Propagation Centrality in Social Networks

• Propagation Centrality is a PageRank-style metric that estimates a node’s steady-state propagation potential on directed interaction graphs.
• It uses an iterative random walk with restart that balances a baseline teleportation component with diffusion from in-neighbor nodes.
• In health misinformation analyses, PC complements MVC and DIC by identifying cascade starters, enhancing node ranking and intervention efficacy.

Searching arXiv for the specified paper and closely related centrality terminology to ground the article in current literature. arXiv_search(query="id:(Sikosana et al., 11 Jul 2025) OR ti:\"Analysing Health Misinformation with Advanced Centrality Metrics in Online Social Networks\" OR \"Propagation Centrality\" PageRank misinformation", max_results=10) Propagation Centrality (PC) is a PageRank-style centrality defined on a directed interaction graph and used to estimate the steady-state propagation potential of each node. In the health-misinformation framework of “Analysing Health Misinformation with Advanced Centrality Metrics in Online Social Networks,” PC is introduced alongside health misinformation vulnerability centrality (MVC) and dynamic influence centrality (DIC) as an “advanced” metric intended to capture diffusion-aware influence more directly than degree, betweenness, closeness, or eigenvector centrality (Sikosana et al., 11 Jul 2025).

1. Definition and intended scope

In the cited formulation, the online social network is a directed graph $G=(V,E)$, where nodes are users and edges represent interactions such as retweets and mentions. Propagation Centrality assigns each node $v_i\in V$ a score $x(v_i)$ that estimates how strongly its influence persists after repeated rounds of diffusion over the network.

The metric is explicitly defined by the equation

$$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$

with damping factor $d=0.85$, total number of nodes $n=|V|$, in-neighbour set $N_{in}(v_i)$, and out-degree $d_{out}(v_j)$. In this formulation, PC is exactly the classical PageRank equation instantiated under the name “Propagation Centrality.”

The intended scope of PC is not generic structural prominence but long-range propagation capacity. The metric is introduced to address three limitations attributed to traditional static centralities: they capture structural position rather than diffusion dynamics, they can disagree strongly on which nodes are most important, and they do not directly model the recursive accumulation of influence through neighbours. PC therefore targets the question: which nodes are positioned to trigger large cascades because they sit inside a globally influential interaction structure?

2. Mathematical formulation and diffusion interpretation

The directed network is represented by an adjacency matrix $A \in \mathbb{R}^{n \times n}$ with entries $a_{ij}=w_{ij}$, where $v_i\in V$0 denotes the weight of edge $v_i\in V$1. The paper states that the spectral properties of $v_i\in V$2, or of the derived transition matrix, are used to compute PC through an eigenvector-like fixed-point iteration.

The two terms in the PC equation have distinct roles. The baseline term,

$v_i\in V$3

is the teleportation component: every node receives some centrality even if it has no incoming edges. The propagation term,

$v_i\in V$4

redistributes influence from in-neighbours, with each neighbour $v_i\in V$5 contributing a share of its current score proportional to $v_i\in V$6. Influence is therefore shared among outgoing neighbours and accumulates recursively.

The paper does not treat PC as an explicit epidemic model such as SIR, SI, IC, or LT. Instead, it uses a random walk with restart as a diffusion kernel. In that interpretation, with probability $v_i\in V$7 influence follows outgoing edges, and with probability $v_i\in V$8 it teleports to a random node. The paper explicitly states that PC is “implemented as personalised PageRank” and that it “incorporates a diffusion kernel that better reflects the potential for information spread.”

Equivalently, if $v_i\in V$9 denotes the relevant column-stochastic transition matrix, PC seeks a vector $x(v_i)$0 satisfying

$x(v_i)$1

The paper notes that convergence is supported by the Perron–Frobenius theorem: for the positive, irreducible matrix induced by teleportation, the fixed point has a unique positive solution, and power iteration converges to it.

3. Computation and algorithmic properties

PC is computed by an iterative PageRank-style procedure. Initial scores are uniform,

$x(v_i)$2

At iteration $x(v_i)$3, scores are updated as

$x(v_i)$4

The paper states that nodes with $x(v_i)$5 are handled as dangling nodes “in the normal PageRank fashion.” It does not specify the detailed implementation, but it notes the standard practice of redistributing their mass uniformly. Convergence is checked iteratively; the paper mentions that convergence occurs “typically within 50–100 iterations” on the FibVID network. It also notes that convergence may be monitored by an $x(v_i)$6 difference threshold, although no explicit tolerance is given.

No formal complexity proof is provided in the paper, but the computation is characterized in the standard PageRank way: time complexity per iteration is $x(v_i)$7, and total complexity is $x(v_i)$8 for $x(v_i)$9 iterations, with $$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$0 in the reported experiments. This places PC in the class of scalable spectral centralities, while retaining a global rather than purely local view of diffusion.

4. Empirical role in health misinformation analysis

PC is evaluated on two datasets. The first is FibVID, the COVID-19 Fake News Information-Broadcasting Dataset, covering Jan–Dec 2020 and containing fact-checked COVID and non-COVID claims together with Twitter/X propagation and user features. The second is the Monant Medical Misinformation Dataset, which covers broader medical misinformation topics such as vaccines, pharmaceutical skepticism, and alternative treatments. In both cases, the authors construct a directed interaction network from user interactions, compute PC scores, and rank nodes by $$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$1 (Sikosana et al., 11 Jul 2025).

