Robustness and Algebraic Connectivity of Random Interdependent Networks
aa r X i v : . [ c s . S I] A ug Robustness and Algebraic Connectivity of Random InterdependentNetworks
Ebrahim Moradi Shahrivar, Mohammad Pirani and Shreyas Sundaram
Abstract
We investigate certain structural properties of random interdependent networks. We start by studying a propertyknown as r -robustness, which is a strong indicator of the ability of a network to tolerate structural perturbations anddynamical attacks. We show that random k -partite graphs exhibit a threshold for r -robustness, and that this thresholdis the same as the one for the graph to have minimum degree r . We then extend this characterization to randominterdependent networks with arbitrary intra-layer topologies. Finally, we characterize the algebraic connectivity ofsuch networks, and provide an asymptotically tight rate of growth of this quantity for a certain range of inter-layer edge formation probabilities. Our results arise from a characterization of the isoperimetric constant of randominterdependent networks, and yield new insights into the structure and robustness properties of such networks. Index Terms
Random Interdependent Networks, Robustness, Algebraic Connectivity, Isoperimetric Constant.
I. I
NTRODUCTION
There is an increasing realization that many large-scale networks are really “networks-of-networks,” consistingof interdependencies between different subnetworks (e.g., coupled cyber and physical networks) [1]–[6]. Due to theprevalence of such networks, their robustness to intentional disruption or natural malfunctions has started to attractattention by a variety of researchers [6]–[9]. In this paper, we contribute to the understanding of interdependentnetworks by studying the graph-theoretic properties of r -robustness and algebraic connectivity in such networks.As we will describe further in the next section, r -robustness has strong connotations for the ability of networks towithstand structural and dynamical disruptions: it guarantees that the network will remain connected even if up to r − nodes are removed from the neighborhood of every node in the network, and facilitates certain consensusdynamics that are resilient to adversarial nodes [10]–[14]. The algebraic connectivity of a network is the secondsmallest eigenvalue of the Laplacian of that network, and plays a key role in the speed of certain diffusion dynamics[15].We focus our analysis on a class of random interdependent networks consisting of k layers (or subnetworks),where each edge between nodes in different layers is present independently with certain probability p . Our modelis fairly general in that we make no assumption on the topologies within the layers, and captures Erdos-Renyigraphs and random k -partite graphs as special cases. We identify a bound p r for the probability of inter-layeredge formation p such that for p > p r , random interdependent networks with arbitrary intra-layer topologies areguaranteed to be r -robust asymptotically almost surely. For the special case of k -partite random graphs, we showthat this p r is tight (i.e., it forms a threshold for the property of r -robustness), and furthermore, is also the thresholdfor the minimum degree of the network to be r . This is a potentially surprising result, given that r -robustness isa significantly stronger graph property than r -minimum-degree. Recent work has shown that these properties alsoshare thresholds in Erdos-Renyi random graphs [13] and random intersection graphs [11], and our work in thispaper adds random k -partite graphs to this list.Next, we show that when the inter-layer edge formation probability p satisfies a certain condition, both therobustness parameter and the algebraic connectivity of the network grow as Θ( np ) asymptotically almost surely Ebrahim Moradi Shahrivar is with the Department of Electrical and Computer Engineering at the University of Waterloo. E-mail: [email protected] .Mohammad Pirani is with the Department of Mechanical and Mechatronic Engineering at the University of Waterloo. E-mail: [email protected] .Shreyas Sundaram is with the School of Electrical and Computer Engineering at Purdue University. E-mail: [email protected] .Corresponding author.This material is based upon work supported in part by the Natural Sciences and Engineering Research Council of Canada. (where n is the number of nodes in each layer), regardless of the topologies within the layers. Given the keyrole of algebraic connectivity in the speed of consensus dynamics on networks [15], our analysis demonstratesthe importance of the edges that connect different communities in the network in terms of facilitating informationspreading, in line with classical findings in the sociology literature [16]. Our result on algebraic connectivityof random interdependent networks is also applicable to the stochastic block model or planted partition modelthat has been widely studied in the machine learning literature [17]–[20]. While we consider arbitrary intra-layertopologies, in the planted partition model it is assumed that the intra-layer edges are also placed randomly with acertain probability. Furthermore, the lower bound that we obtain here for λ ( L ) is tighter than the lower boundsobtained in the existing planted partition literature for the range of edge formation probabilities that we consider[17], [19]. Both our robustness and algebraic connectivity bounds arise from a characterization that we provide ofthe isoperimetric constant of random interdependent networks.II. G RAPH D EFINITIONS AND B ACKGROUND
