PARTITIONING WIDE AREA GRAPHS USING A SPACE FILLING CURVE

International Journal of Data Mining & Knowledge Management Process (IJDKP) Vol.11, No.1, January 2021
DOI:10.5121/ijdkp.2021.11102 13
PARTITIONING WIDE AREA GRAPHS USING A SPACE
FILLING CURVE
Cyprien Gottstein1
, Philippe Raipin Parvedy1
, Michel Hurfin2
, Thomas Hassan1
and Thierry Coupaye1
1
TGI-OLS-DIESE-LCP-DDSD, Orange Labs, Cesson-Sevigné, France
2
Univ Rennes,INRIA, CNRS, IRISA, 35000RENNES, France
ABSTRACT
Graph structure is a very powerful tool to model system and represent their actual shape. For instance,
modelling an infrastructure or social network naturally leads to graph. Yet, graphs can be very different
from one another as they do not share the same properties (size, connectivity, communities, etc.) and
building a system able to manage graphs should take into account this diversity. A big challenge
concerning graph management is to design a system providing a scalable persistent storage and allowing
efficient browsing. Mainly to study social graphs, the most recent developments in graph partitioning
research often consider scale-free graphs. As we are interested in modelling connected objects and their
context, we focus on partitioning geometric graphs. Consequently our strategy differs, we consider
geometry as our main partitioning tool. In fact, we rely on Inverse Space-filling Partitioning, a technique
which relies on a space filling curve to partition a graph and was previously applied to graphs essentially
generated from Meshes. Furthermore, we extend Inverse Space-Filling Partitioning toward a new target
we define as Wide Area Graphs. We provide an extended comparison with two state-of-the-art graph
partitioning streaming strategies, namely LDG and FENNEL. We also propose customized metrics to
better understand and identify the use cases for which the ISP partitioning solution is best suited.
Experimentations show that in favourable contexts, edge-cuts can be drastically reduced, going from more
34% using FENNEL to less than 1% using ISP.
KEYWORDS
Graph, Partitioning, Graph partitioning, Geometric partitioning, Spatial, Geography, Geometric, Space
Filling Curve, SFC, ISP
1. INTRODUCTION
In the last decade, graphs have seen extreme attention from IT industry and Computer Science
researchers. Because graphs can fit many use cases induced by the development of services from
Google, Facebook or Amazon, and new marking opportunities based upon graph analysis and
recommendation engine, they have reached an unprecedented scale. Along with the explosion of
social networks graph knowledge and analyse about people has become a very profitable resource
and represent a massive subject of interest. Yet, graphs effective range of application is much
wider. From security and fraud detection to financial risk management, graphs analysis is used
extensively in many domains. As performing such analyses requires heavy computation, it is
essential for those graphs to be widely distributed across a large set of machines, i.e. partitioned.
As K. Andreev and H. Räcke shown, graph partitioning is a NP-Complete problem [1].
At this day, a plethora of graph partitioning solutions have been proposed to solve this problem.
The large range of strategies even makes it difficult to choose the appropriate graph partitioning
solution for a given graph. It has been shown in recent surveys on graph partitioning [2], graph

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research domain intensely focuses on the Online Analytic Processing (OLAP) Context and the
infamous scale-free graphs. Scale-free graphs, including social graphs, are graphs whose degree
distribution follows a power law. The OLAP graph partitioning standard, when it comes to large
scale graphs (including scale-free graphs), is FENNEL [5].We describe FENNEL in section 2.
Nevertheless, scale-free graphs have additional characteristics contextual to their use cases. Very
few scale-free graphs hold precise spatial information onto their nodes and, as a direct
consequence, data spatiality has been left aside in these works. Geometric graphs, graphs whose
nodes and edges are associated to spatial elements placed in a plane, and geometric-aware
partitioning strategy have received little interest in the past decade.
Regardless, we believe geometry's popularity in graph partitioning will swiftly be rebuild through
the growth and construction of the Internet of Things. Through giant companies like Amazon or
Microsoft which have already launched new services such as Microsoft Azure Digital Twins and
other companies working on smart-cities or smart-industry, new digital uses are being
conceptualized and enabled by the IoT. To answer those new use cases, we anticipate a new kind
of graphs will emerge through the aggregate of multiple types of infrastructure. From roads,
power or water supply chain to buildings and equipment’s and ultimately connected object from
the IoT, each modelled as graphs with their own set of properties and combined together would
appear, what we will refer until the rest of the paper, Wide Area Graphs (WAG).
Some major characteristics will be common to all WAGs. WAG is a sub family of geometric
graphs. Recall that geometric graphs are graphs for which vertices or edges are placed on a plane.
Therefore, each node of a WAG has an associated location. Also, for a graph to be considered as
a WAG it must be composed of millions of nodes and edges and cover an area as large as a
country’s surface or even wider. At last, a WAG should be infrastructure related e.g. mixing
power grid networks, road networks, buildings, equipment’s or even supply chains. Furthermore,
it should have layers associating local objects interacting with each other and connected to a
larger network. We do not include anything regarding connectivity and density properties into
this high level characterisation of a WAG because these aspects may vary significantly from one
WAG to another. However we envisiona WAG as a highly polarized version of graphs issued
from Finite Element Meshes (FEM).
FEM are used to simulate real world experiences as wind or water movement (see Figure 1). A
planar graph induces a low connectivity ratio, compared to scale-free graphs, and a low edge
Euclidean distance as edges only exists between nodes that are close to each other on the plane. A
mesh is an aggregate of non-intersecting elements like triangles or tetrahedron. When converted
into a graph either nodes are the elements and edges binds adjacent element together or nodes are
the intersection points between the elements and edges represents their border. Because of this
creation rules, FEM graphs are planar graphs with varying density gradually peaking usually
within a single continuous area. WAGs connectivity does not have such limits and may possess
high connectivity nodes with edges going further than their direct neighbours leading to crossing
edges and higher edge Euclidean distance. In Wide Area Graphs, the covered space may not be
square and embeds lots of empty space with multiple scattered high density points. Alongside
harsh and smooth node density gradation, WAGs would then match more the not-so-well shaped
meshes [3]. At last, in FEM a given location is always matched to a single element so in graphs
issued from meshes there may never be overlapping node, this rules does not hold true in WAGs.
In this paper we tackle the problem of WAG partitioning. Geometry is inherent to WAGs and
makes a high differentiation property to partition the nodes of a WAG. Geometric strategies have
already been studied in the graph partitioning context and we deem appropriate to use geometry

