We study the effect of obstacles on diffusion processes on finite path graphs from a linear algebraic perspective. An obstacle is modeled as a weighted self-loop on the path graph, and diffusion is governed by local and nonlocal Laplacian operators. We first analyze the spectral perturbation induced by the obstacle for the standard graph Laplacian and prove that, as the weight of the self-loop increases, the diffusive particle becomes effectively confined to a subregion of the graph. Surprisingly, we show that the same trapping phenomenon occurs for a broad class of nonlocal operators given by matrix functions, including fractional powers of the Laplacian. Although these operators are nonlocal, they inherit structural properties that prevent them from overcoming arbitrarily strong obstacles. In contrast, we prove that transformed d-path Laplacians—particularly the Mellin-transformed operator—overcome obstacles of arbitrary weight. This distinction is explained through spectral decomposition and block diagonalization techniques. Our results establish a sharp structural difference between matrix functions and transformed d-path Laplacians and provide a general framework for modeling nonlocal diffusion on graphs with barriers.

Consensus protocols (linear and nonlinear) are extensively used for multi-agent systems to perform a wide variety of tasks. Its connections with diffusive models make a natural bridge between engineering and social/natural complex systems. These protocols give rise to conservative processes on the graphs representing the systems, and reach homogeneous (consensus) steady states. Here we develop a model that extends the concept of global consensus (diffusion) to dynamical processes which are nonconservative on the graphs, and reach bounded steady states, which are not necessarily a global consensus. That is, they allow heterogeneous steady states in which different subsets of agents can reach a consensus among them, which may be different from the ones reached by other groups. We prove some mathematical results indicating how the structure of the graphs representing the system determines the final states of the dynamics. Finally, we illustrate the applications of the model by considering a heterogeneous rendezvous of a group of agents, as well as the connections of the model with the Bass model for diffusion of innovation and the Lotka-Volterra equations to model the evolution of groups of species that interact in different ways.

Geometry is typically treated as a primitive background structure within which physical processes unfold. In this work we investigate an alternative possibility: that geometry can arise as a structural consequence of relational dynamics. We study the framework of Graph Dynamical Geometrization (GDG), in which a system is specified by a relational structure encoded by a graph together with a dynamical law acting on that structure. The dynamics defines a symmetric operator whose spectral evolution generates a positive–definite kernel. From this kernel one obtains a squared Euclidean distance matrix that induces a canonical embedding of the relational system into a Euclidean space. Geometry therefore appears not as an assumed background but as a structure generated by the dynamics of relations. This construction motivates a philosophical interpretation that we call relational–dynamical structural realism, according to which the fundamental ontology consists of entities connected by relations and governed by a dynamical law, while geometric structure emerges from the spectral organization of that dynamics. We analyze the status of the induced geometry within several forms of structural realism. The GDG framework shows that invariant metric relations generated by relational dynamics can possess explanatory and predictive significance. More generally, the results suggest a structural principle: geometry may emerge whenever relational dynamics admits a spectral articulation.

Abstract The heat kernel of a graph Laplacian is widely used to describe diffusion, transport, correlations and geometry in complex networks. In this work, we show that the same heat-kernel operator emerges both as the propagator of classical diffusion on a network and as the thermal Green's function of a network of coupled quantum harmonic oscillators restricted to the one-particle sector. We demonstrate that the geometry induced by this operator can be reconstructed from multiple observational experiments without direct access to the microscopic physical substrate generating the dynamics. Consequently, the same experimentally observed heat-kernel geometry may arise from fundamentally different physical mechanisms. This implies that heat-kernel-based observations alone may be insufficient to discriminate between classical and quantum interpretations of hidden dynamics in black-box systems. The results are discussed in the context of complex networks and observational geometries reconstructed from indirect measurements, including potential implications for neuroscience and brain connectivity analysis.

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense.

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex volumes, allowing interference, localization, and spreading to be analyzed within a unified metric-angular formalism. We prove that, in contrast to classical diffusion, which collapses to a spherical geometry, quantum dynamics generate a generically non-spherical affine geometry with persistent anisotropy. Applying this theory to real-world networks—including transportation systems, semantic graphs, and neuronal connectomes—we show that quantum geometry reveals dynamically meaningful backbones, interference-based “communities”, and vulnerability structures that are invisible to classical random-walk and spectral methods. In particular, angular and radial quantum descriptors isolate functional hubs, control cores, and coherence classes without any topological or dimensionality assumptions. Together, these results demonstrate that quantum-walk-induced geometry provides a powerful new lens for understanding structure and function in complex networks.

