15 research outputs found
Updating and downdating techniques for optimizing network communicability
The total communicability of a network (or graph) is defined as the sum of
the entries in the exponential of the adjacency matrix of the network, possibly
normalized by the number of nodes. This quantity offers a good measure of how
easily information spreads across the network, and can be useful in the design
of networks having certain desirable properties. The total communicability can
be computed quickly even for large networks using techniques based on the
Lanczos algorithm.
In this work we introduce some heuristics that can be used to add, delete, or
rewire a limited number of edges in a given sparse network so that the modified
network has a large total communicability. To this end, we introduce new edge
centrality measures which can be used to guide in the selection of edges to be
added or removed.
Moreover, we show experimentally that the total communicability provides an
effective and easily computable measure of how "well-connected" a sparse
network is.Comment: 20 pages, 9 pages Supplementary Materia
Edge modification criteria for enhancing the communicability of digraphs
We introduce new broadcast and receive communicability indices that can be
used as global measures of how effectively information is spread in a directed
network. Furthermore, we describe fast and effective criteria for the selection
of edges to be added to (or deleted from) a given directed network so as to
enhance these network communicability measures. Numerical experiments
illustrate the effectiveness of the proposed techniques.Comment: 26 pages, 11 figures, 4 table
Edge manipulation techniques for complex networks with applications to communicability and triadic closure.
Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks.
Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges.
Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process.
Extensive numerical tests are presented to validate our approaches
Edge manipulation techniques for complex networks with applications to communicability and triadic closure.
Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks.
Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges.
Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process.
Extensive numerical tests are presented to validate our approaches
Sensitivity of matrix function based network communicability measures: Computational methods and a priori bounds
When analyzing complex networks, an important task is the identification of
those nodes which play a leading role for the overall communicability of the
network. In the context of modifying networks (or making them robust against
targeted attacks or outages), it is also relevant to know how sensitive the
network's communicability reacts to changes in certain nodes or edges.
Recently, the concept of total network sensitivity was introduced in [O. De la
Cruz Cabrera, J. Jin, S. Noschese, L. Reichel, Communication in complex
networks, Appl. Numer. Math., 172, pp. 186-205, 2022], which allows to measure
how sensitive the total communicability of a network is to the addition or
removal of certain edges. One shortcoming of this concept is that sensitivities
are extremely costly to compute when using a straight-forward approach (orders
of magnitude more expensive than the corresponding communicability measures).
In this work, we present computational procedures for estimating network
sensitivity with a cost that is essentially linear in the number of nodes for
many real-world complex networks. Additionally, we extend the sensitivity
concept such that it also covers sensitivity of subgraph centrality and the
Estrada index, and we discuss the case of node removal. We propose a priori
bounds for these sensitivities which capture the qualitative behavior well and
give insight into the general behavior of matrix function based network indices
under perturbations. These bounds are based on decay results for Fr\'echet
derivatives of matrix functions with structured, low-rank direction terms which
might be of independent interest also for other applications than network
analysis
Accounting for the Role of Long Walks on Networks via a New Matrix Function
We introduce a new matrix function for studying graphs and real-world
networks based on a double-factorial penalization of walks between nodes in a
graph. This new matrix function is based on the matrix error function. We find
a very good approximation of this function using a matrix hyperbolic tangent
function. We derive a communicability function, a subgraph centrality and a
double-factorial Estrada index based on this new matrix function. We obtain
upper and lower bounds for the double-factorial Estrada index of graphs,
showing that they are similar to those of the single-factorial Estrada index.
We then compare these indices with the single-factorial one for simple graphs
and real-world networks. We conclude that for networks containing chordless
cycles---holes---the two penalization schemes produce significantly different
results. In particular, we study two series of real-world networks representing
urban street networks, and protein residue networks. We observe that the
subgraph centrality based on both indices produce significantly different
ranking of the nodes. The use of the double factorial penalization of walks
opens new possibilities for studying important structural properties of
real-world networks where long-walks play a fundamental role, such as the cases
of networks containing chordless cycles
Optimizing network robustness via Krylov subspaces
We consider the problem of attaining either the maximal increase or reduction
of the robustness of a complex network by means of a bounded modification of a
subset of the edge weights. We propose two novel strategies combining Krylov
subspace approximations with a greedy scheme and an interior point method
employing either the Hessian or its approximation computed via the
limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm (L-BFGS). The paper
discusses the computational and modeling aspects of our methodology and
illustrates the various optimization problems on networks that can be addressed
within the proposed framework. Finally, in the numerical experiments we compare
the performances of our algorithms with state-of-the-art techniques on
synthetic and real-world networks
Functional models and extending strategies for ecological networks
Complex network analysis is rising as an essential tool to understand properties of ecological landscape networks, and as an aid to land management. The most common methods to build graph models of ecological networks are based on representing functional connectivity with respect to a target species. This has provided good results, but the lack of a model able to capture general properties of the network may be seen as a shortcoming when the activity involves the proposal for modifications in land use. Similarity scores, calculated between nature protection areas, may act as a building block for a graph model intended to carry a higher degree of generality. The present work compares several design choices for similarity-based graphs, in order to determine which is most suitable for use in land management
Exploring the “Middle Earth” of network spectra via a Gaussian matrix function
We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs