A Framework of Hierarchical Attacks to Network Controllability
GGENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2020 1
A Framework of Hierarchical Attacks to NetworkControllability
Yang Lou, Lin Wang,
Senior Member, IEEE , and Guanrong Chen,
Life Fellow, IEEE
Abstract — Network controllability robustness reflectshow well a networked dynamical system can maintain itscontrollability against destructive attacks. This paper in-vestigates the network controllability robustness from theperspective of a malicious attack. A framework of hierarchi-cal attack is proposed, by means of edge- or node-removalattacks. Edges (or nodes) in a target network are classifiedhierarchically into categories, with different priorities toattack. The category of critical edges (or nodes) has thehighest priority to be selected for attack. Extensive experi-ments on nine synthetic networks and nine real-world net-works show the effectiveness of the proposed hierarchicalattack strategies for destructing the network controllability.From the protection point of view, this study suggests thatthe critical edges and nodes should be hidden from theattackers. This finding helps better understand the networkcontrollability and better design robust networks.
Index Terms — attack strategy, complex network, criticaledge, network controllability, robustness
I. I
NTRODUCTION M ANY real-world systems can be modeled as complexnetworks, which have gained growing recognition andpopularity since the late 1990s, now becoming a self-containeddiscipline encompassing computer science, systems engineer-ing, statistical physics, applied mathematics, and social sci-ences [1]–[4]. In practical applications, it is essential to deter-mine whether or not a networked system can be controlled forutilization. Consequently, network controllability has becomea focal research topic in network studies [5]–[15]. Same asthe classical concept for systems, controllability here refers tothe ability of a dynamical network being steered by externalinputs from any initial state to any desired target state underan admissible control input within a finite duration of time.On the other hand, random failures and malicious attacks oncomplex networks have become more and more frequent andsevere recently [16]–[21]. Such failures and attacks take placein the form of node- and edge-removals, causing significantconsequences to the systems such as malfunctioning or evencompletely crashing. For example, failures of traffic lights may
Y. Lou and G. Chen are with the Department of Electrical Engineer-ing, City University of Hong Kong (e-mails: [email protected];[email protected]).L. Wang is with the Department of Automation, Shanghai Jiao TongUniversity, Shanghai 200240, China, and also with the Key Laboratoryof System Control and Information Processing, Ministry of Education,Shanghai 200240, China (e-mail: [email protected]).This research was supported in part by the National Natural ScienceFoundation of China under Grant 61873167, and in part by the NaturalScience Foundation of Shanghai under Grand 17ZR1445200.(
Corresponding author: Lin Wang. ) cause traffic congestion in the urban transportation networks;neurological disorders may cause dysfunction or illness tohumans. To resist attacks or failures, strong robustness isdesirable and often necessary for a practical networked system.In different scenarios, there are different definitions and mea-sures for network robustness [22]. Since theoretical analysisseems impossible for large-scale complex networks, at least inthe present time, the correlation between network robustnessand topological features are generally investigated empirically,taking advantages of super-computing power available today[22]–[24]. In this pursuit, it is worth mentioning that thedevelopment of deep learning techniques offers an efficientoption for empirical studies [25]–[27].The notion of random failures and malicious attacks oncomplex networks, as well as the corresponding networkrobustness, covers a broad range of subjects. This paperconcerns with the network controllability robustness , whichrefers to how well a networked dynamical system can maintainits controllability against random failures or, in particular,intentional attacks.The issue of network robustness within different contextsregarding network topologies has been extensively investi-gated, and many edge- and node-removal attack strategies havebeen proposed to destruct the connectedness of the networks.Generally, attack strategies can be categorized into random and targeted attacks. A targeted attack aims at removing anintentionally selected object (e.g., the highest-degree nodeor largest-betweenness edge), while a random attack do theremoval randomly.In the above studies, it is commonly assumed that necessaryknowledge of the network is known and is updated after eachattack. For targeted attacks, it is also assumed that the targetedobject is more important than the others in maintaining thenetwork connectedness. Commonly used measures of impor-tance include degree centrality, betweenness centrality, neigh-borhood similarity [28], branch weighting [29], and structuralholes [30]. However, ranking the importance of nodes oredges is practically intractable for large-scale networks, sincemost measures cannot guarantee that removing the targetedobject will definitely cause a greatest effect of damage on thenetwork.The size of the largest connected component (LCC) iswidely used as a measure for connectedness robustness [18].It is observed that betweenness-based attacks may becomeless effective in the later stage of an attack process. Thisobservation consequently leads to the effective conditionalattack strategy: to remove the global highest-betweenness node a r X i v : . [ phy s i c s . s o c - ph ] A ug GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2020 only if it belongs to LCC; otherwise, to remove the localhighest-betweenness node inside the LCC [31]. In [32], degreeand betweenness are used simultaneously, with predefinedweights for their balance, as