Small World Model based on a Sphere Homeomorphic Geometry
Abstract
We define a small world model over the octahedron surface and relate its distances with those of embedded spheres, preserving constant bounded distortions. The model builds networks with both number of vertices and size , where is the size parameter. It generates longrange edges with probability proportional to the inverse square of the distance between the vertices. We show a greedy routing algorithm that finds paths in the small world network with expected size. The probability of creating cycles of size three (C3) with longrange edges in a vertex is . Furthermore, there are expected number of C3’s in the entire network.
keywords:
computational geometry, combinatorial problems, randomized algorithms, small world networks, generative modelsdfnDefinition \newproofpfProof
1 Introduction
Stanley Milgram milgram1967small () conclude that social networks have a large number of paths with small length. This motivates the proposal of small world graphs models watts1998cds (); Kleinberg:2000 (); Kleinberg:2001:SPD:2980539.2980596 (); LibenNowell16082005 (). Kleinberg Kleinberg:2000 () presents a model that generates a lattice of vertices . He defines the lattice distance between two vertices as . The model has three parameters, , and . It links each vertex with directed edges to the vertices within lattice distance , and for each , it generates directed edges (independent random trials) to with probability proportional to . The probability of each is multiplied by the normalizing factor . Kleinberg calls it as the inverse power distribution. We call longrange edges those random generated edges. Kleinberg proves that, for and , there is a greedy routing algorithm that finds paths with expected length. For each vertex, the algorithm takes constant time and logarithmic space (in bits). Small world networks has routing applications in P2P networks AspnesDS2002 (); ZHANG2004555 (); Manku:2004:KTN:1007352.1007368 (), MANETs 4067680 () and WSN 4678803 (); Liu2009 (). Most uses the ideas of Kleinberg, generating a clustered network and longrange edges with the inverse power distribution.
Manku, Naor and Wieder Manku:2004:KTN:1007352.1007368 () present a greedy routing algorithm that considers the vertex neighbors and the neighbors of neighbors in a routing decision. The algorithm finds paths with expected length in a ring with longrange edges per vertex. Martel and Nguyen Martel:2004:AKS:1011767.1011794 () extend the Kleinberg model to the dimensional lattice. The greedy routing algorithm finds paths with expected length for . In this algorithm, the vertices have positioning information of the longrange edges of the closest neighbors. Fraigniaud, Gavoille and Paul Fraigniaud:2006:ESE:1160297.1160302 () present similar results with an oblivious algorithm. Zeng, Hsu and Wang Zeng:2005:NOR:2098796.2098861 () present a model over a unidirectional ring. Each vertex has one longrange edge generated with the inverse power distribution and two augmented local edges to vertices within lattice distance chosen uniformly at random. They present two algorithms that find paths with expected length. One year later, the first two authors Zeng2006 () generalize the model to dimensions and define a routing algorithm that find paths with expected length. Liu, Guan, Bai and Lu Liu2009 () present a model that generates a lattice subdivided in clusters. Only one vertex of each cluster has a longrange edge to another vertex of other cluster generated with the inverse power distribution. The routing algorithm finds paths with expected length, where and each vertex has positioning information of the closest longrange edges.
Some works Manku:2004:KTN:1007352.1007368 (); Martel:2004:AKS:1011767.1011794 (); Zeng:2005:NOR:2098796.2098861 (); Liu2009 () present models that build a base graph over bounded geometries that are plane in the threedimensional euclidean space. Kleinberg’s model Kleinberg:2000 () is an example that generates a clustered network over the square with vertices , , and . Other work Martel:2004:AKS:1011767.1011794 () presents a model that use the twodimensional torus, which is a solid geometry with genus one. There is a lack of small world models that generate the base graph over genuszero solid geometries, that is, are homeomorphic to spheres, such as, for example, any Platonic solid. Our octahedral small world model (OSW) generates the base graph over the octahedron (Section 3). We relate the base graph with spheres (Section 2.1), providing a connection with global friendship networks. The vertices on the octahedron can be projected on the surface of spheres. We relate the distances between neighbors vertices on both and the size of the octahedron with spheres radii. As in the octahedron, where the distances are bounded by a constant for increasing , there are spheres where the distances are also bounded by a constant for increasing radius. Then, there is a family of spheres that are asymptotically similar to the octahedron in terms of distances on the surface. We also define a greedy routing algorithm that finds paths of size in octahedral small world graphs. Moreover, the expected number of C3’s in these graphs is (Section 4). Theorems 14 and 15 can be used, for example, in running time analysis of algorithms that perform searches of C3’s. Identifying the base graph performing a local search of cycles in each vertex is an approach to label vertices 2018arXiv180601469V ().
2 Octahedrons and Graphs
Let the octahedron be the set , with and . Let for , i.e., is the set of all 3D vectors with integer coordinates in the octahedron. We generate an undirected graph on that “wraps” . The edges do not pass through the “inside” of , linking only the closest vectors on . So, each has an undirected edge to all such that , for . Definition 2 formalizes the octahedral graph. Figure 2 illustrates a twooctahedral graph. Lemma 1 shows the number of vertices of . Note that, is the size of the octahedral graph and is the number of vertices. Lemma 2 shows the number of edges of .
The octahedral graph with size is such that and .
Lemma 1
.
[Proof.] We count all combinations of , and that satisfy the Definition 2. For and , , that are two combinations. For each , for and , , that are two combinations for each . For each and , and , that are two combinations for each and . Then,
∎
Lemma 2
.
[Proof.] By Definition 2, for all , the two neighbors of with the same value of the third coordinate, when , are:

