How to `Glue' a Robust Smart-Grid? A “Finite-Network” Theory for Interdependent Network Robustness

Gyan Ranjan and Zhi-Li Zhang Dept. of Computer Science & Engineering, University of Minnesota, Twin Cities, USA

ABSTRACT their seminal work, Buldyrev et al [2] have demonstrated Smart-grids are made up of two interdependent constituents: that interdependent networks can behave very differently a power production and distribution network catering to a from each of their constituents. In particular, two robust geographic area and a communication network that helps power-law networks when made interdependent via “random regulate and control the power-grid from which it derives coupling” may become more vulnerable to random failures. its electricity. Thus, failures in one of the two networks Their work, and those of others, quantify the structural ro- can lead to failures in the other and in some cases cause a bustness of interdependent networks in terms of asymptotic cascade or domino effect, bringing down the entire system statistical properties such as the existence of giant connected completely. In this work, we present a theoretical frame- components under random failures. As in the case of ro- work to model and study the structural properties of such bustness of single networks, while this the- interdependent networks based on a topological interpreta- ory characterization of network robustness provides valuable tion [4] of the Moore-Penrose pseudo-inverse of the graph insight into the general statistical properties of interdepen- Laplacians (L+). Using this framework, we study how the dences of classes of random graphs/networks, they are not way in which node pairs in two networks are coupled or very useful in practice, as real networks are deterministic “glued” together (thereby introducing interdependence) af- and finite. In particular, engineered infrastructure networks fects the overall robustness of the resulting interdependent such as power-grids and communication networks, are de- networks. Our study leads to some surprising (and some- signed to perform certain specific functions. There topo- what counter-intuitive) results. logical structures reflect and are constrained by the func- tional roles of various nodes as well as their geographical locations that are dictated by, say, user population or other 1. INTRODUCTION resources. In other words, their structures may differ signifi- Modern infrastructure networks are becoming increasingly cantly from (theoretically generated) random networks, and complex and dependent on one another. An example of such the interdependencies between two networks (e.g., communi- an interdependence is that of an electrical power-grid net- cation networks and power grids are not arbitrary, but often work regulated by a communication network which in turn are determined by geographical and other constraints). depends on the same power-grid for its electrical supply. In this work, we propose a theoretical framework for as- Due to such interdependence, failures of elements in one sessing structural robustness of interdependent networks which network, e.g., a small fraction of nodes in a communication is not dependent on specific assumptions of structure like network that is used to control and communicate elements in a power-law distribution. In doing so, we address a smart grid, can induce failures in the other, i.e. the power the following important questions related to vulnerability grid network, which would in turn cause further failures in assessment and protection of interdependent networks: (a) the communication and control network, thus producing a how can we characterize the robustness of the interdepen- cascade of failures in the interdependent networks. In the dent network on a whole? (b) how is the overall robustness recent past, electrical blackouts, like the one in Italy on 28 of two interdependent networks affected by the manner in September 2003 [5], have in fact been caused by such cas- which the two networks are coupled (or “glued”) together? caded failures. (c) how do we judicially select an appropriate coupling func- Clearly, the extent to which random failures or targeted tion, namely, select an appropriate collection of coupled node attacks can lead to a cascaded failure, of course, depends pairs to introduce“interdependence”(i.e., where the two net- on the structural properties of the constituent networks. In works are “glued” together) so that the resulting interdepen- dent networks are more