An undirected graph is denoted by G = ( V, E ) where V is the set of vertices (or nodes) and E ⊆ V × V is theset of edges. We denote the set N i = { v j ∈ V | ( v i , v j ) ∈ E } as the neighbors of node v i ∈ V in graph G . The degree of node v i is d i = |N i | , and d min and d max are the minimum and maximum degrees of the nodes in thegraph, respectively. A graph G ′ = ( V ′ , E ′ ) is called a subgraph of G = ( V, E ) , denoted as G ′ ⊆ G , if V ′ ⊆ V and E ′ ⊆ E ∩ { V ′ × V ′ } . For an integer k ∈ Z ≥ , a graph G is k -partite if its vertex set can be partitioned into k sets V , V , . . . , V k such that there are no edges between nodes within any of those sets.The edge-boundary of a set of nodes S ⊂ V is given by ∂S = { ( v i , v j ) ∈ E | v i ∈ S, v j ∈ V \ S } . The isoperimetric constant of G is defined as [21] i ( G ) , min A ⊂ V, | A |≤ n | ∂A || A | . (1)By choosing A as the vertex with the smallest degree we obtain i ( G ) ≤ d min .The adjacency matrix for the graph is a matrix A ∈ { , } n × n whose ( i, j ) entry is if ( v i , v j ) ∈ E , andzero otherwise. The Laplacian matrix for the graph is given by L = D − A , where D is the degree matrix with D = diag ( d , d , . . . , d n ) . For an undirected graph G , the Laplacian L is a symmetric matrix with real eigenvaluesthat can be ordered sequentially as λ ( L ) ≤ λ ( L ) ≤ · · · ≤ λ n ( L ) ≤ d max . The second smallest eigenvalue λ ( L ) is termed the algebraic connectivity of the graph and satisfies the bound [21] i ( G ) d max ≤ λ ( L ) ≤ i ( G ) . (2)Finally, we will use the following consequence of the Cauchy-Schwartz inequality: k X i =1 s i ≥ (cid:0) P ki =1 s i (cid:1) k , (3)where s i ∈ R for ≤ i ≤ k . A. r -Robustness of Networks Early work on the robustness of networks to structural and dynamical disruptions focused on the notion of graph-connectivity , defined as the smallest number of nodes that have to be removed to disconnect the network[22]. A network is said to be r -connected if it has connectivity at least r . In this paper, we will consider a strongermetric known as r -robustness, given by the following definition. Definition 1 ( [10]):
Let r ∈ N . A subset S of nodes in the graph G = ( V, E ) is said to be r -reachable ifthere exists a node v i ∈ S such that |N i \ S | ≥ r . A graph G = ( V, E ) is said to be r -robust if for every pair ofnonempty, disjoint subsets of V , at least one of them is r -reachable.Simply put, an r -reachable set contains a node that has r neighbors outside that set, and an r -robust graph hasthe property that no matter how one chooses two disjoint nonempty sets, at least one of those sets is r -reachable.This notion carries the following important implications: • If network G is r -robust, then it is at least r -connected and has minimum degree at least r [10]. • An r -robust network remains connected even after removing up to r − nodes from the neighborhood of every remaining node [13]. • Consider the following class of consensus dynamics where each node starts with a scalar real value. At eachiteration, it discards the highest F and lowest F values in its neighborhood (for some F ∈ N ), and updatesits value as a weighted average of the remaining values. It was shown in [10], [12] that if the network is (2 F + 1) -robust, all nodes that follow these dynamics will reach consensus even if there are up to F arbitrarilybehaving malicious nodes in the neighborhood of every normal node.Thus, r -robustness is a stronger property than r -minimum-degree and r -connectivity. Indeed, the gap betweenthe robustness and connectivity (and minimum degree) parameters can be arbitrarily large, as illustrated by thebipartite graph shown in Fig. 1(a). That graph has minimum degree n/ and connectivity n/ . However, if weconsider the disjoint subsets V ∪ V and V ∪ V , neither one of those sets contains a node that has more than neighbor outside its own set. Thus, this graph is only -robust.The following result shows that the isoperimetric constant i ( G ) defined in (1) provides a lower bound on therobustness parameter. Lemma 1:
Let r be a positive integer. If i ( G ) > r − , then the graph is at least r -robust. Proof: If i ( G ) > r − , then every set of nodes S ⊂ V of size up to n has at least ( r − | S | + 1 edges leavingthat set (by the definition of i ( G ) ). By the pigeonhole principle, at least one node in S has at least r neighborsoutside S . Now consider any two disjoint non-empty sets S and S . At least one of these sets has size at most n ,and thus is r -reachable. Therefore, the graph is r -robust.As an example of Lemma 1, consider the graph shown in Fig. 1(a). Graph G has isoperimetric constant of atmost . (since the edge boundary of V ∪ V has size n/ ), but is -robust. It is worth mentioning that it is possibleto construct a graph such that difference between the robustness parameter and isoperimetric constant is arbitrarilylarge. For instance, given any arbitrary t ∈ N and n sufficiently large, assume that the interconnection topologybetween partitions V and V in graph G is t -regular (i.e., each node in V ( V ) is connected to exactly t nodes in V ( V )) and the rest of the graph is the same as the structure shown in Fig. 1(a). Then the isoperimetric constantof the graph G is at most t/ by considering the set V ∪ V . However, one can show that G is t -robust.We have summarized the relationships between these different graph-theoretic measures of robustness in Fig. 1(b).To give additional context to the properties highlighted above, we note the following. Consider an arbitrary partitionof the nodes of a graph into two sets. An r -connected graph guarantees that the nodes in one of the sets collectively have r neighbors outside that set. An r -robust graph guarantees that there is a node in one of the sets that byitself has r neighbors outside that set. A graph with i ( G ) ≥ r guarantees that each node in one of the sets has r neighbors outside that set on average .In the rest of the paper, we will study random interdependent networks and show how these various propertiesare related in such networks. ... ... ... ... V V V V (a) i ( G ) > r − G is r -robust G is r -connected d min ( G ) = rG is a graph on n nodes (b)Fig. 1: (a) Graph G = ( V, E ) with V = V ∪ V ∪ V ∪ V and | V i | = n , ≤ i ≤ . All of the nodes in V ( V ) areconnected to all of the nodes in V ( V ). Furthermore, there is a one to one connection between nodes in V and nodes in V . (b) Relationships between different notions of robustness. III. R
ANDOM I NTERDEPENDENT N ETWORKS
We start by formally defining the notion of (random) interdependent networks that we consider in this paper.
