Generalizing Kronecker Graphs in order to Model Searchable Networks

Transcription

1 Generalizing Kronecker Graphs in orer to Moel Searchable Networks Elizabeth Boine, Babak Hassibi, Aam Wierman California Institute of Technology Pasaena, CA {eaboine, hassibi, Abstract This paper escribes an extension to stochastic Kronecker graphs that provies the special structure require for searchability, by efining a istance -epenent Kronecker operator We show how this extension of Kronecker graphs can generate several existing social network moels, such as the Watts-Strogatz small-worl moel an Kleinberg s latticebase moel We focus on a specific example of an expaning hypercube, reminiscent of recently propose social network moels base on a hien hyperbolic metric space, an prove that a greey forwaring algorithm can fin very short paths of length O((log log n 2 for graphs with n noes I INTRODUCTION There exists a large boy of work exploring the various structural properties of social networks small iameter, high clustering, heavy-taile egree istributions, an searchability; see [], [2], an [] for surveys of this area Many generative moels have been propose that capture some of these properties with varying levels of complexity, but the challenge remains to evelop a simple, quantitative moel that can exhibit all of these properties For example, the simple Erös-Rényi ranom graph maintains a small iameter, but fails to capture many of the other properties [4], [5] The combination of small iameter an high clustering is often calle the small-worl effect, an Watts an Strogatz (see section 22 generate much interest on the topic when they propose a moel that maintains these two characteristics simultaneously [6] Several moels were then propose to explain the heavy-taile egree istributions an ensification of complex networks; these inclue the preferential attachment moel [7], the forest-fire moel [8], [9], Kronecker graphs [0] [], an many others [] As emonstrate by Milgram s 967 experiment using real people, iniviuals can iscover an use short paths using only local information [2] This is terme searchability Kleinberg focuses on this characteristic in his lattice moel an proves searchability for a precise set of input parameters, but his moel lacks any heavy-taile istributions [], [], [4] One promising moel propose recently is Kronecker graphs [0], [], [5] These graphs are simple to generate, are mathematically tractable, an have been shown to exhibit several important social network characteristics In particular, these graphs can have heavy-taile egree an eigenvalue istributions, a high-clustering coefficient, small iameter, an ensify over time Aitionally, Leskovec evelope an algorithm that coul fin an appropriate 2x2 or x initiator matrix to fit real-worl ata However, Kronecker graphs are not searchable by a istribute greey algorithm [5] The contribution of this work is to provie an extension of Kronecker graphs that maintains searchability We provie a general moeling framework that shoul allow the exploration of the interaction of several of the properties mentione above Instea of using the traitional Kronecker prouct operation to grow the network, we introuce a istance - epenent Kronecker operation that iteratively grows the network from an initiator matrix This extension allows for the generation of traitional moels such as the Watts- Strogatz ring, the Kleinberg moel, as well as the original Kronecker graphs The key feature of this new moel is that the likelihoo of long-range contacts is now base on a istance measure, which preserves the natural istinction between local an long-range contacts This aitional structure allows the moel to generate graphs that are searchable To illustrate this new moel, we present a few examples an stuy one particular example in etail: an expaning hypercube This example is chosen to mimic the efining features of tree metrics an hyperbolic space, which are thought to be representative of the Internet an social networks [6], [7], [8] In these moels an in ours, the key is to have exponentially expaning neighborhoos aroun each noe, which leas to O(log n iameter, even without long-range links Aing long range links, the iameter can shrink to O(log log n [9] In our expaning hypercube moel, we use a log n-imensional hypercube as the unerlying lattice, with istance -epenent long-range contacts ae to each noe We grow our moel by aing a imension in each iteration, oubling the number of noes We then prove that local, greey agents can fin paths of length O((log log n 2 This paper is organize as follows Section II escribes in etail our moel an its relation to the original Kronecker graph moel an other traitional moels Section III gives a etaile proof of searchability for the example of an expaning hypercube Section IV conclues with propose future work II DISTANCE-DEPENDENT KRONECKER GRAPHS In this section we escribe the original formulation of stochastic Kronecker graphs, as well as our new istance - epenent extension of the moel We then present a few