the measure of importance. Themodule-based attack strategy [33], [34] aims at attackingthe nodes with inter-community edges, which are crucial tomaintain the connectedness among communities. The damage-based attack [35] uses the degree of damage to measure theeffectiveness of an attack, where the damage of an attack isdefined by the change of the LCC size before and after theattack. It is also observed that the evolution process of attackand defense can enhance the network robustness [36], whichis similar to the process of a mutual improvement of spearsand shields.Although the robustness of connectedness has a certainpositive correlation with the robustness of controllability ona network, they actually have very different measures andobjectives, as illustrated by the simple example shown in Fig.1, where the driver node is a node to be controlled by aninput so as to make the whole network become (or returnto be) controllable (after the attack). This paper is concernedwith the interplay of the connectedness, attack strategies, andcontrollability robustness of general complex networks. (a) (b) remove node input unmatched node matched nodematching non-matching ii Fig. 1 : [color online] Example of controllability and connect-edness robustness: (a) given a star-shaped network, the numberof required driver nodes is ; its LCC is ; (b) after the centralhub is removed, the number of required driver nodes becomes ; its LCC drastically dropped to .Specifically, this paper focuses on the attack strategies thataim at destructing the network controllability. It is observedthat removing highest-degree nodes [37] or highly-loadededges [38] are more effective to degrade the network controlla-bility than random removals. Furthermore, in [39] it is shownthat node-removals are more harmful than edge-removals tothe network controllability, and that heterogeneous networksare more vulnerable than homogeneous networks. Also, it isfound that for many real-world networks the betweenness-based attacks are the most destructive to the controllability[39]. Moreover, it is reported in [40] that degree-based node-removal attacks cause greater damage to local-world networks[41] with a larger local-world size, while networks withsmaller local-world sizes are more robust regarding bothconnectedness and controllability. Notably, the hierarchicalstructure of a directed network enables the random upstream(or downstream) attack, which removes the upstream (ordownstream) node of a randomly-picked one, resulting ina more destructive attack strategy than the simple randomattacks [19]. For a directed network, edges can be categorized into three types [5]: 1) critical edges, whose removal destroysthe network controllability; 2) redundant edges, whose re-moval has no influence on the controllability; and 3) ordinary edges, whose removal will not change the number of neededdriver nodes, but may change the set of driver nodes. Thecritical edge attack strategy [42] collects all the critical edgesfrom the initial network and remove them, thereafter a randomedge attack is performed. This is more destructive than asimple degree- or betweenness-based attack in the early stageof the process.In this paper, a hierarchical framework is proposed for bothnode- and edge-removal attacks, aiming at maximizing thedestruction of the network controllability. The main contri-butions of this work are: 1) the concept of critical node is introduced, quantified and analyzed, as a complement tothe concept of critical edge; 2) a new hierarchical attackframework is proposed, which sorts the destruction of nodesor edges in a descending order, and is updated iteratively; 3)extensive simulations are performed to verify the effectivenessof the proposed methods, revealing that the exposure of criticaledges and nodes is harmful to maintain a good networkcontrollability.The rest of the paper is organized as follows. Section IIreviews the network controllability and its robustness, andseveral existing attack strategies. Section III introduces a newhierarchical attack framework. Section IV evaluates both thehierarchical node- and edge-removal strategies by extensivenumerical simulations, on both synthetic and real-world net-works. Section V concludes the investigation. II. P
RELIMINARY
A. Controllability and Controllability Robustness
A linear time-invariant (LTI) networked system, describedby ˙ x = A x + B u , is state controllable if and only if thecontrollability matrix [ B AB A B · · · A N − B ] has a fullrow-rank, where A and B are constant matrices of compatibledimensions, x is the state vector, u is the control input, and N is the dimension of A . The structural controllability is its slightgeneralization dealing with two parameterized matrices A and B , in which the parameters characterize the structure of theunderlying networked system: if there are specific parametervalues that can ensure the system to be state controllable, thenthe system is structurally controllable. If the system is statecontrollable, its state vector x can be driven from any initialstate to any target state in the state space within finite timeby a suitable control input u . Clearly, without control input u , or B ≡ , the networked system is by no means con-trollable. Likewise, for a network of one-dimensional (scalar)nodes, there exist control inputs to some nodes to ensure itscontrollability. This network controllability is characterized bythe minimum number of nodes with control inputs, calleddriver nodes, needed to maintain the controllability. When thenetwork is put into the above LTI system formulation, howmany and which nodes should be driver nodes are describedby the matrix B .Specifically, the controllability of a network of N scalarnodes is measured by the density of the driver nodes n D , OU et al. : A FRAMEWORK OF HIERARCHICAL ATTACKS TO NETWORK CONTROLLABILITY 3 defined by n D ≡ N D N , (1)where N D is the minimum number of driver nodes needed toretain the network controllability. Smaller n D value representsbetter controllability. Practically, N