Fan Chung
University of California at San Diego, La Jolla CA 92093, USA, fan@ucsd.edu,http://www.math.ucsd.edu/~fan/
Abstract. We give an improved local partitioning algorithm using heat kernel pagerank, a modified version of PageRank. For a subsetS with Cheeger ratio (or conductance)h, we show that there are at least a quarter of the vertices inSthat can serve as seeds for heat kernel pagerank which lead to local cuts with Cheeger ratio at mostO(√h), improving the previously bound by a factor of
p
log|S|.
1
Introduction
With the emergence of massive information networks, many previous algorithms are often no longer feasible. A basic setup for a generic algorithm usually includes a graph as a part of its input. This, however, is no longer possible for dealing with massive graphs with prohibitive large size. Instead, the (host) graph, such as the WWW-graph or various social networks, is usually meticulously crawled, organized and stored in some appropriate database. The local algorithms that we study here involve only “local access” of the database of the host graph. For example, getting a neighbor of a specified vertex is considered to be a type of local access. Of course, it is desirable to minimize the number of local accesses needed, hopefully independent of n, the number of vertices in the host graph (which may as well be regarded as “infinity”). In this paper, we consider a local algorithm that improves the performance bound of previous local partitioning algorithms.
Graph partitioning problems have long been studied and used for a wide range of applications, typically along the line of divide-and-conquer approaches. Since the exact solution for graph partitioning is known to be NP-complete [12], various approximation algorithms have been utilized. One of the best known partition algorithms is the spectral algorithm. The vertices are ordered by using an eigenvector and only cuts which are initial segments in such an ordering are considered. The advantage of such a “one-sweep” algorithm is to reduce the number of cuts under consideration from an exponential number innto a linear number. Still, there is a performance guarantee within a quadratic order by using a relation between eigenvalues and the Cheeger constant, called the Cheeger inequality. However, for a large graph (say, with a hundred million vertices), the task of computing an eigenvector is often too costly and not competitive.
A local partitioning algorithm finds a partition that separates a subset of nodes of specified size near specified seeds. In addition, the running time of a local algorithm is required to be proportional to the size of the separated part but independent of the total size of the graph. In [21], Spielman and Teng first gave a local partitioning algorithm using random walks. The analysis of their algorithm is based on a mixing result of Lov´asz and Simonovits in their work on approximating the volume of convex body. The same mixing result was also proved by Mihail earlier independently [19]. In a previous paper [1], a local partitioning algorithm was given using PageRank, a concept first introduced by Brin and Page [3] in 1998 which has been widely used for Web search algorithms. PageRank is a quantitative ordering of vertices based on random walks on the Webgraph. The notion of PageRank which can be carried out for any graph is basically an efficient way of organizing random walks in a graph. As seen in the detailed definition given later, PageRank can be expressed as a geometric sum of random walks starting from the seed (or an initial probability distribution), with its speed of propagation controlled by ajumping constant. The usual question in random walks is to determine how many steps are required to get close to stationary distribution. In the use of PageRank, the
problem is reduced to specifying the range for the jumping constant to achieve the desired mixing. The advantage of using PageRank as in [1] is to reduce the computational complexity by a factor of logn.
In this paper, we consider a modified version of PageRank called heat kernel pagerank. Like PageRank, the heat kernel pagerank has two parameters, a seed, and a heat constant or temperature. The heat kernel pagerank can be expressed as an exponential sum of random walks from the seed, scaled by the temperature . In addition, the heat kernel pagerank satisfies a heat equation which dictates the rate of diffusion. We will examine several useful properties of the heat kernel pagerank. In particular, for a given subset of vertices, we consider eigenvalues on the induced subgraph onS satisfying Dirichlet boundary condition (details to be given in the next section). We will show that for a subset S with Cheeger ratio h, there are many vertices in S (whose volume is at least a quarter of the volume ofS) such that the one-sweep algorithm using heat kernel pagerank with such vertices as seeds will find local cuts with Cheeger ratioO(√h). This improves the previous bound ofO(√hlogs) in a similar theorem using PageRank [1, 2]. Heresdenotes the volume ofS and a local cut has volume at mosts.