2 examples illustrating how to generate existing network moels using the istance -epenent Kronecker graph A Stochastic Kronecker Graphs Stochastic Kronecker graphs are generate by recursively using a stanar matrix operation, the Kronecker prouct [0] Beginning with an initiator probability matrix P, with N noes, where the entries p i enote the probability that ege (i, is present, successively larger graphs P 2,, P n are generate such that the k th graph P k has N k = N k noes The Kronecker prouct is use to generate each successive graph Definition 2: The k th power of P is efine as the matrix P k, such that: P k = P k = P P P }{{} = P k P k times For each entry p uv in P k, inclue an ege in the graph G between noes u an v with probability p uv The resulting binary ranom matrix is the aacency matrix of the generate graph Kronecker graphs have many of the static properties of social networks, such as small iameter an a heavy-taile egree istribution, a heavy-taile eigenvalue istribution, an a heavy-taile eigenvector istribution [0] In aition, they exhibit several temporal properties such as ensification an shrinking iameter Using a simple 2x2 P, Leskovec emonstrate that he coul generate graphs matching the patterns of the various properties mentione above for several real-worl atasets [0] However, as shown by Mahian an Xu, stochastic Kronecker graphs are not searchable by a istribute greey algorithm [5] they lack the necessary spatial structure that allows a local greey agent to fin a short path through the network This is the motivation for the current paper B Distance-Depenent Kronecker Graphs In this section, we propose an extension to Kronecker graphs incorporating the spatial structure necessary to have searchability We a to the framework of Kronecker graphs a notion of istance, which comes from the embeing of the graph, an exten the generator from a single matrix to a family of matrices, one for each istance, efining the likelihoo of a connection occurring between noes at a particular istance We accomplish this with a new Kronecker-like operation Specifically, whereas in the original formulation of Kronecker graphs one initiator matrix is iteratively Kronecker-multiplie with itself to prouce a new aacency or probability matrix, we efine a istance - epenent Kronecker operator Depening on the istance between two noes u an v, (u, v Z, a ifferent matrix [0] For a escription of eterministic Kronecker graphs, see Leskovec et al, from a efine family will be selecte to be multiplie by that entry, as shown below C = A H = where a H (, a 2H (,2 a nh (,n a 2H (2, a 22H (2,2 a 2nH (2,n a nh (n, a n2h (n,2 a nnh (n,n H = {H i } i Z So, the k th Kronecker power is now G k = G H H }{{} k times In the Kronecker-like multiplication, the choice of H i from the family H, multiplying entry (u, v, is epenent upon the istance (u, v Note that our (u, v is not a true istance measure we can have negative istances Further, (u, v is not symmetric ((u, v (v, u since we nee to maintain symmetry in the resulting matrix Instea, (u, v = (v, u an H (u,v = H (v,u This change to the Kronecker operation makes the moel more complicate, an we o give up some of the beneficial properties of Kronecker multiplication Potentially, we coul have to efine a large number of matrices for H However, for the moels we want to generate, there are actually only a few parameters to efine, as (i, an a simple function efines H i for i > The unerlying reason for this simplicity is that the local lattice structure is usually specifie by H 0 an H, while the global, istance-epenent probability of connection can usually be specifie by an H i with a simple form So, while we lose the benefits of true Kronecker multiplication, we gain generality an the ability to create many ifferent lattices an probability of long-range contacts We note in passing that the generation of these lattice structures is not possible with the original formulation of the Kronecker graph moel For example, it is impossible to generate the