Weighted Signed Network Representations

• Weighted signed network representations are frameworks that encode interactions with magnitude (weight) and polarity (sign), enabling analysis of both direct and indirect network effects.
• They generalize classic centrality measures such as Katz centrality, PageRank, and Hubbell centrality by incorporating directionality and variable edge strengths.
• This approach applies to domains like ecology and sociology, predicting outcomes like species extinction and social marginalization through integrated influence dynamics.

Weighted signed network representations provide a framework for encoding and analyzing the complex interactions in graphs where edges possess both magnitude (weight) and polarity (sign), as well as potentially directionality. These representations underpin methods of propagation, influence measurement, consensus formation, and structural inference in domains such as ecology and sociology where relationships are not merely binary or unsigned but reflect nuanced combinations of positive and negative, strong or weak, and directed or reciprocal ties. The approach described in (Gómez-Ambrosi et al., 15 Jan 2025) builds on and unifies classical tools such as Katz centrality, PageRank, and their variants, generalizing them to fully accommodate the complexity of signed, weighted, and directed graphs. This framework not only enhances descriptive and predictive analysis but also connects to indirect effect measurement, extinction risk in ecology, and influence and power dynamics in social systems.

1. Network Model and Net Effects Framework

Let $A$ denote the $n \times n$ interaction matrix where $a_{i \leftarrow j}$ encodes the direct effect (possibly signed, weighted, and directed) of node $j$ on node $i$. Rather than only considering immediate (one-step) influences, the framework expresses the net effect as the sum over all walks of all lengths from $j$ to $i$, with each walk’s effect given by the product of edge-weights (including sign and strength). Specifically, the length-$\ell$ walk contribution is $[A^\ell]_{i \leftarrow j}$, and the overall net effect is the convergent Neumann series: $$\mathrm{net\_effect}_{i \leftarrow j} = \sum_{\ell = 1}^\infty [A^\ell]_{i \leftarrow j}$$ The convergence requires that the spectral radius $\rho(A)$ satisfies $\rho(A) < 1$, ensuring that long indirect interactions do not diverge.

This summation of direct and all indirect effects generalizes simple adjacency to a holistic view of influence propagation and is analytically tractable via the summed power series: $$\mathcal{K}(A,\alpha) = -I_n + (I_n - \alpha A)^{-1} = \sum_{\ell=1}^\infty \alpha^\ell A^\ell$$ where $\alpha$ is a global attenuation parameter controlling the effective walk length.

2. Generalized Katz Centrality and Indirect Measures

Katz centrality was originally defined for unsigned graphs. Here, the generalized Katz matrix $\mathcal{K}(A,\alpha)$ and its derived vectors encode the net effect of the whole network on a node (in-Katz) and the reverse (out-Katz):

• In-Katz: $$\mathcal{K}_\text{in}(A,\alpha) = (-I_n + (I_n - \alpha A)^{-1}) \mathbf{1}$$
• Out-Katz: $$\mathcal{K}_\text{out}(A,\alpha) = \mathbf{1}^T (-I_n + (I_n - \alpha A)^{-1})$$

The Katz definition naturally incorporates both route multiplicity and route length, while the sign and weight structure encode antagonism, cooperation, and strength-of-ties. This formulation seamlessly generalizes to signed, weighted, and directed scenarios, avoiding the loss of information common to approaches that ignore sign or threshold negative links.

3. Taxonomy and Unified View of Measures

The framework reveals that classic centrality and ranking methods emerge as special cases or rescalings within this general model:

• PageRank: By rescaling columns of $A$ via a diagonal matrix $D$ (with $d_i$ representing e.g., out-degree or user-defined preferences), one obtains the PageRank matrix:

$$\mathrm{PR}_\text{in}(A, \alpha, d_1,\dots,d_n) = (I_n - \alpha AD^{-1})^{-1}\mathbf{1}$$

Reverse PageRank is similarly generalized by rescaling rows.

• Hubbell Centrality: Obtained by setting $\alpha$ to 1 and rescaling appropriately.

Thus, approaches such as left and right eigenvector centrality, and recent path-sum–derived scores, are unified within the generalized Katz/Neumann framework and extended without algebraic difficulty to the fully signed, weighted, and directed context.