15
to partition Wide Area Graphs. Such kind of strategy comes with a price, raw geometric
partitioning without heuristics may end very costly. To mitigate this problem, Pilkington and
Baden published in 1994 Inverse Space-Filling curve Partitioning (ISP) [4]a much cheaper
geometric partitioning solution relying on a Space-Filling Curve.A space-filling curve is a
mapping from a multidimensional plane to a single dimension line. A curve construction
algorithm is based on recursion and self-similarity through the levels, each curve is associated to
a level of zoom to fix its granularity. The interesting perk about space-filling curve is their
aptitude to translate the spatial proximity from the multidimensional plane into the single
dimension line. ISP is central to our research and we essentially propose a context extension of
the original paper to cope with possibly upcoming Wide Area Graphs. Our foremost objective is
to figure out for which kind of Wide Area Graphs ISP is relevant.
This research belongs to Thing'in the future, an IoT oriented research project direct by Orange,
France largest ISP provider and Telecommunication Company. Thing'in the future main idea is to
develop a platform able to map connected and unconnected objects of the real world into a
unified graph representing their interactions. Because the large majority of data directly represent
physical objects, the resulting graph is highly connected to geography and can be interpreted as a
WAG prototype. As our objective is to study the WAG family and not only the single specific
Thing’in the future graph, we will also consider synthetic graphs during our experiments.
Through the content of this paper we bring the follow contributions. First, we give an update to
ISP by confronting it to the context of Wide Area Graphs which are basically a tougher graph
partitioning problem than FEM graphs. Next, we expose to what extent edge Euclidean distance
is critical to ISP performance. We propose and establish analysis metric and produce a matrix
showing the limits of ISP helping decide whether or not to use ISP to partition a graph. Finally
we measure its benefits and drawbacks through a set of metrics, mainly comparing our results
against FENNEL algorithm [5].
Figure 1. Illustration of a graph issued from a mesh.
2. RELATED WORK
Graph partitioning is a humongous field of research. The graph partitioning, or the balanced k-
partitioning, problem is described as follows: given a graph G and a number k corresponding to
the number of partitions, we seek to cut G into k balanced pieces while minimizing the number of
edges cut. Edge-cut is the default performance measure to any graph partitioning strategy; it
represents the ratio of edges for which both ends are stored on separate partitions. Since
concretely, each partition is stored in different servers, it is important to have a minimum amount

16
of edge-cuts as it dramatically increases edge traversal time (due to network latencies): a basic
operation for a graph algorithm. Obviously, Load-Balancing is also mandatory; a fairly
distributed load is needed to provide horizontal scalability.
In this section, we depict some of the state-of-the-art strategies in the field of graph partitioning.
We do not aim to cover the whole field of research, but instead we will limit ourselves to the
most relevant research related to our target, e.g. partitioning strategies able to partition a WAG.
For example, a strategy such as METIS [6], the most popular example of multi-level strategy, is
considered to be extremely costly if it is intended to run on WAG scale. While METIS can be
applied on any type of graphs, some strategies are specialized for geometric graphs.
A well-known approach for geometric partitioning is Recursive Coordinate Bisection [7]
(RCB).The RCB method sorts the graph nodes based on the most expanded dimensional axis and
then recursively bisects that axis to distribute the load evenly. This approach then evolved with
random sphere [8] and partitioning based on space filling curve: Inverse Space-Filling
Partitioning [4] (ISP). Random sphere strategy performs geometric sampling over the mesh to
find an efficient circle center point to bisect the mesh with an even load; the algorithm is then
executed recursively to reach the given number of partitions. ISP also relies on geometry but with
a much cheaper approach based on SFC. Nodes which are close on the multi-dimensional space
are mapped close on the single dimension, then they are grouped together and form a partition.
We describe how ISP works in Section 3.
Both RCB and Random sphere are geometric strategies which could have been used to partition
WAG. But given that RCB is known to yield partitioning of poor quality [9] and random sphere
works best on “well-shaped” graphs, which is not what should be expected from Wide Area
Graphs, we will not consider those strategies for our comparisons.
More recent works aim to partition graphs based on geometry [10]–[12]. Akdogan proposes to
partition geometric graphs using Voronoi tessellation whereas Volley places data on a 3D sphere
representing the earth and groups data spatially close to reduce edge-cuts. These solutions work
relatively well but require a lot of computations. Delling et al. strategy is based on the presence of
natural cuts within a graph that minimize edge-cuts. Their solution detects such cuts to build
partitions. Unfortunately the natural cut hypothesis seems unlikely to hold within a WAG.
Today, graph partitioning is considered to be divided into two categories: offline which requires
to store the whole graph in memory to perform global partitioning decision and online with low
requirement strategies based on local decision making. Most online strategies are based on
streaming. Streaming is a powerful approach as it handle well the giant scale induced by scale-
free graphs or Wide Area Graphs. To partition a graph using a streaming strategy is in general
done in a single pass and partitions are built through the streaming process. Every nodes of the
graph are streamed exactly once and their placement is decided with a scoring method. A score is
generated with each partition paired with a given node, the partition with the highest score
obtains the streamed node. Upon stream completion, nodes have all been affected and may stay in
their partition, the partitioning is done.
As far as we know, the first scientific study of streaming graph partitioning was realised by Kliot
and Stanton [13]. In their research they tried various nodes placement strategy with different
heuristics. They concluded in their paper that the best heuristic they could establish for graph
streaming partitioning was Linear Deterministic Greedy (LDG). Later on, Tsourakakis et al. [5]
greatly extended their research with FENNEL. FENNEL basic approach is very similar to LDG,
in fact it is a generalized framework capable of reproducing LDG while bringing subtle

