Distance-dependent Kronecker Graphs for Modeling Social Networks

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1 Distance-epenent Kronecker Graphs for Moeling Social Networks Elizabeth Boine-Baron,* Member, IEEE, Babak Hassibi, Member, IEEE, an Aam Wierman, Member, IEEE Abstract This paper focuses on a generalization of stochastic Kronecker graphs, introucing a Kronecker-like operator an efining a family of generator matrices H epenent on istances between noes in a specifie graph embeing. We prove that any lattice-base network moel with sufficiently small istance-epenent connection probability will have a Poisson egree istribution an provie a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expaning hypercube an iscuss the similarities an ifferences of such a moel with recently propose network moels base on a hien metric space. We also prove that a greey forwaring algorithm can fin very short paths of length O((loglogn on the hypercube with n noes, emonstrating that istance-epenent Kronecker graphs can generate searchable network moels. EDICS: OTH- EMRG, SEN-APPL, SPC-APPL I. INTRODUCTION Beginning with the simple Erös-Rényi moel of ranom networks [], network science has attempte to capture the key characteristics of complex networks such as power networks, the Internet, protein interaction networks, an social networks with a simple, mathematically tractable moel. Social networks in particular have generate much interest ue to the consistency of their characteristics. These networks ten to exhibit small iameter, high clustering, scale-free egree istributions, an perhaps most importantly, they are searchable by a local greey algorithm; see [], [4], an [5] for thorough surveys of this area. The Erös-Rényi ranom graph maintains a small iameter but fails to capture many of the other key E. Boine-Baron is with the Electrical Engineering Department at the California Institute of Technology, MC 6-9, 00 E. California Boulevar, Pasaena, CA 95. Phone: ( Fax: ( eaboine@caltech.eu B. Hassibi is with the Electrical Engineering Department at the California Institute of Technology, MC 6-9, 00 E. California Boulevar, Pasaena, CA 95. Phone: ( Fax: ( hassibi@caltech.eu A. Wierman is with the Computer Science Department at the California Institute of Technology, MC 05-6, 00 E. California Boulevar, Pasaena, CA 95. Phone: ( Fax: ( aamw@caltech.eu A preliminary version of many of the results of this paper first appeare in [] properties [6], []. The combination of small iameter an high clustering is often calle the smallworl effect, an Watts an Strogatz (see section III generate much interest when they propose a moel that maintains these two characteristics simultaneously [7]. Several moels were then propose to explain the heavy-taile egree istributions an ensification of complex networks; these inclue the preferential attachment moel [8], the forest-fire moel [9], [0], Kronecker graphs [], [], an many others []. As emonstrate by Milgram s 967 experiment using real people, iniviuals can iscover an use short paths using only local information []. Kleinberg focuses on this searchability characteristic in his lattice moel an proves searchability for a precise set of input parameters, but his moel lacks any heavy-taile istributions [4], [5], [5]. The Kronecker graphs escribe in [], [], an [6] are simple to generate, mathematically tractable, an have been shown to exhibit several important social network characteristics such as heavy-taile egree an eigen-istributions, high clustering, small iameter, an network ensification. However, Kronecker graphs are not searchable by a istribute greey algorithm [6]. In this paper, we exten the moel propose in [], a generalization of stochastic Kronecker graphs that can generate searchable networks. Instea of using the traitional Kronecker operation, we introuce a new Kronecker-like operation an a family of generator matrices, H, both epenent upon the istance between two noes. This new generation metho yiels networks that have both a local (lattice-base an global (istance-epenent structure. This ual structure is what allows a greey algorithm to search the network using only local information. Aitionally, the networks generate have a high clustering (ue to the lattice structure an a small iameter (ue to the aition of long-range links. As part of the analysis of this new moel, we provie a general framework for analyzing egree istributions an the performance of greey search algorithms on a general lattice-base network. We use this framework to stuy one example in etail: an expaning hypercube with istance-epenent long-range connections. We give an explicit escription of its egree istribution, the circum-

