Table of contents for issue 4, volume 6, Journal of Physics: Complexity

Relations, represented by graph edges, are essential for modelling complex systems across various fields, including physics, computer science, and biology. They provide a universal framework for capturing interactions, dependencies, and pathways, with networks encoding these intricate connections. In this context, we review an approach that uses binary relations as the fundamental constituents of the Universe, utilizing them as building blocks for both space and matter. This model, known as combinatorial quantum gravity, is defined by an ultraviolet continuous fixed point of a statistical model on random networks, governed by the combinatorial Ollivier–Ricci curvature, which acts as a network analogue of the Einstein–Hilbert action. The model exhibits two distinct phases separated by this fixed point, a geometric and a random phase, representing space and matter, respectively. At weak coupling and on large scales, the network organizes into a holographic surface whose collective state encodes both an emergent 3D space and the matter distributed in it. The Einstein equations emerge as constitutive relations expressing matter in terms of fundamental network degrees of freedom while dynamics in a comoving frame is governed by relativistic quantum mechanics. Quantum mechanics, however is an effective theory breaking down at the scale of the radius of curvature of the holographic network. On smaller scales, not only relativistic invariance is lost but also the Lorentzian signature of space-time. Finally, the manifold nature of space-time breaks down on the Planck length, where the random character of the fundamental network on the smallest scales becomes apparent. The network model seems to naturally encode several of the large-distance features of cosmology, albeit still at a qualitative level. The holographic property of black holes arises intrinsically from the expander nature of random regular graphs. There is a natural mechanism to resolve the cosmological constant problem and dark matter appears naturally as a metastable allotrope in the network fabric of space-time. In this model, both gravity and quantum mechanics are macroscopic statistical effects reflecting the free energy minimization of fundamental binary degrees of freedom.

Information filtering networks (IFNs) provide a powerful framework for modeling complex systems through globally sparse yet locally dense and interpretable structures that capture multivariate dependencies. This review offers a comprehensive account of IFNs, covering their theoretical foundations, construction methodologies, and diverse applications. Tracing their origins from early network-based models to advanced formulations such as the triangulated maximally filtered graph and the maximally filtered clique forest, the paper highlights how IFNs address key challenges in high-dimensional data-driven modeling. IFNs and their construction methodologies are intrinsically higher-order networks that generate simplicial complexes-structures that are only now becoming popular in the broader literature. Applications span fields including finance, biology, psychology, and artificial intelligence, where IFNs improve interpretability, computational efficiency, and predictive performance. Special attention is given to their role in graphical modeling, where IFNs enable the estimation of sparse inverse covariance matrices with greater accuracy and scalability than traditional approaches like Graphical LASSO. Finally, the review discusses recent developments that integrate IFNs with machine learning and deep learning, underscoring their potential not only to bridge classical network theory with contemporary data-driven paradigms, but also to shape the architectures of deep learning models themselves.

We show that viewing hypernetworks as polyhedral complexes represents a natural and expressive route towards their geometrization, that endows them with natural notions of curvature. We present both intrinsic and extrinsic methods towards equipping networks with such innate curvatures, and we dwell upon the most meaningful ones.

This work introduces formally the concept of statistical asymmetry (SA) of a system as an entropic measure of how much it fails to be fully symmetric under a given group of transformations. It is shown that it is able to provide an alternative classification of one-dimensional elementary cellular automata that closely aligns with known others only by measuring symmetry. The behaviour of SA can also be an useful indicator of complex behaviour on two-dimensional discrete processes by following the dynamics of configurations, which is demonstrated in the case of the Geenberg–Hastings model, Conway’s game of life, and the random evolution of discrete square matrices.

