A Generalized Notion of Time for Modeling Temporal Networks

Konstantin Kueffner

1,2

and Mark Strembeck

1,2,3

Vienna University of Economics and Business (WU Vienna), Austria

Secure Business Austria Research Center (SBA), Austria

Complexity Science Hub Vienna (CSH), Austria

Keywords:

Multilayer Network, Multivalued Path, Temporal Network, Temporal Path, Time, Weighted Network.

Abstract:

Most approaches for modeling and analyzing temporal networks do not explicitly discuss the underlying no-

tion of time. In this paper, we therefore introduce a generalized notion of time for temporal networks. Our

approach also allows for considering non-deterministic time and incomplete data, two issues that are often

found when analyzing data-sets extracted from online social networks, for example. In order to demonstrate

the consequences of our generalized notion of time, we also discuss the implications for the computation of

(shortest) temporal paths in temporal networks.

1 INTRODUCTION

Temporal networks are a means for modeling and

analyzing the temporal dimension of complex (net-

worked) systems. However, most of the literature on

temporal networks either does not explicitly discuss

the underlying notion of time or uses a rather restric-

tive conception of time. In this paper, we discuss a

generalized notion of time allowing for possible de-

viations from a linear ﬂow of time. Thereby, our ap-

proach allows for considering non-determinism and

incomplete data when analyzing temporal networks

(two issues that often appear when dealing with real-

world data, such as data extracted from online social

networks).

We use a variant of the multilayer network concept

to construct temporal networks alongside our gener-

alized temporal abstraction. In particular, we con-

sider a temporal network, as a (temporal) sequence

of networks, a notion that can easily be expressed

via the well-known multilayer network concept. This

approach allows to clearly separate time from other

(temporally varying) attributes that are attached to the

edges or vertices in a network. This strict separation,

however, requires to consider multivalued path length,

the implications of which will be discussed as well.

For the purposes this paper, we use the notion

We provide an extended version of this paper on our

Web page. The extended version includes additional text

and examples that we had to cut from the paper due to the

page restrictions for the proceedings version.

of multilayer networks as speciﬁed in Deﬁnition 1.1,

which understands multilayer networks as a family of

graphs connected with inter-graph edges.

Deﬁnition 1.1. A weighted multilayer network is a

triple M∶=G, R, ωsuch that for some arbitrary set

of labels I

• G∶=(G

)

α∈I

is a family of weighted graphs G

∶=

V

, E

, ω

such that ∀α, β ∈I α ≠β we have V

∩

=∅;

• R∶=(R

αβ

)

α,β∈I

is a family of relations, such that

∀α, β ∈I α ≠β we have R

αβ

⊆V (G

)×V(G

);

• ω ∶I ×I ×E(M)→R is the function

αβ

∶=

(e) α =β

R otw.

We deﬁne V(G) ∶= {V (G)  G ∈ G} and V (G) ∶=

⋃

V ∈V(G)

V . As well as E(G)∶={E(G)G ∈G} and

E(G)∶=

⋃

E∈E(G)

E. Moreover, let E(R)=

⋃

R∈R

Lastly, V (M)∶=V (G)and E(M)∶=E(R)∪E(G).

This is far from the only notion of multilayer net-

works though. For example, one other formalism de-

ﬁnes tuples for representing the layer membership of

different vertices, while yet another represents a mul-

tilayer network solely as a tensor to be understood

as an analogue to the adjacency matrix of ordinary

graphs. For further details, see, e.g., (De Domenico

et al., 2013), (Boccaletti et al., 2014) and (Kivel

et al., 2014).

For G ∶= ⟨V, E⟩ let V (G) ∶= V and E(G) ∶= E

Kueffner, K. and Strembeck, M.

A Generalized Notion of Time for Modeling Temporal Networks.

DOI: 10.5220/0007764000930100

In Proceedings of the 4th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2019), pages 93-100

ISBN: 978-989-758-366-7

A multiplex is commonly understood as a multi-

layer network where all layers share the same set of

vertices, i.e. ∀U, W ∈V (G)U =W (Kivel

a et al., 2014;

Boccaletti et al., 2014). For the purposes of this paper,

we relax the strong equality implied by those charac-

terizations, as apart from some technical issues with

respect to path properties, strong equality seems to be

inappropriate for our purposes. For example, an ob-

ject (vertex or edge) x at time t may not share the same

properties as the same object (vertex or edge) x at time

′

≠t.

Deﬁnition 1.2. Let M=G, R, ≡ be a multilayer

network with equivalence such that ∀α ∈ I∀v, w ∈

V (G

)v ≡w ⇒v =w. Moreover, the equivalence

classes generated by ≡ are denoted as v ∶={w w ∈

V (G)v ≡w}and we write v

∶=v ∈V

∩v. Such a mul-

tilayer network with equivalence is called a multiplex

iff ∀v ∈V (G)v=G.

