Airport Runway Security Risk Evolution Model Based on Complex Network

Transcription

1 Arport Runway Securty Rsk Evoluton Model Based on Complex Network Xanl Zhao School of Management, Wuhan Unversty of Technology, Wuhan , Chna Abstract Arport runway securty rsk assessment and safety testng s of great sgnfcance n felds lke rsk control, navgaton and traffc montorng. An arport runway rsk evoluton mathematcal model and rsk evoluton topology model based on complex network s ndvdually proposed. In the model, nodes represent every possble rsk factor, edges represent rsk nduced process. The arport runway rsk evoluton mathematcal model s descrbed from clusterng coeffcent, average path length and degree dstrbuton. The rsk evoluton topology model s analyzed from n-degree and out-degree, the number of nodes and the number of branches, and the rsk broken chan control scheme s proposed. The results ndcated that securty management system vulnerabltes, lack of staff safety tranng, lack of safety awareness of the ground staff and llegal operatons, bad weather are key nodes of arport runway securty rsk evoluton model. Keywords: complex network, rsk evoluton model, arport runway, safety and securty. 1. INTRODUCTION Arport securty refers to the technques and methods used n protectng passengers, staff and planes whch use the arports from accdental/malcous harm, crme and other threats. Arport runways are safer today than they were n the past. Oster appearng before the US Senate Commerce subcommttee on avaton safety ssues sad that three year average commercal accdent rate s accdents per 100,000 departures meanng accdent rate s the equvalent of one fatal accdent for every 15 mllon passenger carryng flghts (Oster et al., 2013). In today s world ntellgence, securty partnerng and nformaton sharng has help to reduce ncdents and accdents at the arport runways. Fgure 1. The Block Dagram of SHELL Model In spte of all of the actvtes that the arports have undertaken to reduce runway ncursons, the human factor remans the most valuable and the most vulnerable asset n the avaton envronment. The frst human factors 573

2 models n avaton s SHELL model whch s proposed by Elwyn Edwands (1972). The SHELL model s a conceptual model of human factors that clarfes the scope of avaton human factors and asssts n understandng the human factor relatonshps betwee n avaton system resources or envronment (the flyng subsystem) and the human component n the avaton system (the human subsystem). The model s named after the ntal letters of ts components (software, hardware, envronment, lveware) and places emphass on the human beng and human nterfaces wth other components of the avaton system (Johnston et al., 2001). The SHELL model adopts a systems perspectve that suggests the human s rarely, f ever, the sole cause of an accdent (Wegmann and Shappell, 2003). The systems perspectve consders a varety of contextual and task-related factors that nteract wth the human operator wthn the avaton system to affect operator performance (Wegmann and Shappell, 2003). As a result, the SHELL model consders both actve and latent falures n the avaton system. The block dagram of SHELL model s shown n Fgure 1. Each component of the SHELL model (software, hardware, envronment, lveware) represents a buldng block of human factors studes wthn avaton. The human element or worker of nterest s at the centre or hub of the SHELL model that represents the modern ar transportaton system (Hawkns and Orlady, 1993). The human element s the most crtcal and flexble component n the system, nteractng drectly wth other system components, namely software, hardware, envronment and lveware. However, the edges of the central human component block are vared, to represent human lmtatons and varatons n performance. Therefore, the other system component blocks must be carefully adapted and matched to ths central component to accommodate human lmtatons and avod stress and breakdowns (ncdents/accdents) n the avaton system. To accomplsh ths matchng, the characterstcs or general capabltes and lmtatons of ths central human component must be understood. The four components of the SHELL model or avaton system do not act n solaton but nstead nteract wth the central human component to provde areas for human factors analyss and consderaton (Wegmann and Shappell, 2003). The SHELL model ndcates relatonshps between people and other system components and therefore provdes a framework for optmzng the relatonshp between people and ther actvtes wthn the avaton system that s of prmary concern to human factors. In fact, the Internatonal Cvl Avaton Organzaton has descrbed human factors as a concept of people n ther lvng and workng stuatons; ther nteractons wth machnes (hardware), procedures (software) and the envronment about them; and also, ther relatonshps wth other people. Accordng to the SHELL model, a msmatch at the nterface of the blocks/components where energy and nformaton s nterchanged can be a source of human error or system vulnerablty that can lead to system falure n the form of an ncdent/accdent (Johnston et al., 2001). Avaton dsasters tend to be characterzed by msmatches at nterfaces between system components, rather than catastrophc falures of ndvdual components (Wener and Nagel, 1988). There have been a lot of researches about arport runway securty rsk models, but the research on rsk evoluton s relatvely lack. Arport runway rsk s a dynamc evolvng open network, the randomness and regularty coexst. In ths paper, arport runway rsk evoluton mathematcal model and rsk evoluton topology model based on complex network s constructed. The model can reveal the nherent law of rsk evoluton, t s very useful to mprove the safety management level of arport runway. 2. COMPLEX NETWORK THEORY In the context of network theory, a complex network s a graph (network) wth non-trval topologcal features features that do not occur n smple networks such as lattces or random graphs but often occur n graphs modelng of real systems. The study of complex networks s a young and actve area of scentfc research nspred largely by the emprcal study of real-world networks such as computer networks, socal networks (Kuran and Thran, 2006; Barabâs and Albert, 1999; Kossnets, 2006), bran networks and technologcal networks. 2.1 Topologcal features of complex network 574