robust to random failures or targeted attacks? The answer to the last question provides insights as to how we can harden two interdependent networks. Permission to make digital or hard copies of all or part of this work for In the following we demonstrate that these questions can personal or classroom use is granted without fee provided that copies are be answered – at least theoretically – by studying the topo- not made or distributed for profit or commercial advantage and that copies logical properties captured by our “geometry of networks” bear this notice and the full citation on the first page. To copy otherwise, to approach [3], namely, by using the Moore-Penrose pseudo- republish, to post on servers or to redistribute to lists, requires prior specific + permission and/or a fee. inverse of the graph Laplacians (L ) for the individual net- CSIIRW ‘11, October 12-14, Oak Ridge, Tennessee, USA Copyright 2011 works and that of the interdependent network. Based on ACM 978-1-4503-0945-5 ISBN ...$5.00. the topological interpretations of L+ [4], in particular, the the structural roles of nodes in a network, whereas K(G) structural metrics and Kirchhoff index, we de- can be used to measure the overall robustness of a network. velop a (deterministic) “finite-network” theory to study the In the following, we provide yet another interpretation – a robustness of interdependence. This theory enables us to topological interpretation – of L+ in terms of (connected) mathematically quantify the structural centrality and roles bi-partitions and spanning (sub-)trees. of coupled nodes (as well as uncoupled nodes) in the interde- Definition 1. A (connected) bi-partition P = (S,S′) of pendent networks as well as the robustness of interdependent ′ networks as a whole. More importantly, this theory allows us a graph G consists of two sub-graphs, S and S , where V (S)∩ V (S′) = φ and V (S) ∪ V (S′) = V (G), and each sub-graph to explicitly study how the way node pairs in two networks ′ are coupled or “glued” together (thereby introducing inter- S (resp. S ) is connected. dependence) affects the overall robustness of the resulting Let E(S,S′) denote the sets of “cross-edges” of the partition interdependent networks. Our study leads to some surpris- P = (S,S′), i.e., E(S,S′) := {(u, v) ∈ E, u ∈ S, v ∈ S′}. ing (and somewhat counter-intuitive) results: i) simply “glu- We see that a partition P =(S,S′) represents a state of the ing” together of structurally most central node pairs in the network in which E(S,S′) edges have failed, leaving the net- two constituent networks (when considered independently) work partitioned into two mutually exclusive sub-networks. does not always result in the most structurally robust in- It is this view that provides a significant insight into the true terdependent network ii) coupling a large number of struc- nature of structural centrality and Kirchoff index. turally least central node pairs in the two constituent net- Let P(G) denote the set of all (connected) bi-partitions works often leads to more robust interdependent networks of the graph, and given a bi-partition P = (S,S′) ∈ P(G), than coupling the same number of structurally most central let TS and TS′ represent the sets of spanning trees defined node pairs. Intuitively, this result suggests that by diffusing over the nodes of S and S′, respectively (since S and S′ are and distributing inter-dependencies among a large number of connected subgraphs of G, both TS and TS′ are non-empty). (geographically dispersed) node pairs in the two constituent It can be shown [4] that networks produce more robust interdependent networks. ′ ′ X |T (S)||T (S )||V (S )| + P ∈P(G),i∈V (S) 2. GEOMETRY OF NETWORKS AND lii ∝ ′ ′ (1) NETWORK ROBUSTNESS X |T (S)||T (S )||E(S,S )| P ∈P(G) In this section, we provide a brief overview of our “geom- etry of networks” approach [3], namely, via the L+ embed- and ′ ′ ding, for studying the robustness of a single network. In |T (S)||T (S )||V (S)||V (S )| + X particular, we present the topological interpretations of L n − 1 P ∈P(G) with respect to the bi-partitions of a network (see [4] for K(G)= (2) n ′ ′ details). X |T (S)||T (S )||E(S,S )| We model our network N as a simple graph G(V,E). Let P ∈P(G) |V (G)| = n be the number of vertices/nodes, henceforth + + From eq.