Improving Network Connectivity Using Trusted Nodes and Edges

Transcription

1 Improing Network Connectiity Using Trsted Nodes and Edges Waseem Abbas, Aron Laszka, Yegeniy Vorobeychik, and Xenofon Kotsokos Abstract Network connectiity is a primary attribte and a characteristic phenomenon of any networked system. A high connectiity is often desired within networks; for instance to increase robstness to failres, and resilience against attacks. A typical approach to increasing network connectiity is to strategically add links; howeer adding links is not always the most sitable option. In this paper, we propose an alternatie approach to improing network connectiity, that is by making a small sbset of nodes and edges trsted, which means that sch nodes and edges remain intact at all times and are inssceptible to failres. We then show that by controlling the nmber of trsted nodes and edges, any desired leel of network connectiity can be obtained. Along with characterizing network connectiity with trsted nodes and edges, we present heristics to compte a small nmber of sch nodes and edges. Finally, we illstrate or reslts on arios networks. I. INTRODUCTION Connectedness among arios components is a central theme and a characteristic phenomenon in any networked system. It signifies a large nmber of other network attribtes, sch as network reliability, robstness to failres, resilience against attacks, and other strctral properties of the network. As a reslt many notions defining connectedness hae been proposed. In particlar, network connectiity in terms of ertex and edge connectiities are one of the most widely considered concepts with significant ramifications. In essence, network connectiity captres the ability of the network to retain a connection between any two nodes nder a certain nmber of node or edge failres. A higher network connectiity is desirable in general to ensre that any two nodes remain connected een nder a large nmber of node or edge failres. Not only for strctral robstness, highly connected networks are more sitable to achiee arios other network objecties sch as information spread, resilient consenss [], and sriable network designs nder node or edge failres [2]. Network connectiity can be improed by strategically adding links (edges) between nodes, i.e., by incorporating redndancy, a techniqe commonly referred to as the connectiity agmentation [3], [4]. Althogh effectie, improing network connectiity by adding edges is not always practical de to economic and feasibility isses. Moreoer, adding more links also increases the attack srface for the potential attackers to case edge failres. These concerns inspire s to deise an alternatie strategy to improe network connectiity. Here, we propose an alternatie approach that does not inole strategic addition of edges or links to improe network connectiity. The basic idea is to ensre the aailability and operational integrity of a ery small sbset of nodes and edges at all times by protecting them from failres/attacks. We call sch a sbset of nodes and edges trsted. They correspond to the network components that are more reliable owing to more resorces and higher secrity measres. For instance, in terms of protection against physical attacks, defense mechanisms against jamming, protected memory, and sophisticated athentication mechanisms. We then show that the oerall network connectiity is significantly improed by making a ery small sbset of nodes and edges trsted, een if the original network is sparse. Moreoer, by controlling the nmber of trsted nodes, any desired leel of connectiity cold be obtained. Ths, instead of the idea of redndancy to improe connectiity, we exploit the notion of reliability (trstedness) of a small sbnetwork to improe connectiity. Or main contribtions are as follows: ) We propose and characterize the notion of network connectiity sing trsted nodes and edges, and show that network connectiity can be significantly improed by selecting a small sbset of trsted nodes or edges. 2) We proide ways to compte network connectiity in the presence of trsted nodes and edges, and also gie heristics to select the minimm set of trsted nodes and edges to achiee any desired ale of network connectiity. 