On FibVID, the top-10 PC nodes are compared with the top-10 nodes under degree, eigenvector, betweenness, and closeness. The reported result is that nine of the ten PC leaders coincide with degree/eigenvector hubs, a 90% overlap. This shows that PC largely validates structurally prominent hubs while still surfacing a small PC-exclusive set not captured by the traditional top-10 lists.

Across the broader comparison between traditional and advanced metrics, traditional centralities identified 29 influential nodes, while the new metrics uncovered 24 unique nodes, yielding 42 combined nodes, an increase of 44.83%. The paper presents this as evidence that traditional and advanced centralities are complementary rather than mutually exclusive.

On Monant, the distinction is sharper. The paper reports that only 53 nodes are common between the traditional and advanced top sets; 314 are unique to traditional metrics and 247 are unique to the advanced metrics. It further notes that several impactful users, including author IDs 204095 and 202800, appear only when advanced metrics are used. This suggests that PC is most informative when used as part of a multi-metric framework rather than as a replacement for all prior centralities.

PC is also used in intervention simulations. A baseline strategy removes or neutralises nodes identified by traditional centralities. An enhanced strategy additionally targets nodes identified by PC, MVC, and DIC. The reported reduction in misinformation volume rises from 50% under the baseline to 62.5% under the enhanced strategy, which the paper describes as a 25% improvement relative to baseline.

5. Position relative to traditional centralities, MVC, and DIC

The paper distinguishes the three advanced metrics by the type of signal they are designed to capture.

Metric	Intended quantity	Temporal status
PC	steady-state propagation potential	static snapshot
MVC	susceptibility-weighted connectivity	vulnerability-weighted
DIC	cumulative, time-dependent influence	explicitly time-indexed

PC differs from degree by not restricting importance to immediate neighbourhood size. It differs from eigenvector centrality by incorporating teleportation and a random-walk viewpoint, making it less tied to local reinforcement and more sensitive to global stationary flow. The paper characterizes PC as measuring who can propagate information far and wide through direct and indirect paths, and as identifying potential cascade starters.

MVC addresses a different question. It weights structural connectivity by vulnerability to misinformation, using a vulnerability score updated as

$$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$2

Because fine-grained credibility data were unavailable, vulnerability was simulated with random values $$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$3. MVC therefore identifies nodes that are structurally exposed and vulnerable, rather than nodes with purely structural propagation strength.

DIC is the explicitly temporal complement to PC. Its update rule is

$$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$4

with $$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$5, iteration over a finite number of time steps, and later normalisation. Whereas PC summarizes a steady-state influence field on a fixed snapshot, DIC models accumulation over time through repeated exposures. Empirically, the paper reports that all 10 DIC leaders are entirely disjoint from the traditional and PC top-10 lists, which it interprets as evidence that DIC detects persistent long-tail spreaders that a steady-state metric does not foreground (Sikosana et al., 11 Jul 2025).

The practical implication is a layered interpretation: PC captures high-throughput structural hubs, MVC captures influential yet vulnerable amplifiers, and DIC captures persistent time-extended spreaders.

6. Interpretation, limitations, and broader terminology

A high PC value indicates that a node occupies a structurally central position, is tied directly and indirectly to other influential users, and can trigger large-scale cascades when it amplifies content. In the specific health-misinformation setting, the paper describes high-PC users as priority targets for early detection, monitoring, algorithmic dampening, prioritized fact-checking, and counter-messaging. A plausible implication is that PC is most operationally useful where intervention resources are scarce and the goal is to identify structurally efficient spreaders rather than merely visible accounts.

The paper also identifies several limitations. PC is computed on a static snapshot and therefore does not directly model time-varying edges or evolving network structure. Its diffusion kernel is a PageRank random walk, which is a proxy rather than a full behavioural or epidemiological model; actual misinformation spread may involve threshold effects, user fatigue, and content semantics. PC does not incorporate susceptibility or content features, which is precisely why MVC and DIC are introduced as complementary metrics. Finally, although PageRank is scalable, very large online social networks still pose computational challenges, motivating future work on sparse and GPU implementations.

The term “Propagation Centrality” is not standardized across the arXiv literature. In the 2025 misinformation paper, it denotes the PageRank-style steady-state quantity described above (Sikosana et al., 11 Jul 2025). Elsewhere, propagation-oriented centrality has been defined differently: as perturbation centrality, the reciprocal of silencing time under a communicating-vessels dynamics (Szalay et al., 2013); as dynamic centrality, which counts attenuated time-respecting paths in evolving networks (Lerman et al., 2010); as transmission centrality, which estimates how many nodes receive diffusion through a link (Zhang et al., 2018); and as a broader family of influence-based centralities defined as expectations over cascading-distance profiles (Chen et al., 2018). These usages are not identical. They indicate that “propagation centrality” is best understood as a class of diffusion-oriented centrality constructions rather than a single universally fixed metric.

In the specific sense established in health-misinformation analysis, however, Propagation Centrality is precise: it is the PageRank fixed point

$$x(v_i) = \frac{1-d}{n} + d \sum_{v_j \in N_{in}(v_i)} \frac{x(v_j)}{d_{out}(v_j)},$$6

interpreted as a steady-state estimate of a node’s capacity to sustain network-wide propagation.