=Paper= {{Paper |id=Vol-3946/TGD_paper3 |storemode=property |title=Transforming Time Series into Graphs and Vice Versa with HyGraph |pdfUrl=https://ceur-ws.org/Vol-3946/TGD-3.pdf |volume=Vol-3946 |authors=Mouna Ammar,Shubhangi Agarwal,Angela Bonifati,Erhard Rahm }} ==Transforming Time Series into Graphs and Vice Versa with HyGraph== https://ceur-ws.org/Vol-3946/TGD-3.pdf

Transforming Time Series into Graphs and Back with HyGraph
Mouna Ammar1,* , Shubhangi Agarwal2 , Angela Bonifati3 and Erhard Rahm1
1
Leipzig University and ScaDS.AI, Leipzig, Germany
2
Lyon 1 University, LIRIS, Lyon, France
3
Lyon 1 University, LIRIS and IUF, Lyon, France

Abstract
Existing graph data management systems still provide limited support for evolving and temporal data. In addition, time-series data often
reside outside graph engines, hindering unified analysis. HyGraph is a new hybrid approach to manage and analyze both temporal
graph and time series data in a unified manner. In particular, it supports rich transformations between graph and time-series data. We
discuss two novel operators on HyGraph to illustrate such transformations, a time-series-based graph operator and a graph-based
time-series operator. The first ingests time-series data and produces a new graph (or a subgraph) that captures relationships among time
series based on correlation values. The second operator, in contrast, generates a time series based on the evolution of temporal graph
metrics, such as aggregated edges or changes in node degree. The transformation operators allow the augmentation of derived values to
the hybrid structure for self-enrichment. We also outline open challenges of dynamic transformations within the hybrid model.

Keywords
hygraph, hybrid graph, property graph, temporal graph, time-series, multi-model

1. Introduction elements and computing similarity-based transformations.
We start by formally introducing the HyGraph model in
Graphs are a powerful tool for modeling interconnected real- Section 2, followed by its UML and system architecture in
world data, widely used in domains such as social networks, Section 3. We present the two key transformation operators
knowledge graphs, and urban mobility. Many of these ap- in Section 4 and illustrate the applicability of our model
plications inherently involve temporal dynamics, where through a micro-mobility use case (Section 5). Broader im-
graph elements evolve. For instance, sensor networks con- plications and future directions are discussed in Section 6.
tinuously generate time-series data [1, 2], and ride-sharing
platforms track vehicle metrics over time [3, 4]. Existing
graph database systems are often limited in their ability to 2. HyGraph Data Model
natively manage and analyze such evolving temporal data.
By contrast, the representation of time series data falls short Analyzing data that combines graph structures and time
of preserving interaction between the entities. The time se- series offers deeper insights than separate analyses. For
ries databases (TSDBs) are designed to efficiently store and instance, in micro-mobility applications, tracking how usage
analyze temporal data and are not optimized for capturing patterns evolve alongside spatial station layouts can predict
complex graph structures. They primarily focus on sequen- demand and uncover efficient vehicle distribution strategies.
tial data retrieval and aggregation [5], lacking native support Although there have been efforts to unify graph and time-
for graph traversal, multi-hop, or relationship-based analyt- series data, they often rely on graph representations for both,
ics. Further, high-dimensional time series data challenges limiting the depth of time series analysis and relegating time
traditional mining techniques, motivating graph-based rep- series to a secondary role, primarily representing property
resentations as a powerful tool for analysis and visualization evolution [8, 9]. Such approaches essentially extend the
[6]. As a result, time-series data in graph applications is of- property graph model rather than creating a truly unified
ten stored separately, either in side systems or as attributes model. As a result, time-series capabilities are limited in
in graph databases, leading to inefficient data management. terms of analysis and querying, and there remains a dearth
HyGraph aims at addressing these limitations with a of operators and algorithms that fully leverage both data
unified model that seamlessly integrates property graphs types in tandem. Although some domain-specific machine
with time-series data. HyGraph directly represents time- learning models combine those data types [10, 11, 12, 13], a
dependent attributes and supports new transformation op- general-purpose approach is lacking.
erators for evolving graph analytics. A broader discussion Through HyGraph we aim to provide a unified system
of HyGraph’s vision and related work can be found in [7], that handles the complexities of integrating graph and time
where we also outline the motivation behind the approach series data, offering flexible functionalities. The core of Hy-
and its high-level goals. In contrast, this paper provides a Graph is a novel data model designed with equal emphasis
detailed exploration of the HyGraph data model and trans- on graph and time series data, enabling the development of
formation operations, like extracting time series from graph hybrid operators, algorithms, and data mining techniques
specifically tailored for this combined data structure. This
model, detailed below, lays the foundation for a flexible
Published in the Proceedings of the Workshops of the EDBT/ICDT 2025 approach to analyzing graph and time series data in unison.
Joint Conference (March 25-28, 2025), Barcelona, Spain
*
Corresponding author.
Let 𝒦 be the set of property keys, 𝒩 the set of property
$ ammar@informatik.uni-leipzig.de (M. Ammar); values, ℒ the set of labels and 𝒯 the set of timestamps.
shubhangi.agarwal@liris.cnrs.fr (S. Agarwal);
angela.bonifati@univ-lyon1.fr (A. Bonifati); Definition 1. Temporal Property Graph (TPG). We refer-
rahm@informatik.uni-leipzig.de (E. Rahm) ence the property graph model defined in [14] and extend
0009-0005-8959-3643 (M. Ammar); 0009-0004-4405-4833 it by adding a validity period for each element. A TPG 𝒢
(S. Agarwal); 0000-0002-9582-869X (A. Bonifati); 0000-0002-2665-1114 can be represented by a tuple as, 𝒢 = (𝑉𝑝𝑔 , 𝐸𝑝𝑔 , 𝜌, 𝜑, 𝜆, 𝜂),
(E. Rahm)
Copyright © 2025 for this paper by its authors. Use permitted under Creative Commons License where:
Attribution 4.0 International (CC BY 4.0).

CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
• 𝑉𝑝𝑔 and 𝐸𝑝𝑔 : Sets of vertices and edges respectively.
• 𝜌: 𝐸𝑝𝑔 → 𝑉𝑝𝑔 × 𝑉𝑝𝑔 maps an edge to its source and
target vertices.
• 𝜑 : (𝑉𝑝𝑔 ∪𝐸𝑝𝑔 )×𝒦 → 𝒩 is a property function mapping
each graph element and property key 𝑘 ∈ 𝒦 to a property
value in 𝒩 .
• 𝜆 : (𝑉𝑝𝑔 ∪ 𝐸𝑝𝑔 ) → ℒ associates each graph element with
a unique label from the set of labels ℒ.
• 𝜂 : (𝑉𝑝𝑔 ∪ 𝐸𝑝𝑔 ) → 𝒯 × 𝒯 retrieves the start and the end
timestamps between which the graph element is valid.
Let {𝑡start , 𝑡end } ∈ 𝒯 represent the two timestamps, then
𝑡start ≺ 𝑡end , where the symbol ≺ specifies ordering, (𝑡end
is initialized to 𝑚𝑎𝑥(𝒯 )).
Figure 1: Example HyGraph snapshot of bike-sharing data
Definition 2. Time series. A time series 𝑡𝑠 (univari-
ate or multivariate) is an ordered set of tuples 𝑡𝑠 =
{(𝑡1 , 𝑦1 ), (𝑡2 , 𝑦2 ), . . . , (𝑡𝑛 , 𝑦𝑛 )|𝑛 ∈ N}, with timestamp
𝑡𝑖 ∈ 𝒯 , such that 𝑡𝑖 ≺ 𝑡𝑗 if 𝑖 < 𝑗, and 𝑦𝑖 represents a tuple Definition 5. HyGraph Model. The HyGraph model is
of real values 𝑦𝑖 = (𝑣𝑎𝑙𝑖1 , 𝑣𝑎𝑙𝑖2 , . . . , 𝑣𝑎𝑙𝑖𝑘 ). denoted by a tuple,