Unfortunately, when modeling the progression of

time, equivalence is necessary but not sufﬁcient. For

achieving the required expressiveness we have to im-

pose an additional order onto our structure. That is, if

a graph G

precedes a graph G

according some order

⪯, then the vertex v

should precede v

Deﬁnition 1.3. Let M=G, R, ≡, ⪯be a multilayer

network with equivalence and order. If ⪯is a partial

ordering on G. Moreover, G

⪯G

⇐⇒(∀v ∈G

∀w ∈

(v ≡w ⇒v ⪯w)∧(v ≢w ⇒v ∦w)).

Having deﬁned the fundamental language of mul-

tilayer networks used in this paper, we shall now dis-

cuss the semantics and structure of time underlying

the deﬁnition of temporal networks as presented in

Section 3.

2 GENERALIZED FLOW OF

TIME

We abstract from the notion of a temporal graph to

discuss some more general observations regarding ob-

jects in a temporal dimension. When studying the no-

tion of time we distinguish between two classes of

properties. Informally, the properties concerned with

the density of time and the ones determining the struc-

ture of time. Many concepts mentioned in this section

can be found in and/or build on the ideas presented

by (Venema, 1998), (Markosian et al., 2016) and

(Goranko and Galton, 2015) and are heavily based on

Kripke semantics (Van Ditmarsch et al., 2007).

Deﬁnition 2.1. Let Φ =T, R, ιbe a structure repre-

senting the ﬂow of time, where T is a set of points, R is

a relation over T , i.e. R ⊂T ×T and ι ∶T →℘(L)is a

function assigning each point in time a set of proper-

ties P described in some language L. If the properties

at a point in time are not relevant we write Φ =T, R.

The function ι, corresponding to the evaluation

function of Kripke models, is best understood as a

function that assigns each point in time a description

of the world. R simply puts the points in time into

relation. For example, one can embed a graph into a

point in time, by attaching some axiomatisation of the

desired graph theory together with a characterization

of the graph itself.

There are some interresting conceptions of how

points in time may relate, e.g. cyclic or bidirectional

notions of time. Hence, in its most general form R

should be open to interpretation. For example, in

(Taylor et al., 2017) time is conceived as an undi-

rected, discrete ﬂow. However, as we intend to dis-

cuss a class of speciﬁc notions of time, allowing an

arbitrary relation R is cumbersome. Hence, we re-

strict ourself for all subsequent discussions to the set

of structures building upon a directed notion of time.

Deﬁnition 2.2. (Venema, 1998) Let Φ =T, ⪯ be a

structure representing the ﬂow of time, where T is a

set of points and ⪯is a partial order over T.

For more on partially orders see (Matthews,

1994). Moreover, we write x ≺y ∶=x ⪯y ∧x ≠y as a

shorthand.

By imposing further restriction onto ⪯we can de-

velop certain notions of time. For example, a ﬂow

of time is linear if its underlying order is total, i.e.

∀x, y ∈T x ⪯y ∨y ⪯x; it is strictly linear if ∀x, y ∈T x ≺

y∨y ≺x∨x =y; it is backwards-branching if for a point

a representing the present there are two incomparable

points in the past, i.e. ∃x, y ∈T x ⪯a ∧y ⪯a ∧x ∦y; it is

forwards-branching if for a point a representing the

present there are two incomparable points in the fu-

ture, i.e ∃x, y ∈T a ⪯x ∧a ⪯y ∧x ∦y; it is backwards-

serial if there is always another point in the past, i.e.

∀x ∈T ∃y ∈T y ≺x and forwards-serial if there is al-

ways another point in the future, i.e. ∀x ∈T ∃y ∈T x ≺y

(Venema, 1998).

The concept of linear time is highly intuitive:

Time ﬂows within a straight line, there are no alterna-

tive time lines, no branching and no cycles, allowing

us to work in a deterministic fashion. For example,

one encounters this notion of time when dealing with

normal time series. Moreover, a good part of the liter-

ature regarding temporal networks, is concerned with

linear time. However, one can easily conceptualize

scenarios, where we deal with some kind of uncer-

tainty or non-determanism. Here, the notion of pos-

sible worlds can guide our reasoning. Kripke mod-

els, which provide the foundation for the semantics of

COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk

modal logic, heavily rely on the concept of possible

worlds (Venema, 1998; Van Ditmarsch et al., 2007).