2
Preliminaries
In a graph G, the transition probability matrixW of a typical random walk on a graphG = (V, E) is a matrix with columns and rows indexed byV and is defined by :
W(u, v) =
1
du if{u, v} ∈E, 0 otherwise
where thedegreeofv, denoted bydv, is the number of vertices thatvis adjacent to. We can writeW =D−1A whereAdenotes theadjacency matrixofGandDis thediagonal degree matrix.. A random walk on a graph Ghas a stationary distributionπifGis connected and non-bipartite. The stationary distributionπ, if exists, satisfiesπ(u) =du/
P
vdv.
The PageRank we consider here is also called personalized PageRank (see [13, 14]), which generalized the version first introduced by Brin and Page [3]. PageRank has two parameters, apreference vector f (i.e., the probabilistic distribution of the seed(s)) and ajumping constantα. Here, the functionf : V →Ris taken to be a row vector so that W can act onf from the right by matrix multiplication. The PageRank prα,f, with the scale parameterαand the preference vectorf, satisfies the following recurrence relation:
prα,f =αf+ (1−α)prα,fW. (1)
An equivalent definition for PageRank is the following: prα,f =α
∞ X
k=0
(1−α)kf Wk. (2)
For example, if we have one starting seed denoted by vertexu, thenf can be written as the (0,1)-indicator function χu of u. Another example is to takef to be the constant function with value 1/nat every vertex as in the original definition of PageRank in Brin and Page [3].
The heat kernel pagerank also has two parameters, with (the temperature)t≥0 and a preference vector f, defined as follows: ρt,f =e−t ∞ X k=0 tk k!f W k . (3)
We see that (3) is just an exponential sum versus (2) is a geometrical sum. For many combinatorial problems, exponential generating functions play a useful role. The heat kernel pagerank as defined in (3) satisfies the following heat equation:
∂
∂tρt,f =−ρt,f(I−W). (4) Let us define L = I−W. Then the definition of the heat kernel pagerank in (3) can be rewritten as follows: ρt,f =f Ht, whereHtis defined by Ht=e−t ∞ X k=0 tk k!W k =e−t(I−W) =e−tL = ∞ X k=0 (−t)k k! L k .
From the above definition, we have the following facts forρt,f which will be useful later. Lemma 1. For a graphG, its heat kernel pagerankρsatisfies the following:
(i) ρ0,f =f. (ii) ρt,π =π.
(iii) ρt,f1∗=f1∗= 1if f satisfies
P
vf(v) = 1 where1denotes the all1’s function (as a row vector) and 1∗ denotes the transpose of 1.
(iv) W Ht=HtW.
(v) DHt=Ht∗D andHt=Ht/2Ht/2=Ht/2D−1Ht/∗2D.
Proof. SinceH0=I, (i) follows. (ii) and (iii) can be easily checked. (iv) follows from the fact that Ht is a polynomial ofW. (v) is a consequence of the fact thatW =D−1A.
3
Dirichlet eigenvalues and the restricted heat kernel
For a subsetSofV(G),there are two types of boundary ofS — thevertex boundaryδ(S) andedge boundary
∂(S). The vertex boundaryδ(S) is defined as follows:
δ(S) ={u∈V(G)\S : u∼v for somev ∈S}.
For a single vertexv, thedegreeofv, denoted bydv is equal to|δ(v)|(which is short for|δ({v})|.
TheclosureofS, denoted byS∗, is the union ofS andδS. For a functionf :S∗→R, we sayf satisfies theDirichlet boundary conditioniff(u) = 0 for allu∈δ(S). We use the notation f ∈D∗S to denote thatf satisfies the Dirichlet boundary condition and we require thatf 6= 0.
