Absolute Influence Centrality Measure

• The paper defines the absolute influence centrality as the total absolute effect of perturbing a node’s initial opinion on the network’s final opinion.
• It employs a signed Friedkin–Johnsen model on directed, weakly connected graphs, integrating agent stubbornness and signed interactions to compute an exact influence matrix.
• The measure differentiates between cooperative, balanced, and unbalanced sinks, linking network topology and steady‐state sensitivity to initial opinions.

Searching arXiv for the cited papers and recent related work to ground the article. Absolute Influence Centrality Measure denotes, in its most explicit recent formulation, a node-level influence score derived from signed Friedkin–Johnsen opinion dynamics on arbitrary weakly connected directed graphs. In that formulation, the centrality of agent $i$ is the total absolute effect of perturbing $i$’s initial opinion on the network’s final opinion vector, computed as the absolute column sum of an exact influence matrix $\Theta$: $\widetilde{\Theta}^T\mathbb{1}_n$, where $\widetilde{\Theta}=[|\theta_{ij}|]$ (Shrinate et al., 2024). In adjacent literatures, the phrase is not standardized. Some works distinguish an underlying “absolute” node centrality $c(i)$ from relative or subnetwork-normalized variants, while others propose process-specific influence scores based on diffusion, cooperation breakdown, or long-range activation without naming them “absolute influence centrality” (Cerqueti et al., 2019, Chen et al., 2018, Simko et al., 2013). The resulting concept is therefore best understood as a family resemblance centered on direct quantification of a node’s network-wide effect, with the signed Friedkin–Johnsen construction providing the clearest formal definition (Shrinate et al., 2024).

1. Formal definition in signed Friedkin–Johnsen dynamics

The explicit absolute influence centrality measure is introduced for a signed, weighted digraph

$\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$

where $\mathcal{V}=\{1,2,\dots,n\}$, $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$, and $A=[a_{ij}]\in\mathbb{R}^{n\times n}$ is the signed weighted adjacency matrix. A directed edge $i$0 means agent $i$1 influences agent $i$2, positive and negative edges represent cooperative and antagonistic interactions, the graph has no self-loops, and the analysis is carried out on weakly connected graphs that may contain multiple SCCs (Shrinate et al., 2024).

Each agent $i$3 has self-belief $i$4 and stubbornness $i$5 for non-sinks, with $i$6 indicating a stubborn agent. For a sink in the graph, an agent may be fully stubborn with $i$7; a sink and a fully stubborn agent are equivalent in the sense that its opinion never changes (Shrinate et al., 2024). The scalar signed FJ update is

$i$8

and in vector form

$i$9

with

$\Theta$0

The admissibility condition is that for a non-sink node $\Theta$1, $\Theta$2, ensuring the network structure is not altered by the model parameters (Shrinate et al., 2024).

Within this model, absolute influence centrality is defined through a perturbational question: how much does a change in one agent’s initial opinion alter the entire final opinion profile? The paper formalizes that question by perturbing agent $\Theta$3’s initial opinion by $\Theta$4 and measuring the induced $\Theta$5-change in the final steady state: $\Theta$6 This quantity is identified as the absolute influence centrality of agent $\Theta$7 (Shrinate et al., 2024).

2. Opinion leaders, followers, and the role of stubbornness and balance

The measure is inseparable from the paper’s classification of agents through the condensation graph $\Theta$8. Let $\Theta$9 be the set of sinks of $\widetilde{\Theta}^T\mathbb{1}_n$0. Opinion leaders are the nodes in the sinks of $\widetilde{\Theta}^T\mathbb{1}_n$1, and followers are all other nodes. Opinion leaders are then partitioned into $\widetilde{\Theta}^T\mathbb{1}_n$2, consisting of leaders that are the only node in their sink, and $\widetilde{\Theta}^T\mathbb{1}_n$3, consisting of leaders in sinks with two or more opinion leaders (Shrinate et al., 2024).

For sinks containing multiple leaders, the sign structure matters. The paper distinguishes $\widetilde{\Theta}^T\mathbb{1}_n$4 for sinks whose opinion leaders interact cooperatively only, $\widetilde{\Theta}^T\mathbb{1}_n$5 for sinks with at least one negative edge but structurally balanced, and $\widetilde{\Theta}^T\mathbb{1}_n$6 for sinks that are structurally unbalanced. A key result is that under signed FJ dynamics some opinion leaders can become non-influential if they are in a structurally unbalanced sink or if a stubborn leader dominates the sink (Shrinate et al., 2024).

The convergence theory is equally central. The matrix $\widetilde{\Theta}^T\mathbb{1}_n$7 is semi-convergent but not convergent iff $\widetilde{\Theta}^T\mathbb{1}_n$8, where $\widetilde{\Theta}^T\mathbb{1}_n$9 is the set of effectively balanced sinks with no stubborn opinion leaders; otherwise, $\widetilde{\Theta}=[|\theta_{ij}|]$0 is convergent. The steady state is

$\widetilde{\Theta}=[|\theta_{ij}|]$1

when $\widetilde{\Theta}=[|\theta_{ij}|]$2, and

$\widetilde{\Theta}=[|\theta_{ij}|]$3

when $\widetilde{\Theta}=[|\theta_{ij}|]$4 (Shrinate et al., 2024).

