# Questions tagged [network-theory]

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### Netting edges of a graph

Consider a weighted directed graph. Is there any known algorithm to compute the associated graph with "netted edges"? By edge netting I mean a function such that the (sub-) branch ...
• 101
42 views

### integer network flow with symmetry

Suppose we have a weighted directed graph $G=(V,E,f)$. Each $e\in E$ is associated with $f_e\in \mathbb{N}$. There is a source node $s$, which only has outgoing edges, and a sink node $t$, which only ...
• 1,365
141 views

### Real world applications of graph limits

It is well known that the area of graph limits (initiated by Lovász and coauthors) had provided a very powerful framework to deal with problems arising, for instance, in extremal combinatorics and ...
• 1,296
21 views

### Finding paths in subgraphs of tripartite graphs such that any two paths have at most one common vertex

Definition (Minimum T-Transversal): Let $T$ be the set of all triangles in a simple graph $G=(V,E)$. A T-transversal is a subset $S\subseteq E(G)$ such that every triangle in $T$ contains at least one ...
• 1
235 views

### What are the right mathematical tools / language to analyse complex networks over time?

In this article about human physiology as a complex network the authors say that: "Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse ...
• 1,463
1 vote
43 views

### Is a function of a centrality measure a centrality measure?

I have been trying to wrap my head around the following question. Suppose you a have a centrality measure for a weighted, undirected network. Let's call the calculated centralities with $\pmb{x}>0$....
• 31
44 views

### Correlation of centrality measure random vectors [closed]

Let's assume that we have 2 random vectors A=(a1,a2,a3) and B=(b1,b2,b3). Each of these elements is a centrality measure of a network. For instance a1 and b1 are the centrality measures of the same ...
• 109
250 views

### Centrality measures in a network with negative correlations

I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as ...
• 109
1 vote
63 views

### Is there a theory for forcing "sludge" through a network, analogous to electric current flows?

I'm familiar with the correspondence between reversible Markov chains, random walks, and electric current flows, as described in Probability on Trees and Networks by Lyons and Peres. Is there an ...
• 23
243 views

### Small world network regime

I have recently read Watts, D., Strogatz, S., Collective dynamics of ‘small-world’ networks, Nature 393 (1998) pp. 440–442, doi:10.1038/30918, on small-world networks, and is still not very clear to ...
• 33
105 views

### Sources of information on algorithms for finding Hamiltonian cycles (Pósa)

I research various algorithms in complex networks and I am quite new in this field. I am currently focusing on random geometric graphs - Pósa's algorithm for finding a hamiltonian cycle. Can you ...
1 vote
78 views

### How do I fit flow values to connections in a known network?

This is not my area and I'm new to its terminology, and am posting my problem in the hope that someone will be able to direct me to where it has been solved, or who has written about it. I have a flow ...
• 11
284 views

### Searching for an early, highly theoretical, even philosophical, math paper on models or small-world networks

All I can remember is that it was very high-level / abstact and kind of philosophical, explaining (the discovery or interdependence of) small world networks. I assume that it was +50 years old and '...
• 103
476 views

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

$\DeclareMathOperator\deg{deg}\DeclareMathOperator\ndeg{ndeg}\newcommand\abs{\lvert#1\rvert}$The friendship paradox goes most people have fewer friends than their friends have on average. The ...
• 343
1 vote
187 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 ...
• 373
1 vote
26 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 ...
• 345
132 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 ...
• 3,503
3k views

### Adjacency definition for a directed graph

For an undirected graph, we know that nodes are adjacent to each other if there is a link that connects them. What about adjacency for directed graphs? Is it based on: outgoing links: node $n$ is ...
50 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 ...
• 510
79 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 ...
• 21
1 vote
139 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, ...
• 220
1 vote