Tailored Random Graph Ensembles

Roberts, E. S.; Annibale, Alessia; Coolen, A. C. C.

doi:10.1088/1742-6596/410/1/012097

Cited by 3 publications

(6 citation statements)

References 4 publications

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“…One avenue has been the investigation of the networks or ensembles of networks that result from imposing certain constraints over the exponentially huge space of possible networks [8,9]. A fundamental contribution has come from approaches that extend ideas and concepts from statistical mechanics and information theory to complex network ensembles [9,10]. A precursor of this approach is the configuration model, which focuses on an ensemble of networks that have the same degree sequence [8].…”

Section: Introductionmentioning

confidence: 99%

The scaling of the minimum sum of edge lengths in uniformly random trees

Esteban¹,

Ferrer-i-Cancho²,

Gómez-Rodríguez³

2016

J. Stat. Mech.

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Abstract. The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.

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Section: Introductionmentioning

confidence: 99%

The scaling of the minimum sum of edge lengths in uniformly random trees

Esteban¹,

Ferrer-i-Cancho²,

Gómez-Rodríguez³

2016

J. Stat. Mech.

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“…The differences between the empirical network and its randomized counterparts may imply some significant functional or evolutionary properties of the empirical network. Stated more technically, randomized networks serve as null models for empirical networks [31,32,[36][37][38][39][40]. This approach was utilized in the study of network motifs, which are over-represented in empirical networks compared to the corresponding randomized networks [41,42].…”

Section: Discussionmentioning

confidence: 99%

“…It is found that on the giant component the abundance of nodes of degree k = 1 is depleted with respect to their abundance in the whole network, while the abundance of nodes of higher degrees is enhanced.InFig. 4we present the mean degree K (dashed line), obtained from Eq (40),. of a configuration model network whose giant component exhibits an exponential degree distribution with mean degree c = E[K|1], as a function of c. The mean degree, c, of the giant component (solid line) is also shown for comparison.…”

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confidence: 99%

Generating random networks that consist of a single connected component with a given degree distribution

et al. 2019

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We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations. Moreover, they are found to be disassortative, namely high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for singlecomponent networks with ternary, exponential and power-law degree distributions. PACS numbers: 64.60.aq,89.75.Da 1 I. INTRODUCTIONNetwork models provide a useful description of a broad range of phenomena in the natural sciences and engineering as well as in the economic and social sciences. This realization has stimulated increasing interest in the structure of complex networks, and in the dynamical processes that take place on them [1-9]. One of the central lines of inquiry has been concerned with the existence of a giant connected component that is extensive in the network size. In the case of Erdős-Rényi (ER) networks, the critical parameters for the emergence of a giant component in the thermodynamic limit were identified and the fraction of nodes that reside in the giant component was determined [10][11][12][13]. These studies were later extended to the broader class of configuration model networks [14,15]. The configuration model framework enables one to construct an ensemble of random networks whose degree sequences are drawn from a desired degree distribution, with no degree-degree correlations. The resulting network ensemble is a maximum entropy ensemble under the condition of the given degree distribution. A simple example of a configuration model network is the random regular graph, in which all the nodes are of the same degree, k = c. For random regular graphs with c ≥ 3 the giant component encompasses the whole network [16]. However, in general, configuration model networks often exhibit a coexistence between a giant component, which is extensive in the network size, and many finite components, which are non-extensive trees. This can be exemplified by the case of ER networks, which exhibit a Poisson degree distribution of the formwhere c = K is the mean degree. ER networks with 0 < c < 1 consist of finite tree components. At c = 1 there is a percolation transition, above which the network exhibits a coexistence between the giant component and the finite components. In the asymptotic limit, the size of the giant component is N 1 = gN, where N is the size of the whole network and the parameter g = g(c), which vanishes for c ≤ 1, increases monotonically for c > 1. At c = ln N there is a second transition, above which the giant component encompasses the entire network [16]. In the range of 1 < c < ln N, where the giant and finite components coexist, the...

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“…However, maximum-entropy models allow for an analytical treatment of the problem and simplify the generation of network samples when the considered constraints are increasingly complicated (both at the coding level and at the computational one). This has many implications, including the possibility of computing p values, information-theoreticrelated quantities such as ensemble entropies [13][14][15]18,22] and model likelihoods as well as efficient weighted network pruning algorithms [38,39]. Moreover, this procedure helps in the fast and simple generation of samples of networks with prescribed constraints.…”

Section: Linear Constraints On Aggregated Occupation Numbersmentioning

confidence: 99%

“…A different approach to this problem has its roots in the analogy of networks with classical statistical mechanics systems [11][12][13][14][15], though it was originally proposed by sociologists and also by urban planners [16] under the name of exponential random graphs [17]. It is based on the idea of constructing an ensemble of networks with different probabilities of appear-* osagarra@ub.edu ance, which on average fulfill the considered constraints.…”

Section: Introductionmentioning

confidence: 99%

Role of adjacency-matrix degeneracy in maximum-entropy-weighted network models

Sagarra¹,

Vicente²,

Dı́az-Guilera³

2015

Phys. Rev. E

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Complex network null models based on entropy maximization are becoming a powerful tool to characterize and analyze data from real systems. However, it is not easy to extract good and unbiased information from these models: A proper understanding of the nature of the underlying events represented in them is crucial. In this paper we emphasize this fact stressing how an accurate counting of configurations compatible with given constraints is fundamental to build good null models for the case of networks with integer-valued adjacency matrices constructed from an aggregation of one or multiple layers. We show how different assumptions about the elements from which the networks are built give rise to distinctively different statistics, even when considering the same observables to match those of real data. We illustrate our findings by applying the formalism to three data sets using an open-source software package accompanying the present work and demonstrate how such differences are clearly seen when measuring network observables.

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Tailored Random Graph Ensembles

Cited by 3 publications

References 4 publications

The scaling of the minimum sum of edge lengths in uniformly random trees

The scaling of the minimum sum of edge lengths in uniformly random trees

Generating random networks that consist of a single connected component with a given degree distribution

Role of adjacency-matrix degeneracy in maximum-entropy-weighted network models

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