We introduce a magnetic extension of the Hückel molecular orbital (HMO) method in which the effect of an external magnetic field is incorporated through the Peierls substitution. In this formulation, the π -electron Hamiltonian of a conjugated molecule becomes a complex-weighted hopping operator defined on the hydrogen-depleted molecular graph. Under the standard Hückel assumptions, this operator coincides with the Hermitian adjacency matrix of the graph, providing a physical realization of Hermitian adjacency operators within molecular electronic structure theory. The magnetic-HMO framework enables the calculation of flux-dependent electronic observables such as bond currents, ring currents, orbital magnetization, and orbital susceptibility directly from the spectrum of the magnetic adjacency matrix. Applications to polycyclic aromatic hydrocarbons show that the magnetic response of π -electron systems is strongly controlled by molecular topology. In particular, linearly fused systems exhibit regular flux-dependent oscillations in the orbital magnetization, whereas nonlinear and extended molecules display more complex spectral rearrangements associated with multiple conjugation pathways. MSC (2020): 05C50; 92E10; 81Q10; 05C90.

Structural balance is an important characteristic of graphs/networks where edges can be positive or negative, with direct impact on the study of real-world complex systems. When a network is not structurally balanced, it is important to know how much balance still exists in it. Although several measures have been proposed to characterize the degree of balance, the use of matrix functions of the signed adjacency matrix emerges as a very promising area of research. Here, we take a step forward to using Mittag-Leffler (ML) matrix functions to quantify the notion of balance of signed networks. We show that the ML balance index can be obtained from first principles on the basis of a nonconservative diffusion dynamic, and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions. Finally, we demonstrate the important information in the ML balance index with both artificial signed networks and real-world networks in various contexts, ranging from biological and ecological to social ones.

We introduce several geometric measures for mixed graphs represented by complex-valued Hermitian adjacency matrices. We define the communicability functions based on the exponential of the Hermitian adjacency matrix and define complex-valued position vectors. Then, we define a Euclidean distance as well as complex, and Euclidean angles between these positions vectors for mixed graphs. Further we introduce Kähler and Hermitian angles between different planes between position vectors and holomorphic and projection planes, respectively. We find several mathematical relations and inequalities between all these geometric parameters. To illustrate the usability of some of these indices in the study of real-world networks we study the Kähler angle for finding hierarchies and detecting hierarchical clusters of vertices in ecological food webs, networks of co-purchasing of political books, a neuronal network, an Internet trolls network, and a software collaboration graph. These applications give empirical evidence that the Kähler angle contains important information about the structure of mixed graphs which is relevant for real-world applications in the study of complex networks.

We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This provides a unified framework for determining whether a matrix represents a genuine diffusive operator on a graph. Within this framework, we prove that the standard Laplacian, the fractional Laplacian, the dd-path Laplacians, and the degree-attracting and degree-repelling Laplacians are all proper diffusive Laplacians. In contrast, the in-degree and out-degree Laplacians correspond to advection operators, while the signed, signless, magnetic, and deformed Laplacians are improper, as they cannot be written as the composition of a divergence and a true gradient. The magnetic Laplacian is shown to arise as the Schur complement of an extended proper Laplacian defined on a higher-dimensional space, a property also inherited by the signless Laplacian. The Lerman-Ghosh Laplacian is identified as a nonconservative diffusive operator coupled to an external reservoir. Finally, we prove that the Moore-Penrose pseudoinverse of the Laplacian is itself a proper Laplacian. This classification establishes a rigorous operator-theoretic foundation for distinguishing proper, nonconservative, and improper graph Laplacians.

The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove that the resistance distance is given by a difference of “mass concentrations” obtained at the vertices of a graph by a diffusive process. The nature of this diffusive process is characterized here by means of an operator corresponding to the matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian. We prove also that this operator is indeed the Laplacian matrix of a signed version of the original graph, in which nonnearest neighbors’ “interactions” are also considered. In this way, the resistance distance is part of a family of squared Euclidean distances emerging from diffusive dynamics on graphs.

Signed graphs are an emergent way of representing data in a variety of contexts where antagonistic interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability for signed graphs and explore in depth its mathematical properties. We also prove that the communicability induces a hyperspherical geometric embedding of the signed network, and derive communicability-based metrics that satisfy the axioms of a distance even in the presence of negative edges. We then apply these metrics to solve several problems in the data analysis of signed graphs within a unified framework. These include the partitioning of signed graphs, dimensionality reduction, finding hierarchies of alliances in signed networks, and quantifying the degree of polarization between the existing factions in social systems represented by these types of graphs.

