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Robustness of Information Diffusion Algorithms to Locally Bounded Adversaries
Haotian Zhang, Shreyas Sundaram
TL;DR
The paper addresses information diffusion when malicious nodes are present but normal nodes know only a local bound on adversaries, not the network topology. It introduces r-robustness and uses it to derive conditions for distributed consensus and fault-tolerant broadcast. The results show that local topology-based conditions support these tasks without requiring nodes to know the graph topology, while improving on traditional connectivity and minimum-degree characterizations.
Problem
Information diffusion must remain reliable in networks with malicious nodes even when normal nodes lack global topology knowledge and only know a local adversary bound.
Method
The paper introduces r-robust graphs and applies this topological property to distributed consensus and fault-tolerant broadcast under locally bounded adversaries.
Results
The proposed conditions characterize resilience for consensus and broadcast without requiring nodes to know the graph topology, including broadcast in some networks excluded by prior conditions.
Takeaways & Limitations
r-robustness provides a network property for analyzing local filtering algorithms for consensus and fault-tolerant broadcast beyond classical connectivity and minimum degree.
Abstract
from arXiv · showhide
We consider the problem of diffusing information in networks that contain malicious nodes. We assume that each normal node in the network has no knowledge of the network topology other than an upper bound on the number of malicious nodes in its neighborhood. We introduce a topological property known as r-robustness of a graph, and show that this property provides improved bounds on tolerating malicious behavior, in comparison to traditional concepts such as connectivity and minimum degree. We use this topological property to analyze the canonical problems of distributed consensus and broadcasting, and provide sufficient conditions for these operations to succeed. Finally, we provide a construction for r-robust graphs and show that the common preferential-attachment model for scale-free networks produces a robust graph.
I. INTRODUCTION
The paper studies how networks can diffuse legitimate information while limiting malicious influence when nodes have only local knowledge. It introduces r-robustness to analyze fault-tolerant broadcast and distributed consensus under locally bounded adversaries.
- Motivation: Reliable information diffusion must balance the spread of legitimate innovations against disruption by faulty or malicious nodes.The paper frames this as a fundamental challenge for natural and engineered networks.
- Prior limitations: Existing connectivity-based dissemination methods can tolerate f malicious nodes when connectivity is at least 2f + 1, but require full topology knowledge and high computational overhead.These requirements limit applicability when nodes lack global network information.
- Problem setting: The paper analyzes dissemination when each normal node observes neighbors’ values and knows only a bound on malicious nodes in its own neighborhood.Nodes need not know the rest of the topology, network size, or malicious nodes’ locations and behavior.
- Contribution: The authors introduce r-robust graphs as a topological basis for resilience to malicious nodes.The property is used for fault-tolerant broadcast and distributed consensus under a locally bounded fault model.
- Results: The proposed broadcast conditions can succeed in networks that do not satisfy conditions from prior work, while consensus conditions address both agreement and limiting malicious influence.The consensus result provides sufficient and necessary conditions for normal nodes to reach consensus while constraining the final value’s manipulation.
II. SYSTEM MODEL
The system model represents a directed network whose nodes update information over synchronous time steps, while some nodes may behave maliciously. The paper focuses on distributed consensus and broadcast under locally bounded malicious behavior.
- Network model: The network is modeled as a directed graph G = {V, E}, where an edge indicates that one node can receive information from another.Each node’s in-neighborhood determines the values it can observe.
- Node behavior: Normal nodes follow prescribed update functions at every time step, whereas malicious nodes may deviate from those strategies.The malicious set is M and the normal set is N = V \ M.
- Fault model: The model allows arbitrary malicious behavior but assumes common-neighbor values are transmitted identically under the communication modality considered.The paper distinguishes this model from Byzantine faults, which can send different values to different neighbors.
- Locally bounded faults: An f-local malicious model bounds the number of malicious nodes in each normal node’s neighborhood rather than the total number in the network.The f-total model is treated as a special case of the f-local model.
- Objectives: The paper derives topological conditions for distributed consensus and broadcast that guarantee resilience to locally bounded adversaries.These are the two algorithms examined in the system model.
III. ASYMPTOTIC CONSENSUS WITH LOCALLY BOUNDED ADVERSARIES
This section considers iterative consensus when malicious nodes can update arbitrarily and potentially bias the network’s agreement. It reviews linear neighbor-weighting strategies and the need for conditions ensuring convergence despite adversaries.
- Consensus dynamics: Linear iterative consensus updates each node’s value from weighted values of itself and its neighbors.The weights assigned to neighbors may vary by node and time step.
- Consensus objective: Asymptotic consensus requires |x_i[t] − x_j[t]| → 0 as t → ∞ for all node pairs.This defines agreement of node values in the limit.