17
configuration possibilities. Streaming strategies all share the same simple base, yet they offer a
large number of declinations each ending with different results.
3. INVERSE SPACE-FILLING CURVE PARTITIONING EXTENSION
In this section we discuss how we extend ISP previous work. As said before, geometric
partitioning within a multi-dimensional space is very expensive. Alternatively, ISP [4] proposes
to map the multi-dimensional space on a single dimension thanks to a space filling curve leading
to a simplified one dimension (1D) partitioning of the curve. The curve is divided into cells and
their order is defined by the curve algorithm. Each cell is a point on the curve mapped to a
bounding box on the multi-dimensional space. A cell is to be assigned to a single partition and
has an associated weight defined by the number of nodes it contains. Thus, the weight of a
partition is equal to the sum of the weights of its cells.
When ISP was first proposed, graph partitioning streaming strategies have not yet been identified
as a particular class of strategies. Therefore, ISP is classified as a geometric partitioning strategy.
We believe it is fully streaming compliant and should therefore be considered as such, the only
requirement is to stream the nodes in the order defined by the cells of the curve used to perform
the partitioning. In our implementation of ISP, based on the total number of nodes (denoted n)
and the number of partitions (denoted k), we assign beforehand a capacity to each partition. We
assume for simplicity they are uniform with a capacity of roughly n/k. Uniform partition capacity
is not mandatory, ISP only requires a total capacity large enough to store the whole graph. We
define the load of a partition as its weight divided by its capacity.
A partition corresponds to a set of contiguous cells of the SFC. Partitions are created sequentially.
To create the first partition, we start from the first cell of the SFC and we consume the cells in the
order defined by the curve. We greedily assemble cells until the partition is fully loaded (i.e., its
load is no longer lower than its expected capacity). Then we move onto the next partition and the
next cell. As a result, the curve is segmented into k intervals and each interval is corresponding to
a partition. Partitions are continuous segment over the curve which corresponds to continuous
area over the multi-dimensional space thanks to the space filling curve properties.
We have respected the original design choice of ISP: even if measuring the load of a partition has
seen some evolution like using the number of edges [13] or other custom metric[5], the load
metric of a partition is still based on the number of nodes it actually stores. We view the number
of nodes as the most suited metric because we are targeting the On Line Transaction Processing
(OLTP) context. We also kept the Hilbert curve[14] as it has been proven to be the optimal space
filling curve for spatial proximity preservation by Knoll and Weis[15]. At this point, whenever
we mention a Space Filling Curve (SFC), we will implicitly assume it is the Hilbert SFC.
Our main contribution is about the design of a dedicated ISP [4]to partition WAG which are
harder to handle than graphs issued from meshes. Besides the properties we described for WAG,
we also mentioned that nodes may overlap in a WAG. During the partitioning process partitions
will be assigned on average a few more nodes than expected (except for the last one which will
be less weighted). Indeed, the last cell assigned to a partition may have more nodes than
necessary and, therefore, its assignment to the partition can potentially overload it. On the
contrary, the last partition will consist of the remaining cells and will be potentially be under
loaded. The possible imbalances are due to the fact that cell unit is the granularity adopted when
defining the partitions: whatever the weight of a cell, it is never subdivided to be shared between
two consecutive partitions. This directly relates to how density and node overlapping is handled
in a SFC.

18
SFCs are usually combined with the adaptive refinement technique which allows the SFC to
zoom in and out depending on current needs e.g. local density for graph partitioning. With graph
where nodes never overlap e.g. graph issues from meshes, such method ensures that each cell will
store at most one node. However, in WAGs nodes may overlap e.g. a building with multiple-
storeys stored in a 2D plane in which case the cell's weight will remain unsplittable even with
adaptive refinement. This is a non-trivial issue as instead of assembling cells which have a weight
of exactly one node to build a partition, we may have to cope with high weight cells tougher to
balance.
Figure 2 illustrates this risk. If the cells have different weights (1, 2, 3 or 5 in the example), the
consumption of the cells in the order defined by the SFC sometimes leads to achieve exact load
balance (at the bottom of the figure) or, on the contrary, leads to exceed partition capacity
threshold by adding a last cell with too many nodes (at the top of the figure). As such, the
existence of super cells (SFC cells with extreme load) may or may not be problematic. High
weight cells are the reasons why partitions tend to have slightly more nodes than expected.
4. EDGE DISTANCE AND DENSITY, ISP DECIDING FACTOR
ISP can potentially be applied to any kind of graph as long as each of its nodes has position on
aEuclidean space. We argue in this section that there are precise conditions for ISP to perform
well. The decisive factor is the distance covered on average by an edge in proportion to the
surface covered on average by a partition, we defined it as:Edge Distance To Partition Square
Size(EDTPS).
The idea is the following: the more we increase the number of partitions, the more the space
covered by each partition decreases as the space is finite and the likelier we are to produce edge-
cuts using a geometric partitioning strategy such as ISP. The higher the EDTPS, the closer is an
edge from being as large as the average square surface of a partition. Geometric partitioning with
low EDTPS should therefore behave well.
Let G(V,E) be a directed geometric graph placed on a 2D plan. The graph G has a surface
englobing polygon P composed of points {( j
i y
x , )} so that P= (( 0
0, y
x ), ( 1
1, y
x ), …( 1
1, 
 n
n y
x )).
The points in P are all sorted in order. Let D(e) be the Euclidean distance between the source and
destination for each edge e of E. We first define the function used to convert a polygon to a
surface area.
)
(
2
1
)
(
1
0
1
1







n
i
i
i
i
i y
x
y
x
P
Area Where x
xn 0
 and y
yn 0

Equation 1. Area formula
Then, the EDTPS formula is as follows:
k
P
Area
E
e
D
k
G
EDTPS E
e
)
(
1
*
)
(
)
,
(
 

