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Breakdown of Interdependent Directed Networks Xueming Liu, ∗ † H i i \"SI Appendix"" | 2015/12/23 | 9:13 | page 1 | #1 i i Supporting Information: Breakdown of interdependent directed networks Xueming Liu, ∗ y H. Eugene Stanley y and Jianxi Gao z ∗Key Laboratory of Image Information Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China,yCenter for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, and zCenter for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115 Submitted to Proceedings of the National Academy of Sciences of the United States of America I. Notions related to interdependent networks Multiplex networks. The agents (nodes) participate in ev- Both natural and engineered complex systems are not isolated ery layer of the network simultaneously. The connections but interdependent and interconnected. Such diverse infras- among these agents in different layers represent different tructures as water supply systems, transportation networks, relationships [14, 15, 28, 29, 13, 30, 31]. For example, a fuel delivery systems, and power stations are coupled together user of online social networks can subscribe to two or more [1]. To study the interdependence between networks, Buldyrev networks and build social relationships with other users on et al. [2] developed an analytic framework based on the gener- a range of social platforms (e.g., LinkedIn for a network of ating function formalism [3, 4] and discovered that the interde- professional contacts or Facebook for a network of friends) pendence between networks sharply increases system vulner- [11, 14]. ability, because node failures in one network can lead to the Temporal networks. This is a group of networks that repre- failure of dependent nodes in other networks, and this process sents the same networked system during different time pe- can become recursive and cause a failure cascade and system riods [16, 17]. The system dynamically changes with time: collapse. For example, electrical blackouts that affect large re- some nodes are added or deleted and links become active gions are usually the result of cascading failures between the or inactive during different time periods. For example, in a interdependent communication network and power grid [5]. telephone network, links represent sequences of nearly in- The development of interdependent networks triggered much stantaneous contacts [32, 33]. Telephone networks during research on the topic. Parshani et al. [6] studied a model different time periods form temporal networks [17]. similar to a real-world system: two partially interdependent Multilayer networks. These multi-dimensional multiplex networks. Shao et al. [7] developed a theoretical framework networks can be represented by a quadruplet M = for understanding the robustness of interdependent networks (VM ;EM ; V; L) (see Fig. S1), in which M is a multilayer with a varying number of supports and dependency relation- network, V the union of all the nodes of all layers, L a ships. Baxter et al. [8] studied avalanches in interdependent sequence of the sets of elementary layers (each elementary networks. Gao et al. [9] developed a general framework for layer contains multiple networks within which the nodes studying percolation in a network of networks. Focusing on are connected by different interaction type, time, or other interdependent networks and networks of networks, there are chacteristics), VM a set that contains only the node-layer other approaches to describing complex systems that comprise combinations in which a node is present in the correspond- multiple networks, e.g., interconnected networks [10, 11], mul- ing layer, and EM a set of pairs of possible combinations tilayer networks [12, 13], multiplex networks [14, 15], tempo- of nodes and elementary layers [12, 13]. ral networks [16, 17, 7] etc. They approaches differ from each A network of networks. This is a general concept describ- other and are not interchangeable. Here we describe these ap- ing systems composed of multiple networks where the net- proaches and in Fig. S1 provide schematic diagrams for the works may be of different types and may be interdependent different multiple networks. [34], be interconnected [35], may share interactions that are antagonistic (the functioning of one node causes the failure Interdependent networks. The nodes within one network of another node) [36, 37], or be coupled with other complex are connected by connectivity links, and the nodes of differ- relations [38]. A network of networks model has been used ent networks are adjacent to one another via dependency in the study of a wide range of topics, including the per- links [2, 6, 18]. Connectivity links, e.g. the