Watts-Strogatz moel with conventional Kronecker graphs However, it can be one with the current generalization This is illustrate in our examples below Example : Original Kronecker Graph The simplest example is that of the original Kronecker graph formulation For this case, the istance can be arbitrary, an the family of matrices, H, is simply G, the same G use in the original efinition Thus, we efine G k = G H H }{{} k times = G G G }{{} k times Example 2: Watts-Strogatz Small-Worl Moel The next example we consier, the Watts-Strogatz moel, consists of a ring of n noes, each connecte to their neighbors within istance k on the ring The probability of a connection to any other noe on the ring is then P (u, v = p [6] To generate the unerlying ring structure with k =,

3 start with an initiator matrix K, representing the graph in figure (a Fig Generating the Watts-Strogatz Moel In orer to obtain the sequence of matrices representing the graphs in Figure, we efine a istance measure as the number of hops from one noe to another along the ring, where clockwise hops are positive, an counter-clockwise hops are negative Recall that the efinition of negative istance is require only to keep the matrix symmetric The negative matrix is ust the transpose of the matrix efine for the positive irection After each operation, the istance between noes is still the number of hops along the ring, though the number of noes oubles each time We then efine the following family of matrices, H: ( ( ( p p H 0 =, H =, H i = i > p Note that H i = H i So, starting from the initiator matrix in Figure (a, we have the following progression of matrices: p p G = p, p G 2 = G H H 0 H p H 2 H H H 0 H p H 2 G 2 = p H 2 H H 0 H H p H 2 H H 0 p p p p p p p p p p p p p p p p p p p p G 2 = p p p p p p p p p p p p p p p p p p p p Note that the W-S moel is not searchable by a greey agent; however, if P (u, v = (u,v, it becomes searchable [], [] It is possible to moel this P (u, v by simply austing H i, i as follows: ( ( 2i 2i+ H 0 =, H i = i, i, i n 2, 2i 2i H i = i ( 2i 2i 2i 2i, i, i = n 2 As in the previous examples, H i = H i The ifferent efinition for the mile noe in the ring is ue to the fact that we nee the probability of a connection to reach a minimum at this point, an then start to rise again We omit showing the resulting matrices for brevity This example alreay illustrates that the generalize operator we have efine allows the generation of searchable networks, but we will provie another more realistic example in the next section Example : Kleinberg-like Moel The final example we consier, Kleinberg s lattice moel, is particularly pertinent as it was shown to be searchable [] In the original formulation, local connections of noes are efine on a k-imensional lattice, an long-range links occur between two noes at istance with probability proportional to α We focus on a Kleinberg-like moel here, where instea of a k-imensional lattice, we have an an expaning hypercube as our unerlying lattice In this example, at any point, the graph is a hypercube with some extra long-range connections, an when it grows, it grows by oubling the number of noes an aing a imension to the hypercube Note that we will have n noes arrange on a k = log n- imensional hypercube This example is of particular interest ue to recent work suggesting that many networks have an unerlying hyperbolic or tree-metric structure [7], [6] The expaning hypercube captures the core of these topologies, as the number of noes at istance grows exponentially in Interestingly, this example is also very naturally represente using our istance -epenent Kronecker operation an a Hamming istance as our istance measure Fig 2 Example: the growth of an expaning hypercube To efine the expaning hypercube, we efine a graph G with n noes, numbere n, where each noe is labele with its corresponing log n-length bit vector We efine the istance between two noes as the Hamming istance between their labels The family of matrices H is as follows: ( ( βi H 0 =, H i =, for all i β i