Equation 2. EDTPS Formula

19
A graph partitioning solution is evaluated over its ability to preserve edge from being cut and
how well it maintains load balance. ISP, as mentioned in Section 3, may have trouble handling
graph with high local density peak, especially without adaptive refinement. To measure those
potential troubles we introduce our second metrics:Cell Density To Partition Capacity(CDTPC).
Figure 2. ISP Imbalance scenario
As shown in the second example of Figure 2, a hyper dense SFC cell may not automatically
trigger imbalance, it needs to be consumed within a specific interval. This interval is the ratio
between the weight of the cell and the space left in the partition. The heavier the cell, the larger
the interval is to build imbalance partition. CDTPC is designed to evaluate such interval and
detect in advance potential load imbalance. The higher it is, the larger the interval will be. While
a high CDTPC does not necessarily imply an imbalance problem, a small ratio most likely
guarantees a very sane balance.
In order to define CDTPC we need first to compute maximum cell density which we will refer as
MCD. To do so we map every nodes position to the SFC and retain the SFC cell which includes
the most nodes. The CDTPC formula is then the following:
k
N
MCD
k
G
CDTPC 
)
,
(
Equation 3. CDTPC Formula
Note that in both formulas we use k, the number of partitions, as EDTPS and rely on it. It is
expected since the performance of the ISP partitioning algorithm vary depending on the number
of partitions.
5. GEOMETRIC GRAPH GENERATOR
In front of the limitation of crawling graphs with interesting geometric properties and complete
absence of datasets that could be close to represent future Wide Area Graphs (a geometric graph,
covering a large area and with a high number of nodes and edges), we resorted early to use a
graph generator. There already exists geometric graph generators [16], [17], yet, none can

20
provide a WAG-like graph (with the desired density or connectivity characteristics).They
partially allow some edge Euclidean distance configuration but not in absolute terms and lacks
the input parameters to specify how density should behave across the surface of the generated
graph. Such features are needed to produce synthetic Wide Area Graphs with precise EDTPS and
CDTPC variation. For all these reasons we have designed our own geometric graph generator.
The generator is designed to create graphs similar to infrastructure or what we call IoT graphs.
Also, we could not incorporate the preferential attachment model [18], therefore, the generator is
unable to produce scale-free graph. The generator is written in Java and integrates its own R-Tree
[19] to spatially index the nodes and enable fast geospatial query on the generated graph, needed
for the edge-binding step. The generation process is performed in three steps: process population
density through a SFC, then create and assign the nodes on the plane and finally build the edges.
The following parameters are needed to produce a graph: the number of nodes and edges, a
surface plane, node density distribution behaviour in the plane and expected edges Euclidean
length. The surface plane must be a square composed of GeoJSON coordinates. Density
distribution is configured through a set of subcategories composed by a surface unit and a density
factor. The surface unit sets the total portion of the plane which will be affected to this density
factor. Then, densities factors determines, relatively to the others, how dense is a subcategory. At
last, edge distance distribution is also defined through categories, each with a ratio, and a
minimum and maximum possible distance. During generation the same number of starting edges
are assigned to all nodes. The edge ratio is used to determine how many of those edges belong to
a given distance category. At the end, nodes will have the same number of outgoing edges but a
different number of incoming edges.
To generate the node density distribution across the surface of the graph, we apply a grid on the
plane and map each cell to a SFC. The definition of the grid (number of columns and rows) is
defined with the SFC level of granularity. Because each density subcategories are affected
surface units we can determine, relatively to the total of surface units, how much of the SFC (and
consequently of the underlying plane) will be assigned to each subcategory. Again, through
relative ratio, surface units combined with density factors allow us to process the amount of
nodes which will end in a given density subcategory and by extension its local cell density (the
number of nodes populated inside a single SFC cell mapped to the plane).
For the purpose of generating a density distribution somewhat representative of the real world,
we split the reserved portion of the plane assigned to a given density subcategory into multiple
segments scattered across the SFC. Starting with the densest subcategory, the generator will
create a segment and for this purpose it first starts at a random available cell on the curve. Then, a
random Pareto function is used to calculate the segment length: number of contiguous cells of the
curve. Density itself is used as a parameter in the Pareto function which leads to create small
hyper dense segments against long established low density segments. Upon completion, the curve
is composed by a multitude of density subcategory segments. All that remains is to populate
those segments with nodes and proceed to the edge binding. We provide a visual example at
Figure 3, the left part show the construction order of density segments with red being the densest
subcategory. On the right we see the curve composed of density segments mapped to the 2D
plane.

