# The anatomy of a population-scale social network

Eszter Bokányi1,∗, Eelke M. Heemskerk1, Frank W. Takes2

(1University of Amsterdam
2Leiden University
<EMAIL_ADDRESS>
###### Abstract

Large-scale human social network structure is typically inferred from digital
trace samples of online social media platforms or mobile communication data.
Instead, here we investigate the social network structure of a complete
population, where people are connected by high-quality links sourced from
administrative registers of family, household, work, school, and next-door
neighbors. We examine this multilayer social opportunity structure through
three common concepts in network analysis: degree, closure, and distance.
Findings present how particular network layers contribute to presumably
universal scale-free and small-world properties of networks. Furthermore, we
suggest a novel measure of excess closure and apply this in a life-course
perspective to show how the social opportunity structure of individuals varies
along age, socio-economic status, and education level. Our work provides new
entry points to understand individual socio-economic failure and success as
well as persistent societal problems of inequality and segregation.

##  1 Introduction

Recent studies on large-scale human social networks successfully utilize the
growing abundance of digital trace data such as online social media or mobile
communication datasets to uncover fundamental insights on human interactions
[1, 2, 3, 4]. However, deriving scientific meaning from large-scale data
originally not created for the purpose of academic research is far from
trivial. In networks created from such data, nodes are a mere sample based on
users of a certain platform or mobile phone provider. Therefore, they are
likely not representative of investigated populations. Moreover, most social
ties ultimately appear as indistinguishable edges after being inferred from
messages [5], calls [6, 7], or friendship nominations [8, 3, 4]. This
inference is often based on non-generalizable research design choices [9], for
example, when mobile calls are thresholded by call volume or call numbers to
arrive at a set of social ties. The formation of edges from online social
networks might also be influenced by platform-specific recommendation
algorithms. On most platforms, edges represent people’s activity rather than
their social connectivity, or may not cover important connections such as
close family. As a result, most large-scale social network datasets are non-
random samples of an underlying social structure [5, 10, 8]. It is therefore
hard to derive generically applicable actionable insights from such data, or
to generalize findings across studies [11]. Despite the above mentioned non-
trivial concerns, online and mobile communication social network data are
still used for instance as a direct proxy for real-world friendships, or to
inform policy measures on crucial issues as combating prevalent societal
problems such as inequality [12, 3].

In this paper, we present a promising way forward in obtaining reliable
insights into large-scale social structures through a population-scale social
network analysis based on government-curated administrative register data.
While such a network is based on national registers that are initially also
not designed for academic research, its coverage of an entire population,
explicit notion of well-categorized edge types, and the presence of high-
quality demographic information provide insights at an unprecedented level of
detail. Most importantly, nodes are people officially enlisted in a country’s
administrative system. This gives a clear geographical boundary and legal
definition of the included nodes resulting in an almost perfect population-
scale coverage and thus, a representative node sample. As opposed to more
informal ties such as retweets, phone calls, or friendships that are typically
captured in aforementioned online social media data, social ties encoded in
official registers are _formal ties_. However, it has been shown that formal
ties do capture the majority of people’s strong connections [13, 14, 15].
Using formal ties, we can describe individuals’ _social opportunity structure_
similarly to, e.g., labour opportunities [16].

The register-based data allows us to empirically take into account how
different edge types contribute to social opportunity structures. The
framework used in this work is hence a node-aligned multilayer social network
[17, 18, 19], in which each edge type is modeled as a separate network layer.
Previous work pointed towards the necessity of distinguishing edge types in
order to adequately understand overlapping community structure of social
networks [20], or the measurement and modeling of weak ties and their
importance in network cohesion [21, 18]. For automatically collected large-
scale networks, it is sometimes possible to infer edge types from the
communication content [22] or to capture different interactions between nodes
by the data collection design [23, 24]. However, these examples remain
exceptions and limited to the particular measurement setting used in these
studies, or include only a small number of nodes (see [25], [26], and [27],
Chapter 3). Instead, we map and analyse the multilayer network properties of
an entire country’s population. An important benefit of register data is that
it provides a wealth of reliable demographic information on the nodes that is
often unavailable when other data sources are used, for instance on income,
education, age, or neighbourhood properties. Therefore, not only does this
approach deliver an efficient way for mapping people’s social opportunity
structures at a high level of granularity, but it also holds the promise of
deriving meaningful conclusions about consequences of social structure for
crucial matters. These include quantifying poverty and inequality [28, 12],
constructing better epidemic models [29], understanding related public health
issues [30, 31], and growing political polarization [32].

We present a network analysis of all 17.2M residents registered in the
Netherlands. The network is sourced from Statistics Netherlands and contains
roughly 827M edges of the following types: close and extended family,
household, next-door neighbor, school, and work relationships. We present a
thorough investigation of the anatomy of this network, and find three
surprising differences with respect to commonly observed or expected
properties of other well-known large-scale social networks. First, the fat-
tail degree distribution of online social networks [33, 34] is conspicuously
absent because the generative logic of connectivity and therewith the degree
distribution is different in a population-scale social network of formal ties.
Second, short average shortest path length is generated not through random
link rewiring [35] or strong edges closing large network distances [36], but
through a particular interaction of the various network layers. Third, closure
measured by the local clustering coefficient [37] reaches unprecedented high
levels, and increases with increasing degree.

To address the challenges in the measurement of closure, we propose a novel
multilayer measure that we call _excess closure_ , which specifically captures
social closure that is the result of overlap between edges in different
layers. An application of this measure on the population-scale social network
reveals fascinating patterns of social opportunity structures in the
Netherlands. We show that across different age groups, higher degree
characterizes higher income and higher education population subgroups, and
that degree peaks for people with a university age. Similarly, excess closure
is high for the youngest and the oldest, but reaches its lowest point for
young adults. Interestingly, this behaviour is markedly different for
different socio-economic groups. Altogether, our results illustrate the need
for new measurement tools tailored to the particular multilayer structure of
population-scale social networks. Moreover, our work brings the field a step
closer to reliably using large-scale social network data for deriving policy-
relevant actionable insights from the network structure underlying our complex
society.

