Questions tagged [network-theory]

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Clustering coefficient in a directed graph

I am wondering what the meaning of the clustering coefficient in a directed graph. I know how to calculate it: local clustering coefficient formula, retrieved from [Wikipedia]1 and I know how to ...
12
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1answer
313 views

Distribution of degree in graphs: when is the friendship paradox the paradox it wants to be?

$\DeclareMathOperator\deg{deg}\DeclareMathOperator\ndeg{ndeg}\newcommand\abs[1]{\lvert#1\rvert}$The friendship paradox goes most people have fewer friends than their friends have on average. The ...
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58 views

total unimodularity of a matrix

Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
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18 views

Number and different kinds of spnning trees in a weighted graph

We know that for a unweighted graph the number $\tau(\mathcal{G})$ of unique spanning trees of $\mathcal{G}$ is $$\tau(\mathcal{G})=\det L_\mathcal{G}^{\{n-1\}},$$ where $L_\mathcal{G}^{\{n-1\}}$ is ...
4
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1answer
111 views

Almost all simple graphs are small world networks

Two days ago it occurred to me that almost all simple graphs are small world networks in the sense that if $G_N$ is a simple graph with $N$ nodes sampled from the Erdös-Rényi random graph distribution ...
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0answers
36 views

Transversal deviation in first passage percolation

Take the lattice $\mathbb{L}^{2}=(\mathbb{Z}^{2},\mathbb{E}^{2})$ with i.i.d. $\text{U}[0,1]$ weights on the edges, and the random variable $D$ giving the maximal transversal deviation of the geodesic ...
2
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1answer
75 views

Literature on the controllability of networks under attack

I would like to request your advice on a problem arising from my research in the life sciences. Consider a modular, sparse weighted network which is partially controllable in the sense that some ...
1
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1answer
85 views

Number of sequences of edges that contain at least one subsequence which is a walk between vertex $i$ and $j$

Typically a walk is defined as a vertex-edge sequence, e.g. $(v_1, e_1, v_2, e_2, v_3)$, but suppose we are working in the undirected simple graph setting. Instead, let's say an edge-sequence $(e_1, ...
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50 views

On the defect of a flow network

This problem in graph theory was actually motivated by some problems in Theory of Fractals. To formulate the problem I need to recall some definitions related to flow network. A flow network is a ...