Network Robustness: Diffusing Information Despite Adversaries — 10/22
value according to the W-MSR algorithm with parameter f . Let {tk } denote the set of time-steps in which D[t] is p-fraction robust, where 2f < p â&#x2030;¤ 1. Then, under the f -fraction local malicious model, resilient asymptotic consensus is achieved if |{tk }| = â&#x2C6;&#x17E; and |tk+1 â&#x2C6;&#x2019; tk | â&#x2030;¤ c, â&#x2C6;&#x20AC;k, where c â&#x2C6;&#x2C6; Z>0 .
Proof. The proof is similar to the proof of Theorem 3. For the proof of necessity, note that the choice of Byzantine nodes should satisfy the F -local and f -fraction local properties, respectively. Further note that the only difference between the sufficient and necessary conditions for the f -fraction local model is p = f . When the network is f -fraction robust, 2.5.2 F -Total, F -Local and f -Fraction Local Byzantine Modwe can choose two sets which are f -fraction reachable and els these two sets contain some node i which has df di e neighbors Our above results have focused on the case of malicious (but outside. Consensus can be reached if f di 6â&#x2C6;&#x2C6; Zâ&#x2030;Ľ1 and cannot not Byzantine) adversaries. The recent paper [22] investibe reached if f di â&#x2C6;&#x2C6; Zâ&#x2030;Ľ1 . gates a similar algorithm in the context of F -total Byzantine faults, and provides necessary and sufficient conditions for Corollary 4. Consider a time-varying network modeled by the algorithm to succeed. While their proof techniques are a graph D[t] = {V, E[t]} where each normal node updates different, the main result can be captured neatly by the notion its value according to the W-MSR algorithm with parameter of robustness as follows. F (or parameter f for the f -fraction local model). Let {tk } denote the set of time-steps in which the normal network of Definition 12. For a network D = {V, E}, define the normal D[t] is either (i) (F + 1)-robust, or (ii) p-fraction robust, network of D, denoted by DN , as the network induced by the where f < p â&#x2030;¤ 1. Then, under the (i) F -local Byzantine normal nodes, i.e., DN = {N , EN }, where EN is the set of model, or (ii) f -fraction local Byzantine model, respectively, edges among the normal nodes. resilient asymptotic consensus is achieved if |{tk }| = â&#x2C6;&#x17E; and Theorem 3 ( [22]). Consider a time-invariant network mod- |t k+1 â&#x2C6;&#x2019; tk | â&#x2030;¤ c, â&#x2C6;&#x20AC;k, where c â&#x2C6;&#x2C6; Z>0 . eled by a graph D = {V, E} where each normal node updates Remark 3. Note that when the original network is (2F + 1)its value according to the W-MSR algorithm with parameter robust (or p-fraction robust, where 2f < p â&#x2030;¤ 1), the normal F . Under the F -total Byzantine model, resilient asymptotic network will be (F + 1)-robust (or p-fraction robust, where consensus is achieved if and only if the topology of the normal f < p â&#x2030;¤ 1). Thus, the results in Section 2.5.1 also hold for network is (F + 1)-robust. related Byzantine models. Further note that, the necessary Proof. To prove sufficiency, besides the method used in [22, conditions described in this section do not apply for the related 44], we can also use the approach proposed in the proof of malicious models, which implies that the restriction of the Theorem 1. Consider the two disjoint sets XM (t , i ) and ability of misbehaving nodes results in extra complexity. Thus, Xm (t , i ) defined in the proof of Theorem 1. If the normal the results in Section 2.5.1 are nontrivial. network is (F + 1)-robust, then one of the two sets (or both) contains some normal node which has at least F + 1 normal 3. Extensions of Network Robustness neighbors outside. To prove necessity, if the normal network is not (F + 1)- 3.1 Introduction robust, we can assign the two disjoint sets that are not (F + 1)- In the previous section, we introduced the concept of network reachable the maximum and minimum values, respectively. robustness and showed that this concept is the key property to Since the Byzantine nodes can send different values to differ- characterize the performance of algorithms using only local ent neighbors, suppose they send the maximum and minimum information, such as W-MSR. In this section, we will study values to the maximum and minimum sets, respectively. Then, two extensions of network robustness â&#x20AC;&#x201C; (r, s)-robustness and nodes in these two sets never use any values from outside their strong robustness. We first introduce the concept of (r, s)own sets and consensus is not reached. robustness, where s represents the total number of nodes in a pair of sets that each have at least r neighbors outside their The following results are straightforward extensions of the own set and characterizes another type of information redunabove result from [22] to the local models and time-varying dancy; we will provide a necessary and sufficient condition networks. for the W-MSR algorithm to achieve resilient consensus under the F -total malicious model using this concept.8 Then we will Corollary 3. Consider a time-invariant network modeled by turn to fault tolerant broadcast, which is another important a graph D = {V, E} where each normal node updates its operation in networks, and use strong robustness to show that value according to the W-MSR algorithm with parameter F broadcast will succeed in certain networks that do not meet (or parameter f for the f -fraction local model). Under the the conditions studied before. F -local Byzantine model, resilient asymptotic consensus is achieved if and only if the topology of the normal network is (F + 1)-robust. Under the f -fraction local Byzantine model, 3.2 (r, s)-Robustness for Resilient Consensus resilient asymptotic consensus is achieved if the normal net- Consider the network modeled by the graph in Fig. 4. One can verify that the graph is 3-robust by checking every possible work is p-fraction robust, where p > f , and a necessary 0 8 This part of work is done in conjunction with H. LeBlanc and X. Koutcondition is for the normal network to be p -fraction robust, 0 where p â&#x2030;Ľ f . soukos from Vanderbilt University.