A Note on Emerging Science for Interdependent Networks

A Note on Emerging Science for Interdependent Networks Junshan Zhang School of ECEE, Arizona State University Network Science Workshop, July 2012 (Based on joint work with Osman Yagan and Dajun Qian) 1

From Individual Networks to Network of Networks • Networked systems: modern world consists of an intricate web of interconnected physical infrastructure and cyber systems, e. g. , communication networks, power grid, transportation system, social networks, … • Over the past few decades, there has been tremendous effort on studying individual networks: • Communication networks, e. g. , Internet, wireless, sensor nets, … • Complex networks, e. g. , E-R graph, small world model, scale-free networks … • Little attention has been paid to interdependent networks: Many networks have evolved to depend on each other, and depend heavily on cyber infrastructure in particular • Focus of this talk: interdependent networks (e. g. , cyber-physical systems) 2

Cyber-Physical Systems (CPS) n n n A networked system consists of physical network and cyber network Emerging as the underpinning technology for 21 th century Applications: smart grid, intelligent transportation system, manufacturing, etc. 3

CPS - Two Interdependent Networks Interdependence: Operation of one network depends heavily on the functioning of the other network Q) what is the impact of interdependence between cyber-network and physical network? I) Vulnerability to cascading failures: node failures in one network may trigger a cascade of failures in both networks, and overall damage on interdependent networks can be catastrophic. II) Acceleration of information diffusion: conjoining can speed up information propagation in interdependent networks.

Part I: Impact of Network Interdependence on Cascading Failures Q) What is the impact of interdependence on cascading failures between cyber-network and physical network? How to design a system with better resilience against cascading failures? More susceptible to cascading failure due to interdependence 5

An Example: Modern Power Grid Network interdependence l Power station operation relies on the control of nodes in cyber infrastructure l Cyber nodes need power supply from power stations Vulnerability to cascading failures Even failures of a very small fraction of nodes may trigger a cascade of failures and result in a large scale blackout, e. g, blackout in Italy 2003 Power systems in Italy [Nature 2010] 6

Case Study on WTC Disaster • Telecommunication: e. g. , Verizon lost 200 K voice lines and 4. 4 M data circuits; 71% volume increase in 911 service and was switched to Brooklyn office • Electric power system lost 3 substations, 5 distribution networks, • … Q) Which parts are most vulnerable and which other parts are most resilient? Where are interdependences? 7

Network Model I Two interconnected networks need mutual support • Initial setting: a fraction 1 -p of A-nodes failed. • Approach: To quantify ultimate functioning giant component size and critical threshold p Net B: Cyber infrastructure Inter-edge Net A: Power grid 8 2021/2/22

Giant Connected Component (GCC) Model “one-to-one correspondence” [Nature 2010] Inter-edge: specify interdependence between two networks Intra-edge: connections between nodes in same network Net. A Net. B Assumption: a node can “function” only if l belongs to the giant connected component of its own network l has at least one inter-edge (support) from the other network 2021/2/22 9

An Illustration of Cascading Failures Step 1 Step 2 Step 3 Functioning giant component attack step 1 step 3 l After a 4 is removed, a 3 stops fails since it is no longer in the giant component in A l Cascading failure step 2 l The intra & inter edges associated with a 3 and a 4 will be removed l b 4 and b 3 will be removed due to losing inter-edges from A 2021/2/22 10

Allocation Strategies for Inter-Edges Q) How to allocate inter-edges against cascading failures? Random allocation Number of inter-edges each node random; following binomial distribution Direction of inter-edge Uni-directional: unilateral support from a (support from nodes in the node in the other network) Our strategy Uniform: the same for all nodes Bi-directional: mutual support between two connected nodes Metric for robustness: n Critical threshold pc: minimum p that ensures the existence of functioning giant component after cascading failures; higher pc means less tolerant to network failures (lower robustness) and vice versa 11

Analysis of Cascading Failures Uniform Allocation of Bi-directional Inter-Edges Stage 2: Cascading effect of A-node failures on network B Stage 1: Node failures in Network A Random failures of 1 -p of nodes inter-edge can be disconnected w. p. 1 -p. A 1 functioning giant component A 1 p. A 1=p. PA(p) Removal of inter-edges The remaining fraction of nodes with inter-edges: p’B 2= 1 -(1 -p. A 1)k functioning giant component B 2 p. B 2=p’B 2 PB(p’B 2) Notation: PA(p), PB(p): After a fraction 1 -p of A-nodes (B-nodes) failed, the giant component fraction out of remaining p. N nodes 13

Uniform Allocation of B-directional Edges (Cont’d) Stage 3: Network A ’s further fragmentation due to B-node failures inter-edge can be disconnected w. p. 1 -PB(p’B 2) The remaining fraction of A 1: 1 -(1 -PB(p’B 2))k For A, the joint effect of Stage 1 & 3 on A amount to node failures in A with fraction 1 -p’A 3=1 - p+p(1 -PB(p’B 2))k functioning giant component A 3 p. A 3=p’A 3 PA(p’A 3) 14 Key step: further node failures in A 1 at Stage 3 has the same effect as taking out equivalent fraction of nodes in A

Uniform Allocation of Bi-directional Edges (Cont’d) functioning giant component size in dynamics of cascading failures Stage 1 network A p. A 1=p. PA(p) Stage 3 network B Stage 2 p. B 2=p’B 2 PB(p’B 2) Stage 4 p. A 3=p’A 3 PA(p’A 3) p. B 4=p’B 2 PA(p’B 2) n The recursive process reaches stead state n By calculating the equilibrium point, we can get the ultimate giant component size and critical threshold 15 ….