Definition 2:
An interdependent network G is denoted by a tuple G = ( G , G , . . . , G k , G p ) where G i = ( V i , E i ) for i = 1 , , . . . , k are called the layers of the network G , and G p = ( V ∪ V ∪ . . . ∪ V k , E p ) is a k -partite networkwith E p ⊆ ∪ i = j V i × V j specifying the interconnection (or inter-layer) topology.For the rest of this paper, we assume that | V | = | V | = · · · = | V k | = n and the number of layers k is at least 2. Definition 3:
Define the probability space (Ω n , F n , P n ) , where the sample space Ω n consists of all possible inter-dependent networks ( G , G , . . . , G k , G p ) and the index n ∈ N denotes the number of nodes in each layer. The σ -algebra F n is the power set of Ω n and the probability measure P n associates a probability P ( G , G , . . . , G k , G p ) toeach network G = ( G , G , . . . , G k , G p ) . A random interdependent network is a network G = ( G , G , . . . , G k , G p ) drawn from Ω n according to the given probability distribution.Note that deterministic structures for the layers or interconnections can be obtained as a special case of the abovedefinition where P ( G , G , . . . , G k , G p ) is 0 for interdependent networks not containing those specific structures;for instance, a random k -partite graph is obtained by allocating a probability of to interdependent networks whereany of the G i for ≤ i ≤ k is nonempty. Through an abuse of notation, we will refer to random k -partite graphssimply by G p in this paper. Similarly, Erdos-Renyi random graphs on kn nodes are obtained as a special case ofthe above definition by choosing the edges in G , G , . . . , G k and G p independently with a common probability p .In this paper, we will focus on the case where G p is independent of G i for ≤ i ≤ k . Specifically, we assumethat each possible edge of the k -partite network G p is present independently with probability p (which can be afunction of n ). We refer to this as a random interdependent network with Bernoulli interconnections . We will becharacterizing certain properties of such networks as n gets large, captured by the following definition. Definition 4:
For a random interdependent network, we say that a property P holds asymptotically almost surely (a.a.s.) if the probability measure of the set of graphs with property P (over the probability space (Ω n , F n , P n ) )tends to 1 as n → ∞ . IV. R OBUSTNESS OF R ANDOM INTERDEPENDENT NETWORKS
We will first consider random k -partite networks (a special case of random interdependent networks, as explainedin the previous section), and show that they exhibit phase transitions at certain thresholds for the probability p ,defined as follows. Definition 5:
Consider a function t ( n ) = g ( n ) n with g ( n ) → ∞ as n → ∞ , and a function x = o ( g ( n )) whichsatisfies x → ∞ as n → ∞ . Then t ( n ) is said to be a (sharp) threshold function for a graph property P if1) property P a.a.s. holds when p ( n ) = g ( n )+ xn , and2) property P a.a.s. does not hold for p ( n ) = g ( n ) − xn .Loosely speaking, if the probability of adding an edge between layers is “larger” than the threshold t ( n ) , thenproperty P holds; if p is “smaller” than the threshold t ( n ) , then P does not hold. The following theorem is oneof our main results and characterizes the threshold for r -robustness (and r -minimum-degree) of random k -partitenetworks. Theorem 1:
For any positive integers r and k ≥ , t ( n ) = ln n + ( r −
1) ln ln n ( k − n is a threshold for r -robustness of random k -partite graphs with Bernoulli interconnections. It is also a threshold forthe k -partite network to have a minimum degree r .In order to prove this theorem, we show the following stronger result (see Fig. 1(b)) that i ( G ) > r − when theprobability of edge formation is above the given threshold. Lemma 2:
Consider a random k -partite graph G p = ( V ∪ V ∪ · · · ∪ V k , E p ) with node sets V i = { ( i − n +1 , ( i − n + 2 , . . . , in } for ≤ i ≤ k . Assume that the edge formation probability p = p ( n ) is such that p ( n ) = ln n +( r −