Forf ∈D∗S, we define a Dirichlet Rayleigh quotient: RS(f) = P xP∼y(f(x)−f(y))2 x∈Sf2(x)dx . (5)
where the sum is to be taken over all unordered pairs of verticesx, y∈S∗such thats∼y. The Dirichlet eigenvalue of an induced subgraph onS of a graphGcan be defined as follows:
λS = inf f∈D∗SRS(f) = inf f∈D∗S hf,(DS−AS)fi hf, DSfi = inf g∈D∗S hg,LSgi hg, gi ,
where XS denotes the submatrix of a matrix X with rows and columns restricted to those indexed by vertices inS, and the LaplacianL andLS are defined byL=D−1/2(D−A)D−1/2 andLS =D−1
/2
S (DS− AS)D−1
/2
S , respectively. Here we callfthe combinatorial Dirichlet eigenfunction ifRS(f) =λS. The Dirichlet eigenfunctions are the eigenfuctions of the matrixLS. (The detailed proof of these statements can be found in [5].)
The Dirichlet eigenvalues ofS are the eigenvalues ofLS, denoted by λS,1≤λS,2≤ · · · ≤λS,s,
where s = |S|. The smallest Dirichlet eigenvalue λS,1 is also denoted by λS. If the induced subgraph on S is connected, then the eigenvector of LS associated with λS is all positive (using the Perron-Frobenius Theorem [20] onI− LS). The reader is referred to [5] for various properties of Dirichlet eigenvalues.
For a subset S of vertices in G, the edge separater (or the edge boundary) whose removal separates S consists of all edges leavingS. Namely,
∂S={{u, v} ∈E : u∈S andv6∈S}.
How good is the edge separator? The answer depends both on the size of the edge separator and thevolume
ofS, denoted by vol(S), isPu∈Sdu. The volume of a graphG, denoted by vol(G), isPudu. TheCheeger ratioofS, denoted byhS, is defined by
hS =
|∂S| min{vol(S),vol( ¯S)}
where ¯S=V\Sdenotes the complement ofS. TheCheeger constantof a graphGishG= minS⊆V hS.The Cheeger constant of a graphGis often called the conductance ofG. For a given subsetS, we define thelocal Cheeger ratio, denoted byh∗S, as follows:
h∗S = inf T⊆ShT. Note that in generalhS is not necessarily equal toh∗S.
The Dirichlet eigenvalue λS and the local Cheeger ratio h∗S are related by the following local Cheeger inequality:
h∗S ≥λS ≥ (h∗S)2
while the proof can be found in [6].
In [8], the following weighted Rayleigh quotient: Rφ(f) = sup c P x∼y(f(x)−f(y))2φ(x)φ(y) P x∈S(f(x)−c)2φ2(x)dx , (6)
whereφis the combinatorial Dirichlet eigenfunction which achievesλS . Then we can define λφ= inf
f6=0Rφ(f).
Here we state several useful facts concerningλS andλφ (see [8]).
Lemma 2. For an induced subgraphS of a graphG, the Dirichlet eigenvalues ofS satisfy
λS,2−λS =λφ≥λS.
Theorem 1. For an induced subgraph S of G, the combinatorial Dirichlet eigenfunction φ which achieves
λS satisfies X x∈S φ(x)dx !2 ≥1 2vol(S) X x∈S φ2(x)dx.
Proof. We note that
λS,2= inff
P
xP∼y(f(x)−f(y))2 x∈Sf(x)2dx
,
wheref ranges over all functions satisfyingPx∈Sf(x)φ(x) = 0. Now we use the fact that λS = P xP∼y(φ(x)−φ(y))2 x∈Sφ(x)2dx . We then have λS,2≤ P x∼y(φ(x)−φ(y))2 P x∈S(φ(x)−c)2dx = λS P x∈Sφ2(x)dx P x∈Sφ2(x)dx−c2vol(S) , where c= P x∈Sφ(x)dx vol(S) . This implies P x∈Sφ(x)dx 2 vol(S)Px∈Sφ2(x)dx ≥ λS,2−λS λS,2 = λφ λφ+λS ≥ 1 2 by using Lemma 2.
4
A lower bound for the restricted heat kernel pagerank
For a given set S, we consider the distribution fS with choosing the vertex u with probability fS(u) = du/vol(S) if u∈S, and 0 otherwise. Note thatfS can be written as vol(1S)χSD where χS is the indicator function forS. For any functiong : V →R, we define g(S) =Pv∈Sg(v).
In this section, we wish to establish a lower bound for the expected value of heat kernel pagerank ρt,u=ρt,χu overuin S. We note that
E(ρt,u) =X u∈S
du vol(S)ρt,u =fSHt(S).