Definition 3. Dynamic Subgraph. Let 𝑠 ∈ 𝑆 be a sub- 𝐻𝐺 = (𝑉, 𝐸, 𝑆, 𝑇 𝑆, 𝜌, 𝜑, 𝛿, 𝜓, 𝜆, 𝜂)
graph where 𝑆 represents a set of subgraphs. The function
where, 𝑉 is the set of vertices, 𝐸 the set of edges, 𝑆 the set
𝜓 : 𝑆 × 𝒯 → 𝒫(𝑉𝑝𝑔 ) × 𝒫(𝐸𝑝𝑔 ) maps a subgraph at a time
of logical subgraphs and 𝑇 𝑆 the set of time series.
𝑡 ∈ 𝒯 to a set of constituent vertices and edges from 𝑉𝑝𝑔
and 𝐸𝑝𝑔 , respectively, while 𝒫(·) denotes the power set. • 𝑉 : A union set of property graph vertices (𝑉𝑝𝑔 ) and time
Further, two subgraphs may overlap at any point in time, series vertices (𝑉𝑡𝑠 ), i.e., 𝑉 = 𝑉𝑝𝑔 ∪ 𝑉𝑡𝑠 .
𝑡 ∈ 𝒯 . The overlap between two subgraphs {𝑠1 , 𝑠2 } ∈ 𝑆
• 𝐸: Similar to 𝑉 , it is defined as a union set of property
can then be captured as the set of vertices and edges common
graph and time series edges, i.e., 𝐸 = 𝐸𝑝𝑔 ∪ 𝐸𝑡𝑠 .
to both the subgraphs at 𝑡, i.e., 𝛼(𝑠1 , 𝑠2 , 𝑡) = {(𝜋v (𝑠1 , 𝑡) ∩
𝜋v (𝑠2 , 𝑡)), (𝜋e (𝑠1 , 𝑡) ∩ 𝜋e (𝑠2 , 𝑡))}. Here, 𝜋v : 𝑆 × 𝒯 → • The function 𝛿 : (𝑉𝑡𝑠 ∪ 𝐸𝑡𝑠 ) → 𝑇 𝑆, maps a time-series
𝒫(𝑉𝑝𝑔 ) and 𝜋e : 𝑆 × 𝒯 → 𝒫(𝐸𝑝𝑔 ) are projection func- vertex and edge to a multi-variate time series in 𝑇 𝑆.
tions that retrieve the set of constituent vertices and edges,
respectively, for a subgraph at any given time. All the mapping functions are adapted to include both prop-
We extend the property-graph model to incorporate time- erty graph and time series graph objects.
series data, such that any vertex, edge, or subgraph can hold
time-series properties. Formally, we expand the scope of • 𝜌 : 𝐸 → 𝑉 × 𝑉 maps edges to source and target vertices.
sets of property keys and values to include both static and
• 𝜑 : (𝑉 ∪𝐸 ∪𝑆)×𝒦 → 𝒩 . The map function is modified
dynamic, thus embedding time series as a natural property.
to include a subgraph, which is treated as a logical graph
Definition 4. Property. The property of a graph element is object and can have associated properties.
represented by a key-value pair, where the key and value be-
long respectively to the set of keys 𝒦 and values 𝒩 , respec- • The subgraph mapping function is adapted to allow a sub-
tively. The map function 𝜑 : (𝑉𝑝𝑔 ∪𝐸𝑝𝑔 ∪𝑆)×𝒦 → 𝒩 maps graph to have edges and vertices of both types, property
a vertex, edge or a subgraph and a property key to a prop- graph, and time series, as, 𝜓 : 𝑆 × 𝑇 → 𝒫(𝑉 ) × 𝒫(𝐸).
erty value in 𝒩 , where 𝒩 = {𝒩Σ ∪ 𝒩TS | 𝒩Σ ∩ 𝒩TS = ∅}. • The label function 𝜆 : (𝑉 ∪ 𝐸 ∪ 𝑆) → 𝒫(ℒ) associates
The set 𝒩Σ is the set of all possible static property values an entity with labels from the set of labels ℒ.
and the set 𝒩TS contains the dynamic property values, i.e.,
time series. Dynamic properties are further classified into • Finally, the function 𝜂 : (𝑉 ∪ 𝐸 ∪ 𝑆) → 𝒯 × 𝒯 retrieves
two categories: the start and the end timestamps between which a graph
element is valid.
• Regular Properties. These store external data associated
with the object, representing attributes that evolve based This new extension ensures that time series are treated as
on external updates or observations. structured entities that can be queried, connected to other
• Meta-Properties. These store internal data derived from time series, and analyzed within a TPG framework.
the graph itself, such as the evolution of graph metrics Figure 1 shows an example HyGraph created from a snap-
(e.g., node degree, centrality measures) or aggregated shot of a bike-sharing system (NYC "CitiBike" [15]). In this
properties over edges (e.g., traffic volume between nodes). representation, stations are modeled as property graph ver-
These meta-properties provide insights into the graph’s tices (𝑉𝑝𝑔 ), while trips between stations are represented as
internal structure and dynamic behavior. property graph edges (𝐸𝑝𝑔 ). The edges, shown in green
in Figure 1, encode the trips connecting two station nodes.
Now that we have established the fundamental defini- The AvailBikeSim edge set is derived to represent the sim-
tions, we formally introduce the HyGraph model, detail- ilarity in bike availability patterns between station nodes.
ing its structural components and integration of property An AvailBikeSim edge is generated when the computed time
graphs and time series data models.
series similarity between the corresponding dynamic proper- Gr aphElement

ties (𝑛𝑢𝑚_𝑏𝑖𝑘𝑒𝑠_𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 and 𝑛𝑢𝑚_𝑒𝑏𝑖𝑘𝑒𝑠_𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒) 0..1
+ oid
+ label
of two stations meets or exceeds a predefined threshold. M etadataTimeser ies + static_properties