1) ln ln n + x ( k − n , where r ∈ N is a constant such that r ≥ and x = x ( n ) is some function satisfying x = o (ln ln n ) and x → ∞ as n → ∞ . Define S r as the property that for any set of nodes with size of at most ⌊ kn/ ⌋ , we have | ∂S | > ( r − | S | . Then G p has property S r a.a.s. In other words, we have i ( G p ) > r − a.a.s. The proof of the above lemma is provided in Appendix A. The idea behind the proof is to first use a union-boundto upper bound the probability that for some set S of size ⌊ kn/ ⌋ or less, we have | ∂S | ≤ | S | ( r − . Via the useof algebraic manipulations and upper bounds, we then show that this probability goes to zero when the inter-edgeformation probability p ( n ) satisfies the conditions given in the lemma.Using the above lemma, we can prove Theorem 1. Proof of Theorem 1:
Consider a random k -partite graph G p of the form described in the theorem with edgeformation probability p ( n ) = ln n +( r −
1) ln ln n + x ( k − n , where r ∈ N is a constant and x = x ( n ) is some function satisfying x = o (ln ln n ) and x → ∞ as n → ∞ . By Lemma 2, we know that i ( G ) > r − . Therefore, by Lemma 1, therandom k -partite graph G p is at least r -robust a.a.s. This also implies that it has minimum degree at least r a.a.s.(by the relationships shown in Fig. 1(b)).Next we have to show that for p ( n ) = ln n +( r −
1) ln ln n − x ( k − n , where x = x ( n ) is some function satisfying x = o (ln ln n ) and x → ∞ as n → ∞ , a random k -partite graph is asymptotically almost surely not r -robust. Let G p = ( V ∪ V ∪ · · · ∪ V k , E p ) be a random k -partite graph. Consider the vertex set V = { v , . . . , v n } , and definethe random variable X = X + X + · · · + X n where X i = 1 if the degree of node v i is less than r and zerootherwise. The goal is to show that if p ( n ) = ln n +( r −
1) ln ln n − x ( k − n , then Pr ( X = 0) → asymptotically almost surely.This means that for the specified p ( n ) , there exists a node in V with degree less than r with high probability.Since any r -robust graph must have minimum degree of at least r , we will have the required result.The random variable X is zero if and only if X i = 0 for ≤ i ≤ n . The random variables X i and X j areidentically distributed and independent when i = j and thus we have Pr ( X = 0) = Pr ( X = 0) n = (1 − Pr ( X = 1)) n ≤ e − n Pr ( X =1) , (4)where the last inequality is due to the fact that − p ≤ e − p for p ≥ . Now, note that n Pr ( X = 1) = n r − X i =0 (cid:18) n ( k − i (cid:19) p i (1 − p ) n ( k − − i ≥ n (cid:18) n ( k − r − (cid:19) p r − (1 − p ) n ( k − − r +1 ≥ n (cid:18) n ( k − r − (cid:19) p r − (1 − p ) n ( k − , (5)where the last inequality is obtained by using the fact that < (1 − p ) r − ≤ for r ≥ . Using the fact that (cid:0) n ( k − r − (cid:1) = Ω (cid:0) n r − (cid:1) for constant r and k , and (1 − p ) n ( k − = e n ( k −
1) ln(1 − p ) = Ω( e − n ( k − p ) when np → (which is satisfied for the function p that we are considering above), the inequality (5) becomes n Pr ( X = 1) = Ω (cid:16) n r p r − e − n ( k − p (cid:17) . Substituting p = ln n +( r −
1) ln ln n − x ( k − n and simplifying, we obtain n Pr ( X = 1) = Ω (cid:18) (ln n + ( r −
1) ln ln n − x ) r − (ln n ) r − e x (cid:19) = Ω( e x ) . Thus we must have that lim n →∞ n Pr ( X = 1) = ∞ , which proves that Pr ( X = 0) → as n → ∞ (from (4)).Therefore, there exists a node with degree less than r and thus the random k -partite graph G p is not r -robust for p ( n ) = ln n +( r −
1) ln ln n − x ( k − n .As described in the introduction, the above result indicates that the properties of r -robustness and r -minimum-degree (and correspondingly, r -connectivity) all share the same threshold function in random k -partite graphs,despite the fact that r -robustness is a significantly stronger property than the other two properties. In particular,this indicates that above the given threshold, random k -partite networks possess stronger robustness properties thansimply being r -connected: they can withstand the removal of a large number of nodes (up to r − from everyneighborhood), and facilitate purely local consensus dynamics that are resilient to a large number of maliciousnodes (up to ⌊ r − ⌋ in the neighborhood of every normal node). A. General Random Interdependent Networks
With the sharp threshold given by Theorem 1 for random k -partite graphs in hand, we now consider generalrandom interdependent networks with arbitrary topologies within the layers. Note that any general random interde-pendent network can be obtained by first drawing a random k -partite graph, and then adding additional edges tofill out the layers. Using the fact that r -robustness is a monotonic graph property (i.e., adding edges to an r -robustgraph does not decrease the robustness parameter), we obtain the following result. Corollary 1:
Consider a random interdependent graph G = ( G , G , . . . , G k , G p ) with Bernoulli interconnections.Assume that the inter-layer edge formation probability satisfies p ( n ) ≥ ln n +( r −
1) ln ln n + x ( k − n , r ∈ Z ≥ and x = x ( n ) is some function satisfying x = o (ln ln n ) and x → ∞ as n → ∞ . Then G is r -robust.The above result shows that if the inter-layer edge formation probability between all layers of the k -layer graph G is higher than the threshold for r -robustness of a k -partite network, then G is a r -robust network, regardless ofthe probability distribution over the topologies within the layers. B. Unbounded Robustness in Random Interdependent Networks
The previous results (Theorem 1 in particular) established inter-layer edge formation probabilities that causerandom interdependent networks to be r -robust, and demonstrated that the properties of r -minimum-degree, r -conne -ctivity and r -robustness share the same probability threshold in random k -partite graphs (see Fig. 1(b)).Here, we will investigate a coarser rate of growth for the inter-layer edge formation p , and show that for suchprobability functions, the isoperimetric constant and the robustness parameter have the same asymptotic rate ofgrowth. This will also play a role in the next section, where we investigate the algebraic connectivity of randominterdependent networks. We start with the following lemma. Lemma 3:
Consider a random k -partite graph G p = ( V ∪ V ∪ · · · ∪ V k , E p ) with node sets V i = { ( i − n + 1 , ( i − n + 2 , . . . , in } for ≤ i ≤ k . Assume that the edge formation probability p satisfies lim sup n →∞ ln n ( k − np < .Fix any ǫ ∈ (0 , ] . There exists a constant α (that depends on p ) such that the minimum degree d min , maximumdegree d max and isoperimetric constant i ( G p ) a.a.s. satisfy αnp ≤ i ( G p ) ≤ d min ≤ d max ≤ n ( k − p √ (cid:18) ln n ( k − np (cid:19) − ǫ ! . (6)The proof of the above lemma is given in Appendix B. The above result leads to the following theorem. Theorem 2:
Consider a random k -partite graph G with Bernoulli interconnections. Assume that the inter-layeredge formation probability p = p ( n ) satisfies lim sup n →∞ ln n ( k − np < . Then i ( G ) = Θ( np ) a.a.s., and furthermore, G is Θ( np ) -robust a.a.s. Proof:
For inter-layer edge formation probabilities satisfying the given condition, we have i ( G ) = Θ( np ) a.a.s. from Lemma 3. From Lemma 1, we have that the robustness parameter is Ω( np ) a.a.s. Furthermore, sincethe robustness parameter is always less than the minimum degree of the graph, the robustness parameter is O ( np ) a.a.s. from Lemma 3, which concludes the proof.Once again, since adding edges to a network cannot decrease the isoperimetric constant or the robustnessparameter, the above result immediately implies that for random interdependent networks with inter-layer edgeformation probability satisfying lim sup n →∞ ln n ( k − np < , we have i ( G ) = Ω( np ) and that the robustness is Ω( np ) a.a.s. This condition on the inter-layer edge formation probability has further implications for the structure ofrandom interdependent networks. In next section, we use Lemma 3 to show that the algebraic connectivity ofrandom interdependent networks scales as Θ( np ) a.a.s. for all p that satisfy the condition, again regardless of theprobability distributions over the topologies of the layers.V. A LGEBRAIC C ONNECTIVITY OF R ANDOM I NTERDEPENDENT N ETWORKS
The algebraic connectivity of interdependent networks has started to receive attention in recent years. Theauthors of [4] analyzed the algebraic connectivity of deterministic interconnected networks with one-to-one weighted symmetric inter-layer connections. The recent paper by Hernandez et al. studied the algebraic connectivity of amean field model of interdependent networks where each layer has an identical structure, and the interconnectionsare all-to-all with appropriately chosen weights [5]. Spectral properties of random interdependent networks (underthe moniker of planted partition models ) have also been studied in research areas such as algorithms and machinelearning [17], [18], [20], [23], [24]. Here, we leverage our results from the previous section to provide a bound onthe algebraic connectivity for random interdependent networks that, to the best of our knowledge, is the tightestknown bound for the range of inter-layer edge formation probabilities that we consider.