These distinctions are not ancillary. They determine which initial opinions survive asymptotically and therefore which agents can register nonzero absolute influence. Cooperative sinks yield consensus, balanced sinks yield bipartite consensus, unbalanced sinks yield zero opinions, and a stubborn leader in a sink remains influential and can suppress others in the same sink (Shrinate et al., 2024). A plausible implication is that absolute influence centrality in this setting is less a purely topological score than a steady-state sensitivity index conditioned by topology, sign pattern, and stubbornness.

3. Influence matrix, signal flow graphs, and the absolute column-sum formula

The measure is computed from an exact decomposition of final opinions into contributions from initial opinions. The steady-state equation

$\widetilde{\Theta}=[|\theta_{ij}|]$5

is rewritten as a linear system over an augmented signal vector

$\widetilde{\Theta}=[|\theta_{ij}|]$6

satisfying

$\widetilde{\Theta}=[|\theta_{ij}|]$7

This system is represented as a signal flow graph (SFG) $\widetilde{\Theta}=[|\theta_{ij}|]$8, where each node is a signal or state and each branch gain equals an entry of $\widetilde{\Theta}=[|\theta_{ij}|]$9. The direction of branches is reverse to the original information flow (Shrinate et al., 2024).

The SFG is reduced by collapsing non-stubborn opinion leaders in cooperative sinks into one source, collapsing balanced sinks into two sources corresponding to the two bipartitions, keeping stubborn agents as separate sources, and removing unbalanced non-stubborn sinks because they contribute zero asymptotic influence. Source-to-node gains in the reduced SFG are then computed by Mason’s gain formula,

$c(i)$0

where $c(i)$1 are forward-path gains and $c(i)$2, $c(i)$3 are the usual loop-determinants over all loops or loops not touching the $c(i)$4-th path (Shrinate et al., 2024).

The outcome is an influence decomposition

$c(i)$5

where $c(i)$6 is the influence of agent $c(i)$7’s initial opinion on agent $c(i)$8’s final opinion. Because signed interactions can make $c(i)$9 positive or negative, the paper defines

$\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$0

This is the “absolute” step: it removes sign cancellation and measures the magnitude of influence regardless of whether the effect is cooperative or antagonistic (Shrinate et al., 2024).

Using $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$1, a unit perturbation in agent $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$2’s initial condition changes the final opinion vector by column $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$3 of $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$4. The paper’s Theorem 5 therefore gives the central formula

$\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$5

Equivalently, the centrality of agent $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$6 is

$\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$7

This makes the measure an exact global sensitivity score over final opinions rather than a heuristic function of degree, path count, or eigenstructure alone (Shrinate et al., 2024).

4. Absolute versus relative centrality in adjacent formulations

A separate line of work distinguishes “absolute” centrality from relative or contextualized centrality without defining a named absolute influence centrality measure. In a connected, undirected graph $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$8 with a normalized nonnegative centrality $\mathcal{G}=\{\mathcal{V},\mathcal{E},A\},$9, the absolute centrality $\mathcal{V}=\{1,2,\dots,n\}$0 is the usual node score in the whole graph. Relative centrality is defined by normalizing with the network average,

$\mathcal{V}=\{1,2,\dots,n\}$1

and this preserves ranking: $\mathcal{V}=\{1,2,\dots,n\}$2 For a subgraph $\mathcal{V}=\{1,2,\dots,n\}$3, the decomposition

$\mathcal{V}=\{1,2,\dots,n\}$4

separates node-within-group prominence from group-within-network prominence (Cerqueti et al., 2019).

That framework is specialized to betweenness and eigenvector centrality. For betweenness,

$\mathcal{V}=\{1,2,\dots,n\}$5

and for eigenvector centrality,

$\mathcal{V}=\{1,2,\dots,n\}$6

The paper explicitly states that it does not define a separate formal measure called “absolute influence centrality”; if one seeks an absolute object there, it is simply the underlying $\mathcal{V}=\{1,2,\dots,n\}$7, such as $\mathcal{V}=\{1,2,\dots,n\}$8 or $\mathcal{V}=\{1,2,\dots,n\}$9, before relative normalization (Cerqueti et al., 2019).

A related contrast appears in long-range interaction centrality. The LRIC framework proposes threshold-aware, group-aware, long-range influence measures on weighted directed graphs, but it also states that it does not define a single “absolute influence centrality” as a final standalone measure with raw absolute scale. Instead, it uses normalized influence quantities at several stages: direct influence entries in $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$0, path-based totals capped or selected to remain in $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$1, simulation-based ratios, and normalized node weights $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$2 (Aleskerov et al., 2016). This suggests that the phrase “absolute influence centrality” is not a stable cross-paper term; in some works it denotes an exact global effect measure, while in others the absolute quantity is only the pre-normalized centrality from which contextual or normalized scores are derived.