The stored values represents the evolution of the similarity
StaticProper ty
Timeser ies + ownerID + dynamic_properties
+ attributes update_graph_element() 0..* + key
+ oid
score (e.g., 0.67, 0.79), computed using a time series simi- + timestamps 1 update_metadata() get/add_property_type() + value

larity method [16] (e.g., Pearson correlation). The edges of + data get/add_static_property()
get/add_dynamic_property()
get_value()
set_value()
+ metadata
this edge set are modeled as time series edges (𝐸𝑡𝑠 ) and are +variables
1..*

depicted in blue color in Figure 1. Each Station node has: aggregate_timeseries()
1..*
DynamicProper ty
sum(), mean(), min(), 0..* + key
max(), count()
1. An id, e.g., 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐴; get_value_at_timestamp()
+ series
get_time_series()
get_timestamp_at_value() Edge
Node Subgr aph set_value()
2. A validity interval, e.g., 𝜂(𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐴) = ⟨2000, ∞⟩; last_value(), first_value()
subset_timeseries() + membership + start_time
+ membership apply_aggregation()
+ start_time + end_time + source get_timestamp()
compute_similarity()
3. Static properties, e.g., 𝜑(𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐴, 𝑛𝑎𝑚𝑒) = Christ Hos- + end_time get_type() + target
+ start_time
get_first_value()
get_last_value
pital, 𝜑(𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐴, 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦) = 2 and;
get_membership() apply_filter()
0..* + end_time
get_neighbors()
get_membership()
0..*
4. Dynamic properties (time-series), the time series proper-
HyGr aphQuer ying
ties are represented as an object with a list of times-
HyGr aph
+hygraph TSNode PGNode + graph
tamps and associated data values for the variable at + node_matches + series get_type() + time_series TSEdge PGEdge

each timestamp. For instance, for 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐴 the dy- + edge_matches
+ pattern
get_type()
0..*
+ subgraphs
+ query
+ series
get_type()
get_type()

namic attribute 𝑛𝑢𝑚_𝑏𝑖𝑘𝑒𝑠_𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 has a variable + conditions 0..*
0..*
0..*
𝑡𝑖𝑚𝑒𝑠𝑡𝑎𝑚𝑝𝑠 = [“21 : 00”, “22 : 00”] which stores a
+ return_elements add/get/delete_tsnode/tsedge
+ groupings add/get/delete_subgraph
list of timestamps, and the associated 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 name + aggregations add/get/delete_timeseries

𝑏𝑖𝑘𝑒_𝑎𝑣𝑎𝑖𝑙 holds 𝑑𝑎𝑡𝑎 = [[1], [2]]. Note that this is a
+ ordering 0..* get_node/edge_by_static_property
+ limit_count get_node/edge_by_dynamic_property
regular dynamic property as its value change due to + subquery_results get_subgraph_by_temporal_property

external factors, like a trip being undertaken. find_path()
get_node_degree_over_time()
add/get/delete_pgnode/pgedge
Each edge in Trips consists of:
1. A label, e.g., 2 ; Figure 2: HyGraph UML conceptual diagram