Theorem 3:
Consider a random interdependent graph G = ( G , G , . . . , G k , G p ) with Bernoulli interconnectionsand assume that the probability of inter-layer edge formation p satisfies lim sup n →∞ ln n ( k − np < . Then λ ( G ) =Θ( np ) a.a.s. Proof:
In order to prove this theorem, we need to show that there exist constants γ, β > such that γnp ≤ λ ( G ) ≤ βnp a.a.s. We start with proving the existence of constant β .Consider the set of nodes V in the first layer. The number of edges between V and all other V j , ≤ j ≤ k isa binomial random variable B ( n ( k − , p ) and thus E [ | ∂V | ] = n ( k − p . By using the Chernoff bound [25]for the random variable | ∂V | , we have (for < δ < ) Pr ( | ∂V | ≥ (1 + δ ) E [ | ∂V | ]) ≤ e − E ( | ∂V | ) δ . (7)Choosing δ = √ √ ln n , the upper bound in the expression above becomes exp (cid:16) − n ( k − p ln n (cid:17) . Since ln n < n ( k − p for n sufficiently large and for p satisfying the condition in the proposition, the right hand side of inequality (7)goes to zero as n → ∞ . Thus | ∂V | ≤ (1 + o (1)) E [ | ∂V | ] a.a.s. Therefore i ( G ) = min | A |≤ nk ,A ⊆ V ∪ V ∪···∪ V k | ∂A || A | ≤ | ∂V || V | ≤ (1 + o (1)) n ( k − pn , a.a.s. Using (2), we have the required result.Next, we prove the lower bound on λ ( G ) . Consider the k -partite subgraph of network G which is denoted by G p . By Lemma 3 and the inequality (2), we know that λ ( G p ) ≥ γnp for some constant γ asymptotically almostsurely. Since adding edges to a graph does not decrease the algebraic connectivity of that graph [26], we have λ ( G ) ≥ λ ( G p ) ≥ γnp asymptotically almost surely.Theorem 3 demonstrates the importance of inter-layer edges on the algebraic connectivity of the overall networkwhen lim sup n →∞ ln n ( k − np < . This requirement on the growth rate of p cannot be reduced if one wishes to stayagnostic about the probability distributions over the topologies of the layers. Indeed, in the proof of the Theorem1, we showed that if lim sup n →∞ ln n ( k − np > , a random k -partite graph will have at least one isolated nodeasymptotically almost surely and thus has algebraic connectivity equal to zero asymptotically almost surely. In thiscase the quantity ln n ( k − n forms a coarse threshold for the algebraic connectivity being , or growing as Θ( np ) .On the other hand, if one had further information about the probability distributions over the layers, one couldpotentially relax the condition on p required in the above results. For instance, as mentioned in Section III, wheneach of the k -layers is an Erdos-Renyi graph formed with probability p , then the entire interdependent network isan Erdos-Renyi graph on kn nodes; in this case, the algebraic connectivity is Ω( np ) asymptotically almost surelyas long as lim sup n →∞ ln nknp < [27]. This constraint on p differs by a factor of kk − from the expression inTheorem 3. VI. S UMMARY AND F UTURE W ORK
We studied the properties of r -robustness and algebraic connectivity in random interdependent networks. Westarted by analyzing random k -partite networks, and showed that r -robustness and r -minimum-degree (and r -connectivity) all share the same threshold function, despite the fact that r -robustness is a much stronger propertythan the others. This robustness carries over to random interdependent networks with arbitrary intra-layer topologies,and yields new insights into the structure of such networks (namely that they can tolerate the loss of a large numberof nodes, and are resilient to misbehavior in certain dynamics). We also provided tight asymptotic rates of growthon the algebraic connectivity of random interdependent networks for certain ranges of inter-layer edge formationprobabilities (again, regardless of the intra-layer topologies), showing the importance of the interdependencies between networks in information diffusion dynamics. Our characterizations were built on a study of the isoperimetricconstant of random interdependent and k -partite graphs.There are various interesting avenues for future research, including a deeper investigation of the role of theintra-layer network topologies, and other probability distributions over the inter-layer edges (outside of Bernoulliinterconnections). A PPENDIX AP ROOF OF L EMMA Proof:
Suppose the edge formation probability is p = ln n +( r −
1) ln ln n + x ( k − n where x = o (ln ln n ) and x → ∞ when n → ∞ . We have to show that for any set of vertices of size m , ≤ m ≤ nk/ , there are at least m ( r − edges that leave the set a.a.s.Consider a set S ⊂ V ∪ V ∪ · · · ∪ V k with | S | = m . Assume that the set S contains s i nodes from V i for ≤ i ≤ k (i.e., | S ∩ V i | = s i ≥ ). Define E S as the event that m ( r − or fewer edges leave S . Note that | ∂S | is a binomial random variable with parameters P kl =1 s l (cid:16)P kt =1 ,t = l ( n − s t ) (cid:17) and p .We have that k X l =1 s l k X t =1 ,t = l ( n − s t ) = k X l =1 s l ( n ( k − − m + s l )= n ( k − m − m + k X l =1 s l . (8)Then we have Pr ( E S ) = m ( r − X i =0 (cid:18) n ( k − m − m + P kl =1 s l i (cid:19) p i (1 − p ) n ( k − m − m + P kl =1 s l − i (9) ≤ m ( r − X i =0 (cid:18) n ( k − mi (cid:19) p i (1 − p ) n ( k − m − m + P kl =1 s l − i ≤ m ( r − X i =0 (cid:18) n ( k − mi (cid:19) p i (1 − p ) n ( k − m − ( k − m k − i . The first and the second inequality follow from the inequalities ≤ m − P kl =1 s l ≤ ( k − m k which is astraightforward result of the Cauchy-Schwartz inequality in (3). Next note that k ≥ and for ≤ i ≤ m ( r − and sufficiently large n , we have (cid:0) n ( k − mi (cid:1) p i (1 − p ) n ( k − m − ( k − m k − i (cid:0) n ( k − mi − (cid:1) p i − (1 − p ) n ( k − m − ( k − m k − ( i − = n ( k − m − i + 1 i × p − p ≥ n ( k − m − m ( r −
1) + 1 m ( r − × p − p ≥ n ( k − − r + 1 r − × p − p> . Thus there must exist some constant
R > such that Pr ( E S ) ≤ m ( r − X i =0 (cid:18) n ( k − mi (cid:19) p i (1 − p ) n ( k − m − ( k − m k − i (10) ≤ R (cid:18) n ( k − mm ( r − (cid:19) p m ( r − (1 − p ) n ( k − m − ( k − m k − m ( r − . Define P m as the probability that there exists a set of nodes T such that | T | = m and | ∂T | ≤ m ( r − . Thenusing the inequality (cid:0) nm (cid:1) ≤ ( nem ) m yields P m ≤ X | S | = m,S ⊂∪ ki =1 V i Pr ( E S ) ≤ R (cid:18) nkm (cid:19)(cid:18) n ( k − mm ( r − (cid:19) p m ( r − (1 − p ) n ( k − m − ( k − m k − m ( r − ≤ R (cid:18) nkem (cid:19) m (cid:18) n ( k − mepm ( r − (cid:19) m ( r − (1 − p ) n ( k − m − ( k − m k − m ( r − (11) = R (cid:18) ke r ( r − r − (1 − p ) r − n (1 − p ) n ( k − ( n ( k − p ) r − m (1 − p ) ( k − mk (cid:19) m ≤ R c n (1 − p ) n ( k − ( n ( k − p ) r − m (1 − p ) ( k − mk ! m . In the last step of the above inequality, c is some constant satisfying ke r ( r − r − (1 − p ) r − ≤ c < ke r ( r − r − , for sufficiently large n . Recalling the function p ( n ) = ln n +( r −
1) ln ln n + x ( k − n and using the inequality − p ≤ e − p yields P m ≤ R (cid:18) c ne − n ( k − p ( n ( k − p ) r − m (1 − p ) ( k − mk (cid:19) m = R (cid:18) c (cid:18) ln n + ( r −
1) ln ln n + x ln n (cid:19) r − e − x m (1 − p ) ( k − mk (cid:19) m ≤ R c e − x m (1 − p ) ( k − mk ! m . Due to the fact that ln n +( r −
1) ln ln n + x ln n < for sufficiently large n , we can consider a constant upper bound c for c (cid:16) ln n +( r −
1) ln ln n + x ln n (cid:17) r − such that < c < c r − . Next, we substitute the Taylor series expansion ln(1 − p ) = − P ∞ i =1 p i i for p ∈ [0 , in the above inequality to obtain P m ≤ R c e − x e − ( k − mk ln(1 − p ) m ! m = R c e − x e ( k − mpk exp { ( k − mk p P ∞ i =2 p i − i } m ! m . Since we have P ∞ i =2 p i − i < P ∞ i =2 p i − = − p and ( k − mk p → as n → ∞ , there must exist a constant c suchthat < ( k − mk p P ∞ i =2 p i − i < c < for sufficiently large n . Therefore, P m ≤ R c e c e − x e ( k − mpk m ! m = R c e − x e ( k − mpk m ! m , where < c = c e c < ke r +1 r ( r − r − .Consider the function f ( m ) = e ( k − mpk m . Since dfdm = e ( k − mpk ( ( k − mpk − m , we have that dfdm < for m < k ( k − p and dfdm > for m > k ( k − p . Therefore, f ( m ) ≤ max { f (1) , f ( nk/ } for m ∈ { , , . . . , ⌊ nk/ ⌋} . We have f ( nk/