4 where β = a normalizing constant, β i = P (i P (i+ The graph may or may not be searchable epening on P (i To mimic ( Kleinberg s moel, we let P (i = i α, so that α i β i = i+ Thus, for the sequence of graphs shown in the figure above, we have the following sequence of matrices: ( G =, G = β β G 2 = β, β β β β β β 2 β β β β 2 β β β β β 2 β β β β 2 β β β β β β 2 β β β β 2 β β β β β 2 β β β β 2 β β β From the matrix, we can tell that in each step, { if (u, v = 0, P (u, v = (u, v α otherwise In the original k-imensional lattice, a istribute algorithm (as efine in Section III, can fin paths of length O(log n only if α = k []; in the moifie case presente above, we will see in the next section that we nee a ifferent probability of connection to fin short paths III PROVING SEARCHABILITY FOR AN EXPANDING HYPERCUBE While our moel is more complicate than the original Kronecker graphs, it can capture several existing network moels, an it incorporates istance into the probability of connection, allowing for several parameters in which searchability can be proven Given the moel of Example in the previous section, we will show that is searchable by a simple greey, istribute algorithm if the probability of a connection is inversely proportional to the number of noes in an exponentially expaning torus We efine a ecentralize algorithm A similar to [] In each step, the current message-holer u passes the message to a neighbor that is closest to the estination, t Each noe only has knowlege of its aress on the lattice (given by its bit vector label, the aress of the estination, an the noes that have previously come into contact with the message For the graph to be searchable, we nee to have this istribute algorithm A be able to fin short paths through the network of O((log log n 2 = O(log 2 k Recall that the hypercube itself provies paths of length log n; the aition of ranom long-range links reuces the iameter, similar to [9] We prove that for a particular efinition of β i, the algorithm A can fin these paths Specifically: Theorem : If β 0 =, β = [2 log k ln ], [ (k 2i ] [ (k 2(i+ β i = i (i + ] i 2 i i+ such that the probability of a connection is if (u, v = 0, P (u, v = [ (k 2 ] log k ln if (u, v = then the ecentralize algorithm A will fin paths of length O((log log n 2 Proof: We will say that the execution of A is in phase when 2 < (u, t 2 + ( Thus, the largest value of is log k Suppose we are in phase ; we want to fin out the probability that the phase ens at this step This is equivalent to the probability that the message enters a set of noes B where B = {v : (v, t 2 } Pr(msg enters B = v B ( P (u, v : v B (2 = u +2 = u 2 ( P ( Nu,t( u +2 ( = u 2 ( P ( min N( (4 where N u,t ( = {v : (v, t 2, (u, v = } an min N( = min N u,t( u,t,(u,t= In any network moel, enforcing searchibility boils own to etermining this min N(, the minimum number of noes at a istance from a given noe u within a ball of noes centere aroun the estination, t, as illustrate in Figure Once this min N( is foun, if we set the Fig Relative positions of noes u,v, an t in phase

5 probability of a connection between two noes istance apart to be inversely proportional to min N(, with an appropriate constant log k ln, we will fin that each phase escribe above will en in approximately log k steps, an, as there are only log k such phases, our greey forwaring algorithm will fin be able to fin very short paths of length O(log 2 k To etermine min N( in our case, since the istance measure is a Hamming istance, we must count the number of possible bit vectors that are at a specific istance from a noe u, while still being within a certain istance of the estination We show in Lemma 5 in Appenix A that min N( = ( k 2 Continuing, we have Pr(msg enters B u +2 = u 2 ( P ( min N( e u +2 = e log k ln (5 = u 2 min N( P ( u +2 (6 = u 2 (7 e u +2 log k ln ln u 2 (8 e 2 log k ln ln 2 (9 = e log k (0 log k ( where line (6 is true in the limit of large n (lim n ( x/n n = e x, an line ( comes from the power series expansion of e x Let X enote the total number of steps spent in phase So, EX = Pr[X i] (2 i= i= ( i ( log k = log k (4 Let X enote the total number of steps taken by the algorithm A log k X = X (5 an =0 log k EX = EX (6 =0 ( + log k(log k (7 δ(log k 2, for a large enough δ (8 Since the expecte number of steps in phase is log k, an there are at most log k phases, the expecte amount of steps taken by the algorithm A is at most δ log 2 k So, with this efinition of P (, the istribute algorithm provies searchabilty IV CONCLUSION We have presente a generalization of Kronecker Graphs that maintains several of their avantageous properties while aitionally allowing the generation of graphs that are searchable In general, by efining a family of istance - epenent matrices, use with a Kronecker-like operation to grow a network, we can efine both local regular structures an global istance-epenent long-range connections Even though this moel is more complicate than the original Kronecker graph formulation, we gain