2 stances uner which it will be searchable by a local greey algorithm, an a lower boun on its iameter. We support our finings with simulations. This example is chosen because it mimics the efining feature of tree metrics an hyperbolic space exponentially expaning neighborhoos which are thought to be representative of both the Internet an social networks [7], [8], [9], [0]. Exponentially expaning neighborhoos lea to very small iameters (O(log log n as oppose to O(log n an we can show that, as in [], a local greey algorithm on the hypercube will fin ultra-short paths, O((loglogn. This paper is organize as follows. Section II briefly efines some key concepts frequently use in social network literature. Section III escribes in etail our moel an its relation to the original Kronecker graph moel an other traitional moels. Section IV explores the connection between a Kleinberg-like expaning hypercube example an the hien metric space moels propose in [7]. Section V escribes a general analysis of egree istributions for lattice-base networks an gives a theorem showing that all such networks will have a Poisson egree istribution provie that P( is sufficiently small, an gives the relevant egree istribution for the expaning hypercube example. Section VI gives a general framework for proving searchability of a latticebase istance-epenent network moel an recovers the searchability result of [4] an finally proves that the expaning hypercube is in fact searchable. Section VII explores the iameter of the expaning hypercube example an Section VIII conclues with propose future work. The appenices support the proof of searchability for the expaning hypercube example in Section VI. II. PRELIMINARIES Before continuing further, it will be useful to efine several terms commonly use in social network literature. A social network is represente by a graph G (V,E, where V an E are the sets of vertices an eges, respectively. There is one vertex for each agent, or person, in the network, an the eges represent the relationships between iniviuals. These relationships can be summarize in an aacency matrix A where { if noes i an are connecte A i 0 otherwise. We note that while we will be working with unirecte an unweighte graphs, in general, the eges in an aacency matrix representing a social network can be both irecte an weighte, showing the irection an the values of ifferent relationships. The neighborhoo N i of a noe i is efine as the set of its immeiately connecte neighbors. The egree k i of a noe is efine as the size of its neighborhoo. We efine the geoesic between two noes u an v as the shortest path connecting them. The iameter of a network, for our purposes, is the length of the maximum geoesic for that network. Note that in some cases, what is meant by iameter is the average of all geoesics; however, for this paper we focus on the maximum. In social an most complex networks, the iameter of the network grows logarithmically with the number of noes int the network [7], []. Another useful an commonly use term is clustering, which measures the amount of community structure present in a network. For an iniviual noe, we efine a clustering coefficient C i where C i {e k} k i (k i : v,v k N i,e k E The clustering coefficient for the entire graph is then the average of the clustering coefficients over all n noes [7]. C n C i n i Finally, we call a network searchable if a istribute search algorithm can fin paths through the network of length on the orer of the iameter. For example, in Kleinberg s lattice moel, a network has iameter O(log n, an is calle searchable if a istribute algorithm can fin paths of lengtho((logn [4]. For more etails on the istribute search algorithm, see section VI. III. 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 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 []. 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,...,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.: 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 a escription of eterministic Kronecker graphs, see Leskovec et al, [].

3 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 []. In aition, they exhibit several temporal properties such as ensification an shrinking iameter. Using a simple x P, Leskovec emonstrate that he coul generate graphs matching the patterns of the various properties mentione above for several realworl atasets []. However, as shown by Mahian an Xu, stochastic Kronecker graphs are not searchable by a istribute greey algorithm [6] 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 from a efine family will be selecte to be multiplie by that entry, as shown below. C A H a H (, a H (,... a n H (,n a H (, a H (,... a n H (,n where a n H (n, a n H (n,... a nn H (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 longrange 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 : 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 [7]. To generate the unerlying ring structure with k, 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 repre-

