From Propagation to Protection: Risk-Aware Diffusion for Harm Minimization in Signed Social Networks

Abstract: Real-world social relationships are not uniformly supportive. Information through hostile connections can increase resistance, anxiety, or misinformation rather than adoption. Classical models such as Independent Cascade and Linear Threshold, together with Influence Maximization (IM), which maximizes spread from a limited seed set, treat activation as discrete and irreversible. Its counterpart, Influence Minimization (Inf-Min), limits undesirable spread but similarly relies on simplified activation assumptions. Signed extensions incorporate polarity but largely retain this irreversibility, leaving no room for beliefs to weaken, reverse, or recover under competing influence. Moreover, both objectives typically treat individuals uniformly, without accounting for differences in vulnerability or prioritizing protection of those most at risk. We introduce RASH, a signed, susceptibility-aware diffusion model in which node awareness is continuous, bounded, and non-monotonic, and prove that despite this added expressiveness it remains monotone and γ-weakly submodular where only positive or negative edges exist, preserving tractable greedy approximation guarantees where strict submodularity provably fails. Building on RASH, we formulate Harm Minimization (HM), which maximizes aggregate reach while minimizing the awareness shortfall (harm). We prove HM is NP-hard, yet its harm-reduction formulation inherits the same monotonicity and weak-submodularity structure, admitting a greedy algorithm with a bounded approximation ratio. Across six structurally diverse signed networks, RASH is the only diffusion model tested to our knowledge that ever allows awareness to reverse after activation, letting sustained discouraging influence drive awareness from positive toward negative, and HM achieves the highest harm reduction of any method evaluated, including its own boundary cases (IM and Inf-Min)

• The paper introduces RASH, a continuous and reversible diffusion model that combines signed influence, node susceptibility, nonlinear thresholding, and memory to model awareness growth, erosion, and reversal.
• The paper defines Harm Minimization, which prioritizes vulnerability-weighted awareness shortfalls through the tunable exponent p, and shows that greedy optimization achieves a provable weak-submodular approximation guarantee.
• The paper demonstrates that Harm Minimization outperforms influence maximization, influence minimization, and six practical heuristics across six signed networks, reaching up to 81% harm reduction on Epinions at a budget of 20 seeds.

Motivation and problem setting

The paper addresses two structural limitations that classical diffusion modeling and influence optimization share in signed social networks. First, canonical models such as Independent Cascade (IC) and Linear Threshold (LT), and their signed extensions, treat activation as a discrete, irreversible event: once a node commits to a sign of activation, its state is permanent. This precludes representing phenomena such as belief weakening, reversal of radicalization, or recovery under competing influence. Second, both Influence Maximization (IM) and Influence Minimization (Inf-Min) evaluate campaigns by aggregate spread or aggregate suppression alone, implicitly assigning every unreached individual equal cost and assigning zero cost to failing a vulnerable individual. The authors argue this is inadequate for protective campaigns—public-health advisories, cybersecurity alerts, crisis communications—where failing to protect a highly susceptible individual can be far more consequential than missing a low-risk one.

The paper makes three contributions: (i) the Risk-Aware Signed Heterogeneous (RASH) diffusion model with continuous, bounded, non-monotonic awareness; (ii) the Harm Minimization (HM) objective over vulnerability-weighted awareness shortfall; and (iii) theoretical guarantees showing that despite non-monotonic node dynamics, the relevant set functions are monotone and $\gamma$-weakly submodular, enabling greedy approximation with provable ratios.

The RASH diffusion model

RASH operates on a directed signed graph $G = (V, E^+, E^-, a)$ where each node $i$ maintains a continuous awareness state $x_i(t) \in [-1,1]$. The update pipeline has five stages:

1. Signed influence aggregation: supportive input $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$ and discouraging input $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$ are computed independently so that they do not cancel before being weighed.
2. Susceptibility weighting: these are combined as $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$ with susceptibility $\lambda_i \in [0,1]$ interpolating between resilient ($\lambda_i \to 0$) and highly susceptible ($\lambda_i \to 1$) nodes.
3. Nonlinear normalization via $G = (V, E^+, E^-, a)$0 to bound influence symmetrically across signs.
4. Risk-aware thresholding: updates occur only when $G = (V, E^+, E^-, a)$1, giving natural sparsification against noise.
5. Memory-preserving propagation: $G = (V, E^+, E^-, a)$2 when the threshold clears, and $G = (V, E^+, E^-, a)$3 otherwise.

