Session F28: Statistical Physics of Networks: Theory and Applications to Complex Systems I

Ginestra BianconiLearning the topology of complex systems from their dynamics

From the brain to the climate, complex systems constitute a real challenge for scientists and mathematicians as they are giving rise to dynamical phenomena notoriously difficult to understand, model and predict.

In the last twenty years the scientific community has made unprecedented progress in unveiling the structure of complex systems encoded in their network skeleton describing the set of their pairwise interactions. However networks are not able to characterize the ubiquitous higher-order interactions between  more than two nodes that give rise  to the complex systems topology captured by higher-order networks and simplicial complexes.

Here we reveal how non-linear dynamical processes can be used to learn the topology, by defining Topological Kuramoto model and Topological global synchronization. These critical phenomena  capture the synchronization of topological signals, i.e. dynamical signal defined not only on nodes but also on links, triangles and higher-dimensional simplices in simplicial complexes. Moreover, will discuss how the Dirac operator can be used to couple and process topological signal of different dimensions. Finally we will reveal how non-linear dynamics can shape topology by formulating triadic percolation. In triadic percolation triadic interactions can turn percolation into a fully-fledged dynamical process in which nodes can turn on and off intermittently in a periodic fashion or even chaotically leading to period doubling and a route to chaos of the percolation order parameter.  Triadic percolation changes drastically our understanding of percolation and can describe real systems in which the giant component varies significantly in time such as in brain functional networks and in climate.   

Uncovering the microscopic mechanism of the devil's staircase in the Kuramoto model with inertia

We originally propose a phase diagram of the synchronization transitions of the second order Kuramoto oscillator system. Emergence of the secondary clusters and the devil's staircase is highlighted and focused. The staircase consists of one giant cluster, a pair of secondary frequency clusters, and higher order clusters. We uncover the microscopic mechanism underlying the formation of the devil's staircase; applying a modified Melnikov's method to the secondary sync clusters, we set up a couple of self-consistent relations to determine its step widths and step heights of the secondary sync clusters. Furthermore, higher order steps are derived from the n-to-m mode locking to the secondary cluster pulsing around the giant cluster at a nontrivial angular speed. As a consequence, each stair step is characterized by a collection of oscillators locked into the rational multiple of the angular speed of the secondary cluster in the long time average. This microscopic results may provide useful insights in stabilizing power grids.   

Shortest-path percolation on complex networks

In various infrastructural systems like power grids and airline networks, path-based interactions serve a crucial role in sustaining the flow dynamics in the system. Thus, well-designed infrastructures should be robust not only under failures/attacks upon single components but also under path-based perturbations. To tackle this problem, we propose a percolation model based on a protocol which removes the edges that constitute the shortest path of a random pair of nodes on networks: the shortest-path percolation model. In this talk, we present our results obtained from extensive Monte Carlo simulations on synthetic networks and discuss the results. In addition, we discuss how the shortest-path percolation model can be applied to design robust infrastructure networks under path-based attacks.   

Extra annealing dimensions for improved performance in physical optimizers

Emerging hardware platforms for combinatorial optimization, like the coherent Ising machine (CIM), have established a new class of network dynamics defined by the controlled annealing of an energy landscape from a trivial convex shape to a rugged shape encoding the problem of interest. Their classical and quantum physics have been studied with tools from nonlinear dynamics and disordered systems theory to understand the behavior of both small and large instances. As an optimizer of nonconvex objectives, the CIM faces the challenge of getting trapped in suboptimal local minima. We study two extensions that may help avoid this: adding phase-insensitive gain and allowing asymmetric couplings. The former expands the space of states by providing access to each unit's full complex phase space, while the latter expands the space of connections by making the network a directed graph. Each extension introduces an extra annealing parameter that can be tuned through novel dynamical regimes, recovering conventional CIM operation at one end of its domain. We show how these parameters can be used to redraw bifurcation and phase diagrams and provide paths to global minima in cases where the conventional CIM would get trapped.   

Opinion Dynamics Entropy Generation via Complex Network Structures

Sociophysics is a compelling and dynamic research field that leverages the principles of physics and sociology to delve into collective behaviors, such as the dynamics of opinions, misinformation spread, financial crises, and political radicalization. Recent scientific investigations have uncovered the role of entropy production in the intricate dynamics of collective systems, fundamental processes underpinning phase transitions, and polarization. The majority-vote model stands out as a straightforward framework for elucidating the dynamics of group opinions in elections, debates, and financial markets. In this model, each individual holds one of two opinions on a given subject, influenced by their social connections. A parameter q introduces a social anxiety capacity, enabling a behavioral social disorder. With probability (1-q), individuals tend to conform to the prevailing opinion within their social interaction network, while with chance q, they act in nonconformity. This work explores the social entropy production of majority-vote opinion dynamics in complex networks, including Erdös-Rényi random graphs, scale-free and small-world networks via Monte Carlo simulations and mean-field analysis.    