D can be calculated intwo ways, for structural controllability and for exact (state)controllability, respectively. It was shown in [5] that iden-tifying the minimum number of driver nodes to achieve afull control of a directed network requires searching for amaximum matching of the network, which quantifies thenetwork structural controllability. When a maximum matchingis found, N D is determined by the number of unmatchednodes, i.e., nodes without control inputs, given by N D = max { , N − | E ∗ |} , (2)where | E ∗ | is the number of nodes in the maximum matching E ∗ . As for exact controllability [6], N D is calculated by N D = max { , N − rank ( A ) } . (3)The controllability robustness is evaluated after some nodesor edges are removed, one by one, yielding a sequence ofvalues (represented by a controllability curve ) that reflecthow robust (or vulnerable) a networked system is against adestructive attack. The controllability curve under a node-removal attack is calculated by n ND ( i ) ≡ N D ( i ) N − i , i = 0 , , . . . , N − , (4)where N D ( i ) is the number of driver nodes needed to retainthe network controllability after i nodes have been removed,and N represents the number of nodes in the original network.Note that, given an N -node network, one can remove at most N − nodes, excluding the trivial empty case. Similarly,the controllability curve under an edge-removal attack iscalculated by n ED ( i ) ≡ N D ( i ) N , i = 0 , , . . . , M, (5)where N D ( i ) is the number of driver nodes needed to retainthe network controllability after i edges have been removed,and N and M represent the numbers of nodes and edges inthe original network. Here, n ND (0) = n ED (0) represents thecontrollability of the original network, of which no node oredge has been removed.To measure the overall controllability robustness of a net-work, the controllability curves are averaged: R Nc = 1 N N − (cid:88) i =0 n ND ( i ) , (6)and R Ec = 1 M + 1 M (cid:88) i =0 n ED ( i ) . (7)Lower R Nc and R Ec values mean better overall controllabilityagainst node- and edge-removal attacks, respectively. B. Existing Attack Strategies
The most frequently used measures of importance are the degree and betweenness centralities. A weighted measure isgiven by p i = α × k i (cid:80) Ni =1 k i + β × b i (cid:80) Ni =1 b i , (8)where k i and b i represent the degree and the betweennessof node i , p i represents the probability of removing it, and α and β are weights, which are set manually in [32] with β being replaced by − α in [43]. Similarly, in [44] threeparameters, α , β and γ , are used to control the weightsof degree, betweenness and harmonic closeness, respectively.These measures have been used in the strategies to attackinterdependent networks [43]–[47], networks of networks [48],[49], and weighted networks [50].
12 3 45 2 3 4 remove edge (2,3) remove edge (3,4) remove node remove node node non-criticaledge i i Fig. 2 : [color online] An example of critical edges and nodeschanging during the attack process: (a) edge (2 , is non-critical in the initial network, but becomes critical after someedge-removals; (b) nodes , and are critical initially, butnode is removed, all of them become non-critical, and afternode is removed, node becomes critical again.The edge-removal attack strategy proposed in [42] aims atremoving the critical edges of the initial network, and after allthe initial critical (IC) edges have been removed a randomattack is performed. This IC attack strategy is specificallydesigned to degrade the network controllability, where the term‘critical’ is defined for controllability. This attack is especiallydestructive in the early stage of the process, but becomes lesseffective in latter stages, because critical edges are changingduring the process due to the removal of some other edges.An example is shown in Fig. 2 (a), where a non-criticaledge (edge (2 , ) becomes critical after some edge-removals.Therefore, critical edges need to be updated throughout theattack process such that the damage to network controllabilitycan be maximized. C. Critical Edges and Nodes
In this paper, the concept of critical edge defined in [5]is adopted. An edge is critical if and only if its removalincreases the number of driver nodes needed to retain thenetwork controllability; otherwise, it is non-critical. Inspiredby this, the concept of critical node is introduced here. Anode is critical if and only if its removal increases the number
GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2020 of driver nodes needed to retain the network controllability;otherwise, it is non-critical. An example of critical node isshown in Fig. 2 (b), where the blue-colored nodes are criticalnodes.The critical nodes and edges are the most important ele-ments in the concern of network controllability, in the sensethat their removal will cause the maximum possible destructionto the network controllability. Therefore, in an efficient attackstrategy, critical nodes and edges should be removed with thehighest priority. It should be noted that, through the attackprocess, both critical nodes and edges will be dynamicallychanged, as illustrated by the example shown in Fig. 2.Therefore, in analyzing the attack strategy and its effect, thelist of critical nodes and edges must be updated iteratively,step by step, after each attack.
III. H
IERARCHICAL A TTACK F RAMEWORK
A. Hierarchical Edge Attack
The proposed framework classifies all edges hierarchicallyinto three types: 1) critical edges, as defined above; 2) sub-critical edges, whose removal does not increase the number ofneeded driver nodes, but increases the number of unmatchednodes; and 3) normal edges, which are the rest edges. Thesubcritical and normal edges are non-critical edges. In ahierarchical attack, the priorities of attacking these three typesare in descending order, namely, selecting the critical edgeswith the highest priority to attack, followed by the subcriticalones, and finally the normal ones. An example of these threetypes of edges is shown in Fig. 3.
