Phase transition in the family of p-resistances - Semantic Scholar
Phase transition in the family of p-resistances Morteza Alamgir
Ulrike von Luxburg
Max Planck Institute for Intelligent Systems Tübingen, Germany
Resistance distance R(s, t) Consider the electrical network corresponding to a graph. R(s,t): The effective resistance between s and t. P i = (ie )e∈E is a unit s-t flow. R(s, t) = mini e∈E re ie2 Pro: In small graphs, it captures the cluster structure!
Small resistance distance
Large resistance distance
Con: (von Luxburg et al. 2010) In large geometric graphs, it converges to the trivial limit R(s, t) ≈
1 1 + ds dt
p-Resistance 30
How we can cure this problem?
25
20
15
p-Resistance : For p ≥ 1, define
10
5
0
0
5
10
15
20
25
30
25
30
25
30
p=2 Rp (s, t) := min i
X
re |ie | p
e edge
30
25
20
15
10
5
Theorem (Special cases of Rp (s, t)) p = 1: Shortest path distance
0
0
5
10
15
20
p = 1.33 30
25
p = 2: Standard resistance distance
20
15
p → ∞: Related to s-t-mincut
10
5
0
0
5
10
15
20
p = 1.1
Phase transition in the family of p-resistances Main Theorem: For large random geometric graphs in R d : 1
If p < 1 + 1/(d − 1), then the “global” contribution dominates the “local” one. ; meaningful distance ,
2
If p > 1 + 1/(d − 2), then all “global” information vanishes. ; useless distance /
Ulrike von Luxburg
Max Planck Institute for Intelligent Systems Tübingen, Germany
Resistance distance R(s, t) Consider the electrical network corresponding to a graph. R(s,t): The effective resistance between s and t. P i = (ie )e∈E is a unit s-t flow. R(s, t) = mini e∈E re ie2 Pro: In small graphs, it captures the cluster structure!
Small resistance distance
Large resistance distance
Con: (von Luxburg et al. 2010) In large geometric graphs, it converges to the trivial limit R(s, t) ≈
1 1 + ds dt
p-Resistance 30
How we can cure this problem?
25
20
15
p-Resistance : For p ≥ 1, define
10
5
0
0
5
10
15
20
25
30
25
30
25
30
p=2 Rp (s, t) := min i
X
re |ie | p
e edge
30
25
20
15
10
5
Theorem (Special cases of Rp (s, t)) p = 1: Shortest path distance
0
0
5
10
15
20
p = 1.33 30
25
p = 2: Standard resistance distance
20
15
p → ∞: Related to s-t-mincut
10
5
0
0
5
10
15
20
p = 1.1
Phase transition in the family of p-resistances Main Theorem: For large random geometric graphs in R d : 1
If p < 1 + 1/(d − 1), then the “global” contribution dominates the “local” one. ; meaningful distance ,
2
If p > 1 + 1/(d − 2), then all “global” information vanishes. ; useless distance /