3) Finally, we ealate or reslts nmerically on arios networks inclding Erdős-Rényi, preferential attachment, and random geometric networks. The problems related to adding the minimm nmber of edges to attain the desired network connectiity are referred to as the connectiity agmentation problems. The isse was inestigated in detail for the first time in [4]. Since then, new techniqes hae been deeloped to optimally increase the ertex and edge connectiities (e.g., [], [6], [7]). For a comprehensie list of papers in the area, we refer readers to an earlier srey by Frank [8] and Chapter 8 in a more recent work of Nagamochi and Ibaraki [3]. A closely related field is the area of sriable network design, in which the basic premise is to design and analyze algorithms that exploit strctral properties of the network to make them srie nder failre of network components (e.g., [2], [9]). To improe strctral properties of networks, adding protected resorces or defending a sbset of network components has been a sefl approach. Similar to trsted nodes, the notion of anchor nodes was sed in [0] in the context of maximizing the size of a special strctre, known as the k-core, in graphs. In a recent work, Dzibiński and Goyal [] explore trade-offs between the cost of adding links and defending nodes against attacks for network connectiity. II. PRELIMINARIES Let G(V, E) be an ndirected graph with a ertex set V and edge set E. An edge between ertices and is denoted by

2 U, and ertices and are called neighbors of each other. The neighborhood of a ertex, denoted by N, is the set of all neighbors of. The neighborhood of a sbset of ertices U V is N U = N. A path P of length n is a nonempty graph with the ertex set V = { 0, 2,, n }, and the edge set { 0, 2,, n n }. The ertices 0 and n are the end ertices, whereas, all remaining ertices are the inner ertices of the path. In a graph G, two paths are ertex-independent if they do not hae any common inner ertex. Similarly, two paths are edge-independent if they do not hae any common edge. We se the terms ertex and node interchangeably throghot the paper. If W is a sbset of ertices (edges), then G \ W is the sbgraph indced by the remaining ertices and edges of G. If G \ W has at least two components, then W separates G. Similarly, if and belong to two different components, then W separates and. Sch a set W is referred to as the ertex ct (edge ct). Next, we state the definitions of ertex and edge connectiity. A graph is k-ertex connected if there does not exist a set of k ertices whose remoal disconnects the graph. Likewise, the graph is k-edge connected if there is no sbset of k edges whose remoal disconnects the graph. Vertex connectiity of G, denoted by κ(g), is the maximm ale of k for which G is k-ertex connected. Similarly, the maximm ale of k for which G is k-edge connected is the edge connectiity of G, denoted by λ(g). The ertex connectiity of a complete graph with n nodes is defined to be n althogh no ertex ct exists. A classical theorem of Menger relates the notion of connectiity to the nmber of ertex and edge-independent paths between any two nodes. Theorem 2.: (Menger s theorem [2]) - Let and be distinct, non-adjacent ertices of G. The minimm size of a ertex ct separating and is eqal to the maximm nmber of ertex-independent paths between and. - Let and be distinct ertices of G. The minimm size of an edge ct that separates and is eqal to the maximm nmber of edge-independent paths between and. As a conseqence, for any k 2, a graph is k-ertex connected (k-edge connected) if and only if any two ertices hae k ertex-independent (k edge-independent) paths between them. Seeral ersions and proofs of this fndamental reslts hae been reported (e.g., see [3]). III. CONNECTIVITY WITH TRUSTED NODES AND EDGES In the traditional k-ertex (k-edge) connectiity notion, the idea is to ensre that the graph remains connected if any k nodes (k edges) are remoed from the network. By adding more edges between nodes, the ertex and edge connectiities can be improed. Howeer, if we fix a small sbset of nodes (edges) sch that they cannot be remoed from the network, then the minimm nmber of nodes (edges) from the remaining set that are reqired to disconnect the network also increases. Ths, instead of adding more edges or links, we get an alternatie way to improe network connectiity. A merit of this approach is that by making only a ery small fraction of the oerall nodes (edges) inssceptible to remoals, or as we call trsted, the oerall node (edge) connectiity can be significantly improed. In practice, trstedness can be achieed by making sch components more resorcefl and highly secre against physical attacks, tampering, and malicios intrsions throgh sophisticated secrity mechanisms. Next, we define the new notion of connectiity as follows: Definition (k-vertex Connected with T ) An ndirected graph G(V, E) is said to be k-ertex connected with T V, if there does not exist a set of fewer than k ertices in V \ T whose remoal disconnects the graph. The maximm ale of k for which the graph is k-ertex connected with T is denoted by κ T (G) and is referred to as the ertexconnectiity with