ,
2. We removenodes with higher K i first (we undo ties at random) andre-evaluate all K i s after each removal. To measure thepercolation state of the multiplex, we compute its mutu-ally connected component (MCC) as the largest fractionof nodes that are connected by a path in every layer usingonly nodes in the component [4].Figure 1 shows results for the real arXiv collabora-tion [16], C. Elegans [17], Drosophila [18], and SaccPomb [18] (see Table I, SM Section I, and SupplementaryVideos I-IV) as well as for their reshuffled counterparts(an illustration of a targeted attack sequence is shown inFig. 2a-d). To create the reshuffled counterpart, we ran-domly reshuffled the translayer node-to-node mappingsby selecting one of the layers and randomly interchangingthe internal IDs of the nodes in that layer. This processdestroys all correlations between the layers without al-tering the layers’ topologies (see SM Section I for furtherdetails). We quantify the vulnerability of the real andreshuffled multiplexes by calculating the critical numberof nodes, ∆ N . The removal of this critical number re-duces the size of the MCC from more than aM to lessthan M β , where M is the initial size of the MCC beforeany nodes are removed, a ≤ β < a = 0 . , β = 0 .
5. The largerthe ∆ N , the more robust (less vulnerable) the system is.For the real arXiv multiplex we find that ∆ N ≈
25, whilefor its reshuffled counterpart ∆ N rs = 1. In fact, in thereshuffled system, the removal of a single node reduces a r X i v : . [ phy s i c s . s o c - ph ] A p r OriginalReshu ffl ed0.80 0.85 0.90 0.95 1.000.000.250.500.751.00 p M CC Arxiv OriginalReshu ffl ed0.5 0.6 0.7 0.8 0.9 1.00.000.250.500.751.00 p M CC CElegansOriginalReshu ffl ed0.80 0.85 0.90 0.95 1.000.000.250.500.751.00 p M CC Drosophila OriginalReshu ffl ed0.85 0.90 0.95 1.000.000.250.500.751.00 p M CC Sacc Pomb a) b ) c) d ) Figure 1. (a)
Relative size of the mutually connected com-ponent (MCC) against the fraction p of nodes remaining inthe system for the arXiv (layers 1, 2) collaboration multiplex(green lines) and for its reshuffled counterpart (red dashedlines). Different lines correspond to different realizations ofthe targeted attacks process. (b) shows the same for the C.Elegans multiplex (layers 2 and 3), (c) for Drosophila (layers1 and 2), and (d) for Sacc Pomb (layers 3 and 4). the relative size of the MCC from 73% to only 0 . a = 40%and √ M /M = 3 . i is represented by its radial (popularity) and angu-lar (similarity) coordinates, r i , θ i , which are both signifi-cantly correlated in different layers, while hyperbolicallycloser nodes in each layer are connected with higher prob-ability (see SM Section I for further details).Radial correlations are equivalent to interlayer degreecorrelations [23]. Angular correlations, instead, lead tosets of nodes that are similar—close in the angular sim-ilarity space—in each layer of the multiplex [15]. Thereshuffling process explained earlier destroys both radial Dataset MCC ∆ N ∆ N rs NMIarXiv Layers 1, 2 790 25.2 1.0 0.58Physicians Layers 1, 2 104 6.0 1.0 0.41Internet Layer 1, 2 4710 81.4 14.1 0.34C. Elegans Layers 2, 3 257 14.0 1.1 0.34SacchPomb Layers 3, 4 426 4.2 1.5 0.17Drosophila Layers 1, 2 449 8.4 2.0 0.26Brain Layers 1, 2 74 7.0 1.0 0.19Rattus Layers 1, 2 158 4.0 1.0 0.18Air/Train Layers 1, 2 67 3.0 3.0 0.10Table I. Analyzed datasets for selected layer pairs (see SMSection I for all layer pairs). MCC denotes the initial size ofthe MCC, ∆ N denotes the critical number of nodes whoseremoval reduces the MCC from 40% to √ M/M (in relativesize), and ∆ N rs the same for the reshuffled system. Valuesare averages over 100 realizations of the removal process. NMIdenotes the normalized mutual information (see SM SectionIX) and gives a measure of the strength of angular correlationsbetween the layers of the considered real systems. and angular correlations between the layers. The extremevulnerability of the reshuffled counterparts in comparisonto the real systems raises fundamental questions: Arethe radial (i.e., interlayer degree) correlations, or angu-lar (i.e., geometric) correlations, or both, responsible forthe robustness of real systems, and which of these corre-lations can help to avoid catastrophic cascading failurewhen multiplexes are under targeted attack?To investigate these questions, we use the geometricmultiplex model (GMM) (see SM Section III) to generatesynthetic two-layer multiplexes, which resemble the realequivalents. The model produces multiplexes with layersembedded into hyperbolic planes, whereby the strengthof interlayer correlations between the radial and angularcoordinates of nodes that simultaneously exist in bothlayers can be tuned by varying the model parameters ν ∈ [0 ,
1] and g ∈ [0 , ν ( ν = 0 for no radial correlations, whereas ν = 1 for maximal radial correlations). Similarly, angu-lar correlations increase with parameter g ( g = 0 for noangular correlations, while g = 1 for maximal angularcorrelations).We find that synthetic multiplexes without angularcorrelations exhibit an extreme vulnerability to targetedattacks (see Fig. 2e, SM Section III, and SupplementaryVideo V), similarly to the reshuffled counterparts of realsystems (cf. Fig. 1 and SM Section II). In particular,if the multiplex is sufficiently large, then the removal ofonly a single node can reduce the size of the MCC from40% to the square root of its initial size, thus destroyingthe connectivity of the system, see Fig. 2f. The abruptcharacter of the transition is also reflected in the distri-bution of mutually connected component sizes. In the a) bcd e) h)f) g)c) d)b) Figure 2. Targeted attacks on synthetic multiplex networks generated by the GMM model (see text). Each layer has power lawdegree distribution with exponent γ = 2 .