generality (as we can generate other social network moels an searchability (as shown by our example This general moeling framework shoul allow us to explore the interaction of properties such as searchability an heavy-taile egree istributions Other recent moels have attempte to explore this interaction by imposing a power-law egree istribution onto existing moels like the Kleinberg lattice [20] Distance-epenent Kronecker graphs, in contrast, allow for these properties to emerge naturally from the efinition of the family of istanceepenent matrices The expaning hypercube example in Section III bears a resemblance to recent moels propose base on hien hyperbolic metric spaces, which have been shown to be representative of real-worl complex networks [6] In [6], the observe network topology is base upon a hien metric space, assume to be a non-eucliean hyperbolic space As in our moel, this leas to scalefree topologies an very efficient greey forwaring, though searchability has only been shown via simulations [9] This paper shoul be viewe as a first step towars the analysis of istance -epenent Kronecker graphs There are many interesting questions that remain, incluing how to parameterize our moel from real-worl atasets Ieally, given any ataset, we woul like to be able to fin an appropriate family of istance -epenent matrices to match any esire characteristic of the ataset We woul also like to use our moel to examine the interaction of searchability an other properties, particularly egree istributions As our moel was not esigne to emonstrate one characteristic in particular, but rather a range of characteristics, it is ieally suite to this sort of analysis Finally, we woul also like to investigate the ynamics of complex networks within this moel etermining how we coul incorporate growth an mobility into our moel REFERENCES [] MEJ Newman, The structure an function of complex networks, SIAM Review, 200 [2] R Albert an A Barabási, Statistical mechanics of complex networks, Reviews of Moern Physics, vol 74, 2002 [] J Kleinberg, Complex networks an ecentralize search algorithms, in Proc of International Conference of Mathematicians, 2006 [4] B Bollobás, Ranom Graphs, Acaemic Press, Inc, 985 [5] P Erös an A Rényi, On ranom graphs, Publicationes Mathematicae 6, p , 959 [6] DJ Watts an SH Strogatz, Collective ynamics of small-worl networks, Nature, vol 9, pp 440, 998 [7] A Barabási an R Albert, Emergence of scaling in ranom networks, Science, vol 286, pp , 999

6 [8] J Leskovec, J Kleinberg, an C Faloutsos, Graphs over time: ensification laws, shrinking iameters an possible explanations, in Proc of ACM SIGKDD conf on knowlege iscovery in ata mining, 2005, pp [9] P Bak, K Chen, an C Tang, A forest-fire moel an some thoughts on turbulence, Phys Lett A, vol 47, pp 29700, 990 [0] J Leskovec, D Chakrabarti, J Kleinberg, an C Faloutsos, Realistic, mathematically tractable graph generation an evolution, using kronecker multiplication, in Conf on Principles an Practice of Knowlege Discovery in Databases, 2005 [] J Leskovec, Dynamics of Large Networks, PhD in Computer Science, Carnegie Mellon University, 2008 [2] S Migram, The small-worl problem, Psychology Toay, vol 2, pp 60, 967 [] J Kleinberg, The small-worl phenomenon: An algorithmic perspective, in Proc of the 2n ACM Symposium on Theory of Computing, 2000, pp 6 70 [4] C Martel an V Nguyen, Analyzing Kleinberg s an other smallworl moels, in In PODC 04: Proc of ACM symposium on Principles of istribute computing, 2004, pp [5] M Mahian an Y Xu, Stochastic kronecker graphs, in In WAW07: Proc of Workshop On Algorithms An Moels For The Web-Graph, 2007, pp [6] D Krioukov, F Papaopoulos, M Boguna, an A Vahat, Greey forwaring in scale-free networks embee in hyperbolic metric spaces, in Proc of MAMA Workshop at Sigmetrics, 2009 [7] A Cvetkovski an M Crovella, Hyperbolic embeing an routing for ynamic graphs, in Proc of Infocom, 2009 [8] V Ramasubramanian an D Malkhi, On the treeness of internet latency an banwith, in Proc of ACM Sigmetrics, 2009, pp 6 72 [9] M Boguna an D Krioukov, Navigating ultrasmall worls in ultrashort time, Physical Review Letters, vol 02, pp 05870, 2009 [20] P Fraigniau an G Giakkoupis, The effect of power-law egrees on the navigability of small worls, in 28th ACM Symposium on Principles of Distribute Computing (PODC, 2009 V APPENDIX - CALCULATING THE SIZE OF N u,t ( In this appenix, we show a lower boun for N u,t (, the number of noes at istance from a given noe u, still within istance 2 of the estination, t