friendships be- colation of a network of interdependent networks [9], the tween individuals in a social network, the business connec- spreading of an epidemic on a network of interconnected tions in a financial network, or the cables between Internet networks [27], and the robustness and restoration of a net- routers [19, 20, 21, 22, 23] enable the nodes to function co- work of ecological networks [35]. operatively as a network. Dependency links represent the functional dependency relations between two elements: if one fails, the other will also fail. For example, the function- ing of a router in a computer network depends on a power station in a power grid. If a power station stops functioning, Reserved for Publication Footnotes the dependent router also stop functioning [2, 23, 24, 25]. Interconnected networks. The nodes within one network are connected by connectivity links, and the nodes of dif- ferent networks are also connected by connectivity links. Alternately, interconnected networks can be regarded as interconnected communities within a single and larger net- work [10, 11, 26, 27]. For example, the transportation sys- tem in a city may contain a network of bus stops and sub- way stations. People may use both when moving from one place to another within the city [11]. www.pnas.org | | PNAS Issue Date Volume Issue Number 1{8 i i i i i i \"SI Appendix"" | 2015/12/23 | 9:13 | page 2 | #2 i i II. Generating functions in a directed network III. Analytic framework of cascading failures The generating function of a directed network is built for the The system of interdependent directed networks contains two joint probability distribution of in- and out-degrees [39, 40], directed networks, network A and network B, consisting of NA denoted as and NB nodes respectively. Network A is characterized by the 1 degree distribution PA(kin; kout), where kin and kout are the X kin kout Φ(x; y) = P (kin; kout)x y : [S1] in- and out- degree of a node respectively, whose generating kin;kout function is When computing the size of the giant weakly connected com- 1 ponent (GWCC), the directness of links can be ignored. The X kin kout ΦA(x; y) = PA(kin; kout)x y ; generating function takes the form Φ(w)(x) = Φ(x; x) and its kin;kout (w) (w)0 (w)0 normalized derivative is Φ1 (x) = Φ (x)=Φ (1). The 0 GWCC exists if Φ(w) (1) > 1 and the size of the GWCC W and the generating functions for the branching processes are can be obtained from the relations (w) @yΦA(x; y)jy=1 W = 1 − Φ (tc); [S2] ΦA1(x; 1) = ; (w) @yΦA(1; 1) where tc = Φ1 (tc). The computation of the final GWCC in the interdependent directed networks is the same as the and computation of final connected component in the interdepen- @xΦA(x; y)jx=1 dent undirected networks [6, 18], which can be covered by our ΦA1(1; y) = : analytic framework. @xΦA(1; 1) The giant strongly connected component (GSCC) is the interception of the giant in-component and the giant out- Analogously, network B is characterized by the degree distri- component [3]. The giant in- and out-component are respec- bution PB (kin; kout), whose generating function is tively characterized by the in- and out-degree distributions, 1 whose generating functions are Φ(x; 1) and Φ(1; y) respec- X kin kout tively. The normalized derivatives for them are respectively ΦB (x; y) = PB (kin; kout)x y ; kin;kout @yΦ(x; y)jy=1 @xΦ(x; y)jx=1 Φ1(x; 1) = ; Φ1(1; y) = : @yΦ(1; 1) @xΦ(1; 1) and the generating functions for the branching processes are [S3] The size of the giant in- I and out-component O can be de- @yΦB (x; y)jy=1 termined in the similar way as the study of the GWCC [39], ΦB1(x; 1) = @yΦB (1; 1) which are respectively I = 1 − Φ(xc; 1) and O = 1 − Φ(1; yc), where x = Φ (x ; 1) and y = Φ (1; y ). The nontrivial so- c 1 c c 1 c and lutions xc < 1 and yc < 1 respectively mean the probability that the connected component obtained by moving against the @xΦB (x; y)jx=1 ΦB1(1; y) = : link directions starting from a randomly chosen link is finite, @xΦB (1; 1) and the probability that the connected component obtained by moving along the link directions starting from a randomly Network A and network B are coupled by dependency links, chosen link is finite. with the coupling strength qA and qB ranging from 0 to 1, The probability that a node belongs to the giant in- where qA is the fraction of nodes in network A (A-nodes) that kin kout depend on the nodes in network B (B-nodes), and qB is the component is 1 − xc , and 1 − yc is the probability that a node belongs to the giant out-component. Each node belong- fraction of B-nodes that depend on A-nodes. ing to the GSCC is the node belonging to the giant in- and out-component at the same time.
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