Modifying Heat Kernel Equation for Graphs
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In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely: $left( W_{ii} = frac{1}{2}, W_{ij} = frac{1}{deg(v_i)+deg(v_j)} right)$
$$ W^t u = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega^t u = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
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show 2 more comments
$begingroup$
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely: $left( W_{ii} = frac{1}{2}, W_{ij} = frac{1}{deg(v_i)+deg(v_j)} right)$
$$ W^t u = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega^t u = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
$endgroup$
$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
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How does the heat kernel equation in the title come into play in the question?
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– mathreadler
Jan 5 at 8:59
1
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48
|
show 2 more comments
$begingroup$
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely: $left( W_{ii} = frac{1}{2}, W_{ij} = frac{1}{deg(v_i)+deg(v_j)} right)$
$$ W^t u = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega^t u = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
$endgroup$
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely: $left( W_{ii} = frac{1}{2}, W_{ij} = frac{1}{deg(v_i)+deg(v_j)} right)$
$$ W^t u = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega^t u = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
edited Jan 9 at 9:31
Zachary Hunter
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
1,007313
1,007313
$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
$begingroup$
How does the heat kernel equation in the title come into play in the question?
$endgroup$
– mathreadler
Jan 5 at 8:59
1
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48
|
show 2 more comments
$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
$begingroup$
How does the heat kernel equation in the title come into play in the question?
$endgroup$
– mathreadler
Jan 5 at 8:59
1
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48
$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
$begingroup$
How does the heat kernel equation in the title come into play in the question?
$endgroup$
– mathreadler
Jan 5 at 8:59
$begingroup$
How does the heat kernel equation in the title come into play in the question?
$endgroup$
– mathreadler
Jan 5 at 8:59
1
1
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48
|
show 2 more comments
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$begingroup$
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
$endgroup$
– Zachary Hunter
Jan 4 at 20:08
$begingroup$
How does the heat kernel equation in the title come into play in the question?
$endgroup$
– mathreadler
Jan 5 at 8:59
1
$begingroup$
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
$endgroup$
– Zachary Hunter
Jan 5 at 9:09
$begingroup$
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
$endgroup$
– mathreadler
Jan 5 at 9:40
$begingroup$
Heh, I was just waiting for him to go over to electrical flows, and he did.
$endgroup$
– mathreadler
Jan 5 at 9:48