23 4 (a) (b) (c) input unmatched node matched node matchingnon-matching ii remove edge (2,3) remove edge (6,4) remove edge (7,8) Fig. 3 : [color online] Example of edge hierarchy: (a) edge (2 , is critical, whose removal will increase the number ofneeded driver nodes by ; (b) edge (4 , is subcritical, whoseremoval will not change the number of driver nodes but willincrease the number of unmatched nodes by ; (c) edge (7 , is normal, whose removal does not change the numbers ofdriver nodes and unmatched nodes.Algorithm 1 shows the pseudo-code for hierarchical edgeselection. Given a network with M edges, represented by itsadjacency matrix A , Algorithm 1 returns the index of the edgeto be removed with the highest priority. Lines – initializethree empty lists for the three types of edges. Lines and calculate the numbers of needed driver nodes and unmatchednodes of the original network before being attacked. The for-loop between Lines – categorizes each edge into a type list. In Lines – , the non-empty list with the highest priorityis assigned to L , which is then sorted according to a certainfeature F (e.g., degree centrality). Finally, L (1) represents theindex of an edge that is with the highest priority to be removed,and meanwhile it has the highest value of feature F (e.g.,highest degree). B. Hierarchical Node Attack
Different from edges, nodes are hierarchically classified intofour types in descending order of priorities: 1) critical nodes;2) subcritical nodes, whose removal does not increase thenumber of needed driver nodes but increases the number ofunmatched nodes; 3) normal nodes, whose removal does notaffect the numbers of driver nodes and unmatched nodes; and4) redundant nodes, whose removal enhances the controllabil-ity contrarily. The subcritical, normal, and redundant nodes arenon-critical nodes. An example of these four types of nodesis shown in Fig. 4. (a) (b) (c) ii (d) input matchingnon-matching unmatched node matched node remove node remove node remove node remove node Fig. 4 : [color online] Example of node hierarchy: (a) node iscritical, whose removal increases the number of needed drivernodes by ; (b) node is subcritical, whose removal does notchange the number of driver nodes, but increases the numberof unmatched nodes by ; (c) node is normal, whose removaldoes not change the numbers of driver nodes and unmatchednodes; (d) node is redundant, whose removal decreases thenumber of needed driver nodes by .Algorithm 2 shows the pseudo-code for hierarchical nodeselection. Given a network with N nodes, represented by itsadjacency matrix A , Algorithm 2 returns the index of the nodeto be removed with the highest priority. Lines – initializefour empty lists for the four types of nodes. Lines and calculate the numbers of needed driver nodes and unmatchednodes of the original network before being attacked. The for-loop between Lines – categorizes each node into a type list.In Lines – , the non-empty list with the highest priorityis assigned to L , which is then sorted according to certainfeature F . Finally, L (1) represents the index of a node thathas the highest priority to be removed, and meanwhile it hasthe highest value of feature F .Source codes of both hierarchical node and edge attackalgorithms are available for the public . https://fylou.github.io/sourcecode.html OU et al. : A FRAMEWORK OF HIERARCHICAL ATTACKS TO NETWORK CONTROLLABILITY 5 Algorithm 1:
Hierarchical Edge Selection
Input : adjacency matrix A ; feature F ; number ofedges M Output: index j of the edge to be attacked L ← []; // highest priority L ← []; L ← []; // lowest priority n A ← number of driver nodes needed for A ; u A ← number of unmatched nodes needed for A ; for i ← to M do A ← A ; Delete edge i from A ; n A ← number of driver nodes needed for A ; u A ← number of unmatched nodes needed for A ; if n A > n A then L .insert ( i ) ; else if u A > u A then L .insert ( i ) ; else L .insert ( i ) ; end end end if L is not empty then L ← L ; else if L is empty and L is not empty then L ← L ; else L ← L ; end Sort L according to feature F , in descending order ; j ← L (1) ; C. Extra Computational Complexity
The computational complexity of calculating the networkcontrollability, mainly in searching for the number of neededdriver nodes, is O ( M · √ N ) , by the Hopcroft–Karp algorithm.In a hierarchical node or edge attack, to identify whether anode or edge is critical, the number of needed driver nodes tobe calculated iteratively introduces a non-negligible amountof extra computational cost. This extra computational costfor hierarchical edge attack is O ( (cid:80) Mi =1 ( i · √ N )) , and forhierarchical node attack is O ( (cid:80) N − i =1 ( i · √ N )) . IV. E
XPERIMENTAL S TUDIES
In this section, the hierarchical attack framework is evalu-ated by extensive simulations. Network features will be takeninto account. For node attacks, betweenness, out-degree andcloseness are used as feature F , respectively; for edge attacks,betweenness and degree are used, respectively. To verifythe effectiveness of the proposed hierarchical framework, thehierarchical feature-based attack strategies are compared to thefeature-based attack strategies, respectively. For example, thehierarchical degree-based attack is compared to the degree-based attack, under the same conditions. Algorithm 2:
Hierarchical Node Selection
Input : adjacency matrix A ; feature F ; number ofnodes N Output: index j of the node to be attacked L ← []; // highest priority L ← []; L ← []; L ← []; // lowest priority n A ← number of driver nodes needed for A ; u A ← number of unmatched nodes needed for A ; for i ← to N do A ← A ; Delete node i from A ; n A ← number of driver nodes needed for A ; u A ← number of unmatched nodes needed for A ; if n A > n A then L .insert ( i ) ; else if n A = n A then if u A > u A then L .insert ( i ) ; else L .insert ( i ) ; end else L .insert ( i ) ; end end if L is not empty then L ← L ; else if L is empty and L is not empty then L ← L ; else if L , L are empty and L is not empty then L ← L ; else L ← L ; end Sort L according to feature F , in descending order ; j ← L (1) ;Nine typical directed synthetic network models are adoptedfor simulation, namely the Erd¨os–R´enyi random-graph (ER)network [51], Newman–Watts small-world (SW) network [52],generic scale-free (SF) network [37], [53], [54], q -snapback(QS) network [55], q -snapback network with redirected edges(QR) [56], random triangle (RT) network [24], and randomrectangle (RR) network [24], extremely homogeneous (HO)network [57], and onion-like (OL) network [58].Recall that the HO networks were empirically verified withoptimal controllability robustness before [57]. Given an N -node and M -edge configuration, the HO network satisfies (cid:98) M/N (cid:99) ≤ k in,outi ≤ (cid:100) M/N (cid:101) , i = 1 , , . . . , N . This meansthat both of its in- and out-degrees are distributed identicallyor nearly identically with a small difference less than . TheOL network is generated from an SF network via simpleedge-swapping with degree reservation [58], thus its degreedistribution follows the same power-law distribution as the SF. GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2020
The detailed generation methods and parameter settingsof these synthetic networks are provided in SupplementaryInformation (SI) . The network size is set to N = 500 , ,and , respectively. The average degree is set to (cid:104) k (cid:105) = 3 , , and , respectively.In addition, nine real-world networks are used for sim-ulations, with data taken from Network Repository . Theirparameters and brief descriptions are presented in Table I. TABLE I : Basic information of the real-world networks.