We consider the restricted heat kernelHt0 for a fixed subsetS, defined as follows: Then the definition of the heat kernel pagerank in (3) can be rewritten as follows:
ρ0t,f =f Ht0 whereHt0 is defined by Ht0=e− t ∞ X k=0 tk k!W k S =e−t(IS−WS) =e−tLS = ∞ X k=0 (−t)k k! L k S.
Also,Ht0 satisfies the following heat equation: ∂ ∂tH
0
t=−LSHt0. (7)
From the above definition, we immediately have the following: Lemma 3.
Ht(0 x, y)≤Ht(x, y)
for all verticesxandy. In particular, for a non-negative function f :V →R, we have
f Ht0(v)≤f Ht(v)
for every vertex v inG.
Therefore, it suffices to establish the desired lower bound for the expected value of restricted heat kernel pagerank.
Following the notation in Section 3, we consider the Dirichlet eigenvalues λS,i ofS and their associated Dirichlet combinatorial eigenfunctionφi withR(φi) =λS,i, fori= 1, . . . ,|S|. Clearly,φiD1
/2
eigenfunction ofLS and form a basis for functions defined onS. Here we assume that
P
u∈Sφi(u)2du= 1. To simplify the notation, in this proof we writeλ0i=λS,i andλ01=λS.
We expressf =pvol(S)fS in terms ofφi as follows: f DS−1/2=X i aiφiD1 /2 S , where ai= X u∈S φi(u)du p vol(S). Sincekf D−S1/2k=kf D −1/2 S k2= 1, we have X i a2i = 1.
From the above definitions we have
fSHt(0 S) =kfH0t/2k2,
whereH0t=D1 /2
S Ht0D−1 /2
S . From Theorem 1, we know the fact that
a21= P u∈Sφ1(u)du 2 vol(S) ≥1/2 ≥1−X j6=1 a2j. Since fSHt0(S) = X i a2ie− λ0it , we have fSHt0(S)≥a21e− λSt ≥ 1 2e −λSt . We have proved the following:
Theorem 2. For a subsetS, the Dirichlet heat kernelHt0 satisfies fSHt0(S)≥
1 2e
−λSt .
As an immediate consequence of Theorem 1, Lemmas 2 and 3 and Theorem 2, we have the following: Corollary 1. In a graphG, for a subset S of vertices the heat kernel pagerank ρt,fS satisfies
E(ρt,u(S)) =ρt,fS(S)≥ 1 2e −λSt≥ 1 2e −h∗St ,
5
A local lower bound for heat kernel pagerank
Corollary 1 states that the expected value ofρt,uis at leaste−h(S)t. Thus, there exists a vertexuinS such that ρt,u(S) is at leaste−h(S)t. However, in order to have an efficient local partitioning algorithm, we need to show that there are many verticesvsatisfying
ρt,v(S)≥cρt,fS(S) for some absolute constantc. To do so, we will prove the following:
Theorem 3. In a graph G with a given subset S of vertices, the subsetT ={u ∈S : ρt,u(S) ≥ 14e−tλS}
satisfies
vol(T)≥1 4vol(S),
if t≥1/λS.
From Lemma 3, Theorem 3 follows directly from Theorem 4:
Theorem 4. In a graph G with a given subset S of vertices, the subsetT ={u ∈S : ρ0t,u(S) ≥ 14e− tλS}
satisfies
vol(T)≥1 4vol(S),
if t≥1/λS.
To prove Theorem 4, we first prove the following lemma along the second-moment methods: Lemma 4. Ift≥1/λS, then X u∈S du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2≤5 4ρ 0 t,fS(S)2.
Proof. We note that
X u∈S du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2 =X u∈S du vol(S)ρ 0 t,u(S)2−(ρ0t,fS(S))2 =fSH20t(S)−(fSHt(0 S))2.
It suffices to show that
fSH20t(S)≤ 9 4(fSH
0
t(S))2.