##  2 Results

The population-scale network data consists of nodes that correspond to the
roughly 17.2M people (the whole Dutch population) registered in the
Netherlands on October 1, 2018 [38], as well as six sets of edges derived from
various government registers, referred to as network layers. In particular, we
consider Close Family $C$ and Extended Family $E$ derived from the parent-
child and partner registers, and Household $H$, Work $W$, and School $S$,
being projections of people’s affiliations. To avoid confusion between next-
door neighbor relationships and network neighbors, i.e., two adjacent nodes
connected by an edge, we refer to edges from the layer of people’s
connectivity based on spatially close households as Next-door, abbreviated by
$N$. As such, our network is a population-scale node-aligned multilayer
network with a single aspect [17], where each layer contains the same set of
nodes, and edges only connect nodes within the layers. All data was
pseudonymized, and analysis took place in a GDPR-compliant manner, ensuring
that the identity of individuals is not revealed (see Data and Methods section
and the Ethical statement for details).

We start with an exploration of the network topology, revisiting three well-
known structural features of social networks: degree, clustering, and shortest
path length. The purpose is to establish to what extent the topological
features of this population-scale social network resemble those of other
previously studied large-scale social networks, with a special interest in how
the different network layers contribute to these measures. We then use these
insights to characterize the social opportunity structures of different
demographic groups by age, income, education level, and urbanization level of
their home locations.

###  2.1 Degree

Degree distributions of large-scale social networks from the literature
typically are said to exhibit two main features. First, most nodes have a
relatively low degree (number of connections), meaning that the lowest degrees
are the most abundant in these networks. Second, some nodes have extremely
large degrees, i.e., there is a fat-tailed degree distribution [8, 5, 34, 39].
In this section, we revisit whether this holds for our population-scale
network of formal connections, and show how the degree distributions and
generating mechanisms of different layers contribute to the overall shape of
the total degree distribution.

Figure 1 shows the degree distributions for the six different layers, the
total degree distribution, and the distribution of the number of unique layer
types each node is involved in (see Data and Methods section for details on
how missing data was handled). Figure 1B, the left inset, shows the number of
nodes with zero degree in these layers. Approximately 2.5% of the nodes have
no close family members, and 18.6% live in a one-person household having a
degree of 0 in the household layer. The number of nodes with degree 0 in the
extended family layer is almost three times as much, 7.6% of all the nodes.
These nodes mostly correspond to first-generation migrants coming either
alone, or only with their close family members to the country. Apart from
these isolated nodes, the degree distributions of the close family, extended
family, and household layers for nodes with at least one neighbor in Figure 1A
all illustrate that a low number of connections is very common. After this
initial flat regime, all three distributions show a somewhat fat fail
corresponding to a relatively small group of people belonging to quite large
close or extended families, and that there are some large households. These
layer-wise distributions thus resemble fat-tailed distributions from
communication networks or that of online social relationships.

Figure 1: Degree distributions per layer and in total. (A) The degree
distributions for the individual layers, distinguished by colors. (B) Number
of nodes with degree zero in each layer. (C) Distribution of the total degree,
on a double logarithmic scale. (D) Distribution of the number of layers in
which a node has nonzero degree. From all figures, categories containing less
than 10 nodes are omitted for privacy reasons.

The degree distribution of the school layer indicates typical class sizes in
the primary and secondary education that peak around 20, and there is a slowly
decaying tail towards the degrees of large university yeargroups. The number
of colleagues, that corresponds to the number of neighbors in the work layer,
starts off with a flat distribution in the lower regime, and then shows a peak
slightly above 100 colleagues (the latter is because of the underlying data
generation mechanism, see Data and Methods).

Interestingly, when adding up the degrees of different layers for the nodes to
obtain the total degree distribution shown in Figure 1C, there are hardly any
nodes (0.08%) that remain disconnected, while the highest degree peaks at only
844. Between the two extremes, 80% of the nodes are in the range of 28 to 177
connections. The peak of the work layer at a degree of about 100 remains
visible. Most nodes have edges in 4 to 6 layers, as shown in Figure 1D. Unlike
many previously studied social networks, the total degree distribution follows
none of the most commonly assumed shapes, i.e., it is neither a power-law nor
a lognormal distribution. This suggests that in a network where the existence
of an edge does not involve user activity, as expected, preferential
attachment is not a generating mechanism behind degrees.

| $C$ | $E$ | $H$ | $N$ | $S$ | $W$
---|---|---|---|---|---|---
$C$ | 1.00 | 0.52 | 0.10 | 0.03 | -0.14 | 0.08
$E$ |  | 1.00 | 0.06 | 0.01 | -0.04 | 0.08
$H$ |  |  | 1.00 | 0.07 | 0.10 | -0.01
$N$ |  |  |  | 1.00 | 0.04 | 0.07
$S$ |  |  |  |  | 1.00 | -0.05
$W$ |  |  |  |  |  | 1.00
Table 1: Inter-layer degree assortativity. Pearson correlations between
degrees calculated from different layers: close family ($C$), extended family
($E$), household ($H$), neighbors ($N$), school ($S$), and work ($W$). Only
nodes with complete family information are included.