3 Trval networks such as lattces and random graphs have been studes extensvely n the past. Lattces are regular graphs whose drawng corresponds to some grd/mesh/lattce. A random graph s a graph that s generated by a random process. Some of the more promnent graph propertes wll be dscussed. Graph propertes or nvarants depend on the abstract structure only, not on the representaton. They nclude degree, clusterng coeffcent, connectvty (scalars); degree sequence, characterstc polynomal, Tutte polynomal (sequences/polynomals). Complex networks dsplay substantal non-trval topologcal characterstcs, wth patterns of connecton between ther nodes that are nether purely regular nor purely random. Such characterstcs nclude a heavy tal n the degree dstrbuton, a hgh clusterng coeffcent, a small average path length, herarchcal structure and communty structure (Albert, 2002). The degree of a node n a network s the number of connectons or edges the node has to other nodes. The degree dstrbuton P(k) of a network s defned to be the fracton of nodes n the network wth degree k. Thus f there are n nodes n total n a network and n k of them have degree kk, we have P(k)=n k/n. A network s named scale-free (Barabâs, 2009) f ts degree dstrbuton follows a partcular mathematcal functon called a power law. The power law mples that the degree dstrbuton of these networks has no characterstc scale. In a network wth a scale-free degree dstrbuton, allowng for a few nodes of very large degree to exst, these nodes are often called hubs. In complex network, clusterng coeffcent s a measure of the degree to whch nodes n a complex network tend to cluster together. The clusterng coeffcent s based on trplets of nodes. A trplet conssts of three nodes that are connected by ether two (open trplet) or three (closed trplet) undrected tes. A trangle conssts of three closed trplets, one centred on each of the nodes. The clusterng coeffcent s the number of closed trplets (or 3 x trangles) over the total number of trplets (both open and closed). Watts and Strogatz defned the clusterng coeffcent as follows, Suppose that a vertex v has K v neghbours; then at most K v (K v -1)/2 edges can exst between them (ths occurs when every neghbour of v s connected to every other neghbour of v). Let C v denote the fracton of these allowable edges that actually exst. Defne Cas the average of C v overall (Watts and Strogatz, 1998). Average path length s one of the three most robust measures of network topology, along wth ts clusterng coeffcent and ts degree dstrbuton. In a network, the dstance d(v,v ) between two nodes v and v s defned as the number of edges along the shortest path connectng them. Assume that d(v,v )=0 f v cannot be reached from v. Connectvty of a graph s a measure of robustness as a network. Two vertces are connected, f there s a path between them. Otherwse they are dsconnected. A graph s connected f every par of vertces s connected. Vertex connectvty s the smallest vertex cut of a connected graph-the number of vertexes that must be removed n order to dsconnect the graph. Local connectvty s the sze of a smallest vertex cut separatng two vertexes n a graph. Graph connectvty equals the mnmal local connectvty. Edge connectvty s the smallest number of edge cuts that renders the graph dsconnected. Local edge connectvty s the smallest number of edge cuts, whch dsconnect the two edges. Agan, the smallest local edge connectvty n a graph s the graph edge connectvty. The vertex- and edge-connectvty of a dsconnected graph are both 0. The complete graph on n vertces has edgeconnectvty equal to n-1. Every other smple graph on n vertces has strctly smaller edge-connectvty. In a tree, the local edge-connectvty between every par of vertces s 1. Edge connectvty s bounded by vertex connectvty and the latter s bounded by the mnmum degree of the graph. Assortatvty refers to a preference of a node to attach to other nodes that are smlar or dfferent. Smlarty s often but not necessarly expressed n terms of a nodes degree,.e. n socal networks, hghly connected nodes prefer to attach to other hghly connected nodes (assortatvty). The opposte effect can be observed n technologcal and bologcal networks, where hghly connected nodes tend to attach to low degree nodes (dssortatvty). 575