4. Ecological Network Application

In ecological systems, species interactions are commonly modeled by generalized Lotka–Volterra (GLV) dynamics: $$\dot{x}_i = x_i \left( r_i + \sum_j a_{i \leftarrow j} x_j \right)$$ where $a_{i \leftarrow j}$ expresses the direct signed/weighted effect of $j$ on $i$, and self-interactions $a_{ii}$ typically reflect intra-species competition or self-limitation. The framework applies a normalization that "zeros the diagonal" and rescales by self-effects, replacing $A$ by $B = AD^{-1}$ with $D = \mathrm{diag}(-a_{11}, ..., -a_{nn})$ to ensure the off-diagonal structure reflects relative effects.

Empirically, using the normalized Katz in-vector $\mathcal{K}_\text{in}(B^\dagger,\alpha)$ with $\alpha=1/n$ reveals that species with strongly negative net effects tend to go extinct under GLV or replicator dynamics simulations, whereas those with high positive net effects tend to persist. This association reflects the cumulative negative or positive feedback received from all direct and indirect interactions.

5. Social Network Application

The approach extends to social networks with weighted, signed, and directed ties—for instance, Sampson’s monastery paper with "like" and "dislike" relations. Constructing the signed adjacency matrix $A$ from the data and computing the in-Katz vector $\mathcal{K}_\text{in}(A, \alpha)$ ranks individuals by their overall net reception (total effect from network to node), while $\mathcal{K}_\text{out}(A, \alpha)$ ranks individuals by their capacity to affect the network.

The analysis demonstrates that nodes (individuals) with highly negative net effects tend to become marginalized or "expelled" (e.g., Basil, Elias, Simplicius), consistent with negative feedback mechanisms observable in reputation, opinion, or affiliation dynamics. In replicator dynamics, these nodes' influences decline and vanish, while persistently positive nodes remain central.

6. Comparative Taxonomy and Implications

By subsuming PageRank, Hubbell, and their reversed forms through algebraic rescaling, the framework clarifies the relationships among myriad influence and centrality measures, demonstrating that once sign, weight, and possibly direction are incorporated consistently, classical and modern metrics fall under the same umbrella. This yields two main benefits:

• Clearer theoretical connections between influence, power, extinction risk, and other outcomes
• A flexible analytic toolkit for network-level intervention, monitoring, or prediction, since both direct and indirect, positive and negative, strong and weak, reciprocal and asymmetric effects are naturally encoded

Key aspects—such as the convergence condition $\rho(\alpha A) < 1$, the path-based semantics for indirect effects, and the retention of sign and weight structure—make the approach applicable beyond ecology and sociology, e.g., to economic, epidemiological, or engineered networks.

7. Broader Implications and Future Directions

The generalization enables the consistent treatment of indirect and net effects in networks that are signed, weighted, and directed—without need for arbitrary thresholding or sign-ablation that can destroy interpretability or predictive value.

Important implications include:

• In ecology, negative net effects, as measured by the in-Katz vector, structurally forecast extinction risk in multi-species assemblies with complex interaction topologies.
• In social systems, the net effect ranking aligns with power, marginalization, or influence propagation, exposing vulnerabilities or concentrations of dynamical importance.
• Classical descriptors (e.g., centralities, rankings) in non-signed or unweighted settings are recovered as special instances, confirming robustness of the mathematical underpinnings.
• The framework also provides a roadmap for future studies incorporating temporal networks, adaptive or nonlinear edge-weight dynamics, and unification with motif-based network analysis.

Potential avenues for continued research include empirical validation on large-scale real-world networks exhibiting nontrivial structure, investigation of direct vs. indirect effect separation under various dynamical rules, and exploration of possible optimization principles governing network evolution.

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Key Mathematical Structures Table

Measure	Formula	Context
Net Effect (Katz)	$$\mathcal{K}(A,\alpha) = -I_n + (I_n - \alpha A)^{-1}$$	Signed, weighted, directed graphs
In-Katz/Out-Katz	In: $\mathcal{K}_\text{in} = \mathcal{K} \cdot 1$	Net effect received by a node
	Out: $\mathcal{K}_\text{out} = 1^\top \cdot \mathcal{K}$	Net effect exerted by node
PageRank (rescaled Katz)	$$\mathrm{PR}_{\text{in}}(A, \alpha, D) = (I_n - \alpha AD^{-1})^{-1}1$$	Random-walk-based ranking

This comprehensive framework demonstrates how weighted signed network representations, grounded in generalizations of Katz centrality and related walk-based summations, preserve the richness of edge polarity, magnitude, and direction, and can be directly applied to questions of influence, consensus, extinction, or structural vulnerability in complex adaptive systems (Gómez-Ambrosi et al., 15 Jan 2025).