(1), it is not too hard to see that if lii < ljj (thus called the order of the graph, and |E(G)| = m be the number C∗(i) > C∗(j)), then “on the average” node i is more likely Rn×n of edges/links in G. Let A ∈ be the adjacency ma- to lie in the larger of the remaining two sub-networks and trix of G(V,E), with elements aij = aji = 1 if e(i, j) ∈ E(G) node j lies in the smaller of the two after a network partition. and aij = aji = 0 otherwise. Let D be a diagonal matrix th Hence node i is in generally better connected (or structurally with d(i) on the i diagonal, where d(i) is the degree of more central) than node j. Similarly, from eq.(2), we see node i. The (unnormalized) graph Laplacian L is defined that smaller K(G) implies that in general more edges (thus as L = D − A. L is symmetric, doubly-centered and posi- ′ + + larger |E(S,S )|) have to be removed to partition G into two tive semi-definite. Let L = [lij ] denote the Moore-Penrose connected sub-networks. pseudo inverse [1] of L. L+ is also symmetric, doubly- centered and positive semi-definite, and admits an eigen- 3. MODELING INTERDEPENDENT decomposition L+ = ΦΛ+Φ′, where Φ is the matrix whose columns are the mutually perpendicular eigen-vectors of L+ NETWORKS and Λ+ is a diagonal matrix with the eigen-values of L+ (i.e. In this section we briefly describe the basic notations and the reciprocals of the non-zero eigen-values of L). Moreover, a simple graph model for interdependent networks. In par- L+ = ΦΛ+Φ′ = X′X, where the ith column of the matrix ticular, we introduce a coupling function, C = {[u, v], which X represents the coordinates for node i in an n-dimensional specifies how and where two constituent networks are cou- Euclidean space. The squared-length of the position vector, pled or“glued”together to introduce inter-dependencies among + i.e., the of node i from the origin, is given by lii . the two networks and form a single interdependent network. In [3], we define the structural centrality of node i as Given two networks, N1 and N2, we represent them in ∗ + C (i) := 1/lii . The Kirchoff index of a network is therefore terms of their respective graphs: G1(V1,E1) and G2(V2,E2). n + n ∗ given by K(G) = Pi=1 lii = Pi=1 1/C (i). Geometrically, Also, for the sake of simplicity, we assume that |V1| ≈ |V2|, we see that closer a node i is to the origin, the higher its i.e., the two networks are of comparable sizes. Let C denote structural centrality C∗(i) is. Likewise, the smaller K(G) is, a collection of node pairs [u, v], one from each constituent the more compact the embedding of the network is. network, i.e., C = {[u, v], u ∈ V1, v ∈ V2}. We refer to C as Previously in [3], we have provided two interpretations for a coupling function, and each node pair [u, v] in C a coupled ∗ + C (i) := 1/lii , in terms of random walks and electrical net- node pair. Intuitively, the coupled node pairs are where works, and demonstrated that C∗(i) can be used to measure an inter-dependency between two constituent networks are Low-Low High-High introduced. The cardinality of C, k = |C|, represents the 1a 1 a 2 b 2 b 5e 5 e 3 c number of inter-dependencies, i.e., the number of coupled 3 c 4 d node pairs. Hence given a coupling function C, two networks, 4 d 1a 1 a Low-Low High-High 2 b 2b b 5e N1 and N2, form as a whole a single interdependent network, 2 3 4 5 1 2 4 5 5 e 3 c 3 c 1a 3c denoted as Gc(Vc,Ec). d e a b d e 4 d 4d b c 1 a As a graph, the interdependent network Gc(Vc,Ec) can be 1a 2 b 1 4 5 2b b 5e 2 3 4 5e 3c defined as follows. Let [u, v], u ∈ V1 and v ∈ V2, be a cou- 5 e 1a 2b 3c b c d a d e 3c 4d pled node pair in C. When glued/coupled together, u and 4 d 1 a 1 5 1a 3 4 c 5e 1a 2b 5e 2b 3c 4d v will result in a new node u ⊗ v in G such that each edge 2b b d a e 5 e c e(u, x) ∈ E1 will now create an edge e(u ⊗ v,x) ∈ E(Gc), 3c 3 5 2b 3c 4d if x is an uncoupled node in N1; it will create an edge 4d 1a 2b 4d 5e 1a 2b 3c 4d 5e c e e(u⊗v,x⊗y) ∈ E(Gc) if x is also a node in another coupled pair [x,y] ∈ C. Similarly, each edge e(v,y) ∈ E2 will create 1a 2b 3c 4d 1a 2b 3c 4d 5e an edge e(u ⊗ v,y) ∈ E(Gc), if y is an uncoupled node in N2; it will create an edge e(u ⊗ v,x ⊗ y) ∈ E(Gc) if y is also Figure 1: Coupling in stars and paths (n = 5). a node in another coupled pair [x,y] ∈ C. In other words, where there were two vertices u and v in the individual net- works, we create a macro- u ⊗ v in the glued network the κ-lowest ranked nodes from G2 and (c) random i.e. κ with a neighbor set that is a union of the neighbors of u random pairs from G1 and G2. The overall robustness of the and v in the original networks. This representation clearly coupled/interdependent network is then measured in terms