To understand how the degree distributions of the several layers add up into
the total degree distribution of Figure 1C, we show in Table 1 the weighted
correlation between the degrees of people for whom the family information was
complete. As expected, the number of close family members and extended family
members show a strong positive correlation of 0.53. A somewhat weaker but
still positive significant relationship is that the larger the close family,
the greater household a person lives in, which is shown by a correlation of
0.10. The fraction of working-age adults is also higher in larger families,
therefore, the close family degree is positively correlated to work degree,
and negatively to school degree. The same effect is existent but weaker for
the extended family degree. However, large households typically house more
school-aged children or young people. Therefore, despite opposite correlation
signs of household size to family size and school degree, school degree and
household degree are positively correlated. To sum up, the different layers
contribute in their own distinct manner to the total degree distribution,
reflecting the different generating mechanisms in those layers.

Overall, these results show that merely observing total degree distributions
as in Figure 1C without taking into account the multilayer nature of large-
scale social networks is disadvantageous. This is particularly the case
because it poses the risk of attributing meaning to compounded distributions.
Moreover, a deeper understanding of the generating mechanisms of edges and the
correlation between degrees in different layers is necessary for the correct
interpretation of patterns in the total degree distribution.

###  2.2 Shortest path lengths

We continue with looking at how the different layers play a role in realizing
shortest paths. The seminal work of Watts and Strogatz suggests that, starting
from a locally clustered ring-like structure, the addition of a limited number
of randomly rewired edges creates a low diameter and low average shortest path
lengths [35]. For large social networks that cover a significant portion of a
population, [36] argues that the small-world structure is created by so-called
“network wormhole” edges. These are relatively strong ties that connect
previously distant parts of the network. In what follows, we assess whether
the Dutch population-scale social network has small-world properties, and
whether network wormholes are a driving force behind it.

To measure the effect of different layers on the average shortest path length,
we create networks from different subsets of layers (or in some cases, a
subset of edges within a layer, denoted by a subscript). The created networks
range in density from only taking parent-child directed edges from the close
family layer $C_{p}$ to the full six-layer network. We subsequently add layers
to the parent-child edges $C_{p}$, being all close family links $C$, the
extended family layer $E$, the household layer $H$, next-door neighbors $N$,
primary school $S_{p}$ and secondary school $S_{s}$, the entire school layer
$S$, small workplaces ($W_{s}$ where workplace size $\leq$ 50), and at last,
every edge from all six layers ($C+E+H+S+W+N$). Table 2 summarizes the average
shortest path lengths for these networks along with several other key global
network measures.

Layers | Nodes | Edges | Components | $GC$ | $D$ | $\overline{d}$
---|---|---|---|---|---|---
$C_{p}$ | 15.76M | 19.16M | 963.75k | 0.58 | 288 | 75.03
$C+E+H$ | 16.80M | 162.15M | 317.30k | 0.93 | 33 | 9.66
$C+E+H+N$ | 17.25M | 361.95M | 2.77k | 1.00 | 64 | 6.12
$C+E+H+N+S_{p}$ | 17.25M | 435.82M | 2.46k | 1.00 | 63 | 5.61
$C+E+H+N+S_{p}+S_{s}$ | 17.25M | 478.66M | 2.32k | 1.00 | 62 | 5.39
$C+E+H+S+N+W_{s}$ | 17.25M | 496.34M | 1.83k | 1.00 | 42 | 5.25
$C+E+H+S+N+W$ | 17.25M | 826.99M | 1.15k | 1.00 | 22 | 4.64
Table 2: Network properties for various layer combinations. Columns show the
number of non-isolated nodes, number of edges, number of connected components,
relative size of the giant component $GC$, this component’s diameter $D$ and
estimated average shortest path length $\overline{d}$.

First, we observe that average distances in the parent-child layer are quite
large: two randomly selected nodes are roughly 75 steps away from each other.
Adding close family and extended family alongside with household links to this
“backbone” of parent-child links already decreases the average distance to
below 10. This addition corresponds to shortcutting paths of length 2, 3, 4,
or even larger in the parent-child network, to distance 1. For example, a
sister would have been at distance 2, and a cousin would have been at distance
4, but in the direct extended family layer they are directly connected, so at
a distance of 1. These family structures can be seen as long “chains” of
highly clustered bubbles of relatives merged together by marriages,
partnerships and having common children. Adding next-door neighbor
relationships on top of family and household edges shortens the average
shortest path length by an additional 3.54 steps down to around 6.12. Primary
schools decrease this distance to 5.61, with small workplaces and other school
types add subsequent decreases, leading to an average shortest path length of
5.25. Thus, next-door neighbor, school and work relationships are effective in
connecting more distant parts of the network, and deliver an average shortest
path length that go below the characteristic value we would expect in so-
called small-world network [40, 35]. Including even more sporadic potential
work connections, that is, all of the larger workplaces, further decreases
average distances to 4.64, that approaches the value suggested for Facebook by
[41]. It also provides empirical confirmation of the findings of [42], who
suggest that shortest paths in large-scale networks are likely to be mediated
by professional or school connections.

Figure 2: Embeddedness and tie range. (A) Embeddedness distribution of all
edges in the network. (B) Example of an edge with embeddedness 3. (C) Tie
range distribution for edges that are not part of a triangle, so with
embeddedness 0. (D) Example of an edge with tie range 5.

The question that remains is whether the underlying mechanism is through the
aforementioned wormhole edges. This can be investigated by the measures of
edge embeddedness and tie range. Edge embeddedness measures the number of
overlapping neighbors of the pair of nodes that are connected by the edge (see
Figure 2B). Local bridges that shorten average distances are edges for which
the nodes at the two endpoints have no common neighbors, and thus correspond
to an embeddedness of zero (see the Data and Methods section for details).
Figure 2A shows the distribution of embeddedness in our network, with the peak
around 100 corresponding to the workplace sampling process (as explained in
the Data and Methods section). But most interestingly, there are only around
312k local bridge edges (0.02%) in our network having zero embeddedness.