In our case, each branch represents a possible fu-

ture, thereby expressing the non-determinism of the

future. For example, having a linear ﬂow of time in

the past that branches into the future, expresses that

we are certain what happend in the past, but we can

not predict the future with certainty. How those fu-

tures are obtained precicely, may it be through sta-

tistical inference, by consulting experts with domain

knowledge, or being a product of a simulation with

randomness is currently not of our concern. Consider-

ing possible worlds is especially useful when dealing

with discrete objects such as graphs. That is, rather

than introducing fuzzy edges (Sunitha and Mathew,

2013) we can, at least for our purposes, consider alter-

native worlds where an edge exists and some in which

it does not. Similarly, backwards-branching could in-

troduce the notion of unreliable data into our models.

For example, if there two contradicting measurements

of the same phenomenon at the same instance, one

models them as two incomparable elements within

the ﬂow of time. Moreover, regardless of forward-

or backward-branching, we do allow for collapsing

ﬂows of time. That is, two branches could meet at

some point in time. Recall the unreliable data ex-

ample, as an example for the applicability of such a

structure.

When discussing the density of time, three general

notions are common: a ﬂow of time can be discrete,

dense or continuous, thus existing in analogy to N, Q

and R respectively. However, as most measurements

of the real world are processed by inherently discrete

machines and this paper focuses on the analysis of

discrete sequences of discrete objects, we will limit

ourself to a discrete conception of time. However, for

analytical and predictive purposes the other two mod-

els of time, especially the continuous one, should be

investigated further in the context of temporal graphs

(Venema, 1998).

The restriction to discrete notions of time allows

for a concrete conception of successors (we can for-

malize steps in time). In linear time, the succes-

sor predicate S corresponds to a bijective mapping s

where t +1 ∶=s(t)∈T . However, as soon as branch-

ing is permitted, S(a, b) can be satisﬁed by multiple

elements.

Deﬁnition 2.3. Let Φ =T, ⪯be a ﬂow of time, then

t +1 ∶=s(t)is called the (direct) successor function iff

s ∶T →℘(T ) s(t)↦{t

′

t

′

∈T S(t, t

′

)}

for the successor predicate S(a, b)∶=a ≺b ∧∀x ∈T x ⪯

a∨b ⪯x. Moreover, t −1 =p(t)∶={t

′

∀t

′

∈T s(t

′

)=t}

is called the (direct) predecessor function

By taking several steps we obtain a path in time.

Deﬁnition 2.4. Let Φ =T, ⪯be a ﬂow of time, then

• a forward path in time a ↝

b between a and b

is a sequence a ↝

b =t

↝

∶=t

, t

, . . . t

n−1

, t

where ∀i ∈{1, . . . n −1}t

i+1

∈s(t

);

• a backwards path in time a ↝

−

b is deﬁned in ana-

logue to a ↝

b with respect to the predecessor

function p.

• if a ⪯b, a path in time is a ↝b ∶=a ↝

b; and if

b ⪯a, a path in time is a ↝b ∶=a ↝

−

The size of a path in time is determined by the ele-

ments in the path and denoted as a ↝b and can be

understood as the amount of steps in time between a

and b in a ↝b.

Notice, if a ↝

−

b there exists some b ↝

a such

that a ↝

−

b =b ↝

a. Moreover, if a ↝b and b ↝c

for a ⪯b and c ⪯b, the concatenation of those two

paths a ↝b ↝c is no path in time, since it is neither

a forwards, nor a backwards path in time. Hence, by

preventing a path from changing directions there can

not be a path between two branches. The concept of

size can be generalized to obtain a notion of length.

Deﬁnition 2.5. Let Φ =T, ⪯, ωbe a ﬂow of time with

spacing, with the function

ω ∶S(Φ)→R

, x ↦ω(x)

Where S(Φ)∶={(x, y)∀x, y ∈T S(x, y)}contains all

steps in time. The length of a path in time a ↝b with

respect to ω is thus a ↝b

∶=

∑

e∈S(a↝b)

ω(e).

Intuitively, ω stretches time, i.e. it assigns a dura-

tion onto a step in time. However, while the length of

a path in time is a useful concept, it cannot serve as a

measure of distance.

Example 2.6. Consider the ﬂow of time Φ ∶=T, ⪯, ω

2 2 1

1 1

What is the distance δ between a, b and between b, c?

If constructed based on the notion of a path we can

observe the following. For ⋅ we have δ(a, b)=2

or δ(a, b)=4 and for ⋅

we have δ

(a, b)=4 or

(a, b)=4. Analogously, we have δ(b, c)=3 or

δ(b, c)=3 and δ

(b, c)=5 or δ

(b, c)=4.

To resolve this ambiguity we make use of a quasi-

metric with inﬁnity (Schroeder, 2006). A fairly natu-

ral choice for modeling non-cyclic, directed ﬂows of

time. As one can easily travel into the future one in-

stance at a time, but travelling into the past is, at least

∣ ⋅ ∣ is shorthand for ∣x ↝ y∣ for some path x ↝ y

A Generalized Notion of Time for Modeling Temporal Networks

for the common man, impossible. Moreover, by as-

signing an inﬁnite distance to backtracking within our

ﬂow of time, the distance between two chains, also

becomes inﬁnite. One can easily check that this holds

for chains with and without a join. More information

about quasi-metrics can be found in (Matthews, 1994;

Schroeder, 2006).