21
Figure 3. Density distribution of the node population
Populating the nodes only consists in taking a cell from the SFC, checking its local cell density
determined earlier in the generation process, and producing the corresponding number of nodes.
A random GPS location constrained within the cell bounding box is generated and assigned to
each node populated within that cell.
Edge binding is much more computation-heavy. To satisfy the constraint of an edge Euclidean
category we perform oversized geometric square query over the embedded R-Tree. Square have a
diameter twice as large of the maximum edge Euclidean distance required for current distance
category, it gives the generator nodes sufficiently spaced to create even the longest edge required.
A square shape allows us to cover the surface of the generated graph through a simple sliding
strategy. It also enables the generator to work with only limited chunks of the graph and with
appropriate offsets to perform multiple edge binding passes on the same area, we make sure no
connected component is formed because of our approach. At last, to avoid abnormal behaviours
in edge connectivity distribution, the edge building process heavily relies on random selection.
While random selection slows down the edge building process we deem it effective for our
current needs: a stable edge connectivity distribution.
6. EXPERIMENTAL SETUP
6.1. Setup
We evaluate our solution on a high processing computing machine with 92 GB of RAM and 48
CPU cores. The spatial graph generator and the metric processor were written in Java 8,
everything related to visualization was done using Python 3.6. We choose to rely on the Graph X
library of Spark (https://spark.apache.org/) to perform parts of our metrics; Spark is an expensive
overhead in both development and computation but brings high scaling potential. Because Spark
is built to work over Hadoop (See https://hadoop.apache.org/) and HDFS, we deployed a
standalone HDFS service to handle the data read and produced by the metrics jobs. To evaluate
the different partitioning solutions, we measure the amount of edge-cuts produced and the
imbalance of the partitions. Regarding load imbalance, we use the normalized maximum load
metric [5] (defined Section 4.1). We also include the custom heuristics based on EDTPS and
CDTPC defined in section 4.
6.2. Dataset
We used two kinds of graphs in our experimentation: the graph of Thing’In whose characteristics
are presented in the following Section 6.2.1 and synthetic WAGs produced using the graph
generator already described in Section 5. We consider different WAGs with distinct
characteristics as it is impossible to reduce WAGs to a single set of properties. Precisely, we
aimed to first produce WAGs close to Thing’In properties and then gradually deviate to obtain a

22
wide range of results for the application of ISP. We also wanted to find an inflexion point where
ISP becomes better or worse than concurrent strategies.
Figure 4 and 5. (Left) Thing’In Degree Distribution, blue column is for nodes with degree less than 10.
(Right) Thing’In edge euclidean distance distribution
6.2.1. Thing’In Graph
The Thing'in the futuregraph is generic, it relies on web semantic to describe its content. It is the
resulting aggregate of open data sources focusing on real world objects crunched into our
platform. Thing'in the future is not an industrial project yet and contains inoperable data for our
experiments, mainly nodes without GPS location. As we need the graph to be totally geometric,
we removed about half the nodes of Thing'in the future. The rest (~ 27.2 millions of nodes and
~27.5 millions of edges) is a mix of transportation network like trains, buses, roads... and
buildings information.
Given its current content, Thing'in the futureis a spatial graph. It is confirmed with Figure 4, we
see a connectivity typical of a graph in which road data represents a big chunk of the graph.
Therefore, Thing'in the future graph cumulative degree connectivity does not follow a power-law.
Super-nodes (nodes with hyper connectivity) are absents and the most connected node maps to
Rennes (the city where the Thing'in the future main developer team is located. They performed
many experimentations related to this city and, consequently most edges of the graph are related
to this city).The average edge Euclidean distance shown in Figure 5 is extremely low: 97.03% of
the edges are shorter than 2km and very few edges are longer than 10 km (~ 0.0001%).
Figure 6. Thing’in the future node population distribution

23
Nodes are distributed in a uniform manner across the territory of France (See Figure 6). There are
still some regions where no data has been injected, thus with a density of 0, whereas all large
cities show progressive density towards the town center. Based on its properties, we consider
Thing’in the future as a WAG. Consequently, it represents an interesting graph to be partitioned
with ISP.
6.2.2. Synthetic datasets
We created this specific synthetic dataset for the purpose of showing potential correlations
between EDTPS and CDTPC (see Section 4) and ISP performances in edge-cuts and load-
balance. As we believe those measures are intertwined, our strategy is to generate graph similar
to Thing'in the future and iteratively change edge Euclidean distance and density. Thing'in the
future is not a fully fledge WAG thus and its density is not guaranteed to be representative of
WAG. However, we think density in WAG should relates to human density to some degree with
equivalents of deserts, empty area, and metropolises, area with extremely concentrated
population.
Table 1. Density configuration
Category 0 0.5 5.0 20.0 50.0 500.0
HD 1000 1000 100 0 10 1
MD 1000 1000 100 0 10 0
LD 1000 1000 100 10 0 0
Table 1 contains the density configuration we used in our experiments. Columns represents a
type of density characterised by a density strength factor in the header, the rows values are
surface units e.g. how much of this density type will be present in current row (density
distribution). For instance, the High Density distribution (HD) is a mix of 47.37% empty surface,
47.37% scarcely populated surface, 4.73% low density surface, 0.47% high density and 0.05%
extreme density. If no surface units are allocated in a given type of density then it is absent from
this density distribution. We explain how density distribution are used within our generator in
Section 5. The low density distribution (LD) was designed to approach Thing'in the future
density whereas HD is based on a simplified Zipf's
(https://en.wikipedia.org/wiki/Zipf%27s_law) distribution of human density across the Earth. At
last, the medium distribution (MD) act as an intermediary point to provide more CDTPC
variation and hopefully load-balance variation.
Our density distributions are merely relative. To obtain varying effective density on the generated
graphs we also need to set graph volumetry (number of nodes and edges) and plane size. We
mainly kept a fixed volumetry of roughly 50 million nodes and 150 million edges and used two
plane sizes (See Table 3). With the same volumetry and density distribution, the large plane (EU)
is supposed to smooth out density peaks. Instead, those peaks will be accentuated within a small
plane (FR) in which we expect the presence of hyper dense hot spot, especially with HD density.
With this method we produced graphs with varying density and CDTPC metrics allowing us to
identify CDTPC and ISP's load balance correlation.
Table 2. Synthetic plane configuration
Type Latitude longitude
FR (Country) [41.15:50.33] [-5:9.85]
EU (Continent) [41.15:65.33] [-5:40.85]

