Network resilience

Overview

After an initial failure, damage can spread to neighbouring parts and stop much of a network. Resilience asks two questions: how far does performance fall, and how far does it climb back. The shared idea is to track a network performance curve K(t) over time and compare it with a normal-state baseline K0 2609.27903 §3.2 eq. 7. Every variant derives its numbers from that curve, so the baseline must be defined before the numbers mean anything. The card does not define K0 beyond "normal circumstances" 2609.27903 §3.2.

The repository holds one card. It models failure spread with a susceptible-failed-recovered-dead (SIRD) model, adds a per-node risk value, and then measures resilience and tests two responses: protecting healthy nodes and restoring failed ones. The response is timed from an LSTM forecast of the resilience curve 2609.27903 §3.3 2609.27903 §3.4. All results come from simulation on one traffic network, with single runs 2609.27903 §4 2609.27903 tab. 1.

Variants

Resilience measures from a performance curve

Three measures are computed from the real-time resilience curve E(t), which is the ratio of performance under attack to normal performance over a window 2609.27903 §3.2 eq. 7:

• Bearing capacity E_withstand: the minimum of E(t) over the run. It shows how far performance falls at the worst point 2609.27903 §3.2 eq. 8.
• Recovery capacity E_recover: E(t) at a chosen cutoff time T. It shows how far the network has climbed back by then 2609.27903 §3.2 eq. 9.
• Overall resilience E: the time average of E(t) over [0, T] 2609.27903 eq. 12.

Performance itself is a weighted share of nodes still in service. The weight is degree centrality, and nodes that are failed or dead count as unavailable 2609.27903 eq. 4 eq. 5 eq. 6. The paper's numbers depend on its choice of weight, its window and its cutoff. Table 2 shows that the overall value depends strongly on T: the same run gives E = 0.5725 at T = 30 and E = 0.7427 at T = 400, while E_recover levels off near 0.768 after T = 250 2609.27903 tab. 2. Compare values only at the same T and window.

Node protection before failure

The protection variant lowers the risk value f of healthy nodes. A lower f lowers the infection rate and the collapse rate and raises the recovery rate 2609.27903 §3.4.1 fig. 15. In the paper's table, the protection rate of 0.5 raised overall E from 0.7310 without protection to 0.8796 2609.27903 tab. 6. The meaning of the protection rate beyond the table values is not defined, and the figure legend calls it "defense" 2609.27903 tab. 6 fig. 15.

Node recovery of failed nodes

The recovery variant restores failed nodes to working state at a recovery rate 2609.27903 §3.4.2 fig. 16. At a recovery rate of 0.5, overall E rose from 0.7282 to 0.9117 and E_recover reached 1.0000 2609.27903 tab. 7. The "time cost" column in these tables is not defined in the text, so do not read it as a cost in any unit 2609.27903 tab. 6 2609.27903 tab. 7. The forecast time advance decides how early recovery starts. In the paper's table, E was 0.9278, 0.9354 and 0.9559 for advances of 0.5, 0.3 and 0.1, with the unit not stated 2609.27903 tab. 8.

Forecast-driven timing

An LSTM is trained on the E(t) series and predicts the next time point from a fixed number of earlier observations 2609.27903 §3.3. Its reported forecast quality is MRE 0.00639, MSE 0.00016 and R2 0.96842 on the paper's setup 2609.27903 tab. 3. Against an RNN (R2 0.91152) and a CNN (R2 0.85833), the LSTM scores best in the paper's table, but the baseline architectures and training settings are not given 2609.27903 tab. 4. The LSTM's layers, units, window length and split are also not stated 2609.27903 §3.3. For the general forecasting trade-offs, see Time-series forecasting.

When to use which

The paper's combined run gives E = 0.9354 for the traffic network, up from 0.7261 unoptimised 2609.27903 tab. 11. The same table shows gains on aviation, industrial and power networks, whose data are not named 2609.27903 tab. 11. The gains are author-reported and come from simulation with no variance across seeds 2609.27903 §4.

Implementation sketch

The common core is the failure step, the performance share and the three measures. The paper's protection and recovery act on the same state, and the forecast only decides when they run. The risk-scaled step below follows the paper's recovery term in its figure, not its printed equation 2609.27903 §3.1 fig. 4.

# One Euler step of a per-node SIRD model with risk value f  [§3.1] [fig. 4]
def sird_r_step(S, I, R, D, f, beta0, mu0, lam0, dt):
    # Infection scaled by risk: lower risk fails less  [§3.1]
    infected = f * beta0 * S * dt
    # Recovery uses (1 - f); the printed eq. 3 differs  [fig. 4]
    recovered = (1 - f) * mu0 * I * dt
    # Collapse scaled by risk  [§3.1]
    dead = f * lam0 * I * dt
    S = S - infected
    I = I + infected - recovered - dead
    R = R + recovered
    D = D + dead
    return S, I, R, D

# Performance and the three resilience measures  [eq. 4] to [eq. 12]
def performance(weights, available):
    # available is 1 for S or R, 0 for I or D  [eq. 5]
    return dot(weights, available) / sum(weights)  # [eq. 4]

def measures(E_t, times, T):
    E_withstand = min(E_t)                     # [eq. 8]
    E_recover = value_at(E_t, times, T)        # [eq. 9]
    E_overall = integrate(E_t, times, 0, T) / T  # [eq. 12]
    return E_withstand, E_recover, E_overall

Contradictions

• Recovery term. The printed equation for the recovered count uses μ0·I, while the infected count subtracts only (1 − f)·μ0·I. Figure 4 uses (1 − f)·μ0 for recovery. Summed over all states, the printed form does not keep the node total fixed 2609.27903 eq. 3 fig. 4. The authors do not say which form they simulated.
• Forecast quality. The text says R2 is close to 1 under all six tested conditions 2609.27903 §4.2. The table gives R2 from 0.87052 to 0.97014 across those conditions 2609.27903 tab. 5.
• Table 1 against Table 2. Table 1 gives overall E = 0.7474, but the largest value in Table 2 is 0.7427, at T = 400. The cutoff for Table 1 is not stated 2609.27903 tab. 1 2609.27903 tab. 2.
• No-intervention baseline. Bearing capacity without intervention is 0.4014 in Table 1, 0.4241 in Table 6, 0.3996 in Table 7 and 0.4094 in Table 8. The paper gives no seeds or risk values that would explain the gap 2609.27903 tab. 1 2609.27903 tab. 6 2609.27903 tab. 7 2609.27903 tab. 8.
• Scaling. The text says the complexity is O(n) 2609.27903 §4.2. Simulation time in Table 10 rises about 10 times for a 400 times larger network, and the paper gives no derivation 2609.27903 tab. 10.

Papers

• A Resilience Recovery Method for Complex Traffic Network Security Based on Trend Forecasting: defines SIRD-R with a per-node risk value, three resilience measures, an LSTM forecast, and forecast-driven protection and recovery, evaluated by simulation on a traffic network.