For the rest of this paper we use the following

quasi-metric.

Deﬁnition 2.7. Let Φ =T, ⪯, ωbe a ﬂow of time with

spacing. We call

(a, b)∶=











min

a↝b

(a ↝b

) a ⪯b

∞ otw.

the distance between a and b (with respect to ω).

Moreover, we call δ

, where ω is the constant func-

tion 1, the step distance between a and b.

Proposition 2.8. Φ = T, ⪯ and Φ =T, ⪯, ω are

quasi-metric ﬂows of time.

Proof. Since δ

is a special case of δ

we only

consider the latter. Firstly, δ

(x, y) ≥0 holds as

∀(x, y)∈S(Φ)ω(x, y)∈R

. Secondly, δ

(x, y)=0 =

δ(y, x) ⇐⇒ x =y holds because x =y ⇐⇒ x ↝y=0,

especially with respect to ω, and the empty sum is al-

ways 0. Finally, δ

(x, z)≤δ

(x, y)+δ

(y, z). If z ≺x

then δ

(x, z)=∞. Since there are no circles, it must

be that either δ

(x, y)=∞or δ

(y, z)=∞. For y ≺x

or z ≺y we have that in either case backtracking would

be required, leading to a shift in direction, which by

deﬁnition is not a valid path in time. If x ≺y ≺z, as-

sume the contrary. Thus δ

(x, z)>δ

(x, y)+δ

(y, z).

Hence, ∃x ↝y, y ↝z, resulting in x ↝y ↝z such that

x ↝z

>x ↝y ↝z

. But x ↝z

is minimal, thus

arriving at the desired contradiction.

However, while this may allow us to speak about

distance, in one form or another. It creates some se-

mantic inconveniences or inconsistencies. Some of

which are addressed in the following subsection.

2.1 Linear and Homogeneous Flows of

Time

The notion of distance developed above may lead to

unfortunate outcomes (see Example 2.9). Many of

which can be reduced to the fact that, in their gen-

eral form, ﬂows in time can easily be used to express

time as having varying density, i.e. the spacing be-

tween points may vary within a single chain or across

branches. While sometimes useful, e.g. consider rela-

tivity of time, a characterization of time without such

properties is desirable as well.

Example 2.9. Consider the ﬂow of time as in Exam-

ple 2.6. We can ﬁnd two paths from a to b and two

paths from b to c. For the latter, our notion of step

distance causes no issues, as all paths between b and

c have the same size, i.e. ⋅. However, in the case a to

b, two paths of different length, i.e ⋅

, can be found.

Hence, if one wants to consider all points at a certain

step distance from b one obtains δ

(a, b)<δ

(a, p

)

but p

≺b. That is, the successor is closer than its

predecessor. This can be resolved by adjusting the

density of the ﬂow of time across branches, e.g. by

manipulating the spacing between two points.

A ﬂow of time, in which this issue cannot arise is

a ﬂow where every path between a join and a meet

has the same length. Leading us to the deﬁnition of

global-homogeneousness.

Deﬁnition 2.10. Let Φ =T, ⪯, ω be a quasi-metric

ﬂow of time. Then Φ is a globally-homogeneous ﬂow

of time iff ∀x, y, z ∈T

(x, y)+δ

(y, z)=∞∨δ

(x, z)=δ

(x, y)+δ

(y, z)

Notice, that every linear ﬂow of time satisﬁes this

property. Lastly, we have to check whether global ho-

mogeneousness resolves the issue.

Proposition 2.11. Let Φ =T, ⪯, ωbe a ﬂow of time

then ∀x, y, z ∈T (x ≺y ≺z Ô⇒ δ

(x, y)<δ

(x, z))if

it is globally homogeneous

Proof. If δ

(x, y)+δ

(y, z) = ∞, ∀x ↝ z (y ∉ x ↝

z). Hence, we only consider x ⪯ y ⪯ z. If either

of ⪯ are equal, we are done. Otherwise, we have

x ≺y ≺z. Since, δ

(x, z)=δ

(x, y)+δ

(y, z), thus

(x, z)−δ

(y, z)=δ

(x, y) and since 0 <δ

(y, z),

(x, y)<δ

(x, z)follows

However, there is another notion of homogeneous-

ness, namely local homogeneousness.

Example 2.12. Consider the ﬂow of time from Exam-

ple 2.6. Interpret o as the present. We see that the

past is globally-homogeneous, while the future is not.

Even though, in the past there is a higher density of

points in the bottom branch than in the top. While

the inverse case holds in the future. So in some sense

some strains of our time ﬂow are denser than others.