24
Relevant to identifying ISP performances and EDTPS correlation, Table 3 lists the 8 edge
Euclidean distance setup we applied to create our synthetic dataset. Just like density distribution
and plane size, our edge distance settings were designed to force out a wide range of EDTPS
values to evaluate its accuracy. Also, having two kind of planes allows us to prove absolute plane
size is impactless on ISP. Overall, as we wanted to have tough load-balance to manage HD was
set as the default density distribution with the exception of TINY- which combined with medium
density distribution expressly targeted at producing graph similar to Thing'in the future. At last,
the edge distance category LONG was tested with all 3 densities.
Table 3. Average edge distance and EDTPS of synthetics WAGs
Category AVG_Dist EDTPS
FR EU
TINY-_MD 1041.49 0.00192 0.00076
TINY 13521.19 0.02123 0.00967
SHORT 25860.80 0.04773 0.01765
MEDIUM 48902.51 0.09026 0.03395
MEDIUM+ 55766.50 0.10293 0.03562
LONG 84392.80 0.15608 0.0562
LONG_MD 86869.69 0.16034 0.05508
LONG_LD 89290.85 0.14142 0.06008
LONG+ 176261.23 0.32547 0.12684
HUGE 391988.63 0.7235 0.2865
Once we finished the generation of our synthetic datasets, we had to verify if it was satisfying our
initial objectives: producing graphs with varying EDTPS and density. Again, as shown in Table
3, we obtained interesting EDTPS values ranging from 0.1% to 72.35% with 4 partitions on FR
plane. With 16 partition, on FR plane and HUGE category the EDTPS even reaches more than
100% percent and represent the default case where ISP is bound to perform poorly.We failed to
match exactly the EDTPS of Thing'in the future (See the row TINY-) as Thing'in the future is
even lower, however it remain satisfying in absolute terms as EDTPS values are extremely
small.Additionally, we checked on cumulative connectivity distribution and density. As our
generator does not embed the features to produce graph with scale-free properties, graphs
connectivity remained low with no nodes above one hundred connectivity degree.
7. RESULTS
We seek in this experiment to compare existing state of the art streaming graph partitioning
method to ISP. We applied ISP, FENNEL and LDG on each dataset with the same streaming
approach. The only difference aside the decision algorithm is the streaming order. In ISP nodes
were streamed ordered by the cells of the SFC whereas nodes were streamed randomly for
FENNEL and LDG because, given our scale, DFS (Depth First Search) and BFS(Breadth First
Search) were too expensive to apply. We first look at how strategies behave in terms of edge-cuts
and secondly how they handle balance across their partitions. Note that LDG results are not
showed in the tables: LDG is always worse than FENNEL regarding edge-cuts and equivalent in
load-balance.
7.1. Edge-cuts
As it was our objective, we managed to reach a breaking point where the performance of
FENNEL finally exceeds ISP. It is shown in Table 5 (See LONG+) that on a plane with the size

25
of a country like France, with edge Euclidean distance past 175 kilometres on average and 16
partitions, it becomes better to apply FENNEL rather than ISP.
We expected while building our datasets that the performances of ISP would heavily correlates to
EDTPS, it has been confirmed through the results. Each time EDTPS increases so does the
number of edge-cuts produced by ISP without exceptions. We also managed to reach a scale
where ISP is bound to perform extremely poorly (HUGE), in that case up to 95% of the edges are
cut. On the contrary, when edges are extremely short (e.g. TINY-), ISP outperforms 112 times
the results of FENNEL in synthetic graphs and up to 125 times when applied to Thing’In (See
Table4 and6).
We used the same space-filling curve based strategy for graph generation and ISP. Yet this has no
impact on the results for two reasons. First, both SFCs are not matching. For two curves to match,
those needs to use the same algorithm, granularity and map the same multidimensional plan.
Although algorithm and granularity used are identical, the multidimensional plan are different.
ISP covers the whole world map whereas the generator cover only the bounding boxes given in
Table 3 changing both SFC cell bounding box surface and SFC cell ordering. Figure 7 is a visual
example of both curves, their respective mapping and ordering are impossible to match or
correlate. Second, the generator edge binding step ignores the curve presence, edges are created
randomly around a node with respect to edge Euclidean distance configuration. Even if the curves
were matching, it still would not give any form of advantage to ISP.
Figure 7. Visual example of generator (Blue SFC) and ISP (Red SFC) mismatch
Interestingly, we see multiple EDTPS and ISP results correlation, it must however be weighted
with the amount of borders induced by the partitions which is currently ignored by our metric.
Each new partitions produces additional border which may cut edges. As EDTPS does not
include those borders in its estimation, we use the following interpretation rule: two graphs with
identical EDTPS but different number of partitions should have different behaviour with ISP. The
one with less partition should yield better results. We joined a part of the results obtained for the
EU plane size in Table 6 and this interpretation rule does hold true.