For these local bridges, we measure the length of the second shortest path
between nodes corresponding to an edge (see Figure 2D), the so-called tie
range (see Data and Methods section). Note that this value is one less than
the length of the shortest cycle going through the selected edge. Therefore,
edges with a nonzero embeddedness automatically have a tie range of 2. Network
wormholes are relatively strong long-range ties having a range of at least 6
[36]. In the Twitter network of the country of Singapore, these ties amounted
up to 0.46% of all edges. Figure 2B shows the distribution of tie range in our
network, that mostly spans small distances of either 2 or 3. Only 9419 edges
have a range of at least 6, making the proportion of such wormhole edges
$0.0011\%$, two orders of magnitude less than in the case of aforementioned
Twitter network. A closer look at the data shows that 92% of these long-range
edges are either close family, extended family, or household edges. This
implies that such edges in themselves are not expected to create smaller
shortest path lengths in the Dutch population-scale network.

Thus, we conclude that in our population-scale social network, small shortest
path lengths are not brought about by wormhole edges, but instead appear to be
facilitated by a particular multilayer interaction where next-door neighbor,
school, and work edges cut paths of clustered kinship relationships shorter.

###  2.3 Clustering and excess closure

Finally, we turn to closure as an important property of the social opportunity
structure of individuals. Throughout people’s lives and across demographic
groups, closure affects their access to opportunities and information [43, 44,
12]. It is well documented that human social networks are more clustered than
what could be expected from sensible null models [45, 33]. Closure is
typically measured as the average of the local clustering coefficient over all
nodes, which captures the fraction of closed triangles between a node’s direct
neighbors. For Facebook and Twitter, both very large online social networks,
the clustering coefficient has an average value of 0.4 and 0.23 for degree
$k=5$, 0.30 and 0.19 for degree $k=20$, and around 0.14 for degree $k=100$
[46, 39]. In these and similar networks, the local clustering coefficient is
said to decay roughly with $k^{-1}$, thus, clustering mostly disappears for
large degrees, especially for large hubs [37, 5, 8].

Local clustering coefficient.

Contrary to findings described in the literature discussed above, in our
network, the local clustering coefficient [35] reaches unprecedented high
levels. This is shown by the distribution in Figure 3A in blue, where half of
the nodes have a local clustering coefficient value $c\\_{local}$ larger than
0.38, with an average of 0.40. If measured as a function of degree (Figure
3B), another peculiar observation in our network is that the local clustering
coefficient actually increases (with the exception of only two apparent local
minima) with growing degree. This is caused by large degrees originating from
layers that are in fact bipartite projections of particular affiliations
(school, work), where the larger the degree, the more triangles are present in
the ego network structures. Furthermore, close and extended family are also
highly clustered, yet they are so by nature, regardless of degree. The origin
for this lies in the logic of kinship relations and how people maintain them
[47, 48]. High clustering also holds for next-door neighbors, because being
spatially close is often a transitive relationship.

We conclude, based on the observations above, that the local clustering
coefficient is not a useful measure for understanding closure in the
considered population-scale social network. Moreover, it inherently disregards
the multilayer topology of the network. This is problematic because while the
local clustering coefficient is expected to be high inside each layer, we
currently do not capture the “unexpected” closure that comes from the complex
interaction of various layers in the network.

Figure 3: Local clustering coefficient and excess closure. (A) Distribution of
the local clustering coefficient and excess closure. (B) Average local
clustering coefficient and average excess closure for nodes with a certain
degree. Shaded areas corresponds to the 25th and 75th percentiles within that
degree. (C) Local clustering coefficient of dark grey node, calculated on all
network layers (left column, grey edges), and excess closure (right column,
colored edges).

Excess closure. Figure 3C shows three small typical examples of local
structures in the left column, all exhibiting high levels of local clustering.
However, this degree of clustering merely reflects that the neighborhoods of
nodes contain many closed triangles within the layers, that can be _expected_
because of the generative mechanisms behind the edge creation processes. To
discount for this abundance of _expected_ within-layer triangles, we propose a
modification of the local clustering coefficient that we call the _excess
closure_ , denoted by $c\\_{excess}$. The excess closure measure
$c\\_{excess}$ is a value between 0 and 1, where 0 means that there is no
overlap in the neighbors within different layers of an ego. A value close to 1
would happen when the number of triangles with edges from at least two layers
approaches the number of wedges (open triangles) that are not closed by same-
layer edges. It thus particularly measures closure brought about by what could
be called multilayer triangles (see Data and Methods for details).

Excess closure can either exist because neighboring nodes have edges in
multiple layers, for example when a next-door person is also a schoolmate, or
because of triadic closure between edges from different layers. The examples
of the right hand column of Figure 3C illustrate this by highlighting the
layers in different colors. In the first case, clustering only happens within
layers, therefore, excess closure is 0. In the second case, there is an alter
in the ego network connected by two edges from two layers. Thus, there are
some triangles that contain edges from at least two layers, that add to the
excess closure measure value. In the bottom example, there is a local bridge
between the alters connected by edges from two highly clustered layers that
realizes excess closure.

The distribution of excess closure values $c\\_{excess}$ for all 17.2M people
in the population-scale network is shown in red in Figure 3A. Compared to the
local clustering coefficient, excess closure indeed has typically much lower
values. However, it certainly happens that network layers and thus social
opportunity structures overlap or are bridged. Figure 3C illustrates that
contrary to the local clustering coefficient, excess closure does not
counterintuitively increase with degree. All in all, excess closure overcomes
the problem of very large local clustering coefficients that originates from
the generating mechanisms of formal ties. More specifically, it is suitable
for uncovering closure caused by overlap and local bridging in social
opportunity structures.