Lemma 5: min N u,t ( = ( k 2 Proof: We first count exactly the number of noes in N u,t (, the number of noes at a istance from a given noe u within a ball of noes centere aroun the estination, t, as illustrate in Figure Without loss of generality, efine t as the all zero noe, t = (000 Arrange the label of u such that u = ( 0 0 Define v = (v v 0 v 0 v 00 accoring to this partition of u, so that v an v 0 have entries an v 0 an v 00 have 0 entries Let x enote the weight, or number of ones, of the label of noe x We know the following: v + v 0 + v 0 + v 00 = k (9 We can solve in terms of v, yieling v + v 0 = u (20 v 0 + v 0 = (2 v + v 0 = v (22 v 00 =k v (2 v 0 = u v (24 v 0 = u + v (25 We also know that we must satisfy the following: v, v 0, v 0, v 00 0 (26 From these bouns we have 2 < u 2 + (27 u 2 u + 2 (28 v 2 (29 max(0, u v min( u, k, 2 (2 + u (0 Note that the secon an thir bouns o not affect v Counting the number of noes in the ball, we have N u,t ( = v u v =v l ( ( u k u v u + v ( where we have substitute v u an v l, for the upper an lower bouns above, respectively We can now approximate the number of noes in N u,t (, using the entropy approximation for combinations Let u = ak, = bk, 2 = ck, x = v Using this notation, we have where N u,t ( = v u x=v l ( x X = max ah x ak subect to ( ak x ( k( a k( b + x v u 2 k(ah( x ak +( ah x=v l ( b a+ x k a (2 ( 2 kx (4 + ( ah ( b a + x k a (5 k max(0, a b x k min(a, b, (a b+c (6 2 Note that line (4 is true as ( n k = 2 n(h(p+o( when k pn There are two solutions to the optimization problem state above, yieling two ifferent values of min N u,t ( : yieling x = ak abk when c a + b( 2a min N u,t ( = ( k x 2 = k(a b + c when c < a + b( 2a 2 yieling ( k 2 min N u,t ( = The function is concave in x, so the two possible solutions can be seen from the bounary points an the bouns for the region The resulting min N u,t ( are erive in Sections A an B below As the secon solution yiels a smaller min N u,t (, we have an overall min N u,t ( = ( k 2

7 A Solution : c a + b( 2a In this region, the solution to the unconstraine problem, x = ak abk gives us the maximal X Substituting in for the size of N u,t (, we have ( ak abk b a+ k(ah( ak abk k ak +( ah a N u,t ( = 2 (7 = 2 k(ah( b+( ah(b (8 = 2 kh(b (9 ( k (40 bk ( k = (4 where line (4 is true in the limit of large k B Solution 2: c < a + b( 2a In this region, we choose one of the bounary points, x 2 = 2k(a b + c, as the solution to the maximization problem Substituting this solution for x in N u,t (, we obtain N u,t ( = 2 a b+c k(ah( 2a +( ah ( a+b+c 2( a (42 This gives us a function of a, b, c, so we want to fin the worst case a, c that minimizes N u,t ( The new optimization problem is thus f(b = min N u,t ( (4 ( ( a b + c a + b + c = min ah + ( ah a,c 2a 2( a (44 Note that the bouns for this region are: a b c 0 2 a b + c 0 c < a 2c 4 0 c a, b a b c 7 0 a + b c 8 0 a + b c 2ab where an 2 come from the bouns on (u, v, comes from the bouns on u, an 4 an 5 come from the ranges for an the size of the network Note that -5 are always true, not ust in this region 6,7, an 8 come from the fact that our solution x 2 is minimal in this region Note that 8 implies 7 Computing the Hessian of the function in (45 shows that it is concave in both a an b; however, we omit the cumbersome calculation for brevity Since our function is concave, the min N u,t ( is foun from the bounary points of Region 2 Rearranging the bouns from before in terms of a we have: a b + c 2 a b c a > c, a 2c 4 c > 0, c a, a 6 a 2 b c 7 a b + c 8 a c 9 a c 2b 2b b 2b when b b 2 2b when b > 2 Fig 4 Bounaries of f(b when b 2 When b 2, only bouns (,2,,4 apply to f(b, yieling 5 points that we nee to examine, as shown in Figure 4 If b 5, then f(b is minimal at point (, ( b, 2b, yieling 2b k( min N u,t ( = 2 = ( k 2bk bk ( k 2 (45, for large k (46 H ( b 2b (47 If b < 05, then f(b is minimal at point (5, (b, 2b, yieling min N u,t ( = 2 k2b = 4 (48 When b > 2, only bouns (2,,4,an 8 apply to f(b, yieling 4 points that we nee to examine, as shown in Figure 5 For this region, f(b is minimal at point (, matching point Fig 5 Bounaries of f(b when b 2

8 (5 in the previous region, yieling 2b k( min N u,t ( = 2 ( k 2 H ( b 2b (49, for large k (50 Thus, when b < 05, we have min N u,t ( = 4, an when b 05, we have min N u,t ( = ( k 2 Finally, we have that when c < a + b( 2a, we apply Solution 2 from Lemma 5, an we have min N u,t ( = ( k 2 when Solution 2 is vali Comparing the Solution with Solution 2, we have again that min N u,t ( = ( k 2