Network File name Brief description
N M
BMK bn-mouse-kasthuri-graph-v4 brain network 1029 1559ICM ia-crime-moreno interaction network 830 1474IEU inf-euroroad infrastructure network 1175 1417DEL delaunay-n10 DIMACS10 problem 1024 3056DW5 dwt-1005 symmetric connectionfrom Washington 1005 3808DW7 dwt-1007 1007 3784LSH lshp1009 Alan GeorgesL-shape problem 1009 2928OLM olm1000 computational fluiddynamics problem 1000 2996RAJ rajat19 Rajat19 circuitsimulation matrix 1157 4433
A. Comparison of Attack Strategies
1) Node-removal attacks:
Nine node-removal attack strate-gies are compared, namely the betweenness-based (B), out-degree-based (D), closeness-based (C), random (R), hierar-chical betweenness-based (HB), hierarchical out-degree-based(HD), hierarchical closeness-based (HC), hierarchical random(HR), and hybrid (Hy) attacks.The B, D, C strategies aim at removing the node withthe largest betweenness, degree, and closeness, respectively,at every step. The HB, HD, HC strategies aim at removingcritical nodes that are sorted in a betweenness-, degree-, andcloseness-descending order, respectively, at every step. TheHR removes the critical nodes at random.The Hy strategy is designed as follows: First, remove eitherthe node (or edge) with the maximum degree (or betweenness),according to the removal of which node (or edge) will causegreater destruction to the network controllability. If equal, thenchoose either one to attack.
2) Edge-removal attacks:
Eight edge attack strategies arecompared, namely the betweenness-based (B), out-degree-based (D), random (R), hierarchical betweenness-based (HB),hierarchical out-degree-based (HD), hierarchical random (HR),initial critical (IC) [42] and hybrid (Hy) attacks.Here, the ‘out-in’ edge degree is used, which is defined asthe sum of the out-degree of its source node and the in-degreeof its target node [42].For each node and each edge attack, the simulation repeats and independent runs, respectively. B. Simulation Results on Synthetic Networks
Here, the structural controllability (see Eq. (2)) is consid-ered for controllability robustness comparison. The simulation https://fylou.github.io/pdf/hatk_si.pdf http://networkrepository.com/ results of some synthetic networks with N = 1000 and real-world networks are presented. More detailed and completeresults for networks with N = 500 , and are givenin the SI. n D (a) ER h k i =3 E-BE-HB (b) ER h k i =5 (c) ER h k i =10 n D (d) HO h k i =3 (e) HO h k i =5 (f) HO h k i =10 p E n D (g) OL h k i =3 p E (h) OL h k i =5 p E (i) OL h k i =10 Fig. 5 : [color online] Results of edge attacks on ER, HO, andOL ( N = 1000 ): hierarchical betweenness-based (E-HB) andbetweenness-based (E-B) strategies.Fig. 5 shows the results of ER, HO, and OL under E-HBand E-B attacks. It is clear that E-HB is consistently moredestructive than E-B throughout the entire process as shownin Figs. 5 (a,b,d,e,f,g,h,i); while Fig. 5 (c) shows that E-HB ismore destructive than E-B when P E < . , but E-B is slightlymore destructive when P E > . . n D (a) ER h k i =3 E-DE-HD (b) ER h k i =5 (c) ER h k i =10 n D (d) HO h k i =3 (e) HO h k i =5 (f) HO h k i =10 p E n D (g) OL h k i =3 p E (h) OL h k i =5 p E (i) OL h k i =10 Fig. 6 : [color online] Results of edge attacks on ER, HO,and OL ( N = 1000 ): hierarchical degree-based (E-HD) anddegree-based (E-D) strategies.Fig. 6 shows that E-HD is consistently more destructivethan E-D. Fig. 7 shows that E-HR is consistently moredestructive than E-R and E-IC. Figs 5–7 show that HO hasbetter controllability robustness than ER; both HO and ERhave significantly better controllability robustness than OL. OU et al. : A FRAMEWORK OF HIERARCHICAL ATTACKS TO NETWORK CONTROLLABILITY 7 n D (a) ER h k i =3 E-RE-HRE-IC (b) ER h k i =5 (c) ER h k i =10 n D (d) HO h k i =3 (e) HO h k i =5 (f) HO h k i =10 p E n D (g) OL h k i =3 p E (h) OL h k i =5 p E (i) OL h k i =10 Fig. 7 : [color online] Results of edge attacks on ER, HO, andOL ( N = 1000 ): random (E-R), hierarchical random (E-HR)and initial critical (IC) strategies.As the average degree increases, the controllability robustnessimproves. n D (a) ER h k i =3 E-HBE-HDE-HRE-Hy (b) ER h k i =5 (c) ER h k i =10 n D (d) HO h k i =3 (e) HO h k i =5 (f) HO h k i =10 p E n D (g) OL h k i =3 p E (h) OL h k i =5 p E (i) OL h k i =10 Fig. 8 : [color online] Results of edge attacks on ER, HO, andOL ( N = 1000 ): three hierarchical attacks (E-HB, E-HD andE-HR) and hybrid (E-Hy) strategies.Fig. 8 compares the three hierarchical attacks (E-HB, E-HDand E-HR) and hybrid attack (E-Hy). For ER and HO, it isclear that E-HB is more destructive than the other strategies;while for OL, either E-HR or E-HB is most destructive.Figs. 5–8 show that the hierarchical framework increases thedestruction effects on the network controllability. It can alsobe seen that, among the hierarchical attack strategies, E-HBperforms the best.The overall comparison is summarized in Table II, whereeach value is the ratio of the overall destruction (see Eq. (6)for node attacks and Eq. (7) for edge attacks) under the twocorresponding attack strategies. For example, the value . in row ‘ (cid:104) k (cid:105) = 3 , ER’ and column ‘Node Attack, HB/B’represents that, given an ER ( (cid:104) k (cid:105) = 3 ) under node attacks,the overall destruction ratio of N-HB versus N-B is . . Referring to Fig. 5 (a), the value is equivalent to the ratio ofthe area under the blue dashed-line versus the area under thered dotted-line.If HB/B > , it means that HB is more destructive; if HB/B < , it means that B is more destructive; otherwise, if HB/H = 1 , it means that HB and B are equivalently destructive.Here, an equivalent overall destruction does not mean that thetwo controllability curves are overlapped, but means that theareas under the two controllability curves are equal, namelythey are equivalent in the average sense.As can be seen from the ‘Node Attacks’ part in Table II, thehierarchical attack strategies are consistently more destructivethan the non-hierarchical and the hybrid strategies, for the ratiovalues are greater than in the columns of HB/B, HD/D,HC/C, and HR/R.In the columns of HB/Hy and HD/Hy, hierarchical strategiesgenerally outperform the hybrid strategy. It should be notedthat, when the ratio is within [0 . , . , one may considerthe two comparing strategies to have equivalent performances.As for the edge-removal attacks, the hierarchical strategiesare more destructive than the non-hierarchical ones and theIC (which does not update the critical edge list). However,the ratio values in the column of HD/Hy are mostly less than , meaning that HD is less destructive than Hy. This impliesthat edge degree is not a good measure of importance regard-ing destructive attacks. Nevertheless, within the hierarchicalframework, E-HD is more destructive than E-D.The results for cases of N = 500 and N = 1500 are tabledin SI.The attack simulation results on real-world networks areshown in Table III, while the detailed comparison figures fordifferent networks are included in SI.The attack simulations on various synthetic networks andreal-world networks show that the hierarchical strategies areconsistently more destructive to network controllability thanother attack strategies. C. Critical Edges and Nodes
A common phenomenon is observed from the results pre-sented in Sec. IV-B: as the average degree increases, the ratioof areas under the controllability curves subject to hierarchicaland non-hierarchical attacks tends asymptotically to . This isbecause, as the network becomes denser, hence more homo-geneous, fewer critical edges and nodes are exposed. It notonly improves the controllability robustness of the network,but also makes the proposed hierarchical attack strategies lesseffective, thereby becoming similar to non-hierarchical attacks.Table IV shows the minimum (integer) average degree whenthere is no critical nodes or edges found in the initial network.Here, initial means the network that has not been attacked. Foreach topology with a given (cid:104) k (cid:105) value, network instancesare simulated. Given N = 500 , (cid:104) k (cid:105) is set from with anincremental value . If there are no critical nodes or edgesfound in all the instances, then the (cid:104) k (cid:105) value is recorded intoTable IV; otherwise, (cid:104) k (cid:105) increases by and then the processis run again. It can be seen from the table that, for SW andHO, there are no critical nodes or edges found when (cid:104) k (cid:105) = 3 , GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2020