- Convergence conditions: Existing convergence results commonly impose lower bounds on update weights because some graphs and updates without such bounds fail to reach consensus.The lower bound is used to guarantee convergence.
- Adversarial behavior: Malicious nodes may update arbitrarily to prevent consensus or bias the consensus value.The section therefore studies consensus conditions under adversarial deviations from the linear strategy.
A. Previous Results on Resilience of Asymptotic Consensus
Prior work established resilience conditions for consensus and broadcast, but often required full topology knowledge or restricted network structures. These limitations motivate a topology-independent robustness condition for local filtering algorithms.
- Alternative resilience approaches: Other approaches addressed tree networks or continuous-time consensus but required adversary knowledge, special structures, or additional network information.The tree-based rewiring method required nodes to know adversary locations and intentions, while related results required full topology knowledge.
- Complete-network consensus: Approximate Byzantine Consensus used trimming and averaging, achieving agreement in complete synchronous networks with more than 3f nodes and asynchronous networks with more than 5f nodes.The method disregards the largest and smallest f values before averaging a selected subset.
- General topologies: General-topology Byzantine-consensus results provided only local convergence or global convergence for special topologies.The cited literature had limited coverage beyond complete networks.
- Motivation for robustness: Traditional degree and connectivity metrics are not the key determinants for algorithms making purely local filtering decisions.The paper instead introduces r-robustness as a more fundamental characterization for MSR, ARC-P, and fault-tolerant broadcast.
B. Description of the Algorithm
The W-MSR algorithm locally filters extreme neighbor values before applying a weighted update, using only a neighborhood bound on malicious nodes. It extends MSR-style filtering to arbitrary topologies and aims for f-local safety rather than eliminating all malicious influence.
- Information requirements: Normal nodes need not identify malicious neighbors or know the network topology; they require only an upper bound f on malicious neighbors.This local-information requirement is central to the algorithm's distributed operation.
- Filtering rule: Each normal node receives neighbor values, removes up to f largest and f smallest values relative to its own state, and updates using the remainder.The algorithm can discard fewer than 2f values when the node's value lies within the extreme range.
- Algorithm generality: The algorithm permits time-varying weights and operates in arbitrary graph topologies, including effectively state-dependent switching.The discarded-node set can change with relative values even when the physical topology is fixed.
- Safety objective: The algorithm targets f-local safety: normal nodes asymptotically reach consensus within the range of their initial values.This guarantee allows malicious nodes to influence the final value within that range but prevents an f-local set from pushing it outside.
- Application motivation: The paper motivates this bounded-influence objective with sensor networks, where consensus outside the normal measurement range could severely impair network function.The desired consensus remains within the interval represented by normal sensors' initial measurements.
C. Analysis of the Algorithm
The analysis shows that large connectivity or minimum degree alone does not guarantee W-MSR consensus. A constructed graph demonstrates failure, motivating robustness as the more informative structural property.
- Consensus criterion: Consensus occurs when the normal-state spread Φ[t] = MN[t] − mN[t] converges to zero.The maximum normal value is non-increasing and the minimum is non-decreasing.
- Invariant range: W-MSR preserves every normal node's state within the interval [mN[t], MN[t]] at every time step.Filtering removes at most f extreme values on each side, and the remaining values lie within the normal-value range.
- Connectivity counterexample: A graph can have large connectivity yet fail to reach asymptotic consensus under W-MSR with parameter 2f.The paper constructs two densely connected groups whose cross-group values are discarded by filtering.
- Failure mechanism: In the constructed graph, each group retains its initial value because all cross-group neighbors are among the discarded extremes.With c1 < c2, nodes in each group use only values from their own group, so consensus is never reached.
- Implication: Large minimum degree or connectivity is therefore insufficient by itself to analyze W-MSR performance.The analysis uses this failure to motivate a topological notion that better characterizes when the algorithm is f-local safe.
IV. ROBUST GRAPHS
The paper defines robustness through cross-boundary reachability: every pair of disjoint nonempty node sets must include a set with a node having enough neighbors outside. This property supports structural guarantees under edge removal and underlies local-filtering analysis.
- Connection to consensus: The motivating failure occurs because no node has enough neighbors in the opposite group, causing filtering to discard all useful cross-group information and preventing consensus.This connects the counterexample's mechanism to the cross-boundary condition formalized by robustness.
- Reachability: A set S is r-reachable if some node in S has at least r neighbors outside S.This captures whether a subset has sufficient exposure to the rest of the graph.
- Robustness definition: A graph is r-robust when every pair of disjoint nonempty subsets contains at least one r-reachable subset.Robustness requires this cross-boundary condition globally across all such subset pairs.