Gaussian random fields (GRFs) provide a fundamental framework for modeling stochastic spatial phenomena with broad applications in physics and related fields. Using the tools of information geometry, we investigate the geometric structure of the statistical manifold associated with a three-parameter isotropic GRF model. By deriving the Fisher information metric and computing its Christoffel symbols, we formulate the geodesic equations governing the dynamics in parameter space. A hybrid computational approach, combining Markov Chain Monte Carlo estimation of statistical quantities with Runge–Kutta integration, enables us to numerically explore these geodesic flows. Our simulations reveal a phenomenon we term geodesic dispersion, where forward and backward geodesic trajectories deviate in regions of strong curvature gradients. While this effect is not claimed as direct evidence of thermodynamic irreversibility, it provides a geometric marker of sensitivity in the statistical manifold that may underlie asymmetric behavior in driven random field dynamics. Additionally, singularities in the Fisher metric are observed to coincide with parameter regimes reminiscent of phase transitions, suggesting that geometric properties can serve as indicators of critical phenomena. Our study establishes two key insights: 1) time asymmetry in GRFs correlates with local manifold curvature variations during forward and backward simulations; 2) this asymmetry, arising from sharply curvature variations, underscores a fundamental insight: curvature in the statistical manifold acts as a geometric source of asymmetry in stochastic dynamics. These findings establish geodesic dispersion as a useful geometric lens for studying complex stochastic systems, and point to future connections with nonequilibrium thermodynamics, entropy production, and phase transition detection.

This study examines systemic risk within the Iranian economic ecosystem by analyzing the interactions between different sectors from January 2013 to October 2022. A new systemic risk stress indicator is developed based on five key market indicators: money, equity, financial intermediaries, foreign exchange (FX), and oil markets, which are constructed using various sub-indices. Granger causality networks are used to explore the relationships between these indicators, and the results are validated by mutual information (MI)-based networks for sub-index selection. The results are further validated by an improved exponentially weighted moving average indicator that utilizes entropy distance. Additionally, a novel systemic risk stress indicator has been developed based on Granger causality findings. This indicator incorporates MI to capture time-dependent causal relationships and to analyze the interdependencies between various economic and financial indicators. The findings reveal significant bidirectional causality between specific indicators, with strong evidence of cross-market spillovers. Notably, oil price shocks and exchange rate fluctuations have substantial effects on economic stability. The analysis highlights key interconnections within the financial markets, including a significant causal link between the money market and the FX market, primarily driven by central bank interventions. The equity market also influences both the FX and oil markets, with capital outflows and sectoral composition playing a role in these dynamics. Additionally, the financial intermediaries market impacts market sentiment, and a causal relationship between FX and oil prices is established. These results suggest that policymakers and investors should consider these interconnected relationships when formulating strategies to mitigate risks in volatile economic environments. The study also calls for further research into the microstructure of sub-indices across different markets to better understand these complex dynamics.

The promising future of the quantum internet hinges on how we scale entanglement distribution on quantum networks (QNs). Classical entanglement percolation (CEP) offers a simple approach for this but is limited by ‘classical’ scaling laws, while deterministic entanglement transmission (DET) leverages the determinacy of quantum protocols to achieve a more effective ‘quantum’ scaling. This advantage is, however, unexplored for Gaussian-state QNs, where the infinite-dimensional, continuous-variable states are often required to be first prepared into qubits or qudits via additional conversions. This overhead raises a question: can DET’s quantum advantage survive in a practical Gaussian-to-qudit conversion process? To answer this, we analyze both DET and CEP across a spectrum of conversion strategies, from best-case theoretical limits to probabilistic, practical projections. We find that DET’s superiority is robust: even a DET scheme burdened by the overhead of practical projective conversions consistently outperforms CEP endowed with theoretically perfect conversion. Our work thus provides a possible roadmap, demonstrating that the quantum-native approach of DET could be the key to unlocking the potential of long-range quantum communication.

Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player’s goal is to select a subset of vertices of minimum cardinality that eventually leads to all vertices of the graph being colored. A variant of this game, called the q-analogue of zero forcing, was later introduced. In this version, the player again seeks to choose the smallest number of vertices that will eventually color the entire graph, while an oracle attempts to force the player to select a larger subset. In this paper, we exploit the structural properties of several graph classes in order to both derive algorithms to compute the exact value of , and to establish bounds and exact values of the parameter for these graph classes. In particular, we present a SAT-based algorithm to compute the q-analogue zero forcing number, offering optimal strategies for both the player and the oracle. Additionally, we propose a polynomial-time algorithm for computing the q-analogue zero forcing number for q = 1 of cactus graphs. Lastly, we prove the exact value of this parameter for several graph classes such as block graphs. Our work extends previous results about trees by Butler et al (2020 Graphs Comb.36 1401–19) and Blanco et al (2024 arXiv:2305.11748).

