dario fasino university of udine rome, feb. 19, 2019 · notations a: adjacency matrix of...

So, what is the random walk centrality?

Dario FasinoUniversity of Udine

Rome, Feb. 19, 2019

D. Fasino 2ggALN@RM 1/ 16

An influential paper

J. D. Noh, H. Rieger.Random walks on complex networks.Phys. Rev. Letters 92 (2004), 118701 (4pp.).

Introduces the RWC for nodes of an undirected graph

Citations in Scopus: 600+Citations from items in MathRev: 12Occurrences of this ‘RWC’ in MathRev: 1


Notations

A: adjacency matrix of undirected, connected graph.

P = D−1A: (row stochastic) transition matrix of theassociated random walk.Pij is the probability of the i → j transition.

τi→j : first hitting time of node j starting from i .τi→i = 0 and, for i 6= j ,

τi→j = 1 +∑n

k=1 Pikτk→j .

The first step takes us to a neighbor k of i , and then we haveto reach j from there. In matrix form,

(I − P)T = 11T −Diag(τ↪→1, . . . , τ↪→n)

where T = (τi→j) and τ↪→i is the return time of node i .


By Perron–Frobenius theory. . .

. . . there exists exactly one stationary probability vector π suchthat πT = πT P and 1T π = 1.Owing to A = AT it holds π = d/1T d where d = A1. Now,

0 = πT (I − P)T = πT 11T − πT Diag(τ↪→1, . . . , τ↪→n).

From πT 1 = 1 we get τ↪→i = 1/πi . Moreover,

(I − P)Tπ = 11T π −Diag(τ↪→1, . . . , τ↪→n)π = 0.

Then Tπ ∈ Ker(I − P) = 〈1〉, that is,

Random target lemma

(Tπ)i =n∑

j=1

πjτi→j = κ,

where κ is the Kemeny’s constant.


Computing T

Let X1,X2 be any two solutions of (I − P)X = 11T −Diag(π)−1. Then,

(I − P)(X1 − X2) = O X1 − X2 = 1vT , v ∈ Rn.

Hence, given any solution X we have T = X − 1diag(X )T .The matrix L = D1/2(I − P)D−1/2 is the normalized Laplacian. It holdsKer(L) = 〈diag(D1/2)〉 and

LD1/2XD1/2 = D1/2(I − P)XD1/2

= D1/2(11T −Diag(π)−1)D1/2

= D1/211T D1/2 − (1T d)I = −(1T d)LL+.

We can set D1/2XD1/2 = −(1T d)L+ and we obtainX = −(1T d)D−1/2L+D−1/2. In conclusion,

T = (1T d)[1vT −∆L+∆

], ∆ = D−1/2, vi = L+

ii /di .


The symmetric and skew-symmetric parts of T

The symmetric part of T gives us the commute time of i and j ,

Sij = (T + TT )ij = τi→j + τj→i .

We have S = (1T d)R where R is the resistance matrix,

Rij = (ei − ej)T ∆L+∆(ei − ej).

Rij is a distance taking into account the availability of multiple pathsfrom i to j . The number

ri = 1n

∑nj=1 Rij = . . . = 1

n

[tr(∆L+∆) +

L+iidi

]is the average resistance distance of node i .


The symmetric and skew-symmetric parts of T

The symmetric part of T gives us the commute time of i and j ,

Sij = (T + TT )ij = τi→j + τj→i .

We have S = (1T d)R where R is the resistance matrix,

Rij = (ei − ej)T ∆L+∆(ei − ej).

On the other hand, rk(T − TT ) = 2. Indeed,

T − TT = (1T d)[1vT − v1T

].

Here v = diag(∆L+∆) but the substituition v → v + α1 withα ∈ R gives another solution.


The RWC

Using very different notations and arguments,Noh and Rieger (2004) proved the identity T − TT = 1kT − k1T,found a special solution k ∈ Rn and defined the Random WalkCentrality of node i as

RWC (i) = 1/ki .

From τi→j − τj→i = kj − kiwe get τi→j ≶ τj→i iff RWC (i) ≶ RWC (j).That is, RWC ranks nodes according to their accessibility.


The RWC

From T = (1T d)[1vT −∆L+∆

]we have

T − TT = (1T d)[1vT − v1T

]with vi = L+ii /di . Owing to π = d/(1T d) we get

τi→j − τj→i =L+jjπj−L+iiπi.

This gives us one possibile explicit formula for RWC:

RWC (i) = πi/L+ii .

But what is the significance of that number?


Quantifying accessibility

Lemma

L+ii /πi =∑

j πjτj→i .

Indeed,

πT T = (1T d)πT[1vT −∆L+∆]

= (1T d)vT − dT∆L+︸︷︷︸=0

∆ = (1T d)vT .

Then, L+ii /πi = (1T d)vi = (πT T )i =∑

j πjτj→i .

Equivalently, (1T d) diag(∆L+∆) = TT π.



Lemma

L+ii /πi =∑

j πjτj→i .

Theorem

Every solution k ∈ Rn to T − TT = 1kT − k1T has the form

ki =∑

j πjτj→i + α, α ∈ R.



Lemma

L+ii /πi =∑

j πjτj→i .

Theorem

Every solution k ∈ Rn to T − TT = 1kT − k1T has the form

ki =∑

j πjτj→i + α, α ∈ R.

Choose α = κ, the Kemeny’s constant. Then,

ki =∑

j πj(τi→j + τj→i ).

This formula has a clear probabilistic interpretation.


Structures in complex networks

Interesting sub-structures in complex networks: commmunities,(almost-)bipartite subgraphs, and core-periphery

Random walks are powerful tools to discover them.


An experiment: A yeast PPI network

Left to right: adjacency matrix; cumulative degree distribution;ki vs. ri .


An experiment: A yeast PPI network

Left: permuted adjacency matrix (nodes renumbered by increasingv -values). Right: 2D layout (colors represent v -values).


When connectivity is high

Theorem

Let 1 = λ1 > λ2 ≥ . . . ≥ λn be the eigenvalues of ∆A∆. Then

1

1− λn≤ L+ii ≤

1

1− λ2.

Non-bipartite, well connected networks, having many differentpaths joining any node pair, have |λi | ≤ ε < 1 for i = 2 . . . n.In that case ki ≈ 1/πi +O(ε).


In conclusion

The (reciprocated) RWC

quantifies node acessibility from the stationary distribution

is the average resistance distance in disguise

is able to detect ‘peripheral’ nodes in a C-P network

is close to degree, if connectivity is large.


References

D. Bini, J. Hunter, G. Latouche, B. Meini, P. Taylor. Why isKemeny’s constant a constant? J. Appl. Probab. 55 (2018),1025–1036.

D. F., F. Tudisco. A modularity based spectral method forsimultaneous community and anti-community detection. LinearAlgebra and its Applications 542 (2017), 605–623..

D. F., F. Tudisco. The expected adjacency and modularity matricesin the degree corrected stochastic block model. Special Matrices 6(2018), 110–121.

S. Kirkland. Random walk centrality and a partition of Kemeny’sconstant. Czechoslovak Math. J. 66(141) (2016), 757–775.

L. Lovasz. Random Walks on Graphs: A Survey. Combinatorics 2(1993), 1–46.

J. D. Noh, H. Rieger. Random walks on complex networks. Phys.Rev. Letters 92 (2004), 118701 (4pp.).


#Advertising


dario fasino university of udine rome, feb. 19, 2019 · notations a: adjacency matrix of...

Documents