Temperature as a measure of complexity in games and social hierarchies

Patterns of wins and losses in pairwise contests, such as occur in sports and games, social hierarchies, and other contexts, are often analyzed using probabilistic models that allow us to quantify the strength of competitors or to predict the outcome of future contests. For example, the popular Bradley-Terry model, familiar as the basis for the Elo chess ratings, uses a sigmoid "score function" for this purpose. We extend this approach in a physically motivated way: we interpret the model's sigmoid function as a Fermi-Dirac distribution and then infer the corresponding temperature by fitting the model to observational data. Examining data sets from a variety of settings we find that sports and games generically have a higher temperature than human social hierarchies, which in turn have a higher temperature than animal social hierarchies. We argue that the inferred temperature gives insight into the rigidity, depth, or complexity of the various hierarchies. We also generalize the model to include a "luck" component, which represents the inherent randomness of a contest regardless of difference in skill. We find that sports and games mostly have little evidence of "luck" in this formal sense, but animal hierarchies appear to have a pronounced luck component.   

StructuralGT v2.0: Graph theoretic characterization and modeling of complex networked materials

Structural analysis of networked materials is a fundamental component in understanding their mechanical, electrical and optical properties. We introduce StructuralGT 2.0 – a graph theoretic software package for analyzing complex networked materials – and demonstrate its applicability to self-assembled nanostructures. In this release, users may now analyze graphs from 3- and 2-dimensional electron microscopy datasets as well as simulated datasets, and integrate their own fast C routines that can operate on Python graphs. This allows for integration with other data-analytic tools from the Python ecosystem, while enabling a previously impossible scale of networked material characterization. To showcase its new capabilities, we demonstrate how we use StructuralGT 2.0 to accurately predict the conductivities, charge carrying capacities, and anisotropy of multilayer silver nanowire films. We also show the graph theoretic characterization of networks of percolating aramid nanofibers. We expect that this tool, which offers a quantitative basis for graph theoretical analysis, will be critical in discovering and designing structure-property relationships for complex networked materials.   

Scale-Free Networks beyond Power-Law Degree Distribution

Complex networks across various fields are often considered to be scale free---a statistical property usually solely characterized by a power-law distribution of the nodes' degree k. However, this characterization is incomplete. In real-world networks, the distribution of the degree--degree distance η, a simple link-based metric of network connectivity similar to k, appears to exhibit a stronger power-law distribution than k. While offering an alternative characterization of scale-freeness, the discovery of η raises a fundamental question: do the power laws of k and η represent the same scale-freeness? To address this question, here we investigate the exact asymptotic relationship between the distributions of k and η, proving that every network with a power-law distribution of k also has a power-law distribution of η, but not vice versa. This prompts us to introduce two network models as counterexamples that have a power-law distribution of η but not k, constructed using the preferential attachment and fitness mechanisms, respectively. Both models show promising accuracy by fitting only one model parameter each when modeling real-world networks. Our findings suggest that η is a more suitable indicator of scale-freeness and can provide a deeper understanding of the universality and underlying mechanisms of scale-free networks.   

The role of network structure in circadian system adaptation

Circadian rhythms, physiological and behavioral changes following a 24-hour cycle, are a ubiquitous feature of life on Earth. While they persist in constant conditions, they can be entrained to environmental cues such as light. Nearly every cell in animals has an autonomous molecular oscillator, which are synchronized by a network of cellular interactions. 

This complex system has previously been modeled as a globally connected network of limit cycle oscillators. However, this simple model does not account for any internal structure that might influence its robustness and adaptation to external influences. Our goal is to understand the relationship between the structural properties of this network and its response to changes in the environment. 

We focus on the phenomenon of jet lag as an example: adjusting to a new time-zone after flying westward is generally easier than flying eastward. Starting from an all-to-all connected network of Kuramoto oscillators, we make systematic changes to the structure and quantify the east/west adjustment asymmetry in each case. We find that hierarchical networks have the most gradual and asymmetrical re-entrainment and regular networks the least. These insights may also be applied to other adaptable networked systems.    

The counter-intuitive network features of optimal diffusive navigation.

Time is a critical factor in numerous physical systems that rely on complex networks to ensure navigability among their components. From human-made transportation networks that seek to optimize routes between stations, to transport of particles in porous media and biological networks, intricate structures and active motion mechanisms ensure low mean first passage times (MFPT) between the target regions. Yet, random walkers can get lost within complex networks and stray from the desired target. This leads to a fundamental question: What is the best architecture that makes a graph optimally searchable? Traditionally, it has been assumed that small-world architectures with numerous shortcuts between distant nodes are ideal. Counter-intuitively, for diffusive exploration of spatially embedded networks, we find that the addition of a link can worsen the total search time between all pairs of nodes, akin to the well-known Braess paradox where increasing a system's capacity reduces its efficiency. This highlights that the topological advantage gained from the extra pathway can be negated by the diffusive delay required to traverse the length of the shortcut, as dictated by the mean squared displacement of anomalous diffusion. We generalize developing an optimization scheme according to which each edge adapts its conductivity in order to minimize the total mean first passage time. This unveals a crossover in the optimal architecture: for super-diffusive motion, the optimal graph features long-range links, while for sub-diffusive propagation geometrical nearest connections are preferred. Ultimately, we explore how multiple propagation speeds can mitigate the impact of "Braessian edges" inspired by the ingenious strategies of intracellular transport networks that utilize diffusion over short distances and motor-driven ballistic motion over longer ones.