We address the problem of defining connected components in hypergraphs, which are models for systems with higher-order interactions. For graphs with dyadic interactions, connected components are defined in terms of paths connecting nodes along the graph. However, defining connected components in hypergraphs is a more involved problem, as one needs to consider the higher-order nature of the interactions associated with the hyperedge. Higher-order interactions can be taken into consideration through a logic associated with the hyperedges, two examples being OR-logic and AND-logic; these logical operations can be considered two limiting cases corresponding to non-cooperative and fully cooperative interactions, respectively. In this paper we show how connected components can be defined in hypergraphs with OR- or AND-logic. While OR-logic and AND-logic provide the same connected components for nondirected hypergraphs, for directed hypergraphs the strongly connected component of AND-logic is a subset of the OR-logic strongly connected component. Interestingly, higher-order interactions change the general topological properties of connected components in directed hypergraphs. Notably, while for directed graphs the strongly connected component is the intersection of its in- and out-component, in hypergraphs with AND-logic the intersection of in- and out-component does not equal the strongly connected component. We develop a theory for the fraction of nodes that are part of the largest connected component and through comparison with real-world data we show that degree-cardinality correlations play a significant role.

We present a stochastic imitation-based model of opinion dynamics in which agents balance social conformity with responsiveness to an external signal. The model captures how populations evolve between two binary opinion states, driven by peer influence and noisy external information. Through both memory-less and memory-based implementations, we identify a critical threshold of social sensitivity that separates an ergodic phase—where agents collectively track the external signal—from a non-ergodic phase characterised by persistent consensus and reduced adaptability to external changes. Analytical results and simulations reveal that memory in decision-making smooths the transition and lowers the critical threshold for ergodicity breaking. Extending the model to various network structures confirms the robustness of the observed phase transition. We further discuss empirical methodologies for estimating the critical threshold and show how the model may be applied to real-world domains. Our findings contribute to understanding how social conformity, memory effects and randomness jointly shape collective behaviour, with implications for predicting social tipping points and influencing large-scale social dynamics.

Thermal macroeconomics (TM), an axiomatic approach to macroeconomics based on the mathematical structure of thermodynamics, is presented. Within the domain of exchange economies, TM deduces relations between aggregate properties of an economy, concerning quantities and flows of goods and money, without recourse to microeconomic foundations concerning individual economic agents. Despite the restricted scope of this initial treatment, TM has three important payoffs. 1) It provides a new and solid foundation for aspects of standard macroeconomics: the existence of market prices, the value of money, the meaning of inflation, the symmetry and negative-definiteness of the macro-Slutsky matrix, and the Le Chatelier–Samuelson principle, without relying on implausibly strong rationality assumptions over individual microeconomic agents. 2) It generates new results, including implications for money flow and trade when two or more economies are put in contact, using new concepts such as economic entropy and temperature; these concepts can be given economic interpretations as aggregate utility and the inverse marginal aggregate utility of money, respectively. 3) It holds the prospect for extensions of economic interest, such as to include production and consumption, via connection to non-equilibrium thermodynamics. More broadly, we hope that the economic analogue of entropy (governing the possible transitions between states of economic systems) may prove to be as fruitful for the social sciences as entropy has been in the natural sciences.

The behavior of multivariate dynamical processes is often governed by underlying structural connections that relate the components of the system. For example, brain activity, which is often measured via time series is determined by an underlying structural graph, where nodes represent neurons or brain regions and edges represent cortical connectivity. Existing methods for inferring structural connections from observed dynamics, such as correlation-based or spectral techniques, may fail to fully capture complex relationships in high-dimensional time series in an interpretable way. Here, we propose the use of path signatures–a mathematical framework that encodes geometric and temporal properties of continuous paths–to address this problem. Path signatures provide a reparametrization-invariant characterization of dynamical data and, in particular, can be used to compute the lead matrix, which reveals lead-lag phenomena. We showcase our approach on time series from coupled oscillators in the Kuramoto model defined on a stochastic block model graph, termed the Kuramoto Stochastic Block Model (KSBM). Using mean-field theory and Gaussian approximations, we analytically derive reduced models of KSBM dynamics in different temporal regimes and theoretically characterize the lead matrix in these settings. Leveraging these insights, we propose a novel signature-based community detection algorithm, achieving exact recovery of structural communities from observed time series in multiple KSBM instances. We also explore the performance of our community detection on a stochastic variant of the KSBM as well as on real neuropixels of cortical recordings to demonstrate applicability on real-world data. Our results demonstrate that path signatures provide a novel perspective on analyzing complex neural data and other high-dimensional systems, explicitly exploiting temporal functional relationships to infer underlying structure.

The simulation hypothesis has recently excited renewed interest in the physics and philosophy communities. However, the hypothesis specifically concerns computers that simulate physical universes. So to formally investigate the hypothesis, we need to understand it in terms of computer science (CS) theory. In addition we need a formal way to couple CS theory with physics. Here I couple those fields by using the physical Church–Turing thesis. This allow me to exploit Kleene’s second recursion, to prove that not only is it possible for us to be a simulation being run on a computer, but that we might be in a simulation that is being run on a computer – by us. In such a ‘self-simulation’, there would be two identical instances of us, both equally ‘real’. I then use Rice’s theorem to derive impossibility results concerning simulation and self-simulation; derive implications for (self-)simulation if we are being simulated in a program using fully homomorphic encryption; and briefly investigate the graphical structure of universes simulating other universes which contain computers running their own simulations. I end by describing some of the possible avenues for future research. While motivated in terms of the simulation hypothesis, the results in this paper are direct consequences of the Church–Turing thesis. So they apply far more broadly than the simulation hypothesis.

Effective distance, a concept that redefines spatial connectivity by considering human mobility links rather than physical distance, has proven to be a powerful tool for predicting epidemic spread. Traditional effective distance models are constructed using shortest paths on mobility networks, however, in highly connected urban environments, such shortest paths typically reduce to direct one-hop flows, thereby neglecting the essential contribution of multi-hop mobility to infection risk. This study extends the effective distance framework by incorporating multi-hop flows, providing a comprehensive understanding of how infection risk propagates through urban networks. Using detailed mobility data from the Shanghai Omicron BA.2 outbreak, we demonstrate that our extended effective distance model improves the prediction of infection arrival times compared to original effective distance models at fine spatial scales, highlighting the importance of incorporating multi-hop mobility pathways in urban disease modeling.

In this paper, the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree) are examined. Our analytical method is based on generating functions and is exact for the networks we model. We also propose a Monte Carlo graph generation algorithm to draw random networks from the ensemble of graphs with fixed statistics. The proposed model is used to understand the effect that network microstructure has, through the arrangement of inter-subgraph clustering, on the global connective properties of the network. We find that even in k-regular networks, with fixed joint degree distributions and clustering coefficients, the arrangement of clustering has a non-trivial influence on the percolation properties of the network. We find that subgraph disassortativity increases the percolation threshold, whilst assortativity among subgraphs decreases and broadens the transition. Finally, we use an edge disjoint clique cover to represent empirical networks using our formulation, finding the resultant model offers a significant improvement over edge-based theory.

Intra-urban population mobility is influenced by both geographical distance and opportunity accessibility, while being governed by the gravitational pull of urban centers. This phenomenon becomes particularly pronounced during morning rush hours, when population flows exhibit heightened activity and distinct central orientation, profoundly reflecting the city’s spatial structure and functional organization. Building upon high-precision population mobility data, this study constructs movement vectors to systematically analyze the spatial relationships between population flows and urban centers. Our analysis identifies three distinct patterns of morning peak-hour population mobility: strongly centripetal, bidirectional interactive, and self-organizing types. Building upon this foundation, we further investigated how urban centrality shapes population mobility patterns and spatial functionality. Through integrated spatial clustering analysis and interpretable machine learning modeling, we quantitatively validated the distance-decay effect of central influence while systematically elucidating the functional roles of distinct travel modes in macro-scale centripetal flows.

