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We present a network model in which words over a specific alphabet, called {\it structures}, are associated to each node and undirected edges are added depending on some distance between different structures. It is shown that this model can…
The recent discovery of universal principles underlying many complex networks occurring across a wide range of length scales in the biological world has spurred physicists in trying to understand such features using techniques from…
There has lately been increased interest in describing complex systems not merely as single networks but rather as collections of networks that are coupled to one another. We introduce an analytically tractable model that enables one to…
In this paper, we study the problem of constructing a network by observing ordered connectivity constraints, which we define herein. These ordered constraints are made to capture realistic properties of real-world problems that are not…
Mostly acyclic directed networks, treated mathematically as directed graphs, arise in machine learning, biology, social science, physics, and other applications. Newman [1] has noted the mathematical challenges of such networks. In this…
In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and…
Most real-world networks are embedded in latent geometries. If a node in a network is found in the vicinity of another node in the latent geometry, the two nodes have a disproportionately high probability of being connected by a link. The…
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic…
Boolean networks have been used in a variety of settings, as models for general complex systems as well as models of specific systems in diverse fields, such as biology, engineering, and computer science. Traditionally, their properties as…
Heterogeneous and complex networks represent intertwined interactions between real-world elements or agents. Determining the multi-scale mesoscopic organization of clusters and intertwined structures is still a fundamental and open problem…
Network connectivity is usually addressed for convex domains where a direct line of sight exists between any two transmitting/receiving nodes. Here, we develop a general theory for the network connectivity properties across a small opening,…
Random linear network codes can be designed and implemented in a distributed manner, with low computational complexity. However, these codes are classically implemented over finite fields whose size depends on some global network parameters…
We introduce an evolving network model in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability $p$. The resulting network is sparse for $p<\frac{1}{2}$ and dense (average degree…
Many real-world networks, ranging from subway systems to polymer structures and fungal mycelia, do not form by the incremental addition of individual nodes but instead grow through the successive extension and intersection of lines or…
A typical complex system should be described by a supernetwork or a network of networks, in which the networks are coupled to some other networks. As the first step to understanding the complex systems on such more systematic level,…
This paper (parts I and II) provides an expository introduction to monotone and near-monotone dynamical systems associated to biochemical networks, those whose graphs are consistent or near-consistent. Many conclusions can be drawn from…
Complex systems are often driven by higher-order interactions among multiple units, naturally represented as hypergraphs. Understanding dependency structures within these hypergraphs is crucial for understanding and predicting the behavior…
The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper we propose and explore a greedy…
In this paper, we present algorithms for designing networks that are robust to node failures with minimal or limited number of links. We present algorithms for both the static network setting and the dynamic network setting; setting where…
While network science has become an indispensable tool for studying complex systems, the conventional use of pairwise links often shows limitations in describing high-order interactions properly. Hypergraphs, where each edge can connect…