captures the interdependent nature of the two vertices in of its Kirchhoff index i.e. K(Gc). In the following, we first question, whereby the macro-vertex u ⊗ v fails if either u use a simple example (networks with tree topologies) and or v fail in their individual networks. Similarly, the failure then a network with realistic (the Italian of edge e(u, x) in G1, results in the failure of e(u ⊗ v,x) in power grid) to illustrate the coupling process and the effect Gc. Uncoupled nodes and their associated edges (to other of different coupling functions on the robustness of resulting uncoupled nodes) in each of the two individual networks interdependence networks. are transported to the coupled/interdependent Gc(Vc,Ec) as is. Thus, if the number of couplings is κ, then |V (Gc)| = 4.1 When Both Networks are Trees |V (G1)| + |V (G2)|− κ and |E(Gc)| = |E(G1)| + |E(G2)|. So We consider a simple case to illustrate our theory, where the order of the glued network reduces by κ as compared two constituent networks are both trees. We study the cou- to the total of its constituents, but its volume (number of pling of two types of trees: stars and chains/paths of order edges) is the same. n, which represent the most well connected (compact) and Given this definition of an interdependent network Nc := the least connected of all trees for any given n. More im- Gc(Vc,Ec) formed by two constituent networks, N1 and N2, portantly, in the context of power-grids (and to an extent in via the coupling function C, we can directly apply the re- communication networks), a star topology represents a pro- sults in Section 2: using the Moore-Penrose pseudo-inverse duction/distribution center i.e. the root of the star, while + of the graph Laplacians (Lc ) for the interdependent network the pendants represent the consumers/first-hop relay-ers. Gc(Vc,Ec), we define the corresponding structural centrality Similarly, a chain represents a linear sub-network formed metrics and the Kirchhoff index for the interdependent net- by a series of relay-ers to disseminate power or information. work to measure the structural roles of individual (coupled Using eq.(1), we observe that for a star topology: and uncoupled) nodes as well as the overall robustness of 2 the interdependent network. + n − 1 + n − n − 1 l = , and l = (3) root n2 pendant n2 It is easy to see that for n > 2, the root of the star has 4. EFFECTOFCOUPLINGFUNCTIONSON + a lower lii value than the pendants and is therefore more + NETWORK ROBUSTNESS structurally central. Similarly, for a chain of order n, the lii th Of particular importance, and one of the principal con- for the i vertex from the end is given as: tributions of our work, is that our theory enables us to in- 2 2 + 6i − 6(n + 1)i + 2n + 3n + 1 vestigate the effect of the coupling function C on the over- lii = (4) all robustness of the resulting interdependent network. In 6n + other words, it allows us to vary the manner in which we The form in (4) is parabolic in lii and i. Clearly, for a select the node pairs from the two constituent networks to given n, the minima is attained when i = ⌈n/2⌉, i.e. at be glued together, and study the “optimal” way of introduc- the middle node/s of the chain, and the maxima is attained ing interdependencies so as enhance or “harden” the overall when i =1= n i.e. at the pendants. Thus, the structural robustness of the resulting interdependent network. centrality of nodes in a chain decreases as we move from the For this purpose, we adopt a structural centrality based center of the chain towards the pendants on either side. ordering of the nodes for the selection process. First, we rank Next, we demonstrate the low−low and high−high gluing the nodes in the networks N1 and N2 in terms of their struc- strategies for a star and a chain respectively with another tural centrality values in their respective networks. Having star and chain of the same orders in Fig. 1, for increasing obtained the ranks, we then define the following three ways values of κ : 1 ≤ κ ≤ n. Observe that for κ ≥ 2, the of gluing them together: (a) high-high i.e. the κ-highest low − low strategy produces multiple cycles in the interde- ranked nodes from G1 with the κ-highest ranked nodes from pendent networks, thereby providing alternate connectivi- G2, (b) low-low i.e. the κ-lowest ranked nodes from G1 with ties between nodes and safeguarding against eventual edge 14 10 High−High High−High 500 High−High Low−Low 12 Low−Low Low−Low Random Random Random 8 10 400 ) ) ) c c

