Minimum entropy of graphs with given size

Entropy, 2012

This paper presents a taxonomy and overview of approaches to the measurement of graph and network complexity. The taxonomy distinguishes between deterministic (e.g., Kolmogorov complexity) and probabilistic approaches with a view to placing entropy-based probabilistic measurement in context. Entropy-based measurement is the main focus of the paper. Relationships between the different entropy functions used to measure complexity are examined; and intrinsic (e.g., classical measures) and extrinsic (e.g., Körner entropy) variants of entropy-based models are discussed in some detail.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α−polygon in the given graph; Sβ is, in turn, an average uncertainty to find the red β−polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Discrete Dynamics in Nature and Society

The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.

arXiv (Cornell University), 2013

Inspired by applications to theories of coding and communication in networks of nervous tissue, we study maximum entropy distributions on weighted graphs with a given expected degree sequence. These distributions are characterized by independent edge weights parameterized by a shared vector of vertex potentials. Using the general theory of exponential family distributions, we derive the existence and uniqueness of the maximum likelihood estimator (MLE) of the vertex parameters. We also prove consistency of the MLE from a single sample in the limit of large graphs, extending results of Chatterjee, Diaconis, and Sly in the unweighted case (the "β-model" in statistics). Interestingly, our proofs require tight estimates on the norms of inverses of symmetric, diagonally dominant positive matrices. Along the way, we derive analogues of the Erdős-Gallai criterion of graphical degree sequences for weighted graphs.

Symmetry

In this paper we extend earlier results on Hosoya entropy (H-entropy) of graphs, and establish connections between H-entropy and automorphisms of graphs. In particular, we determine the H-entropy of graphs whose automorphism group has exactly two orbits, and characterize some classes of graphs with zero H-entropy.

In this article, we tackle a challenging problem in quantitative graph theory. We establish relations between graph entropy measures representing the structural information content of networks. In particular, we prove formal relations between quantitative network measures based on Shannon's entropy to study the relatedness of those measures. In order to establish such information inequalities for graphs, we focus on graph entropy measures based on information functionals. To prove such relations, we use known graph classes whose instances have been proven useful in various scientific areas. Our results extend the foregoing work on information inequalities for graphs.

Journal of Complex Networks, 2018

The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two 154-164. Springer, 2017.

Physical Review E, 2012

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We firstly define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdős-Rényi networks. By using classical results of Pincus, we show that our entropy measure converges with network size to a certain binary Shannon entropy. On a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs permit to naturally associate to a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

Cornell University - arXiv, 2022

We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young tableaux, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory, but they are easy to find for specific values of the order n and size m. We also give a direct argument to characterize the graphs maximizing the entropy given order and size. We indicate that some of our ideas extend to other degree-based topological indices as well.

Physical Review E, 2011

Entropic measures of complexity are able to quantify the information encoded in complex network structures. Several entropic measures have been proposed in this respect. Here we study the relation between the Shannon entropy and the Von Neumann entropy of networks with a given expected degree sequence. We find in different examples of network topologies that when the degree distribution contains some heterogeneity, an intriguing correlation emerges between the two entropies. This result seems to suggest that this kind of heterogeneity is implying an equivalence between a quantum and a classical description of networks, which respectively correspond to the Von Neumann and the Shannon entropy.