Because reinforcement and degradation act through the same rule, awareness grown under supportive influence can later be eroded by discouraging influence—a behavior structurally impossible under irreversible-activation models. Both $G = (V, E^+, E^-, a)$4 and $G = (V, E^+, E^-, a)$5 admit structure-only estimators (fraction of incoming neighbors that are negative; in-degree relative to maximum in-degree), requiring no metadata beyond the signed graph itself.

Setting $G = (V, E^+, E^-, a)$6 for all nodes recovers pure-positive IM dynamics; setting $G = (V, E^+, E^-, a)$7 recovers pure-negative blocking dynamics. Classical objectives therefore emerge as boundary regimes of a single mechanism rather than separate mechanics.

The Harm Minimization objective

Each node carries a safety threshold $G = (V, E^+, E^-, a)$8, the minimum terminal awareness required for protection, and vulnerability weight $G = (V, E^+, E^-, a)$9. The shortfall is $i$0 under seed set $i$1, and HM minimizes

$i$2

over seed sets of budget at most $i$3. The exponent $i$4 acts as a policy lever: $i$5 penalizes shortfall proportionately, while $i$6 approaches a Rawlsian worst-off-individual criterion. Vulnerability is deliberately coupled to the dynamics via $i$7, and the safety threshold is derived from two structural risk factors—negative exposure ratio and structural isolation—combined through a tunable convex mixture mapped onto $i$8.

Three properties make RASH the minimal substrate for this objective: continuous states allow graded shortfalls (binary models collapse them to two values); reversible dynamics allow capturing interventions that leave a vulnerable individual worse off than no intervention; and $i$9 ties objective-level vulnerability to model-level susceptibility by construction.

Theoretical guarantees

The guarantees are established in the pure-positive regime (with an entirely analogous argument in the pure-negative regime). A central technical device is replacing the hard threshold gate with a smoothed logistic gate $x_i(t) \in [-1,1]$0; the hard rule is discontinuous exactly at $x_i(t) \in [-1,1]$1, making output change unbounded relative to input change. The smoothed gate is $x_i(t) \in [-1,1]$2 with strictly positive derivative on $x_i(t) \in [-1,1]$3, recovering the hard rule as $x_i(t) \in [-1,1]$4.

Under this assumption:

• Boundedness and marginal coupling lemma: states remain in $x_i(t) \in [-1,1]$5 for all seedings, and adding a seed $x_i(t) \in [-1,1]$6 to a smaller set produces a marginal effect at least $x_i(t) \in [-1,1]$7 times its marginal effect on any superset $x_i(t) \in [-1,1]$8, where $x_i(t) \in [-1,1]$9 combines the slowest adaptation rate with the network-wide minimum gate sensitivity.
• Monotonicity of harm reduction $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$0 follows directly from state monotonicity and the fact that shortfall is non-increasing in terminal awareness.
• $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$1-weak submodularity of $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$2: strict submodularity provably fails even in simple positive networks—the paper gives a two-seed example where adding $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$3 to $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$4 yields zero marginal gain but adding it to $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$5 yields approximately $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$6, because the second seed tips node $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$7 over its resistance threshold. Weak submodularity nevertheless suffices for greedy guarantees.
• NP-hardness of HM: shown by reducing from IC-based IM; with $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$8 and no negative edges, minimizing $P_i(t) = \sum_{j \in \Gamma_i^+} a_{ji} x_j(t)$9 is equivalent to maximizing IC spread. Notably, the hardness proof covers only $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$0: the map $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$1 does not preserve seed-set orderings across $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$2 (the counterexample of shortfall vectors $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$3 vs. $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$4 shows $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$5 and $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$6 rank them differently), and hardness for $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$7 is explicitly left open.