We consider the Dirichlet eigenvaluesλS,iofSand their associated Dirichlet combinatorial eigenfunctions φi with R(φi) = λS,i = λ0i, for i = 1, . . . ,|S|. Here, φiD1
/2
S are orthonormal eigenfunctions of LS with
P
u∈Sφi(u)2du= 1.
We can write f =χSDSvol(S)−1/2=fS
p vol(S) by: f DS−1/2= X i aiφiD1 /2 S .
We have X i
a2i = 1,
sincekf D−S1/2k2= 1. From Theorem 1, we know the fact that
a21≥1/2≥1−X j6=1 a2j. Also we have fSH20t(S) =kfHt0k2= X i a2ie−2 λ0it , whereH0t=D1 /2 S Ht0D−1 /2 S .
Therefore we have, for t≥1/λ01,
fSH20t(S) = X i a2ie−2 λ0it ≤a21e−2λ01t+ (1−a2 1)e−2 λ02t =a21e−2λ01t+ (1−a2 1)e−4 λ01t ≤9 8a 2 1e−2 λ01t
sinceλ02≥2λ01 by using Lemma 2. Therefore we have fSH20t(S)≤ 9 8a 2 1e−2 λSt ≤ 9 4a 4 1e−2 λSt ≤ 9 4( X i a2ie− λ0it )2 as desired.
Now, we are ready to proceed to prove Theorem 4 which then implies Theorem 3.
Proof of Theorem 4:
Suppose vol(T) ≤ vol(S)/4. We wish to show that this leads to a contradiction. Let T0 = S\T. We consider X u∈S du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2 =X u∈T du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2 + X u∈T0 du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2 ≥ P u∈T du vol(S)(ρ0t,u(S)−ρ0t,fS(S)) 2 vol(T) vol(S) + P u∈T0 du vol(S)(ρ0t,u(S)−ρ0t,fS(S)) 2 vol(T0) vol(S) ≥ X u∈T0 du vol(S)(ρ 0 t,u(S)−ρ0t,fS(S)) 2 1 vol(T) vol(S) + 1 vol(T0) vol(S) since X u∈S du vol(S)(ρ 0 t,u−ρ0t,fS) = 0.
Therefore we have X u∈S du vol(S) ρ 0 t,u(S)−ρ0t,fS(S) 2≥ vol(T0) vol(S)( 3 4)ρ 0 t,fS(S) 2 1 vol(T) vol(S) + vol(1T0) vol(S) ≥ 4(3 4) 4+ (3 4) 3ρ0 t,fS(S)2 ≥ 27 16ρ 0 t,fS(S)2 > 5 4ρ 0 t,fS(S)2,
which is a contradiction to Lemma 4. Theorem 4 is proved.
6
An upper bound for heat kernel pagerank
For a functionf :V →R, we can order vertices according to theirf values. The ordered list is then called a
sweep. The segmentSiconsists of the firstivertices (by breaking ties arbitrarily). We define as-local Cheeger ratio of a sweepf, denoted byhf,sto be the minimum Cheeger ratio of the segmentSiwith 0≤vol(Si)≤2s. If no such segment exists, then we sethf,sto be 0. We will establish the following upper bound for the heat kernel pagerank in terms ofs-local Cheeger ratios. The proof is similar but simpler than that in [7]. Theorem 5. In a graphGwith a subset S with volumes≤vol(G)/4, for any vertexuin G, we have
ρt,u(S)−π(S)≤
r
s du
e−tκ2t,u,s/4,
whereκt,u,s denote the minimums-local Cheeger ratio over a sweep of ρt,u.
Proof. For a functionf : V →R, we definef(u, v) =f(u)/du ifv is adjacent touand 0 otherwise. For an integerx, 0≤x≤vol(G)/2, we define
f(x) = max T⊆V×V,|T|=x
X
(u,v)∈T
f(u, v).
We can extendfto all realx=k+r, with 0≤r <1 by definingf(x) = (1−r)f(k)+rf(k+1). Ifx= vol(Si) where Si consists of vertices with the i highest values off(u)/du, then it follows from the definition that f(x) =Pu∈Sif(u). Alsof(x) is concave in x.
