2026-353

When analyzing the resilience of complex systems, network science literature establishes a fundamental distinction between static structural robustness and dynamic robustness

Consider Watts' Linear Threshold Model, where a network initialized with all functional nodes undergoes a local shock. In this model, a healthy node \(i\) transitions to a failed state if the fraction \(f_i\) of its inoperable neighbors exceeds a local critical threshold \(\phi\).

Based on the theoretical pillars of Network Robustness and the phase diagrams of this model, select the alternative that correctly describes the relationship between network topology, stability limits, and the propagation of catastrophic cascades:

A) In Watts' cascade model, the occurrence of a global avalanche exhibits a non-monotonic dependence on the network's average connectivity \(\langle k \rangle\). In highly dense networks (high \(\langle k \rangle\)), the system enters a subcritical regime where the failure of a single neighboring node represents a perturbation fraction \(1/k\) that is strictly lower than the critical threshold \(\phi\), thereby locally confining the impact and preventing global cascade propagation.

B) Scale-free networks with a degree exponent \(2 < \gamma \leq 3\) are ultra-robust against random structural failures (\(f_c\) → 1) due to the topological protection provided by hubs. This property automatically guarantees absolute immunity against global dynamic cascades triggered by minor shocks in peripheral nodes, since hubs invariably act as static sinks that absorb the overload and halt the domino effect.

C) The Molloy-Reed criterion (\(\kappa = \langle k^2 \rangle/\langle k \rangle > 2\)), which dictates the existence of a giant connected component in networks under inverse percolation, perfectly defines the dynamic threshold for avalanches. This explains why networks violating this inequality become inherently immune to cascading failures in the branching model, keeping the critical avalanche exponent unaltered.

D) According to the modeling of cascades via branching processes, the exponent α of the avalanche size distribution (\(P(S) \sim S^{-\alpha}\)) is a universal constant fixed at \(\alpha = 3/2\) for any complex network configuration, regardless of the underlying shape of the system's original degree distribution \(p_k\).

 E) None of the above.

Original idea by: Maria Luiza Ramos da Silva 

2026-352

Considering a scale-free network with a degree distribution \(P(k) \sim k^{-\gamma}\) and its critical threshold given by

$$f_c = 1 - \frac{1}{\frac{\langle k^2\rangle}{\langle k\rangle} - 1},$$

select the correct alternative.

1. For \(2 < \gamma < 3\), the second moment of the degree distribution tends to zero, indicating extreme fragility to random failures.
2. For \(\gamma > 3\), the second moment of the degree distribution diverges, making the network completely robust.
3. The value of \(f_c\) is independent of the exponent, depending only on the network size \(N\).
4. For \(2 < \gamma < 3\), the second moment of the degree distribution diverges and \(f_c\) tends to 1, indicating that the network can sustain the random removal of almost all nodes without losing the giant component.
5. None of the above

Original idea by: Julia de Pietro Bigi

2025-294

You are a systems analyst at a major data center, and you are tasked with evaluating the robustness of your facility's network infrastructure against random node failures.

Your analysis of the network topology reveals that the average degree <k> of the nodes is 6.25. Based on your understanding of robustness, what breakdown threshold does the network at the datacenter need to have to display enhanced robustness?

a) 0.85

b) 0.84

c) 0.17

d) 0.16

e) None of the above

Original idea by: Alexandre Petrachini

2025-293

The Galactic Federation maintains an interplanetary communication network connecting \(N = 1000\) colonies through hyperspace routes. Scientists from the Central Observatory have estimated:

\(\langle k \rangle = 4\), \(\langle k^2 \rangle = 100\)

During a cosmic particle storm, 800 colonies lost communication with the network.  Does the Galactic Federation’s network still maintain a giant component or has it collapsed?

1. The network still maintains a giant component, since the critical threshold is \(f_c = 0.96\).
2. The network has collapsed, since the critical threshold is \(f_c = 0.96\).
3. The network has collapsed, since the critical threshold is \(f_c = 0.67\).
4. The network still has a giant component, since the critical threshold is \(f_c = 0.67\).
5. None of the above

Original idea by: Aline Azevedo

2025-292

You are the team leader in your firm’s Cybersecurity Department, responsible for maintaining four critical corporate networks: AegisNet, MercuryGrid, HeliosCloud and VanguardLink 

Early one morning, you notice that several nodes across all four networks have been unexpectedly shut down. A quick inspection of the system logs (graphically depicted below) reveals the exact sequence in which each node went offline. Your mission is to determine whether this incident represents a coordinated cyberattack targeting multiple systems simultaneously, or if it is merely the result of random hardware failures in aging equipment.

Time 0 : 

Time 1:  

Time 2: 

An attack is defined as shutting down nodes in decreasing order of current degree.  Based on the logs, your best assessment is:

A) It was an attack on all networks

B) It was an attack only on VanguardLink

C) It was an attack only on AegisNet and VanguardLink

D) It was an attack only on AegisNet, MercuryGrid, and VanguardLink

E) None of the above.

