Spectral Radius and Other Graph Invariants

Stevanović, Dragan

doi:10.1016/b978-0-12-802068-5.00004-x

Cited by 37 publications

(50 citation statements)

References 64 publications

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“…2. In Table 1 we compare some of the general parameters of both networks (see [28] for definitions and examples): (i) edge density δ = 2m/n (n − 1) where m is the number of edges, (ii) average shortest path length l, (iii) average Watts-Strogatz clustering coefficient C (see also [29]), (iv) degree heterogeneity ρ (see also [30,31]), (v) spectral radius of the adjacency matrix λ 1 (see also [32]), and (vi) degree assortativity r (see also [33]). As can be seen both networks are very similar to each other, except that the network of heterosexual contacts (NHeC) is bipartite and thus it contains no odd cycles, which implies that the clustering coefficient is zero, while the network of homosexual contacts (NHoC) is not bipartite and it has clustering different from zero.…”

Section: Computational Experimentsmentioning

confidence: 99%

From networked SIS model to the Gompertz function

Estrada,

Bartesaghi

2021

Preprint

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The Gompertz function is one of the most widely used models in the description of growth processes in many different fields. We obtain a networked version of the Gompertz function as a worst-case-scenario for the exact solution to the SIS model on networks. This function is shown to be asymptotically equivalent to the classical scalar Gompertz function for sufficiently large times. It proves to be very effective both as an approximate solution of the networked SIS equation within a wide range of the parameters involved and as a fitting curve for the most diverse empirical data. As an instance, we perform some computational experiments, applying this function to the analysis of two real networks of sexual contacts. The numerical results highlight the analogies and the differences between the exact description provided by the SIS model and the upper bound solution proposed here, observing how the latter amplifies some empirically observed behaviors such as the presence of multiple and successive peaks in the contagion curve.

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Section: Computational Experimentsmentioning

confidence: 99%

From networked SIS model to the Gompertz function

Estrada,

Bartesaghi

2021

Preprint

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“…The computation of the geometric potential gain requires to fix δ beforehand, which, in turn, requires to fix an approximation of the spectral radius λ 1 . The literature on the estimation of λ 1 provides some bounds on it [46], [47] but, available upper bounds are often not tight and, thus, uninformative; therefore, an alternate way to approximate λ 1 is to rely on algorithms such as the Power Iteration Method [48]. On the other hand, if we target very large graphs, sampling techniques seem the best option [49].…”

Section: Calculation Of Geometric and Exponential Potential Gainsmentioning

confidence: 99%

A General Centrality Framework-Based on Node Navigability

Meo

Levene

Messina

et al. 2020

IEEE Trans. Knowl. Data Eng.

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Centrality metrics are a popular tool in Network Science to identify important nodes within a graph. We introduce the Potential Gain as a centrality measure that unifies many walk-based centrality metrics in graphs and captures the notion of node navigability, interpreted as the property of being reachable from anywhere else (in the graph) through short walks. Two instances of the Potential Gain (called the Geometric and the Exponential Potential Gain) are presented and we describe scalable algorithms for computing them on large graphs. We also give a proof of the relationship between the new measures and established centralities. The geometric potential gain of a node can thus be characterized as the product of its Degree centrality by its Katz centrality scores. At the same time, the exponential potential gain of a node is proved to be the product of Degree centrality by its Communicability index. These formal results connect potential gain to both the "popularity" and "similarity" properties that are captured by the above centralities.

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“…For fundamental knowledge on the spectral radius of a network, see [38,39]. The SRMP is formulated as follows: given a network = ( , ) and a positive integer , find a set of edges of so that the surviving network obtained by removing the set of edges from the network achieves the minimum spectral radius.…”

Section: Preliminariesmentioning

confidence: 99%

The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

Zhang

Chen

et al. 2017

Discrete Dynamics in Nature and Society

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The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.

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Spectral Radius and Other Graph Invariants

Cited by 37 publications

References 64 publications

From networked SIS model to the Gompertz function

From networked SIS model to the Gompertz function

A General Centrality Framework-Based on Node Navigability

The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

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