In Example 2.12 we regard ”density” of time as

the spacing of observations (Zumbach and M

uller,

2001). Analogously to real world problems, consider

a sequence of graphs (G

)

t∈T

distributed unevenly

along a time line, as a result from inconsistent mea-

surements or a contraction of measurements to obtain

certain properties. Hence, this is a problem that can

COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk

occur even in linear ﬂows of time. Therefore, we in-

troduce the concept of local-homogeneousness, to ex-

press that all points in T are evenly spaced with re-

spect to ω. That is, we use a notion of homogeneity

mentioned in (Zumbach and M

uller, 2001).

Deﬁnition 2.13. Φ =T, ⪯, ωis a local-homogeneous

ﬂow of time iff ω is the constant function x with x ∈R

Thus we arrive at:

Deﬁnition 2.14. Φ =T, ⪯, ωis a homogeneous ﬂow

of time iff Φ is locally and globally homogeneous.

It should be mentioned that those are not the only

possible notions of homogeneity, consider for exam-

ple (Venema, 1998). By using homogeneity we can

now safely, navigate through a ﬂow of time.

Deﬁnition 2.15. Let Φ = T, ⪯, ω be a globally-

homogeneous ﬂow of time, then for some point of ori-

gin a ∈T and some x ∈R we have

Φ(a, x)

=

{y ∀y ∈T δ

(a, y)=x} 0 ≤x

{y ∀y ∈T δ

(y, a)=x} 0 >x

However, more importantly, since global-

homogeneity forces the distance between two points

to be unambiguous, we can understand x to be a point

on a linear time line and Φ(a, x) as a function that

maps into a set of possible worlds. By restricting

oneself to homogeneous ﬂows of time, one ensures

that all worlds across all branches are lined up.

Allowing us to consider additional operations, such

as the collapsing of possible worlds. Unfortunately,

due to the page restrictions the various methods of

doing so cannot be discussed here.

3 TEMPORAL NETWORKS

The notion of a temporal network, intuitively to be

understood as a graph with an additional structure

encoding the dimension of time, has been captured

by multiple formalisms and is known under various

names across different ﬁelds (Holme and Saram

aki,

2012; Casteigts et al., 2012). Therefore, the general

approach of multilayer networks was introduced to

unify formalisms that extend the ordinary notion of

a graph, this includes several formalisms concerned

with capturing the notion of a temporal network. We

are partially supporting this attempt.

In (Holme and Saram

aki, 2012) an important dis-

tinction is made between instance-based temporal

graphs and interval-based temporal graphs. The

prior understands time as a sequence of instances,

where, for example, an edge is labeled with a se-

quence of time stamps indicating the graph instances

that include this edge. Hence, all interactions be-

tween nodes can be modeled by intra-layer edges, and

the inter-level edges corresponding to ”identity” rela-

tions. While the latter allows for the modeling of in-

teractions with a duration, i.e. an interaction starts at

layer α and ends at layer β. We focus our attention on

instance-based contact sequences.

Deﬁnition 3.1. Let T ∶=G, R, ≡, ⪯, ωbe a weighted

multilayer network with equivalence, we call it a

instance-based temporal network iff

• G∶={G

∀t ∈T }for some labeling T .

• R is a collection of all successor relations with

respect to ⪯, i.e. R={R

∀G

, G

∈G R

∶=

{(v, w)∀v ∈G

∀w ∈G

S(G

, G

)∧v ≡w}}.

• ω is deﬁned such that it respects the intra-graph

relation weights while assigning weights to every

inter-graph relation, i.e.

(e)∶=











(e) t

≠t

∧G

⪯G

∞ otw.

Moreover, we observe that a temporal graph can

be understood as a ﬂow of time with additional struc-

ture. That is, consider the ﬂow in time T ∶=G, ⪯, ι, ω.

Now for all G

∈Gwe ﬁx ι(G

)=G

where G

is some

weighted graph G

∶=V

, E

, ω

. By ﬁxing the world

at a certain point in time to being the same as its label

ι becomes redundant. Let ≡ be in Deﬁnition 1.2 and

let ⪯extend to the vertices such that ∀G

, G

∈G∀v ∈

∀w ∈G

⪯G

Ô⇒ (

v =w ⇐⇒ v ⪯w)). Then

Ris just the set of successors with respect to our ex-

tended ﬂow relation, where ω carries over.