###  2.4 Network positions in different stages of life

A unique advantage of register-based population-scale social network data is
that high quality census information is available on the individuals. This
allows us to investigate how social network position of individuals differ
across, for instance, income, school level, or living environment. In
addition, the complete coverage of individuals of all ages allows us to take a
life-course perspective, investigating how social network positions are
different across age groups. We present results of such an analysis as an
illustration of our population-scale network’s potential as a foundation for
detailed computational social scientific modeling of processes such as
persisting inequality [12, 3, 4], segregation [49] or various dynamical
phenomena [19] that may play out on the network structure.

Figure 4: Node properties for different demographic groups by age. Average
degree (red, first column), average excess closure (blue, second column), and
average closeness centrality (green, third column) for people with different
income (A-C), education levels (D-F), or type of living environment (G-I) as a
function of age in the Dutch population-scale social network.

Figure 4 shows how degree, excess closure, and closeness centrality (columns)
of all the nodes change with age for different demographic groups by income,
education, and urbanization level (rows) in the Dutch population. For young
children and young adults, step-wise increases in degree are driven by their
participation in different educational settings (Figure 4A), with for example
primary school and high school starting the ages of 4 and 12, respectively.
After that, it is the size of the workplace that is dominant, with degree from
work colleagues disappearing for retired generations. Throughout all ages,
higher household income means higher degree, with differences in degree being
particularly large between the first few income deciles. The mid-life
difference in degree reflects a connection between income and work, i.e.,
people not working have neither income nor colleagues. Moreover, it indicates
that workplace size might be connected to financial compensation, with larger
companies or organizations offering higher-paid jobs. Similar trends are
observable for degree differences between education levels (Figure 4D) and
level of urbanization (Figure 4G), that are both correlated with income.
Degree patterns align well with the survey results from the meta-analysis of
[14]. Even though degree measures only the local structure in the direct
neighborhood of a node, it seems to be correlated to the global reachability
of all other network parts as measured by closeness centrality, which is
defined as the inverse of the average distance to all other nodes (see Data
and Methods). The shape and variation across the different groups with respect
to closeness are similar to that of the degree.

In parallel to having a high degree, young children have high excess closure
(Figure 4B) since they are only part of family, neighborhood, and household
structures. Subsequent phases of education and working opportunities come with
decreasing excess closure which reaches its minimum around the university age.
Working years are characterized by a slight increase in closure for higher-
income groups, and a steep increase for the lowest income deciles. This is
followed by increased closure in retirement years. High closure for low-income
groups in adult life might be an important indicator signalling less access to
new opportunities or information based on their immediate network structure
[50]. Breakdown by education level and urbanization level for excess closure
also follows the patterns shown by the income deciles, though it seems that
there are much smaller differences in people’s social opportunity structures
by their living environment than by their socio-economic status [51]. Patterns
for excess closure remain the same after controlling for degree, indicating
that the observations are not only consequences of the $c(k)$ curve of Figure
3B.

In summary, we find that degree, excess closure, and closeness centrality vary
across demographic groups as well as for different age groups. These
variations are important factors in understanding the role of social networks
in individual socio-economic failure and success. The social embeddedness of
economic action has been a cornerstone of socio-economic theory at least since
the pioneering work of Granovetter [52]. The rich empirical life-course
network data now allows us to universally test these assumptions beyond
survey-based research approaches and unravel how social structures in which
individuals are embedded help or hamper their socio-economic development.

##  3 Discussion

We mapped and analysed the social opportunity structure of an entire
population based on high quality register data. This novel approach overcomes
the persistent problem of selection bias in large-scale social network studies
and is an efficient encompassing approach for mapping social network
structures. We illustrated how the anatomy – the overall structure – of the
population-scale social network measures meaningful opportunity structures
throughout people’s lives. Specifically, we revisited three of the most
commonly studied structural network measures in the field, being degree,
distance, and closure, and presented markedly different results than expected
based on prior work.

Even though the degree distributions of the different layers resemble the fat-
tailed degree distributions common in large-scale online social networks,
these disappear once we consider the total degree distribution. The total
degree distribution shows that nodes have both a lower and an upper bound on
the number of connections. Almost none of the nodes have a degree of zero, and
97% of the nodes have a degree of at least 20. This reflects the social
reality that it is very hard for people not to have social ties. This stands
in contrast to degree distributions in most large-scale communication networks
or online social networks that measure an abundance of low-degree nodes with
only 1 or 2 connections [34], which arguably reflects perceived node activity
rather than actual node connectivity.

In our approach, we precisely map formal ties but do not have information on
informal social ties such as friendships or sport club memberships. One could
argue that omitting friendship ties in a society where kinship is less and
less important [48] creates a biased view of the actual social structures.
However, a large portion of the strong ties that people have come from the
specific layers that we can capture with our data (see, e.g., [13] or [15] for
Dutch ego networks, and [14] for a meta-analysis on survey-based personal
network research). Moreover, multiple strong friendship ties actually would
come from a longitudinal aspect of formal ties — oftentimes, old high-school
friendships or colleagues are one’s current close friends. Thus, it would be
an interesting future line of research to look at longitudinal population-
scale social networks that would cover these strong ties.