4 4 senting 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 G H H 0 H p H H H H 0 H p H p H H H 0 H H p H H H 0 p p p p p p p p p p p p p p p p p p p p 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 [4], [5]. It is possible to moel this P(u,v by simply austing H i,i as follows: ( ( i i+ H 0,H i i, i,i n, i i H i i ( i i i i, i,i n 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. With this new efinition of H i,i, we have the following progression of matrices: / / G /, / G G H H 0 H / H H H H 0 H / H / H H H 0 H H / H H H 0 / / /4 / / / / /4 / / / / / /4 / / / / / /4 /4 / / / / / /4 / / / / / /4 / / / / /4 / / 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 example. Example : Kleinberg-like Moel. The final example we consier, Kleinberg s lattice moel, is particularly pertinent as it was shown to be searchable [4]. 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 expaning hypercube as our unerlying lattice. In this example, at any point, the graph is a Fig.. Example: the growth of an expaning hypercube 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 lognimensional hypercube. This example is of particular interest ue to recent work suggesting that many networks

5 5 have an unerlying hyperbolic or tree-metric structure [9], [8]. The expaning hypercube captures the key feature of these topologies, as the number of noes at istance grows exponentially in. This example is also very naturally represente using our istance - epenent Kronecker operation an a Hamming istance as our istance measure. 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 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 β i ( i+ α. i Thus, for the sequence of graphs shown in the figure above, we have the following sequence of matrices: ( G, β β G β, β G β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β 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 V, can fin paths of length O(logn only if α k [4]; in the moifie case presente above, we will see in section V that we nee a ifferent probability of connection to fin short paths. IV. CONNECTION TO HIDDEN HYPERBOLIC SPACE MODEL As mentione previously, the expaning hypercube moel in Example resembles moels propose in [7] an extene in [8], [], an []. In [7], every noe in the network has a hien variable - their location in a hien metric space. The probability of a connection between two noes is base upon the istance between them in this hien space. The resulting egree istribution epens on the curvature of this hien space; if the space has negative curvature, the egree istribution will be scale-free with P(k k γ [4]. In the istance-epenent Kronecker graph escribe in this paper an [], the probability of a connection is base on the istance between two noes in the given lattice, efine usually by H 0 an H in the family of matrices H. As a result, the lattice, or metric space, is not really hien since neighbors are explicitly connecte in the lattice. It is important to note that both moels incorporate a istance-epenent probability of connection. As will be efine formally in Section V, a local greey search algorithm can take avantage of this embeing into a hien or physical space to forwar a message to a estination. If a given noe u has a message to forwar to a estinationt, it can use its knowlege of the embeing to forwar the message to its neighbor closest to the estination in the embeing. It is not necessary that the embeing be physical, as shown in [8] an []; rather, what is necessary is that the the probability of a connection between two noes is epenent on the istance between them. In most social networks the abstract istance is a measure of social istance - the likelihoo of two iniviuals being connecte epens on their memberships in various groups, among other factors. In aition, in the moels of [7], a hyperbolic space results in exponentially expaning neighborhoos aroun each noe. In the istance-epenent hypercube example, there are ( k noes at each istance, also resulting in exponentially expaning neighborhoos. However, the hien metric space moel necessarily inclues the notion of a core an periphery of the network, where high-egree noes form the core connecting many lowegree noes at the periphery []. In the hypercube example, all noes are homogeneous in expecte egree - there is no notion of a core. In [8], as noes are locate further from the origin in the hien hyperbolic space their expecte egree ecreases exponentially ( e βr. When this is combine with the exponentially expaning neighborhoos ( e αr, the result is a scale-free istribution with γ + α β. It is important to note that an exponential ecrease in expecte egree is not strictly necessary; to see this, let the number of noes at istance r from a reference origin in the hyperbolic space be n(r e αr Let the average egree of noes at istance r be k(r r δ