TABLE II : Comparison of attack strategies on the nine synthetic networks ( N = 1000 ), where B represents betweenness; Drepresents degree; C represents closeness; R represents random; Hy represents hybrid; IC represents initial critical edges; HBrepresents hierarchical betweenness; HD represents hierarchical degree; HC represents hierarchical closeness; HR representshierarchical random attacks. N=1000 Node Attack Edge AttackHB/B HD/D HC/C HR/R HB/Hy HD/Hy HB/B HD/D HR/R HB/Hy HD/Hy HR/IC (cid:104) k (cid:105) =3 ER 1.210 1.151 1.313 1.469 1.111 1.132 1.112 1.585 1.454 1.123 0.848 1.225SW 1.286 1.070 1.209 1.275 1.065 1.081 1.157 1.339 1.426 1.295 0.997 1.426SF 1.033 1.011 1.029 1.266 1.007 1.009 1.158 1.246 1.157 1.157 1.101 1.060QS 1.303 1.198 1.661 1.344 1.098 1.185 1.123 2.256 1.463 1.123 0.886 1.241QR 1.336 1.160 1.330 1.445 1.138 1.155 1.118 1.721 1.440 1.122 0.874 1.411RT 1.148 1.095 1.164 1.590 1.056 1.070 1.260 1.679 1.393 1.280 1.072 1.202RR 1.245 1.110 1.240 1.527 1.095 1.098 1.217 1.682 1.438 1.231 1.013 1.282HO 1.247 1.170 1.436 1.416 1.141 1.148 1.151 1.336 1.378 1.095 1.143 1.382OL 1.034 1.012 1.028 1.273 1.007 1.007 1.161 1.241 1.161 1.164 1.104 1.060 (cid:104) k (cid:105) =5 ER 1.207 1.209 1.339 1.483 1.149 1.165 1.051 1.377 1.354 1.052 0.627 1.272SW 1.296 1.171 1.337 1.383 1.135 1.150 1.067 1.421 1.261 1.142 0.756 1.292SF 1.051 1.017 1.048 1.371 1.015 1.015 1.179 1.348 1.208 1.191 1.101 1.084QS 1.235 1.268 1.984 1.348 1.167 1.214 1.066 2.097 1.476 1.054 0.732 1.477QR 1.273 1.197 1.345 1.502 1.152 1.156 1.078 1.416 1.366 1.077 0.673 1.370RT 1.201 1.152 1.222 1.630 1.110 1.118 1.106 1.543 1.378 1.176 0.751 1.339RR 1.239 1.154 1.264 1.545 1.109 1.120 1.111 1.435 1.305 1.132 0.756 1.326HO 1.234 1.176 1.399 1.431 1.150 1.136 1.074 1.413 1.258 1.026 0.994 1.246OL 1.048 1.019 1.046 1.367 1.010 1.010 1.205 1.340 1.199 1.199 1.098 1.077 (cid:104) k (cid:105) =10 ER 1.214 1.208 1.320 1.450 1.148 1.117 1.025 1.325 1.202 1.030 0.470 1.221SW 1.217 1.208 1.363 1.376 1.149 1.113 1.064 1.238 1.127 1.102 0.479 1.164SF 1.078 1.036 1.080 1.590 1.025 1.027 1.205 1.608 1.336 1.197 0.916 1.155QS 1.167 1.315 2.875 1.295 1.198 1.195 1.036 1.777 1.395 1.188 0.576 1.369QR 1.228 1.203 1.359 1.427 1.148 1.116 1.029 1.341 1.154 1.044 0.487 1.137RT 1.211 1.208 1.286 1.496 1.121 1.128 1.075 1.332 1.280 1.057 0.515 1.221RR 1.224 1.169 1.300 1.409 1.106 1.119 1.090 1.309 1.214 1.083 0.555 1.187HO 1.201 1.194 1.361 1.376 1.157 1.140 1.034 1.275 1.063 1.078 0.697 1.063OL 1.066 1.034 1.079 1.577 1.021 1.022 1.189 1.619 1.369 1.212 0.932 1.167 TABLE III : Comparison of attack strategies on the nine real-world networks, where B represents betweenness; D representsdegree; C represents closeness; R represents random; Hy represents hybrid; IC represents initial critical edges; HB representshierarchical betweenness; HD represents hierarchical degree; HC represents hierarchical closeness; HR represents hierarchicalrandom attacks.
Node Attack Edge AttackHB/B HD/D HC/C HR/R HB/Hy HD/Hy HB/B HD/D HR/R HB/Hy HD/Hy HR/ICBMK 1.066 1.007 1.010 1.157 1.004 1.005 1.031 1.063 1.056 1.030 0.988 1.027ICM 1.044 1.102 1.183 1.361 1.033 1.041 1.101 1.201 1.187 1.100 1.004 1.088IEU 1.187 1.050 1.137 1.361 1.007 1.040 1.109 1.377 1.291 1.115 1.097 1.096DEL 1.203 1.174 1.191 1.311 1.064 1.057 1.108 1.416 1.248 1.112 0.889 1.163DW5 1.276 1.149 1.456 1.423 1.097 0.995 1.167 1.483 1.381 1.169 0.838 1.237DW7 1.323 1.036 1.597 1.314 1.004 0.994 1.205 2.224 1.284 1.235 1.291 1.269LSH 1.301 1.312 1.977 1.264 1.066 1.086 1.121 1.585 1.301 1.122 0.907 1.217OLM 1.007 1.001 1.009 1.946 1.010 1.019 1.281 1.660 1.611 1.696 1.627 1.649RAJ 1.297 1.076 1.252 1.769 1.064 1.082 1.508 1.749 1.502 1.805 1.403 1.292
TABLE IV : The lowest average out-degree when there is nocritical nodes or edges found in the network.