We regard our network as contact _opportunity structures_ of people: formal
ties that an individual can use as resources at some point of their career or
personal lives. As in any large-scale social network study, it is difficult to
assess how active ties are, e.g., whether two people connected actually meet
or talk on a regular basis, or whether ties can actually provide advice,
support or information. In the close family, extended family, and household
layers, people either meet each other daily, or have a kinship connection that
exists regardless of whether they maintain a very active, or a more distant
relationship with each other. We suppose that some of the next-door
relationships are active, and that in small workplaces, people tend to know
all of their colleagues. Even if large university classes or the large
workplace ties do not capture a realized relationship, in most cases, it is
likely that even in a large workplace, one knows a colleague through another
colleague. Therefore, we can interpret large degree values as an upper bound
for the actual number of social ties. They are thus a proxy of how many formal
network neighbors the nodes are able to reach in either 1 or 2 steps through
their existing direct relationships. It also means that in the low-degree
regime of Figure 1C, the degree distribution coincides with what we would
observe in actually realized formal social relationships. In this network of
formal ties, there is a typical range of numbers of connections that people
tend to have. This is somewhat in line with the 150 connections for people
predicted by Dunbar [53]. The lack of high-degree large hubs has implications
when modelling dynamical phenomena such as epidemic spreading over contact
opportunity networks.

Our new measure of excess closure offers a solution for capturing parts of the
multilayer population-scale network with a particularly strong sense of
community. It can be interpreted as the extent of overlap and local bridging
that is encoded into the multilayer structure of an individual’s social
opportunity structure. Higher values for excess closure thus indicate a
stronger sense of actual community in ego networks. It also resolves the
problem that the local clustering coefficient increases with increasing
degree. Therefore, excess closure is a useful measurement of social closure
structures in the wider opportunity structure, and in a generic manner
applicable whenever it is possible to classify a node’s connections into
multiple different contexts (i.e., layers). The study of the anatomy of our
population-scale social network also revealed that strong network wormhole
edges as suggested by [36] are conspicuously absent in our network. Rather,
co-affiliations in school classes and small workplaces wire together the
fabric of family and household connections and as such create shortcuts
between previously distant network parts.

To conclude, we showed that it is possible to study the social opportunity
study of an entire population based on register data, which presents a novel
and highly efficient way of mapping meaningful patterns of social interaction.
We revisited how degree, shortest paths, and closure look like in this very
particular multilayer network structure and showed that contrary to most
large-scale social networks, both low-degree and high-degree nodes are rare.
We measured how different edge types contribute to the reduction of average
shortest path lengths between nodes. We found that not the individual long-
range edges (wormhole edges), but rather co-affiliations in schools and small
workplaces drive shrinking distances. These in return lead to a small-world
structure, despite lack of large hubs or network wormhole edges. Because local
clustering coefficients in this network tend to be high, we proposed a new
measure called excess closure, that captures the overlap and local bridging
between edges of different types in individuals’ ego networks. The degree
distributions, observed shortest path lengths, and excess closure all imply
that it is crucial to distinguish between edges from different contexts (i.e.,
layers) in social networks, as this knowledge gives fundamental insights into
the social opportunity structure of a whole population. We have shown how
degree and closure evolve throughout an individual’s life, and how it differs
between demographic groups, and suggested a link between measurable network
opportunity structures and socio-economic status. Our work offers insights in
the interpretation of representative and population-scale networks derived
from administrative register data, and shows that a network science viewpoint
is useful in assessing social opportunity structures of different societal
groups. Future comparative work across countries will be required to unravel
the intricacies of the relation between social network structure and socio-
economic opportunities. We therefore hope our work inspires studies of
administrative register data in other countries that have similarly detailed
yet hitherto unexplored population-scale social network.

##  4 Data and Methods

In this section we discuss the construction of the network nodes and layers
and introduce the network measures used throughout the Results section.

###  4.1 Nodes

The data used in this paper is derived from multiple different registers of
Statistics Netherlands (CBS) in which nodes are all 17.2M people registered in
the Netherlands (in the Basisregistratie Personen, BRP) on October 1, 2018.

###  4.2 Layers

The available layers are Close Family $C$, Extended Family $E$, Household $H$,
Work $W$, School $S$, and Neighbors $N$. To avoid possible confusion of next-
door neighbor relationships with network neighbors, i.e., two nodes connected
by an edge, we are going to refer to relationships from the Neighbor layer as
Next-door neighbors, which preserves the layer abbreviation $N$.

#### Close family.

The close family layer denoted by $C$ has been created from the complete
digital parent-child register of the Netherlands and partner links. The latter
are derived from multiple different sources such as marriage registers or tax
declarations, and include married partners, fiscal partners and co-parents
having a common child. For details on the databases, and on the addition of
sibling links, we refer the reader to [38]. Only nodes and corresponding edges
that are actively registered on 1st October 2018 are left in the data. Using
the complete register for the derivation of family links means that even if
the parents are deceased, sibling relationships can be observed. Parent-child
edges are directed, but if A is a parent of B, then B is a child of A,
therefore, including both relationships creates an undirected close family
layer. Sibling and partner relationships are undirected by definition.

#### Extended family.

The extended family layer denoted by $E$ includes further family relationships
that are deterministically added using parent-child relations and partner
links. Extended family includes grandparent/grandchild, aunt-uncle/niece-
nephew, first cousin, mother-father-in-law/son-daughter-in-law, sister-
brother-in-law, stepparent/stepchild, and step-sibling relationships. All
edges in this layer are undirected.

#### Household.

A household is a group of people living and managing their lives together at
the same address. Using the assignment of people to households generated from
different registers (BRP, partnerships, and addresses), housemate links
connect all people assigned to the same household. The data includes the
category of institutional households, for example boarding houses, rescue
homes, jails, orphanages, or care homes. These are of course larger in size
than regular households.

#### Work.

The work layer contains links between colleagues that work for the same
employer. This is based on a process where every person is assigned an
employer of their major source of income. This means that each person is
assigned at most one employer. After that, for companies that have less than
100 employees, every colleague link between these employees is created.
However, if companies are larger than that, colleagues are sampled so that
each person is linked the geographically closest 100 colleagues. We consider
all created links to be undirected.