T. Similarly, we can define the k-edge connectiity with trsted edges T e. Definition (k-edge Connected with T e ) For a gien T e E, G(V, E) is said to be k-edge connected with T e, if there does not exist a set of fewer than k edges in E \ T e whose remoal disconnects the graph. The maximm ale of k for which the graph is k-edge connected with T e is called the edge-connectiity with T e, and is denoted by λ T (G). Analogos to the node-independent paths, we define the notion of node-independent paths with T as follows: If T is the set of trsted nodes, then two paths are node-independent with T if the common inner ertices are only the trsted nodes. Similarly, for a gien set of trsted edges T e, two paths are edge-independent with T e if their common edges are only the trsted edges. For instance, in Figre (a), paths { 2, 2 x, x 3 } and { 2, 2 x, x 3 } are nodeindependent with T = {x}, and paths { 2, 2 x, xy, y 3 } and { 2, 2 x, xy, y 3 } in Figre (b) are edge-independent with T e = {xy}. 2 2 (a) x x y Fig.. (a) Node-independent paths with T. (b) Edge-independent paths with T e. A path with all trsted nodes (edges) is referred to as the node-trsted (edge-trsted) path. If for any pair of nodes, there exists a node trsted path between them, then G is referred to as completely ertex-connected with T. Similarly, G is completely edge-connected with T e if there exists an edge-trsted path between any two nodes. If a graph is completely ertex connected with T, we define its κ T =, and if the graph is completely edge-connected with T e, then we define its λ T =. 2 (b) 3 3

3 A. Vertex Connectiity with Trsted Nodes T Next, we compte and relate ertex connectiity with trsted nodes to the traditional notion of ertex connectiity. Let G (V, E ) be a graph obtained from G(V, E) as follows: for eery non-adjacent pair of nodes and, if there exists a trsted node, or a node-trsted path between them, then add an edge {}. An example is shown in Figre 2, where {,, z} is the set of trsted nodes in G. Since nodes and indce a node-trsted path in G, all neighbors of and are pair-wise adjacent in G. Similarly, neighbors of trsted node z are adjacent in G. z G G Fig. 2. G with the set of trsted nodes {,, z} and the reslting G. Proposition 3.: Let G(V, E) be a graph that is not completely ertex connected with T, then κ T (G) = κ(g ). Proof: Let κ T (G) = k. If nodes and are connected trogh a node-trsted path in G, then and are adjacent in G. Ths, if there is no sbset of k non-trsted nodes in G whose remoal disconnects the graph, then there is no sbset of any k nodes in G whose remoal disconnects G. Similarly, we obsere that eery ertex-ct in G consisting of only non-trsted nodes is also a ertex-ct in G. A direct conseqence of the aboe proposition and Theorem 2. is the following Menger s type reslt. Corollary 3.2: For a graph G(V, E) and T V, the following statements are eqialent: ) G is k-ertex connected with T. 2) For any two distinct, non-adjacent ertices, V; either there exists a node-trsted path between and, or there exists at least k paths between and that are ertex-independent with T. As an example, consider the 2-ertex connected graph in Figre 3(a), which becomes 4-ertex connected with two trsted nodes T = {6, 0} as shown in Figre 3(b) (a) (b) (c) Fig. 3. (a) Graph is 2-ertex and 2-edge connected. (b) The graph becomes 4-ertex connected with T = {6, 0}. Between nodes and 9, there are for ertex-independent paths with T, shown in red, ble, green and yellow. (c) The graph is 3-edge connected with T e = {4}. Three edge-independent paths with T e between nodes and 9 are highlighted To compte the ertex connectiity of G with T, we first obtain G as aboe, and then can se any algorithm to z compte the ertex connectiity of G. There is an extensie literatre on sch algorithms [4]. A typical approach is to tilize the max-flow-min-ct theorem (e.g., see [3]). The idea is to compte the local ertex connectiity between any two non-adjacent nodes s, t V in G, which is the minimm nmber of nodes that need to be remoed to disconnect s and t. The local ertex connectiity is eqal to the maximm flow between s and t in a particlar directed network H G obtained from G (e.g., see []). The ertex connectiity of G, and hence the ertex connectiity of G with T, is simply the minimm of the local ertex connectiities between any two non-adjacent nodes in G. The directed network H G reqired to compte local ertex connectiity between non-adjacent nodes s and t is obtained from G as follows []: For each ndirected edge, create two directed edges; one from to, and the other from to. Remoe all edges incoming to s and otgoing from t. For each node V \ {s, t}, create two nodes i and o, and