2. A validity interval, e.g., 𝜂( 2 ) = ⟨2001, 2025⟩;
with everything handled in memory, using NetworkX [17]
3. Dynamic properties (regular), e.g., 𝑚𝑒𝑚𝑏𝑒𝑟_𝑟𝑖𝑑𝑒𝑠,
and Xarray [18] for graph and time series in-memory stor-
𝑐𝑎𝑠𝑢𝑎𝑙_𝑟𝑖𝑑𝑒𝑠, etc.
age. While scalability to larger storage engines is desirable,
Each time series edge AvailBikeSim consists of: the current focus is on proof-of-concept, emphasizing a
uniform data storage and querying model. We discuss the
1. A label, e.g., 8 ; conceptual architecture of the HyGraph model through
a UML diagram, followed by its system architecture with
2. A validity interval, e.g., 𝜂( 8 ) = ⟨2010, 2025⟩;
some core functionalities.
3. Dynamic attribute represented as an object with a list of
timestamps and associated data values for the variable 3.1. UML Architecture
at each timestamp. For instance, edge 8 represents
similarity score evolution between StationB and StationC In Figure 2, a UML diagram describes the HyGraph system
through a list of timestamps [“2010”, “2011”], and the representation in a conceptual aspect. It illustrates the main
associated 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 name is 𝑏𝑖𝑘𝑒_𝑎𝑣_𝑠𝑖𝑚, which holds classes and their relationships. We capture HyGraph as
data [[0.67], [0.70]]. a set of interrelated classes reflecting property-graph and
time-series functionalities. At the root, an abstract class
Figure 1 also shows two non-overlapping subgraphs, Sub- GraphElement defines fundamental attributes (such as id,
graph #1 and Subgraph #2, depicting partial views of the net- and label). Three main classes inherit from GraphElement:
work at time 𝑡1. Subgraph #1 includes StationA and StationE, Node, Edge, and Subgraph. Each of the classes plays core
and connecting edges, and Subgraph #2 features 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐵, structural roles. Within this architecture, the class Node is in-
𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐶 and 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝐷, and connecting edges. herited by classes PGNode (representing standard property-
Since the subgraphs can evolve with time and are dy- graph nodes) and TSNode (representing time-series nodes),
namic, Figure 1 only depicts a snapshot at 𝑡1 . The logical while the class Edge is similarly inherited by classes PGEdge
subgraphs aggregate vertices that share the same Avail- and TSEdge.
BikeSim patterns. For instance, StationB will continue to be The Timeseries class defines a multivariate time series by
a part of Subgraph #2 till it has a high similarity score, but maintaining five attributes, id, data to capture the value of
as soon as the score decreases it can be placed out or moved multi-variate time series at different points in time, variables
to another subgraph. holds information about each dimension of the multivariate
time series, timestamps to hold an ordered list of timestamps
for all recorded entries in data and the fifth attribute is the
3. HyGraph Architecture metadata. The metadata attribute, which is an instance of a
We aim to seamlessly integrate time series and graph compo- separate class MetadataTimeseries, is optional and facilitates
nents in a single system that allows combined querying and additional descriptive attributes.
transformations over these components. At the moment, A top-level HyGraph class aggregates all these compo-
theHyGraph system is being developed as a Python package nents, ensuring that property-graph elements (PGNode,
PGEdge) and time-series elements (TSNode, TSEdge) coexist
in one coherent framework. Node and Edge and Subgraph
have two attributes start_time and end_time as timestamps,
to represent their time validity. TSNode and TSEdge classes
hold in addition an instance of the Timeseries class as at-
tributes to store the time series. The classes StaticProp-
erty and DynamicProperty represent the two types of prop-
erties. Specifically, an object of class GraphElement can
hold none or multiple property instances. The variables
static_properties and dynamic_properties in class GraphEle- Figure 3: HyGraph System Architecture
ment, respectively represent instances of classes StaticProp-
erty and DynamicProperty in Figure 2. The class StaticProp-
erty captures properties with static values (represented by functionalities, implementing classes like Timeseries and
𝒩Σ ). It simply stores the key and its corresponding value MetadataTimeseries for storing and manipulating temporal
as attributes. While the class DynamicProperty corresponds data. To facilitate the import and export of data, the sys-
to a property whose values evolve (represented by 𝒩TS ). tem is equipped with the HyGraphFileLoader module, which
Thus, in addition to the key, it references an ID of the re- aids in streamlining ETL (Extract–Transform–Load) tasks.
lated time series instance. The GraphElement class manages In our system, we also define a module GraphConstruct.
