Redundancy–Synergy Index (RSI) Overview

• RSI is defined as a measure quantifying the balance between redundancy—overlapping shared information—and synergy—extra joint information emerging only from combined sources.
• Different formulations of RSI, including directed, Gaussian, and network-based approaches, vary in sign conventions and underlying redundancy definitions.
• RSI is applied across fields like neuroscience, multiplex networks, and explainable machine learning to decompose mutual information into actionable components.

Searching arXiv for recent and foundational papers on redundancy–synergy indices and related decompositions. Searching arXiv for multiplex-network redundancy/synergy decomposition and directed/undirected RSI links. Redundancy–Synergy Index (RSI) denotes a family of measures that summarize whether multivariate dependence is dominated by information shared across sources or by information that emerges only from their joint state. Across the cited literature, RSI is not a single standardized formula. In directed information-theoretic form it can be defined as a source–target balance, $$\mathrm{RSI}(\mathbf X;Y)=\sum_j I(X_j;Y)-I(\mathbf X;Y)$$, so that positive values indicate redundancy and negative values indicate synergy (Rosas et al., 2024). In other work, the same balance is written with the opposite sign as net synergy or Whole-Minus-Sum, $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$, so that positive values indicate synergy and negative values indicate redundancy (Barrett, 2014). Closely related constructions also arise in multiplex networks, Granger-causal systems, explainability methods, inequality decompositions, and generalized entropy formalisms, where redundancy and synergy are first computed as separate components and scalar contrasts are then derived (Luppi et al., 2023).

1. Conceptual scope and sign conventions

RSI is best understood as a coarse-grained statistic of the redundancy–synergy balance. “Redundancy” denotes information about a target that is shared by multiple sources, or, in network settings, performance that can be realized equivalently by more than one layer. “Synergy” denotes information or utility available only from a joint configuration, whether that joint configuration consists of multiple predictors, multiple network layers, or multiple added features. What varies across frameworks is not the underlying contrast, but the primitive objects being decomposed: Shannon mutual information, prediction-error reduction, shortest-path efficiency, generalized entropy, or SHAP attribution vectors.

Setting	Representative expression	Sign convention
Directed information theory	$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$ (Rosas et al., 2024)	$>0$ redundancy, $<0$ synergy
Gaussian PID / WMS	$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$ (Barrett, 2014)	$>0$ synergy, $<0$ redundancy
Multiplex networks	$p_{\mathrm{syn}}-p_{\mathrm{red}}$ or $\phi_{\mathrm{syn}}-\phi_{\mathrm{red}}$ from PND (Luppi et al., 2023)	Depends on chosen contrast
Unnormalized Granger causality	$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$0 (Stramaglia et al., 2015)	$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$1 redundancy, $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$2 synergy
Random sequential additions	$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$3 (Shamail et al., 12 Dec 2025)	$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$4 redundancy, $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$5 synergy

A recurrent source of confusion is therefore sign. In some traditions RSI means “redundancy minus synergy”; in others it means “synergy minus redundancy”. Another recurrent source of confusion is ontological status: some papers define RSI explicitly, whereas others expose redundancy, uniqueness, and synergy atoms from which RSI-like scalars can be formed. The multiplex-network framework of partial network decomposition, for example, does not use the name “Redundancy–Synergy Index” explicitly, but its path-count and efficiency-fraction summaries directly support such indices (Luppi et al., 2023).

2. PID foundations and the problem of defining redundancy

The modern RSI literature is rooted in Partial Information Decomposition (PID). For two sources $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$6 and a target $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$7, PID decomposes the joint mutual information into redundant, unique, and synergistic parts: $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$8 with

$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$9

These equations leave redundancy and synergy underdetermined unless an additional redundancy functional is supplied (Barrett, 2014).

This underdetermination is structural, not merely notational. In information gain lattices, redundancy components are invariant across decompositions, but unique and synergistic components are decomposition-dependent. Information loss lattices reverse those invariance properties, and dual gain–loss decompositions were introduced to recover a consistent joint characterization of redundancy and synergy from common incremental terms (Chicharro et al., 2016). A direct implication is that any RSI inherits the assumptions of the redundancy formalism beneath it; it is not independent of lattice choice, invariance criteria, or atomic semantics.