2) = 2 e ( k − nkp k nk = 2 k e ( − ln n +( r −
1) ln ln n + x ) / . Since ln ln n = o (ln n ) , we have that f ( nk/
2) = o (1) . Moreover, < f (1) = e ( k − pk < e and thus for sufficientlylarge n we must have f ( m ) ≤ f (1) < e . Therefore, P m ≤ R ( c e − x ) m . Let P be the probability that there exists a set S with size nk/ or less that it is not r -reachable. Then by theunion bound we have P ≤ ⌊ nk/ ⌋ X m =1 P m ≤ ∞ X m =1 R ( c e − x ) m = Rc e − x − c e − x = o (1) , since x → ∞ as n → ∞ . Thus we have the required result.A PPENDIX BP ROOF OF L EMMA Proof:
The inequality i ( G p ) ≤ d min is clear from the definition of the i ( G p ) . We will show that d max ≤ n ( k − p (cid:16) √ ln n ( k − np ) − ǫ (cid:17) asymptotically almost surely. Let d j denote the degree of vertex j , ≤ j ≤ kn .From the definition, d j is a binomial random variable with parameters n ( k − and p and thus E [ d j ] = n ( k − p .Then, for any < β ≤ √ we have [25], [27] Pr ( d j ≥ (1 + β ) E [ d j ]) ≤ e − E [ dj ] β . (12)Choose β = √ ln n ( k − np ) − ǫ , which is at most √ for p satisfying the conditions in the lemma and for sufficientlylarge n . Substituting into equation (12), we have Pr ( d j ≥ (1 + β ) E [ d j ]) ≤ e − (ln n )( ln n ( k − np ) − ǫ . The probability that d max is higher than (1 + β ) E [ d j ] equals the probability that at least one of the vertices hasdegree higher than (1 + β ) E [ d j ] , which by the union bound is upper bounded by Pr ( d max ≥ (1 + β ) E [ d j ]) ≤ kn Pr ( d j ≥ (1 + β ) E [ d j ]) ≤ ke (ln n ) − (ln n )( ln n ( k − np ) − ǫ ≤ ke (ln n )(1 − ( ln n ( k − np ) − ǫ ) . Since the right hand-side of the above inequality goes to zero as n → ∞ for p satisfying the condition in thelemma, we conclude that d max ≤ n ( k − p √ (cid:18) ln n ( k − np (cid:19) − ǫ ! , asymptotically almost surely.Next we aim to prove the lower-bound for i ( G p ) in (6). In order to prove this, we show that for any set ofvertices of size m , ≤ m ≤ nk/ , there are at least αmnp edges that leave the set, for some constant α that wewill specify later and probability p satisfying lim n →∞ ln n ( k − np < .Consider a set S ⊂ V ∪ V ∪ · · · ∪ V k with | S | = m . Assume that the set S contains s i nodes from V i for ≤ i ≤ k (i.e., | S ∩ V i | = s i ≥ ). Define E S as the event that αmnp or fewer edges leave S . Note that | ∂S | isa binomial random variable with parameters P kl =1 s l (cid:16)P kt =1 ,t = l ( n − s t ) (cid:17) and p . Similarly to the equality (8) andinequality (9), we have that Pr ( E S ) ≤ ⌊ αmnp ⌋ X i =0 (cid:18) n ( k − mi (cid:19) p i (1 − p ) n ( k − m − ( k − m k − i . (13) Next note that k ≥ and for ≤ i ≤ ⌊ αmnp ⌋ , (cid:0) n ( k − mi (cid:1) p i (1 − p ) n ( k − m − ( k − m k − i (cid:0) n ( k − mi − (cid:1) p i − (1 − p ) n ( k − m − ( k − m k − ( i − = n ( k − m − i + 1 i × p − p ≥ n ( k − m − αmnp + 1 αmnp × p − p ≥ k − − αpα × − p ≥ α , for α < which will be satisfied by our eventual choice for α .Now let P m denote the probability of the event that there exists a set of size m with ⌊ αmnp ⌋ or fewer numberof edges leaving it. Then there must exist some constant R > such that by the same procedure as in inequalities(10) and (11), we have P m ≤ R (cid:18) nkem (cid:19) m (cid:18) n ( k − mepαmnp (cid:19) αmnp (1 − p ) n ( k − m − αmnp − ( k − m k ≤ R (cid:18) ( k − eα (cid:19) αmnp e m ln ( nkem ) e − p ( n ( k − m − αmnp − ( k − m k ) = Re mh ( m ) , (14)where h ( m ) = αnp + αnp ln( k − − αnp ln α + ln (cid:18) nkem (cid:19) − p ( n ( k − − αnp − ( k − mk )= 1 + ln k + ( k − pmk − ln m + np α + α ln( k − − α ln α + ln nnp − ( k −
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