Uniform vs. Random Allocation Observation: Uniform allocation leads to higher robustness than random allocation Intuition: Random allocation can result in a non-negligible fraction of nodes with no inter-network support, whereas uniform allocation can guarantee support for all nodes No support Random allocation uniform allocation 16

Uni-directional v. s. Bi-directional Observation The bi-directional inter-edges can better combat the cascading failures than unidirectional inter-edges n The cascading failures are more likely to spread with uni-directional edges n For fair comparison, the total number of uni-directional edges should be twice the number of bi-directional edges 17

Numerical Example Lower pc indicates the higher robustness n Two Erdos-Renyi networks with average intra-degree fixed at 4 n The pc varies over different average inter-degree k n As expected, the uniform & bi-directional allocation leads to the lowest p c under various conditions 2021/2/22 18

Limitation of GCC Model for Physical Network Giant Connected Component (GCC) model [Nature 2010] Assumption: Only the nodes in the largest connected component can work properly Pros: facilitate theoretical analysis Cons: Cannot capture some key features of physical network, e. g. , power grid 19

Shortcoming of GCC Model for Power Grid Features of power grid Properties of GCC-model Even a single node failure may lead to large scale blackout Only a substantial fraction of damages can trigger cascading failures isolated components in the power grid may still operate (microgrid, islanding) Only the largest connected component can function impacts of load redistribution on node functioning depends only on its connection to the giant component 2021/2/22 20

Threshold Model Threshold model [Gleeson 07] n A node would fail if the fraction of its failed neighbors exceeds the threshold; capture the load redistribution feature the more power stations fail, the more load being redistributed to A A: more likely to fail 21 2021/2/22

Network Model II Two interdependent networks with mutual support - GCC-model for cyber-network; - threshold model for physical infrastructure Cyber-network Power grid 22 2021/2/22

GCC Model vs Threshold Model Sparsely connected regime (low average degree) power grid Threshold model § isolated components can still function § the propagation of cascading failure is constrained by isolated components § micro-grids: isolated power stations can still function Defensive Islanding: islanding can prevent further failure spreading GCC-model § All power stations cannot function in subcritical region 23 2021/2/22

GCC Model vs Threshold Model Densely connected regime (high average degree) Threshold model § A small fraction of node failures may lead to network collapse GCC-model § cascading failures cannot happen if initially failed fraction q is small power grid Large scale blackout can be triggered by one station failure, e. g. , Italy black out 2003 Main points: § GCC model underestimates the damages that could be triggered by a small fraction of node failures § Threshold model captures some key features of power grid 24

Robustness of CPS model II Robustness performance (initial failed fraction q=0. 1%) small initial failures that have negligible impact on single physical network may damage overall CPS (with high degree and low threshold) 25

Densely Connected Regime Intuition: Single network (low threshold) - Each node can tolerate more neighbors’ failures - Very few node failures are difficult to incur further failures; although still susceptible to large initial failures Interdependent networks (low threshold) -the scale of node failures can be “amplified” due to cascading failures between two networks -the system is vulnerable to a small fraction of node failures 27 2021/2/22

Part II: Impact of Network Interdependence on Information Diffusion Q) What is the impact of interdependence on information diffusion in overlaying social-physical networks? Information cascade • • interdependence between two networks can facilitate information diffusion 28 2021/2/22 information epidemic real-time information propagation

Social-Physical Networks “A social network is a social structure made up of a set of actors (e. g. , individuals or organizations) and the dyadic ties between these actors (e. g. , relationships, connections, or interactions)” [Wiki] Physical information network -Traditional “physical” interactions: e. g. , face-to-face contacts, phone calls … Online social network Social-physical network: medium for information diffusion 29

Interdependence across Multiple Networks “coupling’’ “Multi-member’’ Different social networks can “overlap” due to “multimember” individuals Individuals can be member of multiple social networks Q): How does information propagate across multiple interdependent networks? 30

Model: Overlaying Social-Physical Networks online connection online membership F: online social network same person W: physical info network § § physical interactions individual n nodes in physical information network; only one online social network Each individual in W participates in F with probability α Each node in W has neighbors with Each node in F has online neighbors with 31

Information Cascade interdependence between multiple networks information diffusion in one network can trigger the propagation in another network and may help information diffusion online social network physical info network 32

