## Return to Question

2 edited title; edited title; added 4 characters in body

# longestpathindiameterof a graph with random edge weights

Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be an exponential random variable with mean $w_{ij}$. My question is: what is the expected diameter of $G'$?

Why I'm interested in this: I was intrigued by the observation that the expected diameter of $G'$ can be quite different from the diameter of $G$. Indeed, consider the following example: define $G$ by taking the complete graph $K_n$, K_{n+1}$, picking an arbitrary vertex$a$, and assigning weight$n$to any edge incident on$a$, and weight$1$to every other edge. Then, the diameter of$G$is$n$. On the other hand, the expected diameter of$G'$is O(1) since we can expect one of the edges incident on$a\$ to have small weight.

1

# longest path in a graph with random weights

Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be an exponential random variable with mean $w_{ij}$. My question is: what is the expected diameter of $G'$?

Why I'm interested in this: I was intrigued by the observation that the expected diameter of $G'$ can be quite different from the diameter of $G$. Indeed, consider the following example: define $G$ by taking the complete graph $K_n$, picking an arbitrary vertex $a$, and assigning weight $n$ to any edge incident on $a$, and weight $1$ to every other edge. Then, the diameter of $G$ is $n$. On the other hand, the expected diameter of $G'$ is O(1) since we can expect one of the edges incident on $a$ to have small weight.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 34, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9447965025901794, "perplexity_flag": "head"}