A Mathematical Framework for Misinformation Propagation in Complex Networks: Topology-Dependent Distortion and Control

TL;DR

The paper develops a tractable mathematical framework to quantify misinformation in complex networks by modeling pathwise, error-prone information propagation along geodesics. A drift–fluctuation decomposition reveals that bias shifts the mean uniformly without distorting fluctuations, enabling closed-form analyses and Gaussian/delta-limit results. Numerical experiments across ER, WS, and BA networks uncover topology-specific distortion signatures: ER exhibits a double-peak profile, WS achieves minimal misinformation at intermediate rewiring, and BA leverages hub-mediated integration to suppress distortion across regimes. The findings offer a unified view of how network topology shapes information fidelity and suggest topology-informed strategies to mitigate misinformation in biological and engineered systems.

Abstract

Misinformation is pervasive in natural, biological, social, and engineered systems, yet its quantitative characterization remains challenging.We develop a general mathematical framework for quantifying information distortion in distributed systems by modeling how local transmission errors accumulate along network geodesics and reshape each agent's perceived global state. Through a drift-fluctuation decomposition of pathwise binomial noise, we derive closed-form expressions for node-level perception distributions and show that directional bias induces only a uniform shift in the mean, preserving the fluctuation structure. Applying the framework to canonical graph ensembles, we uncover strong topological signatures of misinformation: Erdős--Rényi random graphs exhibit a double-peaked distortion profile driven by connectivity transitions and geodesic-length fluctuations, scale-free networks suppress misinformation through hub-mediated integration, and optimally rewired small-world networks achieve comparable suppression by balancing clustering with short paths. A direct comparison across regular lattices, Erdős--Rényi random graphs, Watts--Strogatz small-world networks, and Barabási--Albert scale-free networks reveals a connectivity-dependent crossover. In the extremely sparse regime, scale-free and Erdős--Rényi networks behave similarly. At intermediate sparsity, Watts--Strogatz small-world networks exhibit the lowest misinformation. In contrast, Barabási--Albert scale-free networks maintain low misinformation in sparse and dense regimes, while regular lattices produce the highest distortion across connectivities. We additionally show how sparsity constraints, structural organization, and connection costs delineate regimes of minimal misinformation.

Figures (6)

• Figure 1: Schematic of the misinformation model.Left: Illustration of error accumulation between a pair of agents in a network, where each interaction along the shortest path contributes a random error of size $\pm \varepsilon$. Middle: The binomial distribution of the accumulated errors. Right: Comparison between the true distribution $P(x)$ and the perceived distributions $Q^{(j)}(x)$, with the average misinformation quantified by the node-averaged KL divergence.
• Figure 2: Normalized misinformation ($I_M$) as a function of the link probability ($p$) for an Erdős--Rényi network with $n = 5000$ nodes and a delta-distributed true state. The curve begins at a high misinformation level for the sparsest connected graphs and then drops rapidly to a shallow minimum ($I_M \approx 0.04$) immediately above the connectivity threshold, where the network remains almost tree-like and limits error accumulation. As $p$ increases further, $I_M$ rises again to a second, smaller peak around $p \approx 0.031$, reflecting the coexistence of long geodesic paths and emerging short cycles. Beyond this point, misinformation decreases monotonically, approaching zero as $p \to 1$, where uniformly short paths strongly suppress accumulated binomial errors. This overall pattern highlights a distinct double-peaked, nonmonotonic dependence of misinformation on network density.
• Figure 3: Normalized misinformation ($I_M$) as a function of the rewiring probability for Watts--Strogatz networks with $n = 5000$ nodes and a delta-distributed true state, shown for different average degrees (panels a–d). The curves exhibit a clear non-monotonic trend: misinformation is highest in the highly ordered lattice (very low rewiring), decreases to a minimum at a small intermediate level of randomness, and then rises again before saturating in the fully randomized regime. This narrow small-world region balances local order with a few long-range shortcuts—enough to enable efficient communication while still permitting paths long enough for error accumulation—whereas both very ordered and highly randomized networks show elevated misinformation relative to this intermediate optimum. Panels (e) and (f) show the regimes occupied by real-world small-world networks and their corresponding misinformation levels, based on the effective rewiring estimates reported in Table \ref{['tab:biological_beta_two_col_final']}.
• Figure 4: Normalized misinformation ($I_M$) versus the number of edges added per new node ($m$) in Barabási--Albert networks with $n = 5000$ nodes and a $\delta$-distributed true state. Misinformation initially decreases monotonically with increasing $m$, reflecting the well-known role of hubs in enhancing robustness and stabilizing information flow in scale-free structures barabasi_science_1999albert2000errorbarrat2008. Beyond an intermediate threshold, however, misinformation increases again, indicating a regime in which hub-driven fluctuations accumulate and propagate through the network. In the extremely sparse limit (small $m$), BA networks outperform random and small-world networks of comparable density, owing to their highly connected hubs, which act as efficient integrators and provide redundant communication pathways. Shaded regions show the regimes occupied by real-world scale-free networks and their corresponding misinformation levels, based on the comparable connectivity estimates reported in Table \ref{['tab:bio_SF_final_expanded']}.
• Figure 5: A comparison of misinformation across regular, ER, WS, and BA networks, all constructed with $n = 5000$ nodes.Regular networks consistently exhibit the highest misinformation in the sparse to intermediate-dense regimes due to their long path lengths and absence of shortcuts. In the extremely sparse regime, ER and BA networks behave similarly, with BA showing slightly lower misinformation because early-formed hubs provide local aggregation. As connectivity increases into an intermediate (but still relatively low) range, a clear crossover emerges: WS networks achieve the lowest misinformation as they enter the small-world regime, where the introduction of a modest number of shortcuts sharply reduces characteristic path lengths while largely preserving local structure. In the intermediate-dense to dense-connected regime, BA networks outperform all other topologies, yielding the smallest misinformation among the classes considered. The WS curve shown here corresponds to a rewiring probability of $\beta = 0.1$, which lies within the small-world regime. The black solid lines represent the transition points, where $\langle k \rangle$ denotes the average degree.