3.1 Paths in Temporal Networks

The notion of a path in a static, non-weighted graph,

is a fairly simple concept, with its length being

deﬁned as its size, i.e. its number of edges. In this

case a shortest path can be computed efﬁciently and

has an fairly unambiguous semantics (Tang et al.,

2009; Wu et al., 2014). Unfortunately, this property

is already lost when considering weighted graphs in

general. As negative cycles increase the computa-

tional complexity of this problem, and weights in

general require the distinction between similarity and

dissimilarity measures. When analysing a network

those two interpretations have to be distinguished

carefully, as some measures only provide sensible

results when ω is a similarity measure, while others

require dissimilarity measures. Fortunately, it is

possible to invert the semantic interpretation of the

respective measure. One example of this would be

A Generalized Notion of Time for Modeling Temporal Networks

(x, y)=

1+ω

(x,y)

, with ω

and ω

being some mea-

sure of similarity and dissimilarity respectively. Now

consider a multilayer network, each layer having

(possibly) different semantics. The same holds true

for temporal networks, where we have a dissimilarity

measure on inter-graph edges and another measure

with different semantic on intra-graph eges. Hence,

making the notion of what a shortest path may be

even more difﬁcult (Runkler, 2012; Goshtasby, 2012;

Segarra and Ribeiro, 2016).

Within the context of temporal networks (Wu

et al., 2014) introduce a set of minimum temporal

paths, consisting of the earliest-arrival path, i.e. start-

ing from a

ﬁnd the path ending in the smallest b ∈b,

the latest-departure path, i.e. what is the largest a ∈a,

while still being able to reach b; the fastest path, i.e.

what is the shortest path between a and b minimising

the difference between ending time and starting time

and shortest path, thus the path that is the shortest with

respect to traversal time. In (Tang et al., 2009) a tem-

poral graph is conceptualised as a sequence of graphs.

However, by limiting the amount of hops within each

static graph, they manage to encode some notion of

time into each static graph. Moreover, it is not un-

common to make the distinction between the size of

a path and its duration explicit (Holme and Saram

aki,

2012; Michail, 2016; Casteigts et al., 2012). Analo-

gously to ﬂows in time, this distinction roughly corre-

sponds to ⋅

and ⋅

. As already mentioned, when

dealing with weighted temporal graphs an additional

dimension is introduced. Namely, we distinguish not

only between temporal steps and temporal distance,

but also between intra-level steps and intra-level dis-

tance. One approach would be to use some norm to

collapse those two dimensions. However, motivated

by those examples and the issue of similarity and dis-

similarity we instead use the concept of a path in time

together its notion of distance, as well the notion of a

path in a regular graph.

Deﬁnition 3.2. Let T ∶=G, R, ≡, ⪯, ωbe a instance-

based temporal network and let v ∈G

and w ∈G

then the alternating sequence of regular paths and

forward-paths in time T ⊇(v ↝

w)=(v

↝

)∶=

↝

↝w

⋯↝

↝v

is called the temporal

path from v

to w

. We write λ

(x ↝

y)for the set

of maximal paths in time and λ

(x ↝

y)for the set

of maximal intra-graph paths in x ↝

By distinguishing between the types of paths in

temporal graphs, we allow for a separation of mea-

sures, i.e. inter- and intra-length and size.

a ↝

b is shorthand for a ↝ b ⊆ X .

Deﬁnition 3.3. Let T ∶= G, R, ≡, ⪯, ω be an

instance-based temporal network, let p ∶=v ↝

w be

the temporal path from v to w with v, w ∈V (T). Then

we deﬁne

p

∶=(E(λ

(p))

, E(λ

(p))

)

as the length of p with respect to ω and p

as its size.

By allowing for two dimensions with respect to

path length and size we have to deﬁne a speciﬁc or-

der on those values. A natural choice for this is the so

called product order, i.e. (x

, y

)≤(x

, y

) ⇐⇒ x

≤

∧y

≤y

(Dickson, 1913). However, one issue that

immediately arises when using a product ordering, is

the issue of partiality (see Example 3.4). Moreover,

there can be various natural interpretations of shortest

path problems. For example, the problem a ↝

b ad-

dresses the desire to compute the set of overall short-

est distances between two equivalence classes, while

the problem a

↝

b restricts the same question to a

certain starting point and a ↝

to a speciﬁc arrival

date. Lastly, a

↝

is de facto an ordinary shortest

path problem.

Example 3.4. The temporal graph T

where G

and G

are respectively

While each shortest path problem may lead to differ-

ent results (see Table 1), the more signiﬁcant obser-

vation is that from the shortest path between x and y,

x ↝z ↝y it is not possible to conclude that x ↝z is the

shortest path between x and z. Consider, a ↝w and

a ↝b. Thereby, prohibiting the save use of Dijkstra’s

algorithm for some of the speciﬁed problems.

Table 1: Compute all possible shortest paths between a and

b and all elements within those equivalence classes.

∣⋅ ↝ ⋅∣ b

(0, 1) (1, 1) (2, 1) (0, 1)

(∞, ∞) (0, 4) (1, 2) (0, 4), (1, 2)

(∞, ∞) (∞, ∞) (0, 2) (0, 2)

a (0, 1) (1, 1), (0, 4) (2, 1), (0, 2) (0, 1)

While, in general, the multi-objective shortest

path problem may need exponential runtime, we can

do better due to its unique structure (Tarapata, 2007).