4 Assortatvty s measured as correlaton between two nodes. The most common measures are assortatvty coeffcent and neghbor connectvty. The assortatvty coeffcent s a Pearson correlaton coeffcent between pars of nodes. The range of the measure r s from -1 for perfect assortatve mxng patterns to 1 for completely dsassortatve. 2.2 Two typcal complex network Two well-known and much studed classes of complex networks are scale-free networks (Barabâs amd Bonabeau, 2003) and small-world networks (Watts and Strogatz, 1998) whose dscovery and defnton are canoncal casestudes n the feld. A network s named scale-free f ts degree dstrbuton,.e., the probablty that a node selected unformly at random has a certan number of lnks (degree), follows a partcular mathematcal functon called a power law. The power law mples that the degree dstrbuton of these networks has no characterstc scale. In contrast, networks wth a sngle well-defned scale are somewhat smlar to a lattce n that every node has (roughly) the same degree. Examples of networks wth a sngle scale nclude the Erdős Rény (ER) random graph and hypercubes. In a network wth a scale-free degree dstrbuton, some vertces have a degree that s orders of magntude larger than the average-these vertces are often called "hubs", although ths s a bt msleadng as there s no nherent threshold above whch a node can be vewed as a hub. If there were such a threshold, the network would not be scale-free. Interest n scale-free networks began n the late 1990s wth the reportng of dscoveres of power-law degree dstrbutons n real world networks such as the World Wde Web, the network of Autonomous systems (ASs), some networks of Internet routers, proten nteracton networks, emal networks, etc. Most of these reported "power laws" fal when challenged wth rgorous statstcal testng, but the more general dea of heavy-taled degree dstrbutons whch many of these networks do genunely exhbt (before fnte-sze effects occur) are very dfferent from what one would expect f edges exsted ndependently and at random (.e., f they followed a Posson dstrbuton). There are many dfferent ways to buld a network wth a power-law degree dstrbuton. The Yule process s a canoncal generatve process for power laws, and has been known snce However, t s known by many other names due to ts frequent renventon, e.g., The Gbrat prncple by Herbert A. Smon, the Matthew effect, cumulatve advantage and, preferental attachment by Barabás and Albert for power-law degree dstrbutons. Recently, Hyperbolc Geometrc Graphs have been suggested as yet another way of constructng scale-free networks. Some networks wth a power-law degree dstrbuton (and specfc other types of structure) can be hghly resstant to the random deleton of vertces.e., the vast maorty of vertces reman connected together n a gant exponent. Such networks can also be qute senstve to targeted attacks amed at fracturng the network quckly. When the graph s unformly random except for the degree dstrbuton, these crtcal vertces are the ones wth the hghest degree, and have thus been mplcated n the spread of dsease (natural and artfcal) n socal and communcaton networks, and n the spread of fads (both of whch are modeled by a percolaton and branchng process). Whle random graphs (ER) have an average dstance of order logn between nodes, where N s the number of nodes, scale free graph can have a dstance of loglogn. Such graphs are called ultra small world networks. A network s called a small-world network by analogy wth the small-world phenomenon (popularly known as sx degrees of separaton). The small world hypothess, whch was frst descrbed by the Hungaran wrter Frgyes Karnthy n 1929, and tested expermentally by Stanley Mlgram (1967), s the dea that two arbtrary people are connected by only sx degrees of separaton,.e. the dameter of the correspondng graph of socal connectons s not much larger than sx. In 1998, Duncan J. Watts and Steven Strogatz publshed the frst small-world network model, whch through a sngle parameter smoothly nterpolates between a random graph and a lattce. Ther model demonstrated that wth the addton of only a small number of long-range lnks, a regular graph, n whch the dameter s proportonal to the sze of the network, can be transformed nto a "small world" n whch the average number of edges between any two vertces s very small (mathematcally, t should grow as the logarthm of the sze of the network), whle the clusterng coeffcent stays large. It s known that a wde varety of abstract graphs exhbt the small-world property, e.g., random graphs and scale-free networks. Further, real world networks such as the World Wde Web and the metabolc network also exhbt ths property. In the scentfc lterature on networks, there s some ambguty assocated wth the term "small world." In addton to referrng to the sze of the dameter of the network, t can also refer to the co-occurrence of a small dameter and a hgh clusterng coeffcent. The clusterng coeffcent s a metrc that represents the densty of trangles n 576