SIR Model for Information Diffusion § Message can successfully spread along a link that corresponds to physical interaction or online communication with probabilities and , respectively Only existing links can be used in spreading the information 33

Information Cascade in Overlaying Social-Physical Networks “Giant component”: the largest connected component in the network Questions § When an information epidemic can take place? § What is the size of information epidemic? § When a giant component that occupies a positive fraction of nodes can appear? § What is the fractional size of giant component? 34

Analysis of Information Diffusion Challenge § How to characterize the phase transition behavior (existence condition and size of giant component) in two overlaying graphs? Key idea § Treat the overlaying networks as an inhomogeneous random graph Approaches § Colored degree-driven random graphs with different types of links [Soderberg 2003] • general case: nodes in F and W have arbitrary degree distributions § Inhomogeneous random graph with different types of nodes [Bollobás et al. 2007] • Alternative approach for a special case where nodes in F and W have Poisson degree distributions, i. e. , F and W are Erdős–Rényi graphs 35

General Case: Graphs with Arbitrary Degree Distributions • Original overlaying networks can be modeled as a random graph where nodes are connected by two types of links (online communications and physical interactions). • The phase transition behaviors of the equivalent random graph can be characterized by capitalizing on mean-field approach [Soderberg 2003]. treat as a single node F W overlaying social-physical networks random graph with 2 types of links 36

Main Result I The existence of the giant component is determined by the critical threshold where If the critical threshold , then with high probability there exists a giant component with size ; otherwise then the largest component has size § The critical threshold marks the “tipping point ” of information epidemics. 37

Main result II The fractional size of giant component in the random graph is given by where h 1 and h 2 in (0, 1] are given by the smallest solution of § The fractional size of giant component gives the fractional size of individuals that receive the message. 38

Numerical Result: Critical Threshold overlaying social-physical networks α requirement for the existence of giant component when 0. 1 0. 5 0. 9 single network [Newman 2002] § If the network W and F are disjoint, an information epidemic can occur only if or Main point: § Two networks, although having no giant component individually, can yield an information epidemic when they are conjoined together 39

Special Case: Erdős–Rényi Graph Scenario: overlaying Erdős–Rényi Graphs § graph W has n nodes; each node in W participates F w. p. α § any two nodes in W are connected w. p. § any two nodes in F are connected w. p. Approach: inhomogeneous random graph [ Bollobás 2007] § can quantify the size of the second largest connected component when a giant component exists § gives a tighter bound on the largest connected component when a giant component does not exist 40

Special Case: Erdős–Rényi Graph § Critical threshold: If , then w. h. p. the largest component has size § . Fractional size of giant component: where ρ1 and ρ2 in [0, 1] are determined by the largest solution to 41 . If and the second , then the largest component

Impact of Network Interdependence on Information Diffusion n We focus on information diffusion in an overlaying socialphysical network, where message spreads amongst people through both physical interactions and online communications. We show that even if there is no information epidemic in individual networks, information epidemics can take place in the conjoint social-physical network We show that the critical threshold and the size of information epidemics can be precisely determined using inhomogeneous random graph models. 42 2021/2/22

Phase Transition Behavior Information Diffusion vs. Cascading Failures Information Diffusion Cascading Failures 43

Information Diffusion W F Initial set-up W F F to 3 st hop Propagation to Propagation 1 st hop neighbors - v_1, v_9, v_10 are not Facebook users - Information starts at node v_1 Giant Component of W consists of {v_1, v_2, v_3, v_5, v_6, v_7, v_9 } Giant Component of F consists of {v_2, v_3, v_4, v_5, v_6, v_7, v_8 } Giant Component of FUW consists of {v_1, … , v_10} nodes that receive the information F W W W F Propagation to 2 st hop neighbors Steady state Information does cascade between the two networks, but the eventual cascade size can be computed by the giant component size of the conjoint network H = F U W. Behavior boils down to the phase transition of a single combined network. Second-order (continuous) phase transition

A Cascading Failures B Net A: Only the giant component survives Net B: Only nodes that. Net B: Only the giant have support survive component survives Net A: Only nodes that have support survive - Net A and Net B are defined on disjoint vertex sets. - Initially node v_1 fails. At each stage, only the Giant Component of the functional nodes remain. A giant component computation is required at each stage Giant Component of A consists of {v_1, v_2, v_3, v_5, v_6, v_7, v_9 } Giant Component of B consists of {v_2, v_3, v_4, v_5, v_6, v_7, v_8 } While failures cascade between the two networks, the network reduces to its giant component at each step. the overall dynamics is equivalent to the superimposition of possibly many phase transitions. First-order (discontinuous) phase transition

Conclusions We investigate the impact of interdependence between cybernetwork and physical network: • I) Vulnerability to cascading failures: node failures in one network may trigger a cascade of failures in both networks. • To improve the robustness of interdependent networks, we proposed some strategy for allocating inter-edges. • II) Acceleration of information diffusion: conjoining can speed up information propagation in coupled networks. There are still many open questions on network interdependence. Need a foundation for interdependent networks! 46