Proposition 3.5. Let T ∶= G, R, ≡, ⪯, ω be a

globally-homogeneous, instance-based temporal

multiplex then for v, w ∈V (T)the length (and size) of

all shortest path is comparable and thus the same.

COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk

Proof. We know ∃!G

, G

∈G v ∈G

∧w ∈G

. If

there does not exits a path between v and w we are

done. Otherwise, by global-homogeneity and due to

⪯ being directed, we obtain ∀v ↝

w v ↝

w

(δ

, G

), x). As they only differ in x, all of them

are comparable and by minimality all shortest paths

have the same length.

Proposition 3.6. Let T ∶= G, R, ≡, ⪯, ω be a

globally-homogeneous, instance-based temporal

multiplex. The problem of ﬁnding the shortest path

with respect to ⋅

over T from v to w for v, w ∈V (T)

can be reduced in polynomial time to the problem

of ﬁnding the shortest path between v and w in the

weighted directed graph.

Proof. Consider the construction D ∶=

V (T), E(T), ω. This transformation is lin-

ear. Show for p

∶= (v ↝

w) ⊆ T we have

v ↝

w

is minimal ⇐⇒ for p

∶=(v ↝

w)⊆D

we have v ↝

w

is minimal. We observe

E(λ

))

=δ

(v, w). Moreover, every path be-

tween v and w only differs in E(λ

))

. Since,

p



= δ

(v, w)+E(λ

))

, we obtain p



minimal must be equivalent with p



minimal.

Hence, ﬁnding the shortest path between two dis-

tinct vertices in a temporal graph can be solved by

applying a variant of Dijkstra’s algorithm and it is

thus O((n +m)⋅log(n)), where G≤n =V (T) and

m =E(T)(Barbehenn, 1998). By using this knowl-

edge we can implement algorithms for computing the

other shortest path problems. Even a naive implemen-

tation, i.e. one that computes all distances between v

and every member of w is in O((m +n)⋅n ⋅log(n)).

Hence, searching for all minimal paths with respect

to ⋅

is bounded by the same complexity, as ﬁnd-

ing minimal elements is at most O(n

). By applying

the same algorithm for all v

∈v we obtain at most

O((m +n)⋅n

⋅log(n)). This serves only as a rough

estimate to show polynomial membership and to jus-

tify this approach form a computational complexity

point of view.

4 DISCUSSION

We decided against using the classical notion of mul-

tiplex network as a basis for our approach and instead

deﬁned a tailored variant that explicitly considers our

generalized notion of time. This allows us to classify

a temporal path as a sub-network of the main temporal

network, with the temporal path retaining the proper-

ties of being a temporal network itself. Moreover, as

discussed in (Taylor et al., 2017) restricting oneself

to multiplex networks carries a tolerable loss of ex-

pressiveness, as one can add isolated ”ghost nodes”

to each graph G

to obtain the (classical) multiplex

property. Unfortunately, this implies that for certain

measures the existence of such ghost nodes must be

accounted for. While the relation modeling time is

transitive, the corresponding inter-layer edges are nei-

ther transitive nor symmetric. Intuitively they reﬂect

the movement of a vertex through time. Therefore,

by accepting the multiplex property, node persistence

is ensured, allowing for an easier semantic of a move-

ment through time, i.e. a vertex can not pop in and out

of existence as it pleases.

Future work may discuss possible ramiﬁcations

with respect to common network measures. For ex-

ample, by only considering the multivalued notion of

distance on linear time, one can already detect differ-

ent behaviours of some network measures. That is,

consider the path-based centrality measures closeness

and betweenness. As betweenness relies on counts

of shortest paths its extension to a multivalued path

length is straightforward. Even in linear time the var-

ious shortest path problems result in analogous be-

tweenness measures, i.e. with and without restriction

on the starting and end time. In contrast, while close-

ness centrality has a similar relationship with respect

to time, any ranking of nodes will be a partial order-

ing. That is, since the length of a path is incorpo-

rated into this measure, its multivalued nature will be

carried over. Lastly, while interesting, a discussion

of network measures on non-linear models of time is,

unfortunately, far beyond the scope of this paper.

5 CONCLUSION

Most approaches for the analysis of temporal net-

works do not explicitly discuss the underlying con-

ception of time. Moreover, weighted temporal net-

works are still uncommon in the literature, and a di-

rect discussion about how to reconcile the two seman-

tic dimensions seems to be even more rare. In order

to tackle those issues, this paper discussed time as

a formal structure, thereby explicitly engaging with

some of the underlying assumptions of time on which

a temporal network may be operate.