5 the network. For nstance, sparse random graphs have a vanshngly small clusterng coeffcent whle real world networks often have a coeffcent sgnfcantly larger. Scentsts pont to ths dfference as suggestng that edges are correlated n real world networks. 3. AIRPORT RUNWAY SECURITY RISK EVOLUTION MODEL In ths secton, we ntroduce a rsk evoluton model that generates small-world networks wth scale-free propertes. The BA model s based on a smple prncple of preferental attachment. The probablty that an old node recevng lnks s lnearly proportonal to ts degree. It assumes that every new node has the complete nformaton about the whole network, whch s unrealstc for real network formatons. As a matter of fact, many networks have ther topology nfluenced by the envronmental constrants. In our model, nodes k represent every possble rsk factor, edges e represent rsk nduced process. Intally, the network s composed of m 0 (m 0 2) fully connected nodes. At each subsequent tme, we grow the network accordng to the followng prescrpton: (a) At each ncrement of tme, a new node n s bult or created by one of the exstng vertces, and assume the probablty that the new node establshed by node s proportonal to the degree k, more precse, the probablty of node to buld new node at one step tme s: k Pcr () k (1) Eq. (1) ndcates that nodes wth large degrees have large probablty to buld new nodes. (b) Once the new node s establshed by node, m=m 0 edges are dstrbuted to and ts nearest neghbors. One edge connects node and the other (m-1) edges are attached to the nearest neghbors. Here we assume the (m-1) lnks are dstrbuted to the nearest neghbors of node unformly, thus the condtonal probablty that one of the nearest neghbors connectng the new node establshed by node s: m 1 P( ) k (2) These steps are repeated sequentally, creatng a network wth a temporally growng number of nodes t. We note that, snce the network sze t s ncreased by one n each dscrete tme step, t can be used as a system sze or a tme varable. The growng network model can be shown n Fgure 2. Fgure 2. Illustraton of our growng network model We now consder the dynamcs of degree k of a gven node to calculate the degree dstrbuton n our model. The degree k wll ncrease when a new node s establshed by node or by ts nearest neghbors. Consequently, k satsfes the followng dynamcal equaton: 577

6 dk dt 1 p ( ) p( l) p ( l) cr l () cr (3) Where Г() s the set of nearest neghbors of node. The frst term on the rght-hand sde corresponds to the random selecton of node as a creator of the new node wth probablty P cr(), whle the second term corresponds to the selecton of the nearest neghbors of node as a creator. Substtutng P cr(l) and P( l)=m-1/k lnto Eq. (3), we have dk dt m k k (4) Eq. (4) ndcates that the growth and lnear preferental attachment are preserved n our model. Ths equaton can be solved wth the ntal condton k (t=t)=m, yeldng t k() t m t (5) The evoluton of the degree k s the same as that n the BA model, therefore the degree dstrbuton of our model s dentcal to that of the BA model. Fgure 3 shows k of our model versus that of the BA model for t=10000 and m=2. From Fgure 4 we can see that the dashed lne s a power-law p(k)~k -3. Fgure 3. Degree k of our model versus that of the BA model Fgure 4. Degree dstrbutons for our model and BA mode The clusterng C of node s defned by C 2E k ( k 1) (6) 578