5. Related influence-based centralities and process-specific alternatives

Several nearby proposals quantify influence through a dynamical process rather than through static graph structure, but they operationalize influence in different ways. In online social networks with finite attention, limited-attention Alpha-centrality ($\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$3) modifies Bonacich’s Alpha-centrality by scaling incoming influence by inverse in-degree. Its iterative form is

$\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$4

with starting vector

$\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$5

Node-wise,

$\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$6

This measure is influence-based, but it is not presented as an absolute centrality; rather, it is a broadcasting-and-reception model in which recipients’ ability to process a message depends on their in-degree (Lerman et al., 2013).

A more general axiomatic framework defines influence-based centrality as a mapping from an influence profile $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$7 to node scores, and introduces the stochastic sphere-of-influence family

$\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$8

This family generalizes degree, harmonic, closeness, reachability, and sphere-of-influence centralities to diffusion settings, and the paper proves a characterization theorem: for anonymous $\mathcal{E}\subseteq \mathcal{V}\times\mathcal{V}$9, $A=[a_{ij}]\in\mathbb{R}^{n\times n}$0 is the unique influence-based centrality conforming with the corresponding graph-theoretic centrality and satisfying Anonymity and Bayesian linearity (Chen et al., 2018). Here again, the construction is influence-based and dynamic, but not cast as a named absolute influence centrality.

Game centrality offers a different process-based interpretation. For a given node $A=[a_{ij}]\in\mathbb{R}^{n\times n}$1, all nodes but $A=[a_{ij}]\in\mathbb{R}^{n\times n}$2 initially cooperate, node $A=[a_{ij}]\in\mathbb{R}^{n\times n}$3 initially defects, and repeated spatial games are simulated on the network. The game centrality of node $A=[a_{ij}]\in\mathbb{R}^{n\times n}$4 is

$A=[a_{ij}]\in\mathbb{R}^{n\times n}$5

the proportion of defectors averaged over the last 50 simulation steps. The measure is explicitly dynamic and operational: it quantifies the ability of a single initially defecting node to break overall cooperation (Simko et al., 2013). A plausible implication is that game centrality functions as an absolute influence score only within a tightly specified dynamical protocol; unlike the signed FJ measure, it is not a steady-state linear sensitivity measure over all initial opinions.

6. Empirical interpretation, misconceptions, and scope conditions

A common misconception is that influence centrality is interchangeable with conventional reputation or productivity metrics. In an author collaboration network built from papers published in 2017 at major computer systems conferences, degree centrality correlated highest with betweenness at $A=[a_{ij}]\in\mathbb{R}^{n\times n}$6 and with as_author at $A=[a_{ij}]\in\mathbb{R}^{n\times n}$7, while correlating negligibly $A=[a_{ij}]\in\mathbb{R}^{n\times n}$8 with all other author features; betweenness showed the same two strongest correlations and correlated negligibly $A=[a_{ij}]\in\mathbb{R}^{n\times n}$9 with the remaining features. The paper highlights a particularly important absence of statistically significant correlation between degree centrality and hindex, and it reports that betweenness also did not correlate meaningfully with npubs or hindex (Blanchette, 2019). In that setting, network position captures a dimension of collaborative embeddedness or bridging power that is distinct from scholarly reputation.

Another misconception is that structurally central or leader-like nodes are always the influential ones. The signed FJ analysis explicitly shows otherwise: a non-stubborn opinion leader in a structurally unbalanced sink can become non-influential, while a stubborn follower can become influential and other leaders in the same sink can lose influence (Shrinate et al., 2024). The worked example yields the absolute centrality vector

$i$00

ranking agent $i$01 as the most influential even though the example also contains multiple opinion leaders (Shrinate et al., 2024).

The scope conditions are likewise important. The signed FJ absolute centrality is designed for signed networks with stubbornness and antagonism and is derived from exact steady-state dependence on initial opinions (Shrinate et al., 2024). Relative-centrality decompositions presuppose a baseline absolute centrality $i$02 and a meaningful comparison to network or subnetwork averages (Cerqueti et al., 2019). Limited-attention Alpha-centrality presumes that attention is divided uniformly among all incoming ties (Lerman et al., 2013). LRIC depends on node thresholds, critical groups, and multi-step activation, and its measures can be computationally expensive because they may require all pivotal groups, all simple paths, or all node subsets in simulations (Aleskerov et al., 2016). Game centrality depends on payoff rules, update rules, and the chosen social dilemma (Simko et al., 2013).

Taken together, these results situate absolute influence centrality as a model-dependent notion rather than a single graph-invariant quantity. In the narrow and explicit sense, it is the absolute column-sum centrality derived from $i$03 in signed Friedkin–Johnsen dynamics (Shrinate et al., 2024). In the broader methodological sense, it denotes the uncontextualized or pre-normalized component of influence before relative scaling, or a process-specific score that quantifies total network impact under a specified dynamical law (Cerqueti et al., 2019, Simko et al., 2013).