6, average node degree (cid:104) k (cid:105) ≈
6, and clustering ¯ c = 0 . (a-d) Here, each layer has N = 500 nodes, and we have set g = 1 and ν = 0. (a) relative size of the MCC as a function of the fraction of nodes remainingin the system p . (b) MCC after the removal of 4 nodes (corresponding to the dashed blue line in (a)). (c) the same as in (b)after the removal of 23 nodes (dashed red line in (a)). (d) the MCC after the removal of 42 nodes (dashed yellow line in (a)). (e)
Evolution of the MCC in a two-layer synthetic multiplex with layers of size N = 2 × nodes. The inset shows the samefor 10 nodes. (f ) critical number of nodes, ∆ N , as a function of the system size N when there are no angular correlations, g = 0, and for different radial correlation strengths. The results are averages over 60 realizations (for N < (g) the same as in (f) but for different values of the angular correlations strength g and for fixed ν = 1. (h) shows the largest and second largest cascade size (relative to system size). fragmented phase, the entire network is always split intovery small components, even when the system is veryclose to the transition (see Fig. 3a and SM Section IV).In the percolated phase, only nodes that do not belong tothe MCC remain fragmented into small components (seeFig. 3b and SM Section IV). This behavior is not affectedby the strength of the radial (i.e., interlayer degree) corre-lations in the system. Thus, in contrast to the mitigationeffect for random failures, interlayer degree correlationsdo not avoid an abrupt transition in the case of targetedattacks, and essentially do not affect the robustness ofthe system.On the other hand, this extreme vulnerability is mit-igated if angular correlations are present. In Fig. 2a-dand e, we show the MCC percolation transition for max-imal angular correlations (see also SM Sections II, III,and Supplementary Video V). We observe that the tran-sition does indeed start with a multistep cascading pro-cess for relatively small system sizes. However, as shownin Fig. 2f and Fig. 2g, the critical number of nodes, ∆ N ,scales with the system size in the presence of angular correlations, see also SM Section V, while it always con-verges to one for large system sizes if angular correla-tions are absent. Moreover, as shown in Fig. 2h, therelative size of the largest jump after a single node re-moval decreases with the system size, in stark contrastto the case without angular correlations, where this quan-tity becomes size independent. This suggests that, in thethermodynamic limit, the system undergoes a continuoustransition (see inset in Fig. 2e). Furthermore, the size ofthe second largest component scales with the system sizelike N σ , with σ ≈ .
84 (see Fig. 3d, e and SM Section VI).Finally, at the transition, the distribution of componentsizes follows a power-law (see Fig. 3c and SM Section IV).Thus, we conjecture that angular correlations can lead toa multistep cascading process for relatively small systemsizes, and can give rise to a continuous transition in thethermodynamic limit (happening in a range of parame-ters of the model—including those used in Fig. 2—suchthat the multiplex layers have strong metric structure butdo not loose the small-world property in the targeted at-tack process, see SM Section VII). This behavior is not a) b) c)d) e)
Figure 3. (a-c) shows the distribution of component sizes(PDF) during the evolution of the MCC for two-layer syn-thetic multiplexes constructed with the GMM model. Eachlayer has a power law degree distribution with exponent γ = 2 .