6 6 so that Using we have r(k k δ n(k n[r(k] r (k n(k e αk /δ k /δ which asymptotically behaves like a power law with γ + /δ. In the hypercube example, espite the exponential expansion of neighborhoos, the resulting egree istribution will always be Poisson as long as the probability of connection is sufficiently small, as shown in the next section. Nevertheless, the connection between this moel an those base on tree-metrics an hien metric spaces is important to note, as one key factor emerges: a istanceepenent relation is necessary for a greey algorithm to succee in fining shortest paths. V. DEGREE DISTRIBUTION In this section we escribe a general characteristic function-base analysis of egree istributions for lattice-base networks, an apply it to the expaning hypercube example in Section III. In general, any latticebase network with a istance-epenent probability of connection will have a Poisson egree istribution, as long as the probability of a connection at a istance is sufficiently small. Formally, Theorem 5.: The egree istribution of a general lattice-base network with a istance-epenent probability of connection P( an maximum istance max will have the following egree istribution: P(ν i e α α i ( +max O(P ( i! where α max P(σ( ( an σ( number of noes at istance from a reference noe in the lattice. We note that if lim n max P ( 0, then the egree istribution is Poisson. Proof: Let ν enote the egree of an arbitrary noe u in a general lattice-base network with n noes. Thus, ν v +v + +v n where { if link to noe i, v i 0 otherwise. We efine the characteristic function of the egree istribution as E[e itν ] E[e it(v+v+ +vn ] E[e itv ]E[e itv ]...E[e itvn ] We can then group the expectations max E[e itν ] ( P(+P(e it σ( max max ( P(( e it σ( (e P(( eit +O(P (( e it σ( as e x x+o(x ( Thus, we can pull out the first term an using binomial approximation of (+x c +cx+o(x, we have max ( E[e itν ] e P(( eit σ( + O(P (( e it σ( e P(( eit e ( eit max P(σ( max (+O(P (( e it σ(e P(( eit e α(eit (+ max O(P ( Expaning, we see that the characteristic function is E[e itν ] ( + max O(P ( e α ( +αe it + (αeit From such a representation of the characteristic function, we can clearly see the egree istribution as P(ν i e α α i ( +max O(P ( i! We now turn to a specific lattice-base network, the hypercube istance-epenent Kronecker graph escribe in Example in Section III. In this example, σ( ( k, an the maximum istance in the network [ is k log n. We use a particular P( (k ] logkln optimize for searchability, as etermine in Section VI. Theorem 5.: The egree istribution of the expaning hypercube is given by the following Poisson istribution, P(ν i e α α i i!! where α.699n.470 loglogn logn ( Proof: We use the same framework as in the proof of Theorem 5., an let e it x for simplicity. In this case, the characteristic function becomes E[x ν ] e ( x k P(σ( +...

7 7 so that α k P(σ( [ k (k ] (k logkln To calculate α, we use the entropy approximation ( k kh( k, which hols as ( n k n(h(p+o( when k pn, so that k α kh( ( k (k H k logkln We can approximate the sum by using sale point integration. g(ye kf(y y π k f (y 0 g(y 0e kf(y0 ( ( +O k where y 0 is the sale point of the function f(y, i.e., the point at which f (y 0. We rewrite the sum S(k in nats, leaving out the constants in front, [ ( S(k ] k k H( k ( kh k k ek k an then we let y k, S(y k [ ( y ] H(y ( yh y ek y y (4 so that, with the sale point approximation of line (4, g(y y an f(y H(y ( y H( y y. Using Mathematica, we fin yieling, S(k y f(y g(y 0.4 f (y 0. π.k (.4e0.6k (5 So, our α is now π α logkln.k (.4e0.6k.699n0.470 loglogn logn With the results of Theorem 5., we have a Poisson egree istribution with parameter α. A. Expecte Degree From the characteristic function, we can also etermine the expecte egree. E[ν] x E[xν ] x x [e ( xα ] α x Thus, the expecte egree of the expaning hypercube example is a growing function of n. Fig.. Expecte egree of expaning hypercube Fig. 4. Example histogram with n 4096 B. Simulation of expaning hypercube example Simulating the expaning hypercube with the P( etermine in Section VI yiels results that match well, within a constant, the analysis above. Figure shows