Most prominently, we discussed some of the pit-

falls that arise when dealing with non-deterministic

time. Furthermore, our generalized abstraction of

time promotes a clean separation of the semantics of

time and the semantic interpretation of the network

itself (i.e. the semantics of the vertices and edges in

the corresponding network). To retain a clean separa-

A Generalized Notion of Time for Modeling Temporal Networks

tion of dimensions, we introduced the notion of mul-

tivalued (temporal) paths, a variant of paths that, on

the one hand, enables a more in-depth understanding

of temporal networks without sacriﬁcing semantic in-

tegrity, while on the other hand introducing several

new and interesting technical challenges, such as the

computation of multivalued-centrality measures for

temporal networks.

REFERENCES

Barbehenn, M. (1998). A note on the complexity of di-

jkstra’s algorithm for graphs with weighted vertices.

IEEE transactions on computers, 47(2):263.

Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C. I.,

omez-Gardenes, J., Romance, M., Sendina-Nadal,

I., Wang, Z., and Zanin, M. (2014). The structure and

dynamics of multilayer networks. Physics Reports,

544(1):1–122.

Casteigts, A., Flocchini, P., Quattrociocchi, W., and San-

toro, N. (2012). Time-varying graphs and dynamic

networks. International Journal of Parallel, Emergent

and Distributed Systems, 27(5):387–408.

De Domenico, M., Sol

e-Ribalta, A., Cozzo, E., Kivel

a, M.,

Moreno, Y., Porter, M. A., G

omez, S., and Arenas,

A. (2013). Mathematical formulation of multilayer

networks. Physical Review X, 3(4):041022.

Dickson, L. E. (1913). Finiteness of the odd perfect and

primitive abundant numbers with n distinct prime fac-

tors. American Journal of Mathematics, 35(4):413–

422.

Goranko, V. and Galton, A. (2015). Temporal logic. In

Zalta, E. N., editor, The Stanford Encyclopedia of Phi-

losophy. Metaphysics Research Lab, Stanford Univer-

sity, winter 2015 edition.

Goshtasby, A. A. (2012). Similarity and dissimilarity mea-

sures. In Image registration, pages 7–66. Springer.

Holme, P. and Saram

aki, J. (2012). Temporal networks.

Physics reports, 519(3):97–125.

Kivel

a, M., Arenas, A., Barthelemy, M., Gleeson, J. P.,

Moreno, Y., and Porter, M. A. (2014). Multilayer net-

works. Journal of complex networks, 2(3):203–271.

Markosian, N., Sullivan, M., and Emery, N. (2016). Time.

In Zalta, E. N., editor, The Stanford Encyclopedia

of Philosophy. Metaphysics Research Lab, Stanford

University, fall 2016 edition.

Matthews, S. G. (1994). Partial metric topology. Annals of

the New York Academy of Sciences, 728(1):183–197.

Michail, O. (2016). An introduction to temporal graphs:

An algorithmic perspective. Internet Mathematics,

12(4):239–280.

Runkler, T. A. (2012). Data Analytics. Springer.

Schroeder, V. (2006). Quasi-metric and metric spaces.

Conformal Geometry and Dynamics of the American

Mathematical Society, 10(18):355–360.

Segarra, S. and Ribeiro, A. (2016). Stability and continu-

ity of centrality measures in weighted graphs. IEEE

Transactions on Signal Processing, 64(3):543–555.

Sunitha, M. and Mathew, S. (2013). Fuzzy graph theory:

a survey. Annals of Pure and Applied mathematics,

4(1):92–110.

Tang, J., Musolesi, M., Mascolo, C., and Latora, V. (2009).

Temporal distance metrics for social network analysis.

In Proceedings of the 2nd ACM workshop on Online

social networks, pages 31–36. ACM.

Tarapata, Z. (2007). Selected multicriteria shortest path

problems: An analysis of complexity, models and

adaptation of standard algorithms. International Jour-

nal of Applied Mathematics and Computer Science,

17(2):269–287.

Taylor, D., Myers, S. A., Clauset, A., Porter, M. A., and

Mucha, P. J. (2017). Eigenvector-based centrality

measures for temporal networks. Multiscale Model-

ing & Simulation, 15(1):537–574.

Van Ditmarsch, H., van Der Hoek, W., and Kooi, B. (2007).

Dynamic epistemic logic, volume 337. Springer Sci-

ence & Business Media.

Venema, Y. (1998). Temporal logic. Citeseer.

Wu, H., Cheng, J., Huang, S., Ke, Y., Lu, Y., and Xu, Y.

(2014). Path problems in temporal graphs. Proceed-

ings of the VLDB Endowment, 7(9):721–732.

Zumbach, G. and M

uller, U. (2001). Operators on inhomo-

geneous time series. International Journal of Theoret-

ical and Applied Finance, 4(01):147–177.

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