26
Table 4. EDTPS and performances of ISP and FENNEL over the generated graphs for FR plane
Category K EDTPS FENNEL ISP
TINY-_MD 4
8
16
0.00192
0.00272
0.00384
38.232±0.002
45.811±0.002
50.194±0.002
0.34±0.0
0.55±0.0
0.83±0.0
TINY 4
8
16
0.02123
0.03002
0.04245
47.733±0.0
56.241±0.0
61.074±0.0
2.61±0.003
5.1±0.007
8.05±0.006
SHORT 4
8
16
0.04773
0.0675
0.09546
47.872±0.0
56.337±0.0
61.129±0.0
5.89±0.008
9.86±0.017
15.21±0.007
MEDIUM 4
8
16
0.09026
0.12764
0.18052
48.014±0.001
56.545±0.001
61.345±0.001
10.9±0.018
17.9±0.032
27.59±0.021
MEDIUM+ 4
8
16
0.10293
0.14556
0.20586
47.908±0.001
56.372±0.001
61.16±0.001
11.35±0.013
19.45±0.014
27.37±0.01
LONG 4
8
16
0.15608
0.22074
0.31217
47.914±0.0
56.379±0.0
61.175±0.0
16.77±0.013
29.39±0.017
41.18±0.006
LONG_MD 4
8
16
0.16034
0.22675
0.32067
47.917±0.0
56.38±0.0
61.171±0.0
18.12±0.026
28.21±0.031
41.06±0.04
LONG_LD 4
8
16
0.14142
0.2
0.28285
47.897±0.0
56.361±0.0
61.159±0.0
15.5±0.05
25.99±0.082
38.21±0.116
LONG+ 4
8
16
0.32547
0.46028
0.65093
48.07±0.0
56.564±0.001
61.364±0.001
30.06±0.017
47.71±0.013
67.48±0.004
HUGE 4
8
16
0.7235
1.02318
1.44699
48.056±0.001
56.557±0.002
61.359±0.002
63.88±0.017
86.02±0.007
95.49±0.004
If we compare the results of EU LONG+ and FR MEDIUM with k = 8 and 16 (see Table 4 and
5), they have similar EDTPS but EU LONG+ has consistently better edge-cuts results as it uses
fewer partitions to reach the same EDTPS. The same pattern can be detected comparing EU
LONG_LD to FR MEDIUM k = 8.It can also be applied to graphs within the same plane: take
FR TINY with k = 16 and FR MEDIUM+ k = 4 or EU HUGE k = 4 and EU LONG_LD k = 16.
At last, ISP performances are similar when both EDTPS and number of partitions are matching
(See FR TINY and EU SHORT). The EDTPS values are interleaving and so do the edge-cuts
results with ISP. Finally, it is easy to see EDTPS and edge-cuts correlation (See Figure 8). The
only unmatched curve corresponds to HUGE which is perfectly acceptable as there is a ceiling
effect for the edge-cuts. EDTPS may go further than 100% but edge-cuts are ultimately limited to
100%.

27
Table 5. Partial EDTPS and edge-cuts (EC) of ISP and FENNEL over the synthetic EU graphs
Category K EDTPS FENNEL (EC) ISP (EC)
SHORT 4
8
16
0.01765
0.02495
0.03529
47.279±0.001
55.79±0.001
60.601±0.001
2.49±0.003
4.18±0.004
7.24±0.004
LONG 16 0.1124 61.167±0.001 18.36±0.014
LONG_MD 16 0.11017 61.163±0.0 17.81±0.01
LONG_LD 16 0.12016 61.159±0.0 18.65±0.002
LONG+ 4
8
16
0.12684
0.17937
0.25367
48.365±0.001
56.899±0.001
61.709±0.001
14.37±0.022
25.49±0.018
36.19±0.007
Table 6. EDTPS and edge-cuts (EC) of ISP and FENNEL over Thing’in the future
K EDTPS FENNEL (EC) ISP (EC)
Thing’in the
future
4
8
16
0.00006
0.00008
0.00011
6.349
34.636
37.883
0.132
0.277
0.435
While EDTPS seem appropriate to evaluate ISP performance, it ignores other potential factors
like connectivity and volumetry, we argue that it does not affect its veracity. We did run tests to
check the volumetry impact and we couldn't observe anything significant, the problem is more
about connectivity. Connectivity is ignored in both ISP and EDTPS and our dataset does not
present much connectivity variation. Still, in essence, as connectivity degree rise for a given
node, it has to reach further and further to connect to new nodes as there is a finite number of
nodes in its neighbourhood. Consequently, edge distance tends to be longer. But as edge distance
is already used in EDTPS, ignoring connectivity should not be a problem.
We expected ISP results to be a bit dissonant; nonetheless we are surprised by FENNEL extreme
stability. Again, our most serious guess is based on the fact that all the synthetics graphs share the
same low connectivity with absence of super nodes which FENNEL relies on to build its
partitions.
Figure 8. (Left) EDTPS, (Right) Edge-Cuts percentage of graph partitioned with ISP for each edge distance
category. X-axis is the partition number, bounding box is FR, density information excluded as impact less
on EDTPS and edge-cuts.

28
Contrary to our synthetics graphs, Thing’in the future graph possess less edges but its
connectivity degree distribution has a better scaling, it could explain why FENNEL yields better
results when applied on Thing’in the future. There remains a wide disparity between 4 and 8
partitions with FENNEL on Thing’in the future graph for which we have no appropriate
explanation. Overall, we cannot be sure that the results similarities for FENNEL are really due to
the connectivity. It is however safe to assume that edge distance has no direct impact on this
solution.
7.2. Partition balance
We discuss now about how the different strategies handle balance across the partitions. FENNEL
and LDG are pretty much perfect at preserving load balance across the partition, the same cannot
be said for ISP. The partition imbalance ranges from 2.26% to 11.18% with 16 partitions on FR
plane. Obviously, the results regarding imbalance are better for the EU plane. Its worst imbalance
case has been recorded at 2.31% which is normal: there is the same number of nodes and density
but for a larger plane. Looking at Table 7, we focus on the LONG+ edge distance category which
has been tested with all 3 density distributions, as density gets smoother we see a reduced
CDTPC and reduced load imbalance.
Table 7. CDTPC and load imbalance of ISP and FENNEL over the generated graphs for
LONG+ and FR plane.
Category K CDTPC FENNEL ISP
LONG+ 4 0.04916 0.0±0.0 0.22±0.002
8 0.09831 0.0±0.0 2.13±0.013
16 0.19662 0.0±0.0 7.09±0.022
LONG+_MD 4 0.00836 0.0±0.0 0.16±0.002
8 0.01673 0.0±0.0 0.6±0.003
16 0.03345 0.0±0.0 1.96±0.003
LONG+_LD 4 0.00325 0.0±0.0 0.1±0.001
8 0.0065 0.0±0.0 0.19±0.001
16 0.01299 0.0±0.0 0.61±0.004
The balance provided using the ISP strategy is fragile. It hugely depends on the graph, that is if it
is "well behaved" or not, if the density does not peak too much relatively to the precision of the
SFC. Even though a huge CDTPC ratio will not automatically trigger an imbalance problem, it is
a good risk indicator as a graph with a very low CDTPC will never encounter balance trouble. A
single cell might contain more than 500k of nodes all by itself. This does pose a problem because
in some circumstances with a high number of partitions and extremely peaked density a single
cell may have a load high enough to provoke single-cell partitions.
With the graphs generated based on the density HD given in Table 1 over a plane of a size similar
to France and the Hilbert SFC of zoom 12, as demonstrated in Figure 9, 20% of the nodes in FR
synthetic graphs with HD density resides in super-cells (cells with a weight over 100k nodes). In
comparison, EU synthetics graphs even with HD density never reach the super-cell threshold.
With LD density, the most loaded cells weights less than 10k nodes showing that a lower density
will behave much better for load-balancing with ISP.