and if and ;

and if and ;

and if and ;

and if and ;

and if and or and ;

and if and or and .
This arrangement of edges defines a sequence of cycles with increasing sizes for growing from to zero and decreasing sizes from one to , as Figure 2 shows. There are two similar sequences of cycles for and . Counting one edge for each vertex of each cycle in each coordinate,
∎
2.1 The nOctahedral Graph and Spheres
Let the distance between a pair on be the length in the euclidean space of the line segment bounded by and . Let the distance between and on the sphere with radius be the length in the euclidean space of the smallest circular arc defined by the projections of and on the surface of the sphere with radius and center at . Figure 5 shows the projections of and , the circular arc defined by them and the angle between and in radians. The greatcircle distance is the wellknown relation . Thus for all , where is the distance between and on the sphere with radius .
By Definition 2, the distance on between and is for all and . On the other hand, the distances on the sphere vary according to for all radius . In the case of , diverge as tends to infinity, because increases faster than inversely decrease. On the other extreme, when is a positive constant , converge to as tends to infinity. Theorem 3 shows that all are in the interval when for a positive constant . Therefore, the distances on the sphere are for some values of , which is asymptotically similar to the distances on . Figure 5 shows the radius upper bound function plot for .
Theorem 3
The distances on all spheres with radius are , for all , and a constant .
[Proof.] For for all , . The analysis follows on all planes defined by all , and as shown in Figure 5. Given the legs and of the two right triangles defined in each plane, . The values of all are maximum for , that is . Moreover, is upper bounded when is minimum because is constant, for all and . The minimum value of is the radius of the inscribed sphere in , that is . Then for all and and .∎
3 Octahedral Small World Model
Let be the distance function defined by the length of a minimum path between all pairs of vertices in . We define path as a sequence of distinct vertices which each two consecutive vertices the first is incident to the second, length of a path as the number of vertices of the path minus one and minimum path as a path with the minimum length over all paths. Let be the event of choosing to create the directed edge and be the normalizing factor of all .
[OSW model] The octahedral small world (OSW) model is such that for the octahedral graph :

for all , both directed edges and are included in and;

