Signed Network Models Enable Dimensionality Reduction for Higher-Moment Portfolio Optimization

In computational finance, the mathematical selection of investment portfolios has long struggled to reconcile empirical return statistics with the computational complexity of multi-asset optimization. Classical portfolio theory, introduced by Harry Markowitz in 1952, provides the foundation for balancing expected return against portfolio variance, yet its core formulation assumes Gaussian return distributions. In real financial markets, empirical returns routinely display significant asymmetry and heavy-tailed extremes, necessitating the inclusion of higher-order central moments such as skewness and kurtosis. Incorporating these higher moments transforms portfolio selection into a multi-objective, non-convex problem that is generally NP-hard, creating acute statistical and computational bottlenecks when evaluating broad asset universes. To address this persistent barrier, applied mathematician Bibhas Adhikari has formulated a structural framework that harnesses time-evolving complete signed networks to achieve systematic dimensionality reduction, translating moment-based portfolio objectives into combinatorial graph structures without relying on conventional correlation matrices.

The classical Markowitz framework models an investment universe of N assets over a time window T through a mean return vector and an unobserved covariance matrix estimated from empirical price series. Under standard no-short-selling constraints, an investor selects a non-negative weight vector summing to unity that minimizes portfolio variance while attaining a target return, or optimizes a trade-off governed by a risk-aversion parameter. When extended to incorporate higher-order dynamics, the model seeks to maximize expected return and skewness—the third moment capturing return asymmetry—while minimizing variance and kurtosis, the fourth moment capturing heavy tails and extreme risk. However, expanding portfolio skewness and kurtosis requires computing multidimensional expectation tensors across all asset triplets and quadruplets. This exponential explosion in parameters not only introduces severe non-convexity into the objective function but also magnifies estimation noise, rendering exact high-dimensional optimization computationally intractable without prior filtering of the underlying asset universe.

Traditional network-based methods in computational finance frequently model asset interdependencies by converting pairwise correlation or covariance matrices into weighted graphs, treating assets as vertices and correlation coefficients as edges. Despite their widespread use in asset-price forecasting and risk analysis, empirical correlation matrices calculated over finite time windows remain heavily corrupted by statistical noise. Prior literature demonstrates that this noise effect depends critically on the ratio between the number of assets N and the length of the time series T, as confirmed by comparing empirical eigenvalue distributions against random matrix predictions. To mitigate spurious correlations, researchers historically relied on thresholding to prune weak edges or filtering methods that eliminate noisy eigenvalues. More recently, machine learning approaches have emerged to learn robust correlation representations directly. However, standard covariance calculations compress complex day-to-day interactions into single aggregate scalars, inadvertently obscuring fine-grained temporal dynamics and directional co-movements that occur within localized market trading windows.

To capture localized market dynamics more effectively, the newly proposed methodology departs from conventional correlation strengths by constructing a time-indexed sequence of complete signed graphs for each trading day within a training window. For every day, the model calculates the logarithmic return of each asset relative to its historical mean return across the observational period. An edge connecting any pair of assets is assigned a positive sign if the product of their respective mean-deviations is non-negative, indicating that both assets moved in the same directional alignment relative to their individual baselines. Conversely, a negative edge is assigned when their mean-deviations exhibit opposite signs, signifying divergent behavior. Within this weighted signed graph framework, Theorem 3.2 proves that negative edges act as intrinsic portfolio hedges. Specifically, for any long-only portfolio weight vector, the portfolio variance bounded by the actual covariance matrix is strictly less than or equal to the variance computed using absolute covariance values, confirming that negative co-movements contain risk.

A central mathematical feature of this time-series network formulation is established in Theorem 3.4, which proves that the daily complete signed graph is structurally balanced on every trading day. Grounded in structural balance theory, a complete signed graph is balanced if and only if every induced triangle is balanced. Triangles are categorized by their negative edge counts: configurations with zero or two negative edges are balanced, whereas configurations with one or three negative edges are unbalanced. By analyzing the algebraic signs of relative return deviations across any three assets, the theorem demonstrates by contradiction that neither single-negative nor triple-negative triangles can mathematically occur. Leveraging this structural property, the study defines the hedge score of an asset as its cumulative negative degree across all trading days, normalized by the total number of time steps and the remaining market assets. This normalized score ranges between zero and one, quantifying an asset’s persistent tendency to offset market-wide deviations.

Building upon this structural foundation, the framework establishes a rigorous combinatorial interpretation of higher-order return moments, directly connecting portfolio skewness and kurtosis to the density of specific network motifs within the signed graph. When expanding the portfolio skewness formula across all asset triplets, the third-order interaction term for three distinct assets becomes strictly positive precisely when their induced triangle is balanced. Because every daily signed network in the model is guaranteed to be balanced, the structural properties of the graph naturally favor higher skewness without requiring specialized computational constraints. Conversely, expanding the fourth-order kurtosis expression reveals that kurtosis reduction is driven by negative edges and quadruplets of assets whose induced four-vertex subgraphs match a distinct balanced clique configuration designated as K_4^{B_2}. In this specific motif, one vertex maintains negative edges with the remaining three assets, which are mutually connected by positive edges, thereby acting as a combinatorial engine for suppressing heavy-tailed portfolio risk.