One response is to define redundancy pointwise. The common-change-in-surprisal approach measures redundancy as local overlap in the change of surprisal about the target and then counts only those local co-information terms that admit an unambiguous interpretation as redundant information (Ince, 2016). Another response is operational: the fault-tolerance redundancy $$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$0 defines redundancy as the minimum mutual information that remains under source-fallible instantiations satisfying an antichain-level failure constraint. In that framework, redundant information is the information robust to individual source failures, and the resulting redundancy function satisfies symmetry, self-redundancy, and monotonicity on the Williams–Beer lattice (Milzman, 2024).

These constructions matter for RSI because they determine what the “R” term actually means. If redundancy is defined as minimum single-source information, common local change in surprisal, or worst-case fault-tolerant survivability, the resulting redundancy–synergy balance can differ sharply even when the same joint distribution is analyzed.

3. Gaussian, dynamical, and causal formulations

For jointly Gaussian systems with a univariate target, the Minimum Mutual Information PID yields an especially explicit RSI calculus. Under the condition that redundant and unique information depend only on the pairwise marginals $$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$1 and $$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$2, redundancy reduces to

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$3

the weaker source has zero unique information, and synergy becomes the extra information contributed by the weaker source when the stronger source is already known (Barrett, 2014). In this setting the Whole-Minus-Sum quantity

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$4

is computable directly from Shannon mutual informations and acts as a natural RSI with positive values for net synergy and negative values for net redundancy (Barrett, 2014).

This Gaussian formulation yields several nontrivial consequences. Redundancy is independent of source–source correlation in the MMI setting because it depends only on the smaller of the two target–source mutual informations. Synergy, by contrast, depends nontrivially on source correlation: it can increase or decrease with correlation, can remain positive even for uncorrelated sources, and can be positive even when one source is uncorrelated with the target if that source is correlated with the other source (Barrett, 2014). These facts are often counterintuitive, but they are direct consequences of the Gaussian PID geometry.

In time-series analysis, the same distinction reappears in causal language. For Granger-causal inference, fully conditioned GC is not affected by synergy, whereas pairwise GC fails to reveal synergetic effects; conversely, fully conditioned approaches do not work well in the presence of redundancy, and partially conditioned GC can be effective if the conditioning set is selected appropriately (Stramaglia et al., 2014). A complementary approach replaces the usual log-ratio GC with unnormalized GC,

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$5

and then defines a pairwise synergy index

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$6

This index is exactly zero for independent additive sources, positive for redundancy, and negative for synergy (Stramaglia et al., 2015). In biochemical network motifs, the closely related net-synergy quantity

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$7

was used to show that shared upstream regulators induce redundancy whereas independent jointly regulating inputs induce synergy, and that redundancy-dominated regimes can coincide with higher signal-to-noise ratio (Biswas et al., 2017).

A later Gaussian treatment unified TC, DTC, O-information, TSE complexity, and RSI in covariance form. For Gaussian data,

$$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$8

with closed forms in terms of log determinants of standardized covariance matrices; the same expressions extend to elliptical distributions (Pascual-Marqui et al., 11 Jul 2025). This makes RSI particularly tractable in multivariate Gaussian and cross-spectral settings.

4. Multiplex networks and path-based RSI

In multiplex networks, RSI emerges from a decomposition of network utility rather than from mutual information. The key construction is Partial Network Decomposition (PND), introduced for two layers $$\mathrm{RSI}(\mathbf X;Y)=\sum_{j=1}^n I(X_j;Y)-I(\mathbf X;Y)$$9 and $>0$0 on the same node set. For a node pair $>0$1, pairwise utility is taken as efficiency,

$>0$2

and the global utility is its average across unordered node pairs. PND defines pointwise redundant, unique, and synergistic contributions and proves the decomposition identity

$>0$3

with all four terms non-negative (Luppi et al., 2023).