for all and a , is included in with probability .
Note that is undirected, as shown in Definition 2, and is directed, as shown in Definition 3. All undirected edges correspond to the pair of directed edges . We call longrange edges those generated by the independent random trials of OSW. Next, we design a greedy routing algorithm that finds small paths in . Given a vertex and a message, the algorithm sends the message to the vertex with minimum angle with the target vertex , where is the set of outneighbors of in . The algorithm selects computing argmax, where , and is in the message header.
Theorem 6 proofs that this routing algorithm finds paths with squared logarithmic length in . The proof strategy is inspired by the work of Kleinberg Kleinberg:2000 (). Lemma 4 proofs the upper bound of for all , which is used throughout Section 4. Lemma 5 proofs the lower bound of for all , which is used in the proof of Theorem 6. Let for distances . Figure 7 shows the vertices in , and in the bold cycles, where is the central vertex.
Lemma 4
.
[Proof.] The longest distance in is at most . Moreover, for , as shown in the bold area of the Figure 7, where . As , then
∎
Lemma 5
.
[Proof.] As and , then
∎
Theorem 6
The greedy routing algorithm performs expected number of forwards.
[Proof.] We partition the path from source to target that the greedy routing algorithm finds, such that the algorithm is in phase if , where is a vertex of the path, and is in phase zero when . The phase ends if the current vertex with the message has a longrange edge to any vertex , where . By Definition 3 and Lemma 5, . As there are at least vertices such that (in the bold area in Figure 7), then . Besides, for all and . As all are disjoint, then the probability of the phase ends in is
Let be the random variables that count the number of forwards in the phase . As are geometric random variables, then . Let be the random variable that counts the total number of forwards in the greedy routing. As , then and by the linearity of expectation.∎
4 C3’s not in the nOctahedral Graph
The longrange edges generation in OSW may create new C3’s that do not belong to the base octahedral graph . The octahedral graph is a well structured arrangement of C3’s and the longrange edges “hide” it in . A C3 in and not in has at least one longrange edge. Besides, a C3 is a sequence of three edges where a longrange edge may assume any position. Considering these, the fact that is directed and is the first vertex of the C3, there are seven distinct compositions of edges that define C3’s rooted in , in and not in . Let be a directed edge in and be a longrange edge generated in OSW. These compositions are represented by the events . All events refer to the existence of at least one C3, rooted in , but each specific event having an edge sequence as follows: with edge sequence ; with edge sequence ; with edge sequence ; with edge sequence ; with edge sequence ; with edge sequence ; with edge sequence .
Then, the event of the existence of at least one C3 in , not in and rooted in is . Lemmas from 7 to 13 bound the probability of each and Theorems 14 and 15 proof, respectively, that is and that the expected number of C3’s in and not in is .
Lemma 7
.
[Proof.] Let . Note that . Using union bound, Definition 3, Lemma 4 and the facts that for all and , then
∎
Lemma 8
.
[Proof.] Let and for all . As OSW does not generate parallel edges when and happens, then Using union bound, Definition 3, Lemma 4 and the facts that , and for all and , then
∎
Lemma 9
.
[Proof.] Let and for all . Note that . Using union bound, . As and are mutually independent events, so for all and . By this fact, using Definition 3 and grouping the terms with the same value of ,
Using Lemma 4 and the facts that and for all , and ,
The proof follows because for , ^{1}^{1}1Riemann zeta function with parameter three is a convergent series such that . and .∎
Lemma 10
.
[Proof.] Let . Note that . The proof follows similarly to the Lemma 7 proof, using the fact that for all .∎
Lemma 11
.
[Proof.] Let and for all . Note that . In a similar way of the beginning of the Lemma 9 proof and grouping the terms with the same value of ,
The proof follows using Lemma 4 and the facts that , and for all , and .∎
Lemma 12
.
[Proof.] Let and for all . Note that . The proof follows similarly to the Lemma 11 proof.∎
Lemma 13
.
[Proof.] Let and for all . Note that . Using union bound, the fact that the events , and are mutually independent, Definition 3 and Lemma 4,
We claim that (i) and (ii) .
For (i), we split the sum in three others: for , and . When , the value of is positive and the triangle inequality can be used such that . We group the terms of the sum with the same value of and use the fact that such that
Rearranging the terms,
where the first sum in the parentheses is for . In both sums, and, when ,
When , as , and , then
When , we use the triangle inequality, group the terms of the sum with the same value of and use the fact that such that
Therefore, .
For (ii), we group the sum terms with the same value of and use the facts that and such that
∎
Theorem 14
is .
Theorem 15
The expected number of C3’s in and not in is .
[Proof.] Let be the sets of all C3’s that contain and have at least one longrange edge, for all . Let be the partitions of with all C3’s that have the same edge sequence defined in the event , for all and . Note that the cycles in are sequences of three vertices such that is the first vertex in all. Let be the random variables that count the number of C3’s that contain and have at least one longrange edge, for all . Let be the Bernoulli random variables that are one if the C3 is in or otherwise, for all . Note that and are distinct random variables. Given the definitions, .