We consider the lazy walkW= (I+W)/2. Then fW(S) =1 2 f(S) + X u∼v∈S f(u, v) =1 2 X uorv∈S f(u, v) + X uand v∈S f(u, v) ≤1 2 f(vol(S) +|∂S|) +f(vol(S)− |∂S|) = 1 2 f(vol(S)(1 +hS)) +f(vol(S)(1−hS)) .
This can be straightforwardly extended to real x with 0 ≤ x ≤ vol(G)/2. In particular, we focus on x satisfying 0≤x≤2s≤vol(G)/2 and we chooseft=ρt,u−π. Then
ftW(x)≤ 1
2 ft(x(1 +κt,u,s)) +ft(x(1−κt,u,s))
. We now consider forx∈[0,2s],
∂ ∂tft(x) =−ρt,u(I−W)(x) =−2ρt,u(I−W)(x) =−2ft(x) + 2ftW(x) (8) ≤ −2ft(x) +ft(x(1 +κt,u,s)) +ft(x(1−κt,u,s)) ≤0
by the concavity offt. Supposegt(x) is a solution of the equation in (8) satisfyingf0(x)≤g0(x),ft(0) =gt(0)
and ∂t∂ft(x)|t=0 ≤ ∂t∂gt(x)|t=0. Then, we haveft(x)≥gt(x). It is easy to check thatgt(x)≤e−
tκ2t,u,s/4qx du using−2 +√1 +x+√1−x≤ −x2/4. Thus, ρt,u(S)−π(S)≤ρt,u(s)−π(s) ≤ r s du e−tκ2t,u,s/4, as desired.
7
A local Cheeger inequality and a local partitioning algorithm
Lethsdenote the minimum Cheeger ratiohS with 0≤vol(S)≤2s. Also letκt,2s denotes the minimum of κt,u,2s over allu. Combining Theorem 3 and Theorem 5, we have that the set ofusatisfying
1 2e −t h∗S ≤ 1 2e −tλS ≤ ρt,fS(s)−π(s)≤ √ se−tκ2t,u,2s/4,
has volume at least vol(S)/4, providedt≥1/h2
S≥1/λS.
As an immediate consequence, we have the following local Cheeger inequality:
Theorem 6. For a subsetS of a graphGwith vol(S) =s≤vol(G)/4 and t≥logs/(h∗S)2, with probability
at least1/4 a vertexuinS satisfies
h∗S ≥λS ≥ κ2 t,u 4 − 2 logs t
where the Cheeger ratio κt,u is determined by the heat kernel pagerank with seed u.
Corollary 2. Fors ≤ vol(G)/4,and t ≥ 4 logs/(hS∗)2, and a set S of volume s, the Cheeger ratio κt,u,
determined by the heat kernel pagerank with a random seed uinS, satisfies
h∗S ≥λS≥ κ2t,u
8
The above local Cheeger inequalities are closely associated with local partition algorithms. A local partition algorithm has inputs including a vertex as the seed, the volumes of the target set and a target value φ for the Cheeger ratio of the target set. The local Cheeger inequality in Theorem 5 suggests the following local partition algorithm. In order to find the set achieving the minimums-local Cheeger ratio, one can simply consider a sweep of heat kernel pagerank with further restrictions to the cuts with smaller parts of volume between 0 and 2s.
Theorem 6 implies the following.
Theorem 7. In a graph G, for a setS with volume s≤vol(G)/4, and Cheeger ratio hS ≤φ2, there is a
subsetS0 ⊆S withvol(S0)≥vol(S)/4 such that for anyu∈S0, the sweep by using the heat kernel pagerank
ρt,u, witht= 2φ−2logs, will find a setT with s-local Cheeger ratio at most 2φ.
We note that the performance bound for the Cheeger ratio improves the earlier result in [1] by a factor of logs. In fact, the inequality in Theorem 6 suggests a whole range of trade-off. If we choosetto bet= 2φ−2 instead, then the guaranteed Cheeger ratio as in the above statement will be 2φlogsand we obtain the same approximation results as in [1].
We remark that the computational complexity of the above partitioning algorithm is the same as that of computing the heat kernel pagerank. However, the algorithmic design for heat kernel pagerank has not been as extensively studied as the PageRank. More research is needed in this direction.
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