Original idea by: Carolina Albuquerque

2025-291

The image below shows the probability that a node belongs to the giant component of an Erdos–Rényi network, estimated as the ratio between the size of the largest connected component and the number of the remaining nodes after the removal of a fraction f of nodes, with the red vertical line representing the breakdown threshold for the created network.

Which of the following alternatives best approximates the number of nodes (N) of the network and the probability for edge creation (p), respectively:

A) N = 500 and p = 0.015

B) N = 750 and p = 0.02

C) N = 1000 and p = 0.005

D) N = 1500 and p = 0.002
E) None of the above.

Original idea by: João Medrado Gondim

2025-290

Consider a complex network subject to random failures and targeted attacks. Which of the following statements best characterizes network robustness?

1. Scale-free networks are equally robust against both random failures and targeted attacks due to their heavy-tailed degree distribution.
2. Random networks (Erdős–Rényi type) typically remain connected longer under random attacks than scale-free networks with the same average degree.
3. The robustness of a network is maximized when the degree distribution follows a power law with exponent close to 5.
4. In scale-free networks, robustness to random node removal arises because most nodes have low degree, while vulnerability to targeted attacks results from dependence on high-degree hubs.
5. None of the above.

Original idea by: Yan Prada Moro

2024-248

 Consider the following networks:

 

Which of the following options correctly ranks these networks from most robust to least robust against random attacks?

A. NET1 - NET2 - NET3 - NET4

B. NET3 - NET1 - NET2 - NET4

C. NET3 - NET4 - NET1 - NET2

D. NET3 - NET1 - NET4 - NET2

E. None of the above

Original idea by: Darlinne

2024-247

Chris works as an IT consultant and was talking with his friends about how the WWW network might be easy to attack because of its type. What type is the WWW network and what property this network has that explains it?

Answer:

1. WWW is a Erdős-Rényi network and because of that is highly robust against random node removal but vulnerable to targeted attacks.
2. WWW is a Snobbish network and because of that is vulnerable against random node removal but robust to targeted attacks. Chris is wrong.
3. WWW is a scale-free network and because of that is highly robust against random node removal and to targeted attacks. Chris is wrong.
4. WWW is a scale-free network and because of that is highly robust against random node removal but vulnerable to targeted attacks.
5. None of the above.

2023-222

Consider a scale-free network with average degree 2.9, second moment of the degree distribution 32.3, and degree exponent 2.89. It is correct to state that:

1. The preferential attachment assumption is valid, with a probability Π(k) that a link of a new node links to another node of degree k proportional to the square of k.
2. This network is perfectly assortative, with a degree correlation coefficient equal to 1.
3. This network displays enhanced robustness, since its breakdown threshold is greater than the breakdown threshold of a random network with the same average degree.
4. This network is in a random network regime, since the degree exponent is close to 3.
5. None of the above.

Original idea by: Vitoria D. M. Pinho

2023-221

Which of the following statements correctly describes the robustness of scale-free networks?

1. Scale-free networks are highly robust against random node removal but vulnerable to targeted attacks.
2. Scale-free networks are equally robust against random node removal and targeted attacks.
3. Scale-free networks are highly robust against targeted attacks but vulnerable to random node removal.
4. Scale-free networks are vulnerable to both random node removal and targeted attacks.
5. None of the above.

Original idea by: Arthur Hendricks.

2022-183

Experiments with a random network revealed that its giant component vanishes after random removal of about 70% of its nodes. In this case, what was the average degree of the network?

A. 2.56

B. 4.52

C. 1.98

D. 3.33

E. None of the above

Original idea by: Marcelo Silva

2022-136

Based on the Failure Propagation Model used to describe cascading failures, analyze this network. Consider that each node has the same breakdown threshold \( \varphi = 0.4 \).

1. If a failure starts at node 0, it will not propagate to any other nodes.
2. A failure at node 9 will cascade to nodes 10, 11, and 12.
3. If all nodes have their breakdown threshold changed to \( \varphi = 0.3 \), a failure in any of the network nodes will trigger an avalanche.

Now select the option that contains exactly the true statements:

1. Only 1
2. Only 1 and 3
3. Only 2 and 3
4. All the statements
5. None of the above

Original idea by: Márcia Jacobina

2022-132

Scale-free networks are robust to random node failures but vulnerable to attacks. Consider that the telecom network in the country below is under attack, and the attacker knows the topology.

What region is the most likely to entirely lose communication with the rest of the country:

1. South
2. Northeast
3. Southeast
4. North
5. None of the above

Original idea by: Iury Cleveston

2022-131

Given the network below, extracted from "Networks by Mark Newman", which of the following sequences of node removals represents an attack against largest hubs that break s the network into three components?

 

Options:

1. n5, n8, n9
2. n13, n4, n8
   
3. n13, n9, n5
   
4. n9, n13, n5
   
5. None of the above.

Original idea by: Filipe Maciel