6, average node degree (cid:104) k (cid:105) = 6, and clustering ¯ c = 0 . (a-c) each layer has N = 5 × nodes. (a) distribution ofcomponent sizes directly before the transition ( p = 0 . (b) directly after ( p = 0 . ν = 0 , g = 0. (c) distribution of com-ponent sizes at p = 0 . g = 1, and no radial correlations, ν = 0. (d) absolute size of the second largest MCC as a function of p fordifferent layer sizes N as indicated in the legend ( × ); foreach size, the results are averages over 60 realizations of themultiplex (as in (e) ). (e) scaling of the maximum of the sec-ond largest MCC. The black dashed line shows a fit ∝ N . ,while the inset shows the value of p = p c where the maximumis realized. affected by the strength of radial (i.e., interlayer degree)correlations and cannot be explained by the link overlapinduced by geometric correlations (see SM Section VIII).Taken together, our results suggest that angular (similar-ity) correlations can mitigate the extreme vulnerabilityof real multiplexes against targeted attacks.We can validate this conclusion in real systems. Tothis end, we compare the vulnerability of each of theconsidered real multiplexes (see Table I and SM SectionI) with that of its reshuffled counterpart. We define therelative mitigation of vulnerability asΩ = ∆ N − ∆ N rs ∆ N + ∆ N rs , (1)where ∆ N and ∆ N rs are the number of nodes needed forthe critical reduction of the size of the MCC of the realand reshuffled systems, see Table I and SM Section I. Ω isa measure of how much more resilient the real networksare compared to their reshuffled counterparts. Next, we Arx12Arx42Arx41Arx28Phys12Arx52Arx15Arx26InternetArx34CE23Phys13Phys23Sac13Sac35Sac23Sac12Dro12CE13Sac14Sac24BrainRattusCE12Sac34AirTrain
NMI Ω Datasets
AirTrainSac34CE12Ra � usBrainSac24Sac14CE13Dro12Sac12 Sac23Sac35Sac13Phys23Phys13CE23Arx34InternetArx26Arx15 Arx52Phys12Arx28Arx41Arx42Arx12 Figure 4. Relative mitigation of vulnerability Ω (Eq. (1)) as afunction of the normalized mutual information
NMI , whichis a measure of the strength of angular correlations betweenthe layers of the considered real systems (see SM Section IXfor details). study how Ω behaves as a function of the strength ofangular correlations in the considered real systems. Wequantify the strength of interlayer angular correlationsby calculating the normalized mutual information,
N M I ,between the inferred angular coordinates of nodes in dif-ferent layers (see SM Section IX). A larger
N M I meanshigher angular correlations. We find a strong positivecorrelation ( ρ ≈ .
6) between the strength of angularcorrelations in the real systems and their relative miti-gation of vulnerability, see Fig. 4. This finding validatesour previous arguments with real data, and highlights theimportance of angular correlations in making real multi-plexes robust against targeted attacks.The gain of robustness due to angular correlations canbe understood intuitively by the formation of macro-scopic mutually connected structures on the peripheryof the hyperbolic disc in each layer. After enough nodesare removed, the remaining multiplex resembles a “dou-ble ring” (Fig. 2c), because the higher degree nodeswhich have been removed had lower radial coordinatesand hence were closer to the center of the disc. If an-gular correlations are present, the remaining lower de-gree nodes that are close in one layer tend to also beclose in the other layer. As a consequence, the doublering contains macroscopic mutually connected structures(Fig. 2d) that sustain connectivity in the system. Noticethat the mitigation of the extreme vulnerability of multi-plexes by the effect of angular correlations is directly re-lated to their geometric nature and cannot be explainedby any topological feature. To support this point, wechecked whether interlayer clustering correlations (beingclustering the topological feature which is more directlyrelated to the metric properties of networks [24]) or edgeoverlap induced by geometric correlations are sufficient toproduce the mitigation effect. The results, see SM Sec-tions VIII and X, indicate that in the absence of angularcorrelations, neither clustering correlations nor overlapcan explain the observed mitigation effect. We take thisto be a new validation of the geometric nature of com-plex networks and of the role of geometric correlations inmultiplexes.To conclude, we have shown that the strength of geo-metric (similarity) correlations in real multiplex networksis a good predictor for their robustness against targetedattacks, providing, for the first time, strong empiricalevidence for the relevance of this mechanism in real sys-tems. Using a geometric multiplex network model, wehave shown that multiplex networks are extremely vul-nerable against targeted attacks, exhibiting a discontin-uous phase transition if geometric (similarity) correla-tions are absent. Contrarily, the presence of such corre-lations mitigates this vulnerability significantly, inducinga multistep cascading process in relatively small systemswhich does not destroy the system completely but leadinto an eventually smooth percolation transition, withresults suggesting that it can be fully continuous in thethermodynamic limit. In particular, the critical numberof nodes that has to be removed to disconnect the systemscales with the system size only if geometric correlationsare present. Our results can help when designing efficientprotection strategies and more robust and controllable in-terdependent systems. In addition, the results highlightthat dependent networks without similarity correlationsare extremely vulnerable to targeted attacks. Finally,our findings pave the way for an exact analysis of thepercolation properties of such systems via their hiddengeometric spaces.K-K. K. acknowledges support by the ERC Grant“Momentum” (324247); L. B. has been supported byprojects VEGA 1/0463/16, APVV-15-0179 and FP 7project ERAdiate (621386); F. P. acknowledges sup-port by the EU H2020 NOTRE project (grant 692058).M. A. S. and M. B. acknowledge support from a James S.McDonnell Foundation Scholar Award in Complex Sys-tems, MINECO projects no. FIS2013-47282-C2-1-P andno. 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