add a directed edge from i to o. Next, make all incoming edges to as the incoming edges to i, and make all edges leaing from as the otgoing edges from o. Finally, assign a nit weight (capacity) to each directed edge and compte the maximm flow between s and t. In smmary, to compte the ertex connectiity of G with a gien T, following steps are performed, G G s,t V H G max flow(s,t). () κ T (G) is the minimm of the maximm flow between any two non-adjacent nodes s, t in G. In fact, the ertex connectiity of G with T can be compted by directly obtaining a directed network H G from G as described aboe. In other words, the G G transformation in () can be aoided. The only difference in the constrction of H G from aboe wold be that the edge weights (capacities) of all the edges incoming to and otgoing from the nodes corresponding to the trsted nodes in G wold be infinite (i.e., ery large), rather than one. The edge weights of all other edges wold remain to be one. To compte the ertex connectiity with T, maximm flow between all non adjacent nodes in G is then compted oer H G. An example is shown in Figre 4. s G s i o z t z i z o H G o i Fig. 4. For a gien G, T = {}, and non-adjacent nodes s, t, the constrction of a directed network H G B. Edge Connectiity with Trsted Edges T e Here, we discss the case of edge-connectiity with trsted edges, and ways to compte it. Let G(V, E) be a graph with t

4 trsted edges T e E, then we define G(Ṽ, Ẽ) be the graph obtained by identifying the end nodes of each trsted edge. Here, identifying two nodes, say and 2, means replacing them by a single node sch that an edge exists between some node x and in the new graph if and only if an edge exists between x and or between x and 2 in the original graph. We obtain G from G by identifying the end nodes of all trsted edges one by one. Note that G might hae mltiple edges between nodes, and hence G is a mltigraph. All self loops are ignored. An example is illstrated in Figre. 2 G 2 Fig.. The end nodes of trsted edges { 2 } and { 2 } in G are identified as and respectiely into G. We obsere that edge-trsted paths exist between nodes in G that are identified as a single node in G. Ths, if Ṽ =, it implies that G is completely edge-connected with T e. Since the end nodes of all trsted edges are identified in G, ths, if there is no set of k non-trsted edges that disconnects G, then there is no set of any k edges in G whose remoal disconnects G. Similarly, it is easy to obsere that any edgect of size k in G is also an edge-ct in G. Ths, we get the following: Proposition 3.3: Let G(V, E) be a graph that is not completely edge-connected with T e, then λ T (G) = λ( G). Gien a mltigraph G, if we replace each (edge in a) mltiple edge by a path of length two, then there is a one-toone relation between the paths in G and the reslting graph. Ths, the reslt in the Menger s theorem is immediately applicable to the reslting graph. We can state the following. Proposition 3.4: The edge-ersion of the Menger s theorem holds for mltigraphs. As a reslt, for the notion of edge connectiity with T e, the following Menger s type reslt is directly obtained. Corollary 3.: For a graph G(V, E) and T e E, the following statements are eqialent: ) G is k-edge connected with T e. 2) For any two distinct ertices and in G; either there exists an edge-trsted path between them, or there are at least k paths between them that are edgeindependent with T e. As an example, consider the 2-edge connected graph in Figre 3(a), which becomes 3-edge connected with a single trsted edge as shown in Figre 3(c). To compte the edge connectiity of G with T e, we can first obtain the mltigraph G and can then compte the edge connectiity of G sing known methods, for instance [3], [6], [7]. At the same time, as with the ertex connectiity with T, we can directly compte the edge connectiity of G with T e by first obtaining a directed graph D G with appropriate edge weights. D G is obtained from G by replacing each G ndirected edge in G by two directed edges, one from to and the other from to. All the directed edges corresponding to the trsted edges in G are assigned infinite weights (capacities), whereas all remaining directed edges are assigned nit weight. The edge connectiity of G with T e is simply the minimm of the maximm flow between any two nodes in D G. It is sfficient to compte the minimm of the maximm flow between a fixed node (randomly picked) and all other nodes [4]. IV. COMPUTING TRUSTED NODES AND EDGES In this section, we present heristics to select minimm set of trsted nodes T (trsted edges T e ) to achiee the desired ertex connectiity with T (edge connectiity with T e ). A. Problem Complexity