these properties, exposing the logic to read, insert, delete, This module facilitates (re-)construction of a graph purely
or modify them. from a correlation between time series. This enables trans-
The dynamic nature of subgraphs is captured via the at- formations that spawn graph structures based on temporal
tribute membership in Node and Edge, implemented as an similarity. The module HyGraphQuery is defined to combine
instance of the Timeseries class. Concretely, an object of all types of querying within HyGraph. At the moment it
either class can accumulate multiple membership updates already includes the HybridPatternMatching class, and will
over time - one per subgraph change.As a result, every in- also include as a future work other classes like Subgraph-
clusion or exclusion is appended to membership, effectively Matching to enable searching for patterns corresponding
reflecting how the subgraph’s composition evolves. By tying to a whole subgraph. All these modules interoperate un-
membership changes to a time-series structure, we maintain der the central HyGraph module, which coordinates their
a complete history of when a node or edge was valid in each interactions and maintains the global system state.
subgraph. This allows subgraphs to evolve as entities join, The HyGraph system’s functions are intuitively grouped
leave, and transform with updates to the HyGraph object. into three principal interfaces, as shown in Table 1. The
The HyGraphQuerying class provides a hybrid pattern- interface ModelToHyGraph gathers data from an external
matching mechanism, allowing users to define queries model (graph-only or time-series–only) and ingests it into
that simultaneously reference graph and time-series pat- HyGraph. Here, GraphOperator handles graph injection,
terns. The class supports key concepts like node- or edge- adding nodes, edges, and their properties. In parallel, Time-
based matching, groupings, and aggregations, reminiscent seriesOperator manages time-series injection, creating or
of Cypher-style clauses. updating dynamic properties and elements. This interface
also includes the Graph similarity generation function as
part of GraphConstruct module, which will be explained
3.2. System Architecture and Functionalities later in Section 4.2. Note that Model refers to graph or
Figure 3 shows a high-level architecture of the HyGraph time-series data models.GraphConstruct can also generate
system, illustrating interactions between different layers. It another The second interface, HyGraphToHyGraph, com-
stores and processes both graph and time-series elements prises the core operations and algorithms that transform
in memory. one HyGraph into another, such as hybrid pattern match-
We leverage two specialized processors: NetworkX as ing, dynamic subgraph creation, and HyGraph clustering.
the graph processor, for structural operations (e.g., shortest HyGraph instance from the existing one. Finally, the third in-
paths, subgraph extraction, etc.) and; Xarray as the time- terface, HyGraphToModel focuses on extracting or exporting
series processor, suitable for multivariate data storage and data back into an external format or distinct model. Graph-
computations. We extend the pre-defined object models pro- Operator can provide standalone graph operators such as get
vided by these libraries to implement our custom-defined the neighbors, and shortest path, while TimeseriesOperator
classes (described in Section 3.1). An object-mapping strat- handles isolated time-series operations such as correlation
egy then bridges these customized classes with the under- and feature extraction.
lying library objects, enabling both data persistence and
data retrieval. High-level algorithms such as hybrid pat-
tern matching or graph similarity, are developed on top of 4. HyGraph Transformation
this system, to enable analyses of the hybrid data. In the Operators
implementation of the HyGraph system, we adopt a mod-
ular architecture with six principal modules, orchestrated Data transformations may convert one representation into
by the main HyGraph module. The GraphOperator module another or produce a new instance in the same representa-
handles all graph-centric logic, in addition to integrating tion (augmented, summarized, or updated). These transfor-
graph and time series operations, ensuring that HyGraph mations can be static (one-off) or dynamic (continuously
supports standalone property graph or time series features. adapting as data changes). Within HyGraph, we distinguish
This module implements the classes GraphElement, Node, two primary types of transformations. The first transfor-
Edge, Subgraph, StaticProperty, and DynamicProperty. In par- mation type is from time series to graph. It involves ana-
allel, the TimeseriesOperator module manages time series lyzing correlations and other temporal relationships among
Table 1
HyGraph modules and associated functionalities provided for different interfaces