Diffusion is a ubiquitous process in real-world syetems. In many complex systems, ranging from neuronal networks to traffic in cities, diffusion is nonconservative (NC) in the sense that diffusive particles can be created/annihilated at the entities of the system. Here, we consider the important problem of identifying potential navigational bottlenecks in NC diffusion occurring in the networks representing skeletons of complex systems. We develop a first-principles approach based on an NC diffusion using the Lerman-Ghosh Laplacian on graphs. By solving analytically this NC diffusion equation at two different times, we get an index which characterizes the capacity of every vertex in a network to spread the diffusive particles across the network in a short time. Vertices having such capacity diminished are potential navigational bottlenecks in this kind of dynamics. We solve analytically the situations in which the vertices with the highest degree (hubs) are at different distances in the network, allowing us to understand the structural significance of the index. Using algebraic methods, we derive a Euclidean distance between vertices in the context of NC diffusion with potential navigational bottlenecks. We then apply these indices to study several real-world networks. First, we confronted our theoretical results with experimental data about traffic congestion in a city. Then, we illustrated the application of the new methodologies to the study of a neuronal system, an air transportation network and two urban street networks.

A distance based on the exponential kernel of the adjacency matrix of a graph and representing how well two vertices connect to each other in a graph is defined and studied. This communicability cosine distance (CCD) is a Euclidean spherical distance accounting for the cosine of the angles spanned by the position vectors of the graph vertices in this space. The Euclidean distance matrix (EDM) of CCD is used to quantify the similarity between vertices in graphs and networks as well as to define a local vertex invariant—a closeness centrality measure, which discriminate very well vertices in small graphs. It allows to distinguish all nonidentical vertices, also characterizing all identity (asymmetric) graphs–those having only the identity automorphism–among all connected graphs of up to 9 vertices. It also characterizes several other classes of identity graphs. We also study real-world networks in term of both the discriminating power of the new centrality on their vertices as well as in ranking their vertices. We analyze some dictionary networks as well as the network of copurshasing of political books, remarking some of the main advantages of the new approaches studied here.

A fundamental feature for understanding the diffusion of innovations through a social group is the manner in which we are influenced by our own social interactions. It is usually assumed that only direct interactions, those that form our social network, determine the dynamics of adopting innovations. Here, we test this assumption by experimentally and theoretically studying the role of direct and indirect influences in the adoption of innovations. We perform experiments specifically designed to capture the influence that an individual receives from their direct social ties as well as from those socially close to them, as a function of the separation they have in their social network. The results of 21 experimental sessions with more than 590 participants show that the rate of adoption of an innovation is significantly influenced not only by our nearest neighbors but also by the second and third levels of influences an adopter has. Using a mathematical model that accounts for both direct and indirect interactions in a network, we fit the experimental results and determine the way in which influences decay with social distance. The results indicate that the strength of peer pressure on an adopter coming from its second and third circles of influence is approximately two-third and one-third, respectively, relative to their closest neighbors. Our results strongly suggest that the adoption of an innovation is a complex process in which an individual feels significant pressure not only from their direct ties but also by those socially close to them.

The way of thinking of mathematicians and chemists in their respective disciplines seems to have very different levels of abstractions. While the firsts are involved in the most abstract of all sciences, the seconds are engaged in a practical, mainly experimental discipline. Therefore, it is surprising that many luminaries of the mathematics universe have studied chemistry as their main subject. Others have started studying chemistry before swapping to mathematics or have declared some admiration and even love for this discipline. Here I reveal some of these mathematicians who were involved in chemistry from a biographical perspective. Then, I analyze what these remarkable mathematicians and statisticians could have learned while studying chemical subjects. I found analogies between code-breaking and molecular structure elucidation, inspiration for statistics in quantitative analytical chemistry, and on the role of topology in the study of some organic molecules. I also analyze some parallelisms between the way of thinking of organic chemists and mathematicians in terms of the use of backward analysis, search for patterns, and use of pictures in their respective researches.

Alliances and conflicts in social, political and economic relations can be represented by positive and negative edges in signed networks. A cycle is said to be positive if the product of its edge signs is positive, otherwise it is negative. Then, a signed network is balanced if and only if all its cycles are positive. An index characterizing how much a signed network deviates from being balanced is known as a global balance index. Here we give a step forward in the characterization of signed networks by defining a local balance index, which characterizes how much a given vertex of a signed network contributes to its global balance. We analyze the mathematical foundations and unique structural properties of this index. Then, we apply this index to the study of the evolution of international relations in the globe for the period 1816-2014. In this way we detect and categorize major historic events based on balance fluctuations, helping our understanding towards new mixed approaches to history based on network theory.

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n≤10. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.