8 8 the comparison of the theoretical an simulate expecte egrees, while figure 4 shows an example histogram of the egree istribution, both theoretical an simulate, with n The Poisson nature of the istribution is clearly visible, as is the growth of the expecte egree as a function of n. VI. PROVING SEARCHABILITY While the istance-epenent Kronecker graph moel is more complicate than the original Kronecker graph moel, it can capture several existing network moels, an it incorporates istance into the probability of connection, allowing for several cases in which searchability can be proven. In this section, we first give a general framework within which a lattice-base network can be proven searchable an then procee to the specific cases of the Kleinberg moel [4] an the expaning hypercube moel of Example in Section III. probability that the message enters a set of noes B where B {v : (v,t }. Pr({message enters B } v B ( P(u,v : v B (u,t+ (u,t ( P( Nu,t( (u,t+ (u,t ( P( min N( A. General Searchability Theorem We efine a ecentralize algorithm A similar to [4]. 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 in the case of the expaning hypercube, 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 that the istribute algorithm A is able to fin short paths through the network, which are usually O(D where D is the iameter of the network. Let the current message holer be noe u an the estination noe t. We will say that the execution of a ecentralize search algorithm A is in phase when < (u,t +, where (u,t is the istance between noe u an noe t. Thus, the largest value of in a general lattice-base network is max log max where max enotes the maximum geoesic in the network. For example, in a hypercube, the maximum geoesic is max logn k, so max loglogn logk. We efine N u,t ( {v : (v,t,(u,v } an min N( min u,t,(u,t N u,t (. Theorem 6.: A ecentralize algorithm A will fin short paths of length O(log ( max, when the probability of a connection is P(u,v [cmin N( ] (6 where c log max. Proof: Suppose we are in phase with current message holer noeu; we want to etermine the probability that the phase ens at this step. This is equivalent to the Fig. 5. Relative positions of noes u,v, an t in phase In any network moel, enforcing searchability 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 5. Once this min N( is foun, if we set the probability of a connection between two noes istance apart as in Theorem 6., with an appropriate constant, we will fin that each phase escribe above will en in approximately max steps, an, as there are only max such phases, our greey forwaring algorithm will be able to fin very short paths of length O( max. Thus, we have Pr({message enters B } e (u,t+ e c e c e c e c (u,t+ (u,t ( P( min N( (u,t min N( P( (7 (u,t+ (u,t (u,t+ ln (u,t ln c (8

9 9 where the approximation in line (7 requires that lim n max P ( 0, which hols with the P( as specifie in line (6 (see proof of Theorem 5. for extra orer terms, an line (8 comes from the power series expansion of e x. Let X enote the total number of steps spent in phase. Then, EX Pr[X i] c i ( c i i Let X enote the total number of steps taken by the algorithm A. an X 0 max 0 max EX EX X (+ max (c (+log max log max δ(log max (9 where line (9 hols δ, log max. With this framework, we can explore the searchability of any lattice-base network moel with istanceepenent connection probability. B. Searchability in original Kleinberg moel In the original Kleinberg two-imensional lattice [4], the number of noes at a istance from a reference noe is approximately 4, ignoring ege effects. The maximum istance between any two noes is O(n, so max logn. Aitionally, the iameter of the graph is on the orer of logn. In general, min N( for a fixe, resulting in the probability of connection optimize for searchability, P( [αlog(n ]. Using this P(, Pr({message enters B } (u,t+ (u,t ( P( min N( e αlog n e α log n α logn (u,t+ (u,t (0 ( where line (0 hols for the P( specifie, an line ( comes from the power series expansion of e x. Therefore, an EX α logn logn EX EX 0 δ(logn. ( where line ( hols δ, logn. C. Searchability in expaning hypercube example In the expaning hypercube example of Section III, each noe has log n neighbors from the lattice itself. With the aition of long-range links, we expect the iameter to be O(loglogn, similar to [8]. Note that with this example, max loglogn logk an the number of noes at istance equals ( n. Using Theorem 6., we can prove the following result: Theorem 6.: A ecentralize algorithm A will fin paths of length O((loglogn in the expaning hypercube example when β 0, β [logkln], [ (k i ][ (k (i+ β i i (i+] i i i+ ( such that the probability of a connection is if (u,v 0, P(u,v [ (k ] logkln if (u,v (4 Proof: Using Theorem 6., all that remains is to fin min N( an to etermine the appropriate constants to use. Without loss of generality, we assume that the estination noe t is the all-zero noe (i.e., its label is the zero vector so that we can write (u,t u. 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 prove that min N( ( k in Appenix A. We then let c logkln for reasons that will be clear below. Using the same framework as