Interfaces
Modules
ModelToHyGraph HyGraphToHyGraph HyGraphToModel
GraphOperator Graph injection Subgraph creation, Clustering Standalone graph operators
TimeseriesOperator Time series injection – Standalone Time series operators
HyGraphQuery – Hybrid pattern matching Data extraction and retrieval
GraphConstruct Graph similarity generation Graph similarity generation -

time series to generate new graph entities (nodes, edges, 4.2. From Graph Topology to Time series
or subgraphs) that reflect these relationships. The second
Prior approaches have examined how graph metrics evolve,
transformation type transforms a graph into a time series.
by either implementing algorithms that always catch new
By examining evolving graph metrics (like node in-degree
changes in the graph structure and update the results of the
or edge traffic), construction of new time series that capture
graph operator [23] or by creating time series to analyze
these structural changes over time. Moreover, HyGraph
patterns [24, 25]. However, most solutions stop after gener-
supports the continuous execution of these transformations.
ating standalone time series data and do not further link it
If the graph is updated, the transformed time series can be
with the graph. In HyGraph, we can generate time series
updated simultaneously, and vice versa.
by analyzing the evolution of graph topology and option-
In the following subsections, we illustrate two transfor-
ally, embed them back into the HyGraph, either as dynamic
mations: (i) a time-series–based similarity graph operator
property of existing graph elements or as dedicated element
(Section 4.1), and (ii) an extraction operator to extract time
(as instances of TSNode or TSEdge). This allows transforma-
series from evolving graph metrics (Section 4.2). These
tion of the HyGraph through augmentation of the derived
examples demonstrate the support for flexible and bidirec-
data and enables further transformation operations. We
tional transformations that unify structural and temporal
define the extraction operator as:
data in a coherent ecosystem of HyGraph.
𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑇 𝑆(𝐻, ℱ, 𝑚𝑒𝑡𝑟𝑖𝑐, 𝜏, 𝑓 𝑟𝑒𝑞) → {𝑡𝑠1 , ..., 𝑡𝑠𝑚 }∪𝐻 ′
4.1. Time series based Graph Similarity where,
Several existing methods already construct graphs from • 𝐻 is a HyGraph instance,
time-series data for tasks like clustering or anomaly de-
tection [19, 20, 21, 22]. They typically compute pairwise • ℱ is a filter (or set of filters) that specifies which ver-
similarities or distances among time series and generate a tices/edges or subgraphs to evaluate (e.g., node filter based
static graph whose edges represent these similarities. In on labels),
contrast, HyGraph provides a similar time series similarity- • 𝑚𝑒𝑡𝑟𝑖𝑐 specifies the graph property over which the time
to-graph mechanism and also integrates the newly created series is to be generated, e.g., degree centrality, clustering
HyGraph within the unified hybrid system. This implies coefficient, etc.
that the resulting HyGraph can also maintain static and
dynamic properties on edges, and be used for further pro- • 𝜏 = [𝑡start , 𝑡end ) specifies a time range, {𝑡start , 𝑡end } ∈ 𝒯 ,
cessing by hybrid operators, like pattern matching.
• 𝑓 𝑟𝑒𝑞 indicates sampling frequency (e.g., daily, weekly).
We define our time series-based graph similarity opera-
tor formally as a function 𝐺𝑟𝑎𝑝ℎ𝑆𝑖𝑚(𝑇 𝑆, 𝑚𝑒𝑡ℎ𝑜𝑑𝑠, 𝜃) → We enumerate discrete time steps at 𝑓 𝑟𝑒𝑞 in the time range
𝐻 ′ where 𝑇 𝑆 is a set of time series, 𝑚𝑒𝑡ℎ𝑜𝑑𝑠 is a set of 𝜏 , as {𝑡1 , 𝑡2 , . . .}. At each time step 𝑡𝑖 , we take a snap-
similarity strategies (correlation, shape similarity, etc.) and shot of the HyGraph instance. The value for 𝑚𝑒𝑡𝑟𝑖𝑐 is then
𝜃 ∈ [0, 1] is a similarity threshold. The output 𝐻 ′ is a computed for each snapshot and assembled into a time se-
new HyGraph instance generated by analyzing time series ries. The time series thus generated reflects the evolution of
nodes. Specifically, let {𝑣1 , 𝑣𝑛 } ⊆ 𝑉ts represent time se- 𝑚𝑒𝑡𝑟𝑖𝑐 across the selected nodes/edges over discrete time
ries nodes, then for each edge, represented as a vertex pair intervals. The time series can be processed for further anal-
(𝑣𝑖 , 𝑣𝑗 ), we compute the similarity score of their time series ysis, or injected back into the HyGraph instance as dynamic
as 𝑆𝑖𝑚𝑡𝑠 (𝑣𝑖 , 𝑣𝑗 ). If 𝑆𝑖𝑚𝑡𝑠 (𝑣𝑖 , 𝑣𝑗 ) ≥ 𝜃, an edge is created. properties of the graph elements, to produce an updated
The similarity score is stored as the static or dynamic edge instance, or can simply be returned as a set of time series.
property. If the user only requests a single, fixed value, a
PGEdge is created with a static property. However, if the
evolution of similarity over time is of interest, it is more 5. Use case: Micro-mobility
strategic to store it as a time series in an instance of TSEdge.
The objective of the operator is to either create a HyGraph Micro-mobility has emerged as a cornerstone of sustain-
from scratch when only time-series data is provided, or to able urban transportation. Yet, one of its persistent opera-
further analyze time series in the existing HyGraph instance tional hurdles is rebalancing, ensuring that vehicles such
by applying graph operators to time series. In HyGraph ter- as bicycles or e-bikes are appropriately distributed across
minology, this is a ModelToHyGraph transformation, where docking stations to meet fluctuating demand. Studies fo-
one or more time series (either ingested from an external cusing on bike-sharing systems emphasize that neglected
source or extracted from the current HyGraph) are analyzed rebalancing can lead to chronic station shortages or over-
to produce a HyGraph reflecting their interrelationships. flows, hindering overall service reliability and increasing
user dissatisfaction [26, 27, 28].
To address the rebalancing challenge, we propose a multi- 6. Future Research
step pipeline that leverages HyGraph’s transformation oper-
ators, GraphSim and ExtractTS (described in Section 4). Our HyGraph represents an initial step toward integrating prop-
core objective is to determine, for each station, which other erty graphs with time-series, addressing key challenges in
station(s) serve as ideal rebalancing partners, i.e., whenever maintaining and querying dynamic data. By unifying these