First, we show that finding a minimm set of trsted nodes achieing a certain connectiity is a comptationally hard problem. We begin by formlating this as a decision problem. Definition (Trsted Node-Connectiity Agmentation Problem (TNCAP)) Gien a graph G(V, E), a desired connectiity k, and the nmber of trsted nodes T, determine if there exists a set of trsted nodes T of cardinality T sch that G is k -connected with T. Theorem 4.: TNCAP is NP-hard. We show that TNCAP is NP-hard sing a redction from a well-known NP-hard problem, the Set Coer Problem (SCP). Set Coer Problem (SCP) Gien a base set U, a family F of sbsets of U, and a threshold size t, determine if there exists a sbfamily C F of cardinality t whose nion is U. Proof: Gien an instance of SCP, we constrct an instance of TNCAP as follows: For each element U, create a node. Similarly, for each member F of the family F, create a node F. For each U and F F, create an edge (, F ) if F. For each F, F 2 F, F F 2, create an edge (F, F 2 ). Let nmber of trsted nodes be T = t, and let the desired connectiity be k = F. It is clear that the redction can be performed in polynomial time. As a conseqence, we only need to show that TNCAP has a soltion if and only if SCP does. First, let s sppose that there exists a set coer C of cardinality t. Then, let the set of trsted nodes be T = C. Since C is a set coer of U, eery node corresponding to an element of U is connected to a trsted node in T. Frther, eery node corresponding to a member of F \ C is also connected to a trsted node in T, and the trsted nodes are connected to each other. Conseqently, G cannot be separated by the remoal of any set of non-trsted nodes, which proes that T = C is a soltion for TNCAP. Second, let s sppose that there does not exist a set coer of cardinality t. Now, we will show that the graph G cannot be k -connected with any set of trsted nodes T of cardinality T = t. Let T be an arbitrary set of trsted

5 nodes of cardinality T, and consider the remoal of all non-trsted nodes corresponding to members of F. Since T F cannot be a set coer de to or spposition, there exists an element U that is connected only to nontrsted nodes. Conseqently, the remoal of the non-trsted nodes corresponding to members of F separates from the remainder of the graph, which proes that T cannot be a soltion for TNCAP. B. Heristics In a graph, complete connectiity with T is obtained wheneer any two nodes are connected throgh a nodetrsted path between them. A node trsted path exists between any pair of nodes if and only if T is a connected dominating set, which is defined as Definition (Connected Dominating Set) In a graph G(V, E), a sbset of nodes Γ V is a connected dominating set if and only if (i) for eery V, either Γ, or is adjacent to some Γ, and (ii) the nodes in Γ indce a connected sbgraph in G. The cardinality of the smallest connected dominating set is known as the connected domination nmber, denoted by γ G. For any k, the minimm size of trsted nodes reqired to achiee desired k-connectiity with T is bonded by γ G, T γ G. (2) Ths, starting with a T = Γ, we can iteratiely redce the size of T to get a minimal set of trsted nodes with which the graph remains k-ertex connected with T. The notion of connected dominating set in graphs has been extensiely stdied in both graph theory and sensor network literatre (e.g., see [8], [9]), wherein a wide ariety of applications along with arios distribted algorithms for constrcting small sized connected dominating sets hae been reported. Let V Conn Trst(G, T ) be the procedre to determine the ertex connectiity with a gien T, as otlined in Section III-A, and Conn Dom Set(G) be the procedre to determine the minimal connected dominating set. Then, a minimal T reqired to achiee the desired ertex connectiity k with trsted nodes can be obtained sing Algorithm gien below. Starting with a T = Γ, in each iteration, a node is remoed if the reslting connectiity is at least eqal to the desired ertex connectiity with T. Note that in Algorithm, O( Γ ) calls are made to the sb-rotine that comptes ertex connectiity with trsted nodes. Next, we present a heristic to compte a minimal set of trsted edges T e to achiee the desired edge connectiity with T e. As with the ertex connectiity case, complete edge connectiity with T e is obtained if and only if there exists an edge-trsted path between eery pair of nodes, which is tre if T e consists of edges in a spanning tree. Ths, to achiee complete edge connectiity with T e, we need T e = n, where n is the nmber of nodes in G. At the same time, we get an pper bond on the minimm size of T e to achiee desired k-edge connectiity with trsted edges for any