Combining monotonicity and weak submodularity yields a greedy algorithm with approximation guarantee

$N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$8

Because RASH's update rule is deterministic given $N_i(t) = \sum_{j \in \Gamma_i^-} a_{ji} x_j(t)$9, each marginal gain requires a single rollout rather than Monte Carlo averaging over stochastic cascades—a concrete computational advantage over simulation-based signed IM methods. The total cost is $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$0; CELF-style lazy evaluation is used only as a heuristic accelerator, since its correctness proof assumes exact submodularity.

Experimental results

Evaluation covers one synthetic Barabási–Albert graph and five real-world signed networks (Bitcoin Alpha, Bitcoin OTC, Epinions, Slashdot, Wiki-RfA), spanning roughly 500 to 132k nodes, with $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$1 and parameters estimated from structure where possible.

Phase 1 — diffusion dynamics. Against four signed baselines (PID, SLT, SNIC, PLID), RASH is the only model recording nonzero awareness reversal or suppression counts on all six datasets; the baselines remain flat at zero by construction, since SLT/SNIC enforce irreversible activation and PID/PLID lack suppression mechanisms. On smaller networks, negative spread under greedy blocking budgets $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$2 forms well-separated plateaus consistent with the weak-submodularity guarantee; on the three largest networks, negative spread oscillates persistently, which the authors attribute to the decay of the guarantee constant $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$3 with network size—$I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$4 depends on $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$5 through $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$6, which shrinks monotonically as $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$7 grows.

Phase 2 — harm minimization. HM achieves the highest percentage harm reduction of every method evaluated, including its own boundary cases IM and Inf-Min, at every seed budget on every dataset. IM, which dominates raw activation, plateaus at substantially lower harm reduction (approximately 50–65% on most datasets), while Inf-Min performs weakest throughout since suppressing negative spread provides no positive awareness. Against six cheaper heuristics (Random, High-Degree, Degree Discount, Betweenness, Targeted Protection, Reactive seeding), Greedy HM is the most expensive method but the highest-harm-reducing method in every cell of the comparison. Representative figures: on Epinions, HM's runtime grows from 33.25 s at $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$8 to 128.40 s at $I_i(t) = (1-\lambda_i)P_i(t) + \lambda_i N_i(t)$9 while harm reduction rises from 46% to 81%, a margin no baseline approaches; on BA it reaches 100% harm reduction at $\lambda_i \in [0,1]$0 versus 93% for the best heuristic. On smaller networks HM trails IM's raw activation by only a few percent, while on larger, denser networks the gap widens—the authors interpret this as deliberate reallocation of budget toward protecting vulnerable individuals precisely where harm pathways are more abundant, not as a shortcoming.

Limitations and open questions

The paper concedes several points candidly. The approximation guarantee inherits the weakness of $\lambda_i \in [0,1]$1-weak submodularity: worst-case seed-set instability permitted under weak (rather than strict) submodularity occasionally appears as elevated variance rather than systematic failure. For large networks, candidate pools were restricted to degree-ranked subsets, so reported solutions are optimal only within that subset; incorporating high-vulnerability, low-degree individuals into candidate pools is left as an extension. Safety thresholds and vulnerability weights derive solely from structural signals, although the formulation admits richer risk information when available. On the theory side, NP-hardness is established only for $\lambda_i \in [0,1]$2, and the behavior of greedy selection as $\lambda_i \in [0,1]$3—where the objective shifts from broad harm reduction toward protecting the single worst-off individual—is uncharacterized. Extension to dynamic signed networks, in which edges and vulnerability profiles evolve during a campaign, remains open.

Conclusion

This work replaces irreversible, sign-committed activation with continuous, reversible, susceptibility-weighted awareness, while preserving enough structure—monotonicity and $\lambda_i \in [0,1]$4-weak submodularity in the pure-sign regimes—to retain greedy approximation guarantees where strict submodularity provably fails. The HM objective reorients protective campaigns from aggregate reach toward vulnerability-weighted protection deficits, and empirically dominates IM, Inf-Min, and six practical heuristics in harm reduction across six signed networks at the cost of a quantifiable, objective-driven computational overhead. The main open questions concern hardness beyond $\lambda_i \in [0,1]$5, scalability beyond degree-restricted candidate pools, and dynamics over evolving signed topologies.