10 0 in Theorem 6. we have that 5.5 Average iameter an path length of simulate Kron graphs Pr({msg enters B } u + u ( P( min N( e log k ln e log k logk u + u (5 (6 where line (5 hols for the P( specifie, an line (6 comes from the power series expansion of e x. Therefore, we have Average iameter an path length, over 0 trials n Simulate iameter Path length foun by ist alg an logk EX EX 0 EX logk δ(logk, δ, logk Since the expecte number of steps in phase is logk, an there are at most logk phases, the expecte amount of steps taken by the algorithma is at most δlog k. So, with this efinition of P(, the istribute algorithm provies searchabilty. D. Simulation of istribute search algorithm We simulate the local greey algorithm escribe above in MATLAB for 6 n 4096 with the probability istribution as in Theorem 5. an appropriate floor functions. We foun that the greey algorithm fins a path between two noes with an average length of a constant factor away from the iameter of the simulate network, where iameter is efine as the maximum geoesic in the network. Note that the two noes selecte for the simulation are actually the worst-case noes - the istance between them in the network is exactly the iameter. Figure 6 illustrates the results of the greey algorithm simulations. E. Path length with sub-optimal P( In this section we analyze the performance of the local greey search algorithm on the expaning hypercube when P( is not optimal, as in Theorem 6.. For this example, let P( [logk ( k ], which is clearly not min N( from Lemma 9.. We will show that this suboptimal P( also allows for searchability. Fig. 6. Average path length foun by greey algorithm using local information Using the same framework as in Theorem 6., Pr({msg enters B } e (u,t+ ( P( min N( (u,t (u,t+ (u,t P(min N( (u,t+ P((k (u,t e e log k S(k, e log k mins(k, (7 where line (7 hols for the specifie P( an where ( ( k k S(k, min (k H( k kh( k (8 (k H( k kh( k max (k H( k kh( k where we have use the approximation ( k kh( k, which hols as ( n k n(h(p+o( when k pn, in line (8. Since the exponent is convex in, the maximum will be at either the upper or lower boun of the sum. For 0 logk the lower boun ( yiels the maximal exponent. So, we have Pr({msg enters B } e log k f(k, f(k, logk