7 where E s the total number of lnks between the k neghbors of node. When a new node s establshed at tme t, the total number of edges E (t) between the nearest neghbors of node can ncrease ether f the new node s bult by or by ts nearest neghbors. The evoluton equaton for E s thus gven by de () t k k ( m 1) p( l) p ( k ) 2( m 1) dt k k cr l l () (7) Consderng that there are at least m-1 lnks among the nearest neghbors of node when t s establshed, the ntal condton of E can be wrtten as E (t=) m-1, where the equalty holds only for m=2. Therefore Eq. (7) can be readly ntegrated as 2( m 1) k( t) E ( t) ( m 1) m (8) Fgure 5 shows the clusterng spectrum for t =10000 and m=2, whch agrees well wth the analytcal predcton of Eq. (8). Fgure 5. The degree dependent clusterng coeffcent for m=2 and t=10000 The open crcles n Fgure 6 show L, the shortest path length between pars of nodes averaged over all node pars of sngle growng network realzatons, on a lnear scale versus t on a logarthmc scale. Fgure 6. Sem logarthmc graph of the average path length vs the system sze t for our model and BA model The data shows a lnear trend, demonstratng the average path length ncrease logarthmcally wth the network sze t. In addton, from Fgure 6, we can conclude that the local restrcton of edges makes the average path length of our model larger than that of the BA model. 579

8 Snce the network has a hgh clusterng as t and the average path length scales as L~lnt, our model dsplays the small-world propertes. 4. RISK EVOLUTION TOPOLOGY MODEL There are 27 rsk and 51 evoluton chan n the rsk evoluton model. These rsk factors are more complex, they can be dvded nto rsk from the ar and rsk from the ground. In ths model, n-degree and out-degree, the number of nodes and the number of branches can ndcate the mportance of rsk. These topologcal features of some mportant rsk factors are shown n Table 1. Table 1. Topologcal features of some mportant rsk factors Rsk n-degree out-degree number of nodes number of branches Departmental communcaton problems Loopholes n management Bad weather Brd pest Plot mstakes Runway ncurson Arcraft vehcle collson Lack of safety awareness Ground staff volaton Communcaton equpment fault From Table 1 we can see that Plot mstakes, Lack of safety awareness of the ground staff, Ground staff volaton and Bad weather are the key nodes (factors) of the rsk evoluton topology model. 5. CONCLUSION In ths paper, on the bass of rsk source dentfcaton, the arport runway rsk evoluton model based on complex network s proposed. The arport runway rsk evoluton mathematcal model s descrbed from clusterng coeffcent, average path length and degree dstrbuton. From the perspectve of modelng, ths research only constructs the model of the topology structure of the rsk evoluton, and the model of the relatonshp between the rsk evoluton factors s stll to be mproved. REFERENCES Albert R. (2002). Statstcal mechancs of complex networks, Revews of modern physcs, 74(1), Barabâs A., Albert R. (1999). Emergence of scalng n random networks, Scence, 5439, 286. Barabás A.L. (2009). Scale-free networks: a decade and beyond, Scence, 325(5939), Barabás A.L., Bonabeau E. (2003). Scale-free networks, Scentfc Amercan, 288(5), Edwards E. (1972). The computer study of transport processes under extreme condtons, Journal of Physcs C Sold State Physcs, 15(5), Hawkns F.H., Orlady H.W. (1993). Human factors n flght (2nd ed.), Avebury Techncal, England. Johnston N., McDonald N., Fuller R. (2001). Avaton psychology n practce, Ashgate Publshng Ltd, England. Kossnets G. (2006). Emprcal Analyss of an Evolvng Socal Network, Scence, 5757, 311. Kurant M., Thran P. (2006). Layered Complex Networks, Physcal Revew Letters. 13, 96. Oster C., Strong J., Zorn K. (2013). Analyzng avaton safety: Problems, challenges, opportuntes, Research n Transportaton Economcs, 43(1), Watts D., Strogatz S.H. (1998). Collectve dynamcs of small-world networks, Nature, 393(6684), Wegmann D.A., Shappell S.A. (2003). A human error approach to avaton accdent analyss: The human factors analyss and classfcaton system, Ashgate Publshng Ltd, England. Wener E.L., Nagel D.C. (1988). Human factors n avaton, Academc Press Inc, Calforna. 580