While identifying asset combinations rich in K_4^{B_2} motifs provides an interpretable mechanism for kurtosis minimization, the research demonstrates that exploiting this relationship algorithmically introduces severe theoretical complexity. Theorem 4.3 proves that selecting a subset of K assets from a complete signed network of N vertices to maximize the number of induced K_4^{B_2} configurations is formally NP-hard. The proof establishes a polynomial-time reduction from the classical CLIQUE problem, showing that embedding an undirected graph into an augmented complete signed network makes determining the maximum K_4^{B_2} density equivalent to finding a maximum clique. In contrast to the computational hardness of subset selection, Lemma 4.4 proves that verifying and counting the number of K_4^{B_2} motifs within an already identified subset of size K can be executed deterministically in polynomial time of order O(K^4). This sharp divergence underscores why finding the globally optimal motif-dense asset universe requires structured heuristic approximations rather than exhaustive search.

To overcome the computational intractability of exact motif optimization, the study proposes a practical two-stage dimensionality reduction workflow that operationalizes the Markowitz mean-variance principle through graph theory. In the primary stage, designated as OPT1, the full asset universe is filtered down by ranking each asset according to its hedge-adjusted score, defined as the product of its normalized hedge score and its empirical mean return over the training window. This linear formulation can be computed with an overall complexity of O(TN), efficiently extracting assets that exhibit robust expected returns alongside strong market-hedging characteristics. To determine the target portfolio size K, the methodology incorporates quantitative stopping rules, including an elbow criterion based on marginal score drops and percentile-based retention thresholds. In the secondary stage, designated as OPT2, the pre-filtered candidate universe is further refined by evaluating K_4^{B_2} motif densities across the daily signed networks, potentially utilizing quantum-inspired algorithms or noisy intermediate-scale quantum devices to navigate the combinatorial barrier.

The empirical efficacy of the hedge-score dimensionality reduction strategy was evaluated through rolling out-of-sample backtesting on historical market data from August 2006 to December 2021. The experimental dataset comprised 199 equities from the S&P 500 index whose historical records aligned with Google stock data, obtained from Yahoo Finance and hosted on Kaggle. The backtesting protocol implemented a strict non-overlapping annual walk-forward procedure: candidate portfolios were selected using daily data from calendar year y and evaluated on out-of-sample log returns across calendar year y+1, spanning fifteen consecutive annual evaluation cycles from 2006 through 2020. The researchers compared four distinct portfolio implementations across reduced universe sizes of 20, 30, 40, and 50 assets against full-universe benchmarks. These implementations included Sharpe ratio maximization using sequential least squares programming under long-only constraints and uniform equally weighted allocations, alongside competing baseline screening methods including pure mean-return ranking, individual Sharpe ranking, and thirty independent trials of uniform random asset selection.

Across the out-of-sample test periods, numerical evaluations demonstrated that portfolios constructed from hedge-score-reduced universes consistently achieved higher annual returns than the full 199-asset universe in fourteen out of the fifteen backtested years, with 2009 representing the sole exception. However, this enhanced profitability was accompanied by elevated annual volatility, reflecting a structural trade-off between concentrated hedging exposure and broad-market diversification. In terms of overall risk-adjusted performance, Sharpe ratios showed mixed results, with equally weighted reduced portfolios frequently matching or exceeding alternative strategies while avoiding the extreme performance dispersion characteristic of random selection. Because full second-stage motif optimization remains classically intractable, the authors highlight hybrid quantum-classical algorithms, such as quantum annealing and the quantum approximate optimization algorithm implemented on near-term hardware, as vital avenues for evaluating higher-moment signed cliques at scale. By bridging structural balance theory, combinatorial optimization, and empirical asset pricing, this research provides a rigorous graph-theoretic foundation for modern high-dimensional portfolio selection.

Subject of Research: Signed network models and motif-based dimensionality reduction in financial portfolio optimization

Article Title: Signed network models for dimensionality reduction of portfolio optimization

Article References: Adhikari, B. (2026). Signed network models for dimensionality reduction of portfolio optimization. Applied Network Science, 11(1), Article 52. https://doi.org/10.1007/s41109-026-00808-0

Image Credits: AI Generated

DOI: 10.1007/s41109-026-00808-0

Keywords: portfolio optimization, signed networks, dimensionality reduction, Markowitz theory, higher-order moments, computational finance, structural balance theory, graph motifs, Signed, network, models, dimensionality