11 where we have use the power series expansion of e x an where an f(k, (k + H( k + kh(. (9 k Continuing with the proof of searchability, we have EX Pr[X i] i logk EX EX 0 logk f(k, min f(k, (+logklogk δ(logk, δ, logk since f(k, is convex but its minimum occurs close to logk. As a result, even for suboptimal P(, a local greey algorithm can fin short paths. However, the bouns use in the analysis above are looser than those in previous sections, so the final expecte number of steps taken by A is not as tight. This analysis is supporte by simulation results as shown in the figure below. Average iameter an path length of simulate Kron graphs with suboptimal P( 5.5 Fig. 7. Average iameter an path length, over 0 trials Simulate iameter Path length foun by ist alg n Performance of greey algorithm when P( [logk ( k ] Finally, if P( [logk ( k ], using the same sort of techniques as above we can show that EX δk(logk for a large enough δ. Note that in this case, the paths foun will be O(lognloglogn, which are longer than before. Simulation results with this P( are shown in figure 8. Average iameter an path length of simulate Kron graphs with suboptimal P( 8 Average iameter an path length, over 0 trials n Simulate iameter Path length foun by ist alg Fig. 8. Performance of greey algorithm when P( [logk ( k ] a simple lower boun of the hypercube example here, which can be applie to any ranom graph. If we assume that the expecte egree of the hypercube example in Section III is polynomial in n, say n β, similar to what was foun in Section IV for the expaning hypercube, we can lower boun the iameter as follows. We assume that at each step, every noe has neighbors an that it takesαsteps to reach all n noes. Therefore, to reach all n noes in the network, we have α n (n β α n α β Constant iameter Thus, a simple lower boun for the iameter of a graph with polynomial expecte egree is some constant, β. We can also work backwars, assuming a log log n iameter. In this case, we have α n loglogn n n log log n e log n log log n VII. BRIEF DIAMETER ANALYSIS OF HYPERCUBE In this section, we briefly iscuss the iameter of a general ranom graph. Fining the actual iameter, efine as either the maximum or the average geoesic in the network, can be very complicate. We iscuss which is less than a polynomial in n, but still grows with n. Figure 9 compares the simulate iameter of the expaning hypercube example with the two lower bouns iscusse above. For 6 n 4096, both lower bouns appear to be a goo match.

12 Fig. 9. Diameter, over 0 trials Diameter of expaning hypercube n Simulate Constant, β.5 loglog(n Simulate an theoretical iameter of expaning hypercube VIII. CONCLUSION We have presente a generalization of Kronecker graphs by efining a family of istance -epenent matrices an a new Kronecker-like operation. As a result, the network moel efines both local regular structures an global istance-epenent connections. Though the moel is more complicate than the original Kronecker graph moel, it is more general, as it can generate existing social network moels, an more importantly, networks that are searchable. These properties emerge naturally from the efinition of the embeing of the noes an the probability of connection within the family of matrices H. Any lattice-base network moel with istance-epenent connection probabilities can be analyze using the framework escribe in Sections V, VI, an VII for exploring egree istribution, iameter, an searchability. Most importantly, the searchability analysis shows how to make any network moel searchable by efining the appropriate probability of connection base upon N(. The particular expaning hypercube example explicitly escribe here shares characteristics with those base upon hien hyperbolic spaces [7], [8], though it has one maor ifference - egree homogeneity across noes. Nevertheless, its exponentially expaning neighborhoos an istance-epenent probability of connection make it a goo moel for social networks as people ten to exhibit strong homophily, i.e., associating with other people most like themselves. In aition, in contrast to Kleinberg s lattice-base moel [4], the searchability of the expaning hypercube is not too sensitive to the choice of P(. Though this paper gives a near complete escription of the characteristics of istance -epenent Kronecker graphs, there are many interesting questions that remain. These inclue how to parameterize the moel from real-worl ata sets, an how to incorporate network ynamics. Ieally, given any ata set, we woul like to be able to fin an appropriate family of istance-epenent matrices to match any esire characteristic of the ata set. Aitionally, while the current moel incorporates some measure of growth, growing from a small initiator matrix to a final nxn aacency matrix, we woul like to better incorporate mobility into the moel so that it is not ust a static escription of the network at one point in time. IX. APPENDIX A - 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 of the estination, t. Lemma 9.: min N u,t ( ( k 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 5. Without loss of generality, efine t as the all zero noe, t ( Arrange the label of u such that u ( 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 v +v 0 u v 0 +v 0 v +v 0 v We can solve in terms of v, yieling v 00 k v v 0 u v v 0 u +v We also know that we must satisfy the following: v,v 0,v 0,v 00 0 From these bouns we have < u + u u + v max(0, u v min( u,k, ( + u Note that the secon an thir bouns o not affect v. Counting the number of noes in the ball, we have v u ( ( u k u N u,t ( v u +v v v l 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