ER SW SF QS QR RT RR HO OLNode 9 3 23 6 4 8 6 3 22Edge 10 3 24 5 6 8 6 3 22 meaning that removal of any node or edge in the initial SWor HO will not increase the number of needed driver nodes.Thus, their controllability robustness is better than the others.In contrast, for SF and OL, before (cid:104) k (cid:105) increases to and ,respectively, there were no critical nodes or edges. It meansthat in dense SF or OL networks (e.g., (cid:104) k (cid:105) = 20 ), there arestill critical nodes and edges, and removing any critical nodeor edge will directly destroy its controllability. Thus, SF andOL have much worse initial controllability and controllability robustness than the other networks.Fig. 9 shows the number of critical edges in the initial SFand OL networks, against the increase of average degree. Thecorresponding figure for the case with critical nodes is shownin SI. The initial controllability is plotted for reference in eachsubplot. As shown in Fig. 9, the numbers of critical edges inthe initial SF and OL networks increase as (cid:104) k (cid:105) increases from to ; when (cid:104) k (cid:105) > , the numbers of critical edges dropdrastically. Meanwhile, the initial controllability of both SFand OL becomes better as the average degree increases. When ≤ (cid:104) k (cid:105) ≤ , the additional edges enhance the connectednessand make the networks more controllable. These additionaledges become part of the critical edges. However, when (cid:104) k (cid:105) > , the initial controllability of the SF networks tends to besufficiently optimized, reflected by the lower density of needed OU et al. : A FRAMEWORK OF HIERARCHICAL ATTACKS TO NETWORK CONTROLLABILITY 9 C r i t i c a l E dge s I n i t i a l C on t r o ll ab ili t y (a) SF h k i C r i t i c a l E dge s I n i t i a l C on t r o ll ab ili t y (b) OL Fig. 9 : Number of critical edges (boxplots) and initial con-trollability (stars *) against the average degree of (a) SF and(b) OL networks.driver nodes. In this case, the increased edges cover the criticalnodes and edges, which leads to the drastically drops of thenumbers of (the exposed) critical edges.The exposure of critical nodes and edges sets a clear targetfor the attacker to destroy the network controllability. Incontrast, in the networks with strong controllability robustness,there are rare (or no) critical nodes and edges exposed; forexample, SW, HO, QS and QR. For these networks, theattacker is unable or uneasy to find targets to attack in order todestruct the controllability. This finding is consistent with, andactually extends the applicability of, the previous findings: 1)dense and homogeneous networks have better controllability[5]; 2) extremely-homogeneous topology has the optimal con-trollability robustness [57]. Nevertheless, critical nodes andedges will expose themselves during the attack process, asthe network becomes sparser. To design networks with goodcontrollability robustness, the exposure of critical nodes andedges should be dimmed or avoided, if ever possible. If thereare sufficient numbers of available edges, the networks shouldbe designed as dense and homogeneous as possible [5], [57];otherwise, if the numbers of edges are limited, they shouldbe deliberately assigned in such a way that the exposure ofcritical nodes and edges is minimum.Fig. 10 shows the types of removed nodes and edges duringan attack. As can be seen from Fig. 10 (a), HO and SWdo not expose critical nodes in the early stage, and thus theattacker could only remove the normal or subcritical nodes;later, in the middle stage, both HO and SW expose somecritical nodes; finally, SW has only redundant nodes left. Theother networks expose more critical nodes than HO or SWduring the attack process. Note that, although for OL andSF, there are more normal nodes than critical ones during theattack process, their initial controllability is no good (see thecontrollability curves in SI). V. C
ONCLUSIONS
To better understand the network controllability robustnessfrom the perspective of destructive attacks, a hierarchical at-tack framework is proposed, which can be used for both edge-and node-removal attacks. The hierarchical attack strategiesaim at removing the critical nodes and edges with the highestpriority, and they can be combined together with other com-monly used features (e.g., degree or edge centrality), such that the identified critical nodes or edges can be sorted in descend-ing order according to such features. Extensive experimentson nine synthetic networks with various topologies and ninereal-world networks show the effectiveness of the proposed hi-erarchical attack framework on destructive attacks to networkcontrollability for all kinds of networks that are tested. Fornode attacks, betweenness, out-degree, and closeness are usedas the feature, respectively; for edge attacks, betweenness anddegree are used, respectively. The hierarchical feature-relatedattacks show consistently better destructive performances thanthe common feature-only attacks.It is revealed that the exposure of the critical edges andnodes are disadvantageous in resisting attacks to the networkcontrollability. Therefore, to design networks with strongcontrollability robustness, the critical nodes and edges shouldbe deliberately hidden. This finding is consistent with, and alsoextends the applicability of, the previous findings: 1) denseand homogeneous networks have better controllability [5]; 2)extremely-homogeneous topology has the optimal controlla-bility robustness with fixed numbers of nodes and edges [57]. R EFERENCES [1] M. E. Newman,
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