- Edge-removal property: Removing up to K incoming edges per node from an r-robust graph leaves an (r − K)-robust graph when K < r.Robustness degrades predictably under bounded incoming-edge removal.
- Structural consequence: Every r-robust graph with r ≥ 1 has a spanning tree.The proof reduces the claim to 1-robustness and rules out multiple source strongly connected components.
A. Consensus With Locally Bounded Faults
The section characterizes when W-MSR consensus succeeds under the f-local malicious model using r-robustness, and shows the sufficient condition is tight while a weaker condition is necessary. It also extends the results to time-varying networks and Byzantine adversaries.
- The W-MSR algorithm with parameter 2f is f-local safe if the network is (2f + 1)-robust.
- The (2f + 1)-robust condition is tight: for every f > 0, a 2f-robust network can fail to reach consensus.The constructed graph uses an f-local malicious set, and W-MSR leaves two normal nodes' values unchanged.
- For time-varying networks, W-MSR remains f-local safe when the network is (2f + 1)-robust infinitely often with bounded gaps between such times.
- A necessary condition for W-MSR with parameter 2f to be f-local safe is that the network is (f + 1)-robust.If two disjoint sets are not (f + 1)-reachable, assigning them maximum and minimum values prevents cross-set updates and consensus.
- The consensus results also apply to the f-local Byzantine model because the proofs do not require malicious nodes to send identical values to all neighbors.
V. BROADCASTING WITH LOCALLY BOUNDED ADVERSARIES
The section analyzes CPA broadcasting under locally bounded adversaries and gives a robust-graph condition that guarantees every normal node receives the source value. Strong robustness can succeed even when the earlier neighborhood-count condition fails.
- CPA accepts a value only after a normal node receives the same value from f + 1 neighbors, then rebroadcasts it permanently.Under the f-local assumption, a normal node can accept only the source's actual broadcast value.
- The earlier sufficient condition for CPA is X(G) > 2f, where X(G) is the minimum number of neighbors closer to the source.
- A graph is strongly r-robust when every nonempty subset is r-reachable or contains a node connected to every node outside it.
- CPA succeeds for any source under the f-local malicious model if the network is strongly (2f + 1)-robust.
- Strong (2f + 1)-robustness can guarantee CPA success even when X(G) ≤ 2f.The paper gives graphs with X(G) ≤ 2f that are strongly (2f + 1)-robust and therefore support successful CPA.
VI. CONSTRUCTING AN r-ROBUST GRAPH
The section constructs r-robust graphs by adding nodes with at least r incoming neighbors to an existing r-robust graph. This construction includes preferential attachment and supports robustness in scale-free networks when r is sufficiently large.
- The construction shows that preferential-attachment scale-free networks are robust to locally bounded malicious nodes when r is sufficiently large during network formation.
- Adding a new vertex with incoming edges from at least r existing nodes preserves r-robustness.
- An r-robust graph with n nodes can be built by starting from a smaller r-robust graph and continually adding nodes with at least r incoming neighbors.
- The construction includes preferential attachment, in which existing nodes are selected with probability proportional to their current number of edges.Preferential attachment is associated with the formation of scale-free networks.
VII. CONCLUSION AND DISCUSSION
The conclusion argues that r-robustness better characterizes local-filtering algorithms than connectivity or minimum degree and supports consensus and fault-tolerant broadcast without global topology knowledge. It also identifies open problems concerning necessary conditions and connections to related topology results.
- The paper assumes each normal node lacks knowledge of the network's global topology.
- Connectivity and minimum degree are not particularly useful for characterizing algorithms based on purely local filtering rules.
- r-robustness provides conditions for distributed consensus and fault-tolerant broadcast without requiring nodes to know the graph topology.
- A single necessary and sufficient condition for consensus under the f-local malicious model remains an open problem because an earlier necessary-condition proof does not apply.
- The connection between malicious-node analysis and information-cascade results remains complicated and is left for ongoing research.
APPENDIX
The appendix proves that under (f + 1)-robustness, the maximum and minimum values of normal nodes must converge to the same limit. It uses shrinking value sets whose sizes decrease until one becomes empty, contradicting monotone convergence unless consensus is reached.
- The maximum and minimum values of normal nodes are monotone and bounded, so each converges to limits AM and Am.
- Assuming AM > Am, the proof chooses disjoint sets of nodes near each limit and applies (f + 1)-robustness to one of them.The sets are defined using a positive separation parameter ǫ0 and remain disjoint as the argument proceeds.
- Robustness and the filtering rule force at least one extreme-value set to shrink at every iteration.A reachable node has at least f + 1 normal neighbors outside its set, while removing at most f lower or higher values leaves an outside value in its update.
- Because ǫT > 0 contradicts monotone convergence to AM or Am, the proof concludes that AM = Am and normal nodes reach consensus.