13 entropy approximation for combinations. Let u ak, bk, ck,x v. Using this notation, we have v u ( ( ak k( a N u,t ( x k( b+x where X max x ah ( x ak subect to xv l v u k(ah( ( x b a+ x k ak+( ah a xv l kx +( ah ( b a+ x k a (0 ( kmax(0,a b x kmin(a, b, (a b+c ( Note that line (0 is true as n k n(h(p+o( when k pn. Note that the function X is concave in x, so unconstraine optimization yiels the two solutions below, each giving ifferent values of min N u,t ( : yieling x ak abk when c a+b( a min N u,t ( ( k x k(a b+c when c < a+b( a yieling ( k min N u,t ( 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 A. Solution : c a+b( a In this region, the solution to the unconstraine problem,x ak abk gives us the maximalx. Substituting in for the size of N u,t ( an using the same entropy approximation as before, we have N u,t ( ( ak abk ak abk k(ah( b a+ k ak +( ah a k(ah( b+( ah(b kh(b ( k bk ( k. B. Solution : c < a+b( a In this region, we choose one of the bounary points, x k(a b+c, as the solution to the maximization problem. Substituting this solution for x in N u,t (, we obtain N u,t ( a b+c k(ah( a +( ah( a+b+c ( a 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 ( ( ( a b+c a+b+c min ah +( ah a,c a ( a ( Note that the bouns for this region are: a b c 0 a b+c 0 c < a c 4 0 c 5 0 a,b 6 0 a b c 7 0 a+b c 8 0 a+b c ab where an 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 is minimal in this region. Note that 8 implies 7. Computing the Hessian of the function in line ( shows that it is concave in both a an b; the erivation is in Appenix B. Since our function is concave, the min N u,t ( is foun from the bounary points of Region. Rearranging the bouns from before in terms of a we have: a b+c a b c a > c,a c 4 c > 0,c 5 0 a,a 6 a b c 7 a b+c 8 a c b b b when b 9 a c b b b when b >

14 4 where line (4 hols for large k, using the entropy approximation ( n k n(h(p+o(. Thus, when b < 0.5, we have min N u,t ( 4, an when b 0.5, we have min N u,t ( ( k. Finally, we have that when c < a+b( a, we apply Solution, an we have min N u,t ( ( k when Solution is vali. Comparing the Solution with Solution, we have again that min N u,t ( ( k. Fig. 0. Bounaries of f(b when b When b, only bouns (,,,4 apply to f(b, yieling 5 points that we nee to examine, as shown in Figure 0. If b.5, then f(b is minimal at point (, ( b, b, yieling min N u,t ( k( b b ( k bk ( bk ( k where line ( hols for large k, using the entropy approximation ( n k n(h(p+o(. If b < 0.5, then f(b is minimal at point (5, (b,b, yieling H ( b min N u,t ( kb 4 When b >, only bouns (,,4,an 8 apply to f(b, yieling 4 points that we nee to examine, as shown in Figure. Fig.. Bounaries of f(b when b For this region, f(b is minimal at point (, matching point (5 in the previous region, yieling min N u,t ( k( b ( k H ( b b (4 X. APPENDIX B - CONCAVITY OF f(a,b,c FOR SOLUTION Lemma 0.: The function f(a, b, c ah ( ( a b+c a + ( ah a+b+c ( a is concave in both a an b. Proof: To prove that the function is concave in both a anb, we nee to see if the Hessian is negative efinite. Let ( a b+c f(a,b,c ah a +( ah ( a+b+c ( a Taking erivatives with respect to a, we fin ( f a log c+a b log b+a c a a an +log c a+b ( a +log a b c ( a f a (c b a(c+a b(b+a c ( c b + ( a(c a+b( a b c ( a b+c + a+b c + a + a+b+c + a b c + a From the bouns for this region, we can see that the function is concave in a. Taking erivatives with respect to c, we fin an f c ( log(c+a b+log(a+b c log(c a+b+log( a b c ( f c c+a b + a+b c + c a+b + a b c From the bouns in this region, we can see that the function is concave in c.

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