one station experiences a surplus, the other experiences a two paradigms, HyGraph enables seamless temporal graph
deficit, while simultaneously accounting for neighbor con- transformations, but its implementation also presents sev-
nectivity and distance. We base our analysis on the dataset eral complexities.
provided by [15]. It unifies graph and time-series data by However, one major challenge lies in efficiently updat-
representing a bike station as a vertex with a static property ing and querying time-series data associated with graph
representing the parking capacity of the station and dy- nodes and edges. Indexing strategies in traditional graph
namic properties like the number of available bikes; while databases are not inherently designed to accommodate time-
each edge represents trips between two stations, a time- series data efficiently, leading to potential scalability bottle-
series property tracking daily active trips, member rides vs. necks. A new system that integrates indexing techniques
casual rides, and total trips. tailored for both graph structures and time-series storage,
Using the ExtractTS operator, we first extract two dy- would ensure efficient querying and seamless data evolution.
namic properties for each station node: Maintaining indexing structures that accommodate both
topological changes in the graph and temporal variations
1. For a station 𝑣 at time 𝑡, the imbalance is defined as in time-series data requires novel optimization techniques.
the difference between the number of trips that end at Additionally, the lack of a standardized query language
station 𝑣 (i.e. bikes arriving) until time 𝑡, and the number for seamlessly integrating time-series operations with graph
of rides starting from station 𝑣 (i.e. bikes departing) until traversal necessitates the design of new operators and query
time 𝑡. The imbalance value is for a station is captured execution strategies. Existing graph query languages do not
at different timestamps and is stored as a time series natively support analytic operations commonly found in
property, 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒_𝑡𝑠. time-series databases, such as temporal aggregations, win-
dowed computations, and similarity searches based on se-
2. For a station 𝑣, we also compute its connectivity score
quence patterns or shape-based matching. Future research
to quantify how strongly it is connected to its neighbors.
could explore the development of a unified query language
The connectivity score for 𝑣 is defined as the ratio of
that incorporates time-aware traversal semantics and trans-
weighted sum of edges and the degree of 𝑣 at any time 𝑡.
formation operators to enable efficient interaction between
Similar to imbalance of a station, the connectivity score
graph topology and temporal dynamics.
is also stored as a dynamic property, 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦_𝑡𝑠,
The fast-evolving nature of time-series data necessitates
of the vertex.
low-latency updates and retrieval, making it essential to
In the next step, a similarity graph is constructed us- scale HyGraph for real-time applications. Addressing this
ing the module GraphConstruct, where nodes represent challenge requires investigating efficient data streaming
stations and edges represent the similarity of their time architectures, like designing caching mechanisms for fre-
series property 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒_𝑡𝑠. To capture the complemen- quently queried data and hybrid storage layouts optimized
tary behavior of two stations, i.e., pairing a surplus station for high-throughput ingestion and query concurrency.
with a deficit station, we compute a negative correlation be-
tween 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒(𝑣, 𝑡) for station 𝑣 and 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒(𝑢, 𝑡)
for station 𝑢. The function will augment the HyGraph in-
7. Conclusion
stance with new TSNode objects, created to represent the This paper introduced the UML and sytem architecture of
𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒_𝑡𝑠 of each station and new PGEdge objects rep- HyGraph [7], illustrating a unified approach for integrat-
resenting the similarity between the newly created TSNode ing property graphs and time-series data. We introduced
objects. The negative correlation between the imbalance two novel transformation operators: (i) a time-series-based
time series of two stations 𝑢 and 𝑣, 𝑛𝑒𝑔imb (𝑢, 𝑣) quantifies graph operator, which derives graphs based on correlations
how complementary the two stations are. among time series, and (ii) a graph-based time-series opera-
After building the similarity graph, for every edge con- tor, which extracts time-series representations from evolv-
necting stations 𝑢 and 𝑣, we compute a composite score ing graph metrics.
that will represent the weight of the edge. This weight is a Our micro-mobility case study further demonstrated the
combination of the following: a similarity score based on a practical applicability of HyGraph and the transformation
distance decay function [29], the distance between the two operators for augmented analysis in real-world settings. By
stations, the negative correlation score 𝑛𝑒𝑔imb (𝑢, 𝑣) and the establishing a foundation for hybrid graph-time-series an-
average value of 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦_𝑡𝑠 between the two stations. alytics, HyGraph paves the way for plethora of research
This composite score reflects both the temporal complemen- opportunities in graph data management, temporal reason-
tarity of imbalance and the practical factors of connectivity ing, and dynamic query processing.
and distance. Once the similarity graph is fully augmented, Despite its advantages, several challenges remain and
we apply a maximum weighted matching algorithm [30] to future research should explore scalable indexing and query
select a set of non-overlapping edges and return the set of optimization techniques for hybrid queries.
station pairs (𝑢, 𝑣) that maximize the total composite score.
For each matched pair, the average imbalance difference
is computed to suggest the direction and the number of References
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