c 8 6 300

6 K(G K(G K(G 4 200 4 2 100 2

0 0 0 1 2 3 4 5 1 2 3 4 5 0 10 20 30 40 50 60 70 #Couplings #Couplings #Couplings (a) Star (b) Path (c) Italian power grid

Figure 2: K(Gc) for glued/coupled networks for the three different coupling functions.

48 48 66 66

61 61 67 67 59 60 53 53 59 60 results that only for small κ, pairing and coupling the most 40 40 68 68

54 49 55 50 62 54 49 55 65 50 62 41 65 structurally central nodes (in the constituent networks) pro- 56 64 41 56 64 45 57 63 45 57 63 46 46

51 58 duce more robust (i.e., smaller K(Gc)) networks than other 42 51 37 42 58 37 43 47 43 52 47 34 52 coupling strategies; for κ > 10, the low − low strategy 34 38 39 44 38 31 35 39 44 31 35 produces more robust interdependent network than other 32 36 29 32 36 27 29 27 19 33 strategies. We note that as κ increases, the low − low strat- 20 22 19 33 24 20 22 24 14 30 17 25 30 egy glues peripheral nodes thereby creating longer cycles in 14 17 25 28 23 28 15 26 the network as compared to the other two strategies. Such 23 15 26 18 21 10 18 21 10 longer cycles safeguard network wide connectivities against 8 16 7 13 8 16 5 6 7 13 5 random edge failures, thus resulting in more robust interde- 11 4 12 6

11 1 4 12 3 9 1 pendent network structures. We also observe similar results 3 2 9 2 for the western states power grid network in the US.

Figure 3: The Italian power grid network coupled 5. CONCLUSIONS with itself: Red → T urquoise decreasing structural centrality. In this work, we presented a theoretical framework to as- sess structural properties of interdependent networks, based on the topological properties of the Moore-Penrose pseudo- failures. It is well known that greater the number of cycles inverse of the graph Laplacians. We demonstrated that the in a graph, higher the count of spanning trees which signi- structural centrality of nodes can be used to select nodes fies better redundant connectivities between node pairs. In for gluing the two networks together and the robustness of the interdependent network can be measured in terms of contrast, the high − high strategy produces small loops be- + tween adjacent nodes and leaves the overall structure rather its Kirchhoff index, given by the trace of the L of the tree like. Thus the low − low strategy results in more ro- interdependent network. With the help of example tree bust glued/coupled networks for κ ≥ 2 in both the star and structures and the Italian power grid network, we presented chain topologies.This effect is well captured in the values of comparative results for three gluing strategies based on the structural of the nodes. Of the three strategies, K(Gc) shown for 1 ≤ κ ≤ n in Fig. 2(a) and (b), where the the low − low strategy eventually wins out, producing the low − low strategy attains lowest values of K(Gc) for κ> 2 most robust coupled structures (interdependent networks). (note that smaller K(Gc) is, more robust the network is). We see that when κ is small, pairing and coupling the most Our results suggest that by diffusing and distributing inter- structurally central nodes (in the constituent networks) pro- dependencies among a large number of (geographically dis- duces more robust networks than other coupling strategies. persed) node pairs in the two constituent networks produce However, somewhat counter-intuitively, when κ increases more robust interdependent networks. and becomes sufficiently large, pairing and coupling struc- Acknowledgment This research was supported in part by turally least central node pairs in the two constituent net- DTRA grant HDTRA1-09-1-0050 and NSF grants CNS-0905037, works produces more robust interdependent networks. This CNS-1017647 and CNS-1017092. We also thank the anony- result suggests that distributing interdependencies amongst mous reviewers for helpful comments. (geographically) disparate nodes may result in more robust interdependent networks. We explore this further with the help of a real world power-grid network in the next section. 6. REFERENCES [1] A. Ben-Israel and T. Greville. Generalized Inverses: Theory and Applications, 2nd edition. Springer-Verlag, 2003. 4.2 The Italian Power Grid Network Example [2] S. V. Buldyrev and et al. Catastrophic cascade of failures in The Italian power grid network (see Fig. 3) is a network interdependent networks. Nature, 464:1025–1028, 2010. of order n = 68 and m = 93. The nodes in fig. 3 have been [3] G. Ranjan and Z.-L. Zhang. A geometric approach to colored by their structural centralities. Notice how the struc- robustness in complex networks. In Proc. of SIMPLEX’11 tural centrality reduces as we move towards the periphery (co-located with IEEE ICDCS’11), June 2011. of the network. We now glue the Italian power grid network [4] G. Ranjan and Z.-L. Zhang. Geometry of complex networks and structural centrality. arXiv:1107.0989, 2011. with an exact copy of itself, which represents a communi- [5] V. Rosato and et al. Modelling interdependent cation network (cf. [2]) using various coupling functions C infrastructures using interacting dynamical models. Int. J. for increasing values of κ. Once again, (see fig. 2(c)) we ob- Crit. Infrastructure, 4:63–79, 2008. tain the same surprising (and somewhat counter-intuitive)