For shortest paths in a two-layer multiplex $>0$4, the most efficient path between nodes $>0$5 is classified directly from shortest-path lengths $>0$6. A pair is synergistic if $>0$7, unique if $>0$8 but $>0$9, and redundant if $<0$0 (Luppi et al., 2023). Counting such node pairs yields global proportions $<0$1. Alternatively, the efficiency atoms can be normalized to fractional contributions $<0$2. Either route supports RSI-like contrasts such as $<0$3 or $<0$4 (Luppi et al., 2023).

This framework is operational rather than purely information-theoretic. Redundancy means both layers provide equally efficient paths and therefore backup routes; synergy means the optimal route uses edges from both layers and exists only in the combined topology. In random graphs, sparse layer pairs are synergy-dominated, whereas denser pairs become redundancy-dominated. In a lattice plus rewired lattice, redundancy falls rapidly under rewiring, unique contribution of the rewired layer peaks around $<0$5 rewiring, and synergy continues to grow as small-world propensity rises (Luppi et al., 2023).

Empirically, the same decomposition identifies distinct design regimes. In the London transport system, redundancy is low overall, synergy peaks at intermediate path lengths, and degree sequence strongly affects the balance relative to nulls. In the human structural connectome, splitting edges into equal-density long- and short-range subnetworks yields a strong skew toward unique long-range contribution plus synergy, with less redundancy than degree-preserving nulls. Human functional connectomes show dominant synergy between short- and long-range functional edges. Across 220 animals spanning 125 mammalian species, structural connectomes again show more synergy and more unique long-range contribution, and less redundancy, than degree-preserving random nulls (Luppi et al., 2023).

5. Directed and undirected high-order interdependence

A major formal development links directed RSI to undirected measures of high-order dependence. For predictors $<0$6 and target $<0$7, directed RSI is defined as

$<0$8

which is positive when individual source informations overcount overlapping content and negative when the joint state contains extra synergistic content (Rosas et al., 2024). The same quantity equals

$<0$9

so RSI can be read as the amount of dependence among predictors explained, or created, by conditioning on the target (Rosas et al., 2024).

The undirected counterpart is O-information,

$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$0

which is positive for redundancy-dominated systems and negative for synergy-dominated systems (Rosas et al., 2024). The two quantities are linked by exact identities: $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$1 For three variables, directed RSI, O-information, and interaction information coincide: $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$2 For larger systems, O-information can be decomposed into a sum of conditional RSIs over different partitions, so directed and undirected high-order balances are tightly related but not identical (Rosas et al., 2024).

This relation also admits statistical and geometric interpretations. RSI is the asymptotic per-sample generalized log-likelihood ratio between a tail-to-tail class $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$3, representing redundant source structure mediated by the target, and a head-to-head class $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$4, representing synergistic source structure that becomes coordinated only through the target. Equivalently, RSI is the difference between KL projections of the empirical distribution onto those two model families (Rosas et al., 2024).

A later covariance-based treatment extends this picture to structured groups of variables. For Gaussian data,

$I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$5

for complex data, with a factor $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$6 in the real case, where $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$7 is the standardized covariance and $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$8 the standardized inverse covariance. The same paper defines structured between-group versions $I(X;Y,Z)-I(X;Y)-I(X;Z)=S-R$9, $>0$0, and $>0$1, as well as a structured group localizer

$>0$2

which measures whether group $>0$3 contributes to between-group redundancy or synergy independently of within-group interactions (Pascual-Marqui et al., 11 Jul 2025). This is important because global $>0$4 can miss between-group synergy when within-group redundancy dominates; the structured versions recover that hidden balance (Pascual-Marqui et al., 11 Jul 2025).

6. Extensions beyond classical mutual information

RSI-type decompositions have been transplanted into several formally distinct settings. In inequality theory, f-inequalities derived from f-divergences admit a union-lattice Möbius decomposition into redundant, unique, and synergetic contributions of attributes to total inequality. For two attributes, the decomposition yields a redundant part, two unique parts, and a synergistic part, and game synergy collapses these to “synergy minus redundancy” (Mages et al., 2024). This establishes an explicit analogy between decomposing inequality and decomposing information.