We leverage information theory to develop a novel method of structural coarse-graining for unfolding complex networks across multiple scales, enabling further investigations into multi-scale phenomena. The approach utilizes slow dynamics of random walks on graphs, in which the short-term behavior manifests as movement between individual nodes whereas the long-term behavior, governed by the slow-timescale features, comprises moving between clusters. Specifically, the slow-timescale features are obtained by embedding the walk trajectory into a hidden space to maximize mutual information between historical observations separated by a given time-span. The coarse-grained structures are obtained through aggregating nodes in this hidden space, and exhibit more macroscopic properties as the time-span increases. Our method provides greater control over the resolution of the resultant network. Our results demonstrate that the proposed approach (a) clearly reveals hierarchical community structures in scientific collaboration networks, and (b) illuminates structural and dynamical self-similarity in the human connectome. In addition, we derive an analytic formula for mutual information between the current and historical state of a random walk on a given complex network, which allows us to demonstrate the superiority of neural estimators in determining the mutual information between discrete time series. As well as providing a flexible tool for structural coarse-graining, our work illustrates the practical advantages of neural network-based estimators of information theoretic quantities for addressing problems in network science.

A significant number of synchronization models can be formulated as a system of matrix equations for square matrices , where g_i are the variables and M_i denotes the the mean-fields to which all variables couple. Examples include the Kuramoto model with the possible inclusion of phase lag angles, higher-order systems with three- or four-body interactions, and higher-dimensional generalizations to the unit sphere. The transformation to the matrix form is accomplished for Riccati systems by means of linear fractional transformations, and for models on the sphere with cubic nonlinearities via the unit vector map. Since M_i is constant at fixed points, the matrix equations reduce to a linear system which is solved exactly in terms of the matrix exponential. Stability of the asymptotic trajectories is determined by the eigenvalues of M_i, and the fixed point is related to the eigenvector corresponding to the largest real eigenvalue. This leads to relatively simple conditions which are satisfied by asymptotically stable fixed points. These properties are demonstrated by example for models on the unit circle such as periodic ring oscillators, and for models on the two-sphere with pairwise or three-body interactions.

Atmospheric flow underpins virtually all meteorological and climatological phenomena, yet extracting meaningful features from its dynamics remains a major scientific challenge due to its high dimensionality, multi-scale behaviour, and inherent nonlinearity. In this study, we investigate the potential of a network-based framework to reveal the relationships between distinct flow structures. Specifically, we apply three techniques, independent of any particular phenomenon or model, to explore patterns of coherence and information transfer, vortical interactions, and Lagrangian coherent structures. We assess their utility using a rotating shallow-water model of the stratospheric polar vortex, which reproduces key aspects of wintertime dynamics, including sudden stratospheric warming split events. Our results support three central claims. First, the transformation of fluid flow data into a network representation preserves essential dynamical information. Second, this representation enables a more accessible and structured analysis of the underlying dynamical structures. Third, multiple types of networks can be constructed from atmospheric flow data, each offering distinct yet complementary insights into the system’s collective behaviour. Together, these findings highlight the potential of network-based approaches as valuable tools in atmospheric research.

Randomly wired neural networks (RWNNs) serve as a valuable testbed for investigating the impact of network topology in deep learning by capturing how different connectivity patterns impact both learning efficiency and model performance. At the same time, they provide a natural framework for exploring edge-centric network measures as tools for pruning and optimization. In this study, we investigate three edge-centric network measures: Forman–Ricci curvature (FRC), Ollivier–Ricci curvature (ORC), and edge betweenness centrality (EBC), to compress RWNNs by selectively retaining important synapses (or edges) while pruning the rest. As a baseline, RWNNs are trained for COVID-19 chest x-ray image classification, aiming to reduce network complexity while preserving performance in terms of accuracy, specificity, and sensitivity. We extend prior work on pruning RWNN using ORC by incorporating two additional edge-centric measures, FRC and EBC, across three network generators: Erdös–Rényi model, Watts–Strogatz model, and Barabási–Albert model. We provide a comparative analysis of the pruning performance of the three measures in terms of compression ratio and theoretical speedup. A central focus of our study is to evaluate whether FRC, which is computationally more efficient than ORC, can achieve comparable pruning effectiveness. Along with performance evaluation, we further investigate the structural properties of the pruned networks through modularity and global efficiency, offering insights into the trade-off between modular segregation and network efficiency in compressed RWNNs. Our results provide initial evidence that FRC-based pruning can effectively simplify RWNNs, offering significant computational advantages while maintaining performance comparable to ORC.