29
Figure 9. Node distribution by cell density in synthetic graphs (Left) Europe (Right) France. All curves
used SFC zoom 12 except HD_z15 (Right) done with zoom 15.
Regardless, if this issue does happen it can be solved using SFC with higher zoom as it splits the
load of a single cell into multiple cells. In theory, zoom is not guaranteed to splits the load. In
practice it remains an effective approach, HD_z15 in Figure 9 precisely shows the same
configuration with a higher zoom in which super-cells naturally disappear.
A cell which stores 500k nodes at zoom 12 may be reduced to 64 sub cells at zoom 15 with a load
of at most 10k nodes by cell, dropping effectively the CDTPC from 17% to 1.5%. With such a
small CDTPC, the risk of imbalance becomes almost zero, and in all fairness, zoom 15 is a rather
standard precision level. Unfortunately, we could not afford zoom 15 as our metric system
wouldn't have been able to hold the additional load memory wise. At last, if zooming in 2D is not
efficient because of node overlapping, a third dimension (3D) can be used to split the load as both
ISP and Hilbert can be translated in 3D. However, if any amount of dimensions is not able to split
the load, ultimately, geometry itself cannot be used to distribute the graph which is not an issue.
It only means this given graph is out of the scope of ISP.
8. CONCLUSION
In the present article we discussed and described the potential apparition of a new kind of graph
named WAG. To tackle the partitioning of WAG, we proposed an extension of ISP a graph
partitioning solution based on geometry originally designed to partition FEM issued graphs.
Because choosing a graph partitioning strategy has become an issue due to the wide range of
solutions, we also provided additional metrics to analyze why ISP performs well or not and
identify potential use case. To evaluate ISP we compared it to state of the art graph streaming
partitioning solutions: LDG and FENNEL. The lack of appropriate datasets inclined us to build a
custom geometric graph generator to satisfy our needs. The results showed edge Euclidean
distance and node density distribution have significant impact over ISP performances in both
edge-cuts and load balance. ISP is highly polarized, unsuited for graphs using long range edges,
typical of social graphs, producing very poor results and produces extremely good results for
graphs with short edges, typical of spatial network or WAG. In a future with large scale WAG
composed of tiny edges with a distance under one hundred meters as physical interaction between
objects is characterized to such range, ISP could prove to be the best solution.
As for our next researches we think ISP has room for improvement. For instance, given ISP ring-
like base structure, replication could be very beneficial and used for both resiliency and
optimisation. Also, the analytical context is not the end goal of our research on ISP: database is.
Unlike other streaming strategies, ISP adapted with the right life-cycle method can fit within the
database context and remain efficient, thus we are excited to explore ISP in that direction. At last

30
we would like to extend our WAG generator and mix in the preferential attachment model so that
it’s able to produce WAG which spatially logical connectivity.
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31
AUTHORS
Cyprien Gottstein graduated from the University of Rennes 1 with a master degree
in Software Engineering in 2016, Rennes, France. He worked from 2016 to 2018 as
an engineer at Thales services and started his PhD in large scale graph partitioning for
the Internet of Things (IoT)in 2018 at Orange Labs, Rennes, France.
Philippe Raipin Parvedy has received his PhD from the University of Rennes 1,
Rennes, France, in 2004. He was a temporary lecturer and research assistant in the
university of Rennes 1 from 2004 to 2006 and a post doc researcher in Orange from
2006 to 2007, since 2007, he has been a research engineer in Orange. He is currently
the research program manager of the Web of Things platform in Orange. His research
interests include dependability, large systems and graph databases.
Michel Hurfin holds a Ph.D. in Computer Science from the University of Rennes
(1993). After one year spent at Kansas State University, he is currently conducting his
scientific research activities at the INRIA Rennes Bretagne Atlantique research center.
He defended his HDR (Habilitation à Diriger des Recherches) in 2004. Since 2012 he
is a member of the CIDRE project that aims at addressing security problems. His
research interests include distributed systems, software engineering and middleware
for distributed operating systems. His recent works focus mainly on dependability and
security issues in distributed systems (in particular, intrusion detection mechanisms).
Thomas Hassan is a researcher at Orange Labs, Rennes, France since 2019. He
obtained a Thesis in Computer Science from the University of Burgundy, Dijon,
France in 2017. His main research areas are Semantic Web, Machine Learning and
Big Data. His research was applied to the domains of pervasive computing, Internet
of Things, geographic information system and recommender system.
Thierry Coupaye is head of research on Internet of Things (IoT) inside Orange,
Orange Expert on Future Network and Orange Open Source Referent. Prior to that,
after he completed his PhD in Computer Science in 1996, he had several research and
teaching positions at Grenoble University, European Bioinformatics Institute
(Cambridge, U.K.) and Dassault Systems. He joined France Telecom in 2000 where
he had several research expert, project manager, project and program director
positions in the area of distributed systems architecture, autonomic computing,
cloud/fog/edge computing and networking. He is the author of more than 75 refereed articles and has
participated in multiple program and organisation committees of conferences in these areas. His current
research interests include Digital Twins and Edge Intelligence (AI@Edge).

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