In explainable machine learning, SHAP vector decomposition defines, for each ordered feature pair $>0$5, a synergy vector $>0$6, redundancy vector $>0$7, and independence vector $>0$8 from projections of SHAP and SHAP-interaction vectors. Their normalized squared norms produce

$>0$9

with all three values in $<0$0. These are already RSI-like feature-pair indices, though expressed in attribution geometry rather than Shannon information (Ittner et al., 2021).

In nonlinear statistical physics, generalized two-parameter entropy $<0$1 yields a polyadic synergy measure

$<0$2

defined even when macrovariables are statistically independent but microphysical codependence is encoded through non-extensive entropy parameters $<0$3. Here $<0$4 indicates synergy, $<0$5 indicates redundancy, and $<0$6 itself functions as a signed RSI (Perdigão, 2017).

A different operational extension comes from source-failure semantics. The fault-tolerant redundancy $<0$7 is defined as the minimum mutual information over all source-fallible instantiations that redundantly satisfy an antichain $<0$8. Through an order-reversing correspondence between the PID lattice and collections of source-fallible systems, this redundancy function satisfies the Williams–Beer axioms and yields PID atoms interpretable as information robust, or not robust, to source failures (Milzman, 2024). In that setting, an RSI built from $<0$9 measures the balance between fault-tolerant shared information and failure-fragile synergistic information.

7. Empirical domains, misconceptions, and open issues

Neuroscience has been a major testing ground for RSI-like constructions. In movie-driven fMRI from 13 humans and 8 marmosets, redundancy and synergy were estimated using $p_{\mathrm{syn}}-p_{\mathrm{red}}$0ID rather than a single scalar RSI. The study reported stable high-order redundancy hubs, a synergy-driven shift from low- to high-level visual regions as interaction order increased, and strong cross-species synergy between human peri-entorhinal and entorhinal cortex and marmoset occipitotemporal regions. It also reported that redundancy and synergy were negatively correlated ($p_{\mathrm{syn}}-p_{\mathrm{red}}$1), while correlation and redundancy were strongly aligned ($p_{\mathrm{syn}}-p_{\mathrm{red}}$2) (Li et al., 19 Mar 2025). This is a useful reminder that many empirical studies now work with RSI components rather than with one signed scalar.

Several methodological cautions recur across the literature. First, RSI is not standardized: sign conventions differ, and in some fields the term is only implicit (Barrett, 2014). Second, the redundancy functional is decisive. Gain-lattice PID makes redundancy invariant and synergy decomposition-dependent, whereas loss-lattice PID reverses those roles; dual decompositions were proposed precisely because joint characterization of redundancy and synergy is otherwise asymmetric (Chicharro et al., 2016). Third, pairwise summaries can obscure higher-order structure. Pairwise Granger causality misses synergy unless conditioning is introduced, but fully conditioned GC can suppress redundant influences (Stramaglia et al., 2014). By contrast, the L-score from random sequential additions is computed from pairwise measurements yet can reveal higher-order interactions through consistent cross-pair L-shaped patterns, with $p_{\mathrm{syn}}-p_{\mathrm{red}}$3 denoting perfect redundancy, $p_{\mathrm{syn}}-p_{\mathrm{red}}$4 perfect synergy, and $p_{\mathrm{syn}}-p_{\mathrm{red}}$5 independence (Shamail et al., 12 Dec 2025).

A further misconception is to equate global high-order balance with every meaningful notion of redundancy or synergy in the system. The structured Gaussian results show that global O-information can report redundancy domination even when the between-group organization is synergistic; structured $p_{\mathrm{syn}}-p_{\mathrm{red}}$6 and structured oRSI were introduced specifically to disentangle those cases (Pascual-Marqui et al., 11 Jul 2025). Likewise, in multiplex networks, path-count proportions and efficiency-based fractions are both valid RSI-like summaries, but they answer different questions: one counts how many node pairs fall into each regime, the other measures how much global efficiency is carried by each regime (Luppi et al., 2023).

Taken together, these results suggest a stable core meaning for RSI and an unstable perimeter. The stable core is the comparison between overlapping/shared contribution and irreducibly joint contribution. The unstable perimeter concerns sign orientation, atomic semantics, the mathematical object being decomposed, and whether the goal is global balance, local attribution, fault-tolerant robustness, or between-group structure.