k, Algorithm Trsted Nodes for Vertex Connectiity : Inpt: G(V, E), k 2: Otpt: T V 3: Γ Conn Dom Set(G) 4: T Γ : for i = to Γ do 6: V Conn Trst(G, T \ {Γ(i)}) 7: if k do 8: T T \ {Γ(i)} 9: end if 0: end for T e n. (3) Ths, to obtain a minimal T e to achiee desired edge connectiity with trsted edges, say k, we start with a T e consisting of edges in a spanning tree, and then iteratiely redce the size of T e. If E Conn Trst(G, T e ) is the rotine to compte edge-connectiity with T e, as discssed in Section III-B, and Min Span Tree(G) comptes edges in a spanning tree, then Algorithm 2 comptes a minimal T e to achiee the desired edge connectiity with trsted edges. Algorithm 2 Trsted Edges for Edge Connectiity : Inpt: G(V, E), k 2: Otpt: T e E 3: E Min Span Tree(G) 4: T e E : for i = to E do 6: e E Conn Trst(G, T e \ {E (i)}) 7: if e k do 8: T e T e \ {E (i)} 9: end if 0: end for If n is the nmber of nodes in the graph, then, in Algorithm 2, O(n) calls are made to the rotine that comptes edge connectiity with trsted edges. Next, we present a nmerical ealation of both the algorithms. V. NUMERICAL EVALUATION We ealate or reslts for three different types networks, inclding Preferential attachment networks, Erdős-Rényi networks, and Random geometric networks. These network are freqently sed to model arios networking phenomenon existing in natre and also for arios engineering applications. The details of networks considered for or simlations are stated below. - Preferential attachment (PA) networks with n = 00 nodes obtained by adding a node to an existing network one at a time. Each new node is connected to m = 3 existing nodes with the probability proportional to the degree of the nodes.

6 - Erdős-Rényi (ER) networks consisting of n = 00 nodes in which the probability of an edge between any two nodes is p = Random geometric (RG) networks consisting of n = 00 nodes that are distribted niformly at random in the nit area. An edge exists between any two nodes if the Eclidean distance between them is at most 0.8. Eery single point in the plots in Figres 6 (a) and (b) is an aerage of thirty randomly generated instances. The minimm nmber of trsted nodes (compted by the Algorithm ) sfficient to achiee the desired ertex connectiity with T, and the minimm nmber of trsted edges (compted by the Algorithm 2) sfficient to achiee the desired edge connectiity with T e, are plotted in Figres 6 (a) and (b) respectiely. In the case of the preferential attachment networks, the ertex connectiity with no trsted nodes is 3. To increase the ertex connectiity from 3 to 4, we obsere a big jmp in T, which is almost eqal to the size of the minimm connected dominating set. In the case of or preferential attachment networks, ertex connectiity with T is exhibited as an all-or-nothing type phenomenon, i.e., to increase the ertex connectiity een by one, the nmber of trsted nodes needed are sfficient to make the network completely ertex connected with T. Howeer, in the cases of Erdős-Rényi and random geometric networks, we obsere rather a continos increase in T. The plot of T is plateaed once T is eqal to the size of the connected dominating set. Similar patterns are obsered in the cases of edge connectiity with trsted edges. In the case of preferential attachment networks considered, to increase the edge connectiity from 3 to 4, the nmber of trsted edges needed increases from 0 to almost n/2. For the cases of Erdős- Rényi and random geometric networks, T e increases in a more continos manner to achiee higher ales of edge connectiity with T e. T PA PA ER 20 ER RG RG Vertex connectiity with T Edge connectiity with (a) Fig. 6. (a) Nmber of trsted nodes T as a fnction of the ertex connectiity with trsted nodes. (b) Nmber of trsted edges T e as a fnction of the edge connectiity with trsted edges. Te VI. CONCLUSIONS A typical approach to improing network connectiity is to add links in a strategic manner. We adapted a different approach to achiee any desired ale of ertex and edge connectiity. The basic idea was to make a small sbset (b) T e of nodes and edges trsted, i.e., inssceptible to failres. We then showed that existence of sch nodes and edges has an effect of haing a higher network connectiity. We also presented heristics to select a small sbset of trsted nodes and edges to achiee any desired ale of network connectiity. Using this approach, een the sparse networks can be made highly connected withot adding edges. As opposed to traditional way of improing connectiity by redndancy, or approach relies on making a portion of network more reliable and trsted, and thereby, achieing the desired connectiity. 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