Average path length in random networks

Fronczak, Agata; Fronczak, Piotr; Hołyst, Janusz A.

doi:10.1103/physreve.70.056110

Cited by 267 publications

(221 citation statements)

References 42 publications

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“…Compare this formula to the one in (1.1) and note that this formula yields the first order term if and only if β n = o(1/ log log n), while even in this case it fails to capture the second order term, which is of order log log n. While [51] questions the validity of this formula for τ ∈ (2, 3), [24,25,33] argue that a constant β n ≡ β for the truncation exponent yields bounded typical distances in this regime, in agreement with what (1.2) suggests. This contradicts the arguments in [15] where the authors show that the smallest achievable order for typical distances is log log n.…”

Section: Introduction and Resultsmentioning

confidence: 97%

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When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

J Stat Phys

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In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent τ ∈ (2, 3), up to value n β n for some β n (log n) −γ and γ ∈ (0, 1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n β n . These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log(n β n )/| log(τ − 2)| + 1/(β n (3 − τ )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/β n = o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n f n β n , where f n ∈ (0, 1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two 'end-segments' starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log(n β n )/| log(τ − 2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/(β n (3 − τ ))+tight, and it contains only vertices with degree at least of order n (1− f n )β n , thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n β n , and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.

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Section: Introduction and Resultsmentioning

confidence: 97%

“…This sheds light to a discussion in the physics literature [15,19,24,25,33,51] about the validity of the formula derived using generating function methods or n-dependent branching process approximations, stating that…”

Section: Introduction and Resultsmentioning

confidence: 99%

When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

J Stat Phys

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“…Both have shorter path lengths and perimeters than the random electrode data network (see Table 1). These can also be compared to Random network theory [2,8]. The cluster coefficient for a random graph (Erdos-Renyi) is simply given by C = k /(N − 1), while according to [8], the average path length is given by…”

Section: Network Analysismentioning

confidence: 99%

Complex networks in brain electrical activity

et al. 2007

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This paper is the result of work carried out at Starlab in collaboration with the Neurodynamics Laboratory of the U. of Barcelona (UBNL) focusing on complex networks analysis of EEG data (provided by UBNL). The approach is inspired by the work in [6] with fMRI data and represents a follow up on earlier efforts on analysis of ERP/MMN data using tomography and independent component analysis to characterize brain connectivity and spatial funcionalization [22]. Multichannel EEG measurements are first processed to obtain 3D voxel activations using the tomographic algorithm LORETA. Then, the correlation of the current intensity activation between voxel pairs is computed to produce a voxel cross-correlation coefficient matrix. Using several correlation thresholds, the cross-correlation matrix is then transformed into a network connectivity matrix and analyzed. The resulting analysis highlights significant differences between the spatial activations associated with Standard and Deviant tones, with interesting physiological implications. When compared to random data networks, physiological networks are more connected, with longer links and shorter path lengths. Furthermore, as compared to the Deviant stimulus case, Standard data networks are consistently more connected, with longer links and shorter path lengths-consistent with a "small worlds" character. The comparison between both networks shows that areas known to be activated in the MMN wave are connected. In particular, the analysis supports the idea that supra-temporal and inferior frontal data work together in the processing of the differences between sounds by highlighting an increased connectivity in the response to a novel sound. A major challenge for neuroscience is to map and analyze the spatiotemporal patterns of activity of the large neuronal populations which are believed to be responsible for information processing in the human brain. In this paper, we analyze the complex networks associated with brain electrical activity in a specific experimental context. Our approach uses multichannel EEG data analysis to characterize the spatial connectivity of the brain. Multichannel EEG measurements are first processed to obtain 3D voxel activations using the tomographic algorithm LORETA. Then, the correlation of the current intensity activation between voxel pairs is computed to produce a voxel cross-correlation coefficient matrix. Using several correlation thresholds, the cross-correlation matrix is then transformed into a network connectivity matrix and analyzed. To study a specific example, we selected data from an earlier experiment focusing on the Mismatch Negativity (MMN) brain wave-an Event-Related Potential (ERP)-because it gives an electrophysiological index of a "primitive intelligence" associated with auditory pattern and change detection in a regular auditory pattern. EEGs have an exceptional millisecond temporal resolution, but appear to result from mixed neuronal contributions whose spatial location and relationships are not fully understood. Although...

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“…Disregarding the possibility of the network graph having small-world or scale-free properties, we assume the average path length in random networks as our average distance from start to target node. The average path length in a random network lER (and also the average distance between nodes) is calculated as follows [22]:…”

Section: E Stochastic Scalability Analysismentioning

confidence: 99%