#### School.

The school layer includes information on where somebody goes to school, and
aggregates information from different agencies on education. The data
distinguishes between five levels of education in the Netherlands: elementary,
secondary, secondary special, vocational, and higher. We include connections
between people in the same school, year, location, and type of education. In
addition, the particular study programme is used in the case of universities.

#### Next-door neighbors.

Next-door neighbors are relationships based on the household addresses of
people. For each person, the 10 closest households are assigned as neighbors
within 50 meter distance of their own address. If there are more than 10 such
households, then they are randomly sampled down to 10. We assume all next-door
neighbor edges to be undirected.

###  4.3 Network measures

We introduce the abstract description of our network dataset following the
terminology set out by [17]. We represent the data as an undirected multilayer
graph $G=(V,E,L)$, where $V$ is the set of $n=|V|$ nodes, in our case, people
of the Netherlands ($n=17.2\mathrm{M}$). The set of undirected edges $E$
connects pairs of nodes according to the above described relationships:

$E\subseteq\left\\{\left(\left\\{u,v\right\\},\ell\right):u,v\in V,u\neq
v,\ell\in L\right\\},$

There are six layers $L=\\{C,E,H,S,W,N\\}$, and for the number of links $|E|$
we use $m$. Our multilayer graph is _node-aligned_ : nodes are identical
across all layers, and there are exclusively intra-layer edges. We can
represent the edges with the _reduced adjacency tensor_ $A_{uv\ell}$, that is
a generalization of the adjacency matrix, and encodes edges in the following
manner:

$A_{uv\ell}=\left\\{\begin{array}[]{ll}1&\mbox{if }u\mbox{ and }v\mbox{ are
connected in layer }\ell\in L,\\\ 0&\mbox{otherwise.}\end{array}\right.$ (1)

Because the network is undirected, for all $u,v\in V$ it holds that
$A_{uv\ell}=A_{vu\ell}$. We can also think about our network as an edge-
coloured multigraph, that allows multiple types of relationships between two
nodes. This representation can be viewed as a ”flattened” version of the
multilayer representation.

#### Degree.

The degree $k_{u,\ell}$ of a node $u\in V$ in a given layer $\ell\in L$ is
given by

$k_{u,\ell}=\sum_{v\in V}A_{uv\ell}.$ (2)

The total degree across all layers is

$k_{u}=\sum_{\ell\in L}k_{u,\ell}=\sum_{\ell\in L}\sum_{v\in V}A_{uv\ell},$
(3)

which counts edges that are present in more layers multiple times.

The number of unique neighbors connected to node $u$ is

$k^{\prime}_{u}=\left|\left\\{v\in V:\sum_{\ell\in
L}A_{uv\ell}>0\right\\}\right|.$ (4)

In data used to construct the family layer, there are missing parent-child
records. Parent-child links exist in the dataset only if both people have been
registered at some point in the databases after 1st October 1995. This means
that especially for older people, we miss (derived) family relations. If a
person’s parents were not alive in 1995, we are unable to retrieve the
siblings, nieces, nephews etc., because that parent-child link has not been
recorded in the register. But it also means that those younger relatives do
not have their aunts/uncles or cousins listed in the network. In addition, for
first generation migrants, parents are not registered in the Netherlands and
therefore not in our data. However, it is a realistic assumption that they
have less kins in the country they immigrated into.

With the above considerations, we create a weighing scheme that accounts for
the missing family links that affect 16.6% of the nodes. For first-generation
migrants, native people and second-generation migrants born after the 1995
register cutoff, we set the weights to 1 (36.4% of the nodes). For all other
people (46.9% of the nodes), we create age groups of 10 years, and set the
weight to the fraction of people with complete parent information in the group
and in the population. We create averages and distributions for degrees using
these weights. However, we cannot extend the same inference method for
embeddedness, tie range, or closure, that we measure based on the links given
in the data.

#### Embeddedness and tie range

We measure embeddedness by counting the number of triangles an edge is part of
[54, 55]. For an edge between nodes $u$ and $v$, this is formally given by

$embeddedness_{uv}=\sum_{\ell_{1},\ell_{2}\in L}\sum_{w\in
V}A_{uw\ell_{1}}\cdot A_{wv\ell_{2}}.$

Following [36], the tie range of an edge between nodes $u$ and $v$ is the
length of the second shortest path between $u$ and $v$. The shortest path
length would be 1, since there is a direct edge, thus, we can also reformulate
this distance as the shortest path between nodes $u$ and $v$ if the edge would
not exist between them. Tie range is also one less than the length of the
shortest cycle going through the edge. Any edge having an embeddedness larger
than zero has a tie range of 2, since they are part of at least one triangle.
Also, edges for which either end node has a degree of 1 have a tie range of
infinity.

We measure tie ranges for edges that have zero embeddedness, and for which
both end nodes have a degree larger than zero. We alternate two Breadth-First-
Search (BFS) algorithms starting from the two endpoints with the other
endpoint excluded from the search. On the first occasion when the frontier
node sets of the two BFS’s intersect, we stop the algorithm, and sum the
actual search depths of the two trees.

#### Components and shortest path lengths.

A path connecting two nodes in a multilayer network can consist of intra-layer
edges that connect nodes of different layers, and inter-layer edges that
connect nodes within a layer. Because our network is node-aligned, intra-layer
edges are trivial edges, and we do not count these towards path lengths.

A connected component of a graph is a subgraph, in which any two nodes are
connected by a path. A connected component might consist of a single node,
called an isolated node, in case the node’s degree is 0, i.e., it is not
connected to any other node in the network. The connected component with the
largest number of nodes is called the giant component.

We use the teexgraph library [56] to compute the measures in Table 2,
including the exact diameter, being the longest shortest path length, as well
as the average shortest path length, based on a sample of $1000\cdot n$ node
pairs in the network.

#### Closeness centrality.

Closeness centrality is defined as the inverse of the average distance of a
node $u$ to all other nodes:

$closeness\\_centrality_{u}=\frac{n-1}{\sum_{v\in V,u\neq v}d_{uv}},$ (5)

Here, $d_{uv}$ is the length of the shortest path between nodes $u$ and $v$.
The shortest path is calculated using all edges from all layers. We employ a
fast approximation for a 0.03% node sample of the largest connected component
[57].

#### Local clustering coefficient.

The the local clustering coefficient of a node $u$ is given by:

$c\\_local_{u}=\frac{\left|\left\\{(u,v,w):\left\\{u,v\right\\},\left\\{v,w\right\\},\left\\{u,w\right\\}\in
E^{\prime}\right\\}\right|}{k^{\prime}_{u}\cdot(k^{\prime}_{u}-1)},$

which is the number of closed triangles over the number of neighbor pairs
around node $u$, if $E^{\prime}=\left\\{\left\\{u,v\right\\}:\sum_{\ell\in
L}A_{uv\ell}>0\right\\}$ (essentially omitting edge layer identifiers for
readability), meaning that there is at least one edge between nodes $u$ and
$v$.

#### Excess closure.

Just measuring the local clustering coefficient would capture mostly the share
of intra-layer triangles, since we expect edges from the same layer to form
clustered structures around the egos. Kinship networks up to the second degree
are the merges of extremely highly clustered structures through partnerships.
Household, school and work relationships are complete networks because they
come from bipartite projections, and next-door person relationships are
clustered because of spatial proximity. Triangles are considered from the
viewpoint of the ego in an ego network, and triangle types are characterized
by a triplet $\\{\ell_{1},\ell_{2},\ell_{3}\\}$, where $\ell_{1}$ and
$\ell_{3}$ correspond to layers of edges to two selected neighbors of the ego
(ordered by ascending node labels), and $\ell_{2}$ corresponds to the layer
type of the third, closing edge of the triangle.

Figure 5: Three sample ego networks illustrating the concept and the
calculation of excess closure. Ego is the central node shaded darker, edges
come from three layers color-coded by red, blue, and yellow. The values of
$T\\_{pure}$, $T\\_{unique}$, $c\\_{actual}$, $c\\_{pure}$, $P$, and
$c\\_{excess}$ are in the table below the figures.

For this, we first sum up the number of triangles $T_{\ell\ell\ell}$ around an
ego for which all three edges come from the same layer $\ell$:

$T\\_{pure}=\sum_{\ell\in L}T_{\ell\ell\ell}.$

Note that this definition might count some node triplets twice if they exist
in multiple layers. Next, we divide this sum by the number of possible
different alter tie pairs around the ego $P$. This is different from the
number of unique neighbor pairs, since as in Figure 5b, an alter can be
connected to the ego in multiple layers, thus, the same neighbor pair can be
formed using only intra-layer edges, or also edges from different layers. But
$P$ also accounts for the fact that a neighbor pair cannot be formed from two
edges that lead to the same alter.

Formally, for a node $u$, the number of alter tie pairs $P_{u}$ can be
calculated as

$P_{u}={k_{u}\choose 2}-\sum_{v\in{neighbors_{u}}}{\sum_{\ell\in
L}A_{uv\ell}\choose 2},$

where $k_{u}$ is the total degree of node $u$, and $neighbors_{u}=\left\\{v\in
V:\sum_{\ell\in L}A_{uv\ell}>0\right\\}$.

We define the pure clustering coefficient $c\\_{pure}=T\\_{pure}/P$, which
would actually be a theoretical minimum clustering coefficient, if there would
be no multi-edges between the ego and any of its neighbors, and there would
not be any cross-edges between neighbors that are linked by edges from
different layers. To put it differently, it is the case when the ego’s
neighbors of different layers are never overlapping and never linked to each
other.

Then, we calculate the number of unique closed triangles $T\\_{unique}$ around
the ego by listing each node triplet only once, even if there are multiple
possible layer combinations that close wedges between the same nodes. This
would include to some extent the previously calculated $T\\_{pure}$ triangle
count, and in addition, all other triangles of mixed layer types. We divide
this number also by the number of possible neighbor pairs $P$, thus
calculating the clustering coefficient $c\\_{actual}=T\\_{unique}/P$. We
obtain the excess closure $c\\_{excess}$ by comparing where this clustering is
between $c\\_{pure}$ and the theoretical maximum clustering of 1:

$c\\_{excess}=\frac{c\\_{actual}-c\\_{pure}}{1-c\\_{pure}}.$ (6)

Figure 5 shows the concept for three example egos with edges from three
different layer types. The table below the figure enumerates the number of
unique closed triangles $T\\_{unique}$, $T\\_{pure}$, $P$, and then
$c\\_{actual}$, $c\\_{pure}$, and $c\\_{excess}$.

From the viewpoint of the ego, excess closure relates to the number of closed
triangles that consist of edges from different layers. However, we still
cannot be sure whether this measure captures overlap or cohesion in the ego’s
different opportunity structures, or it actually captures bridges between two
very different social circles, such as a family member and a colleague being
neighbors of each other. Having a multilayered social network, we can actually
characterize the triangle types from this angle. Each triangle can contribute
to cohesion, and/or to bridging. Cohesion happens when the triplet
$\\{\ell_{1},\ell_{2},\ell_{3}\\}$ encoding the triangle has
$\ell_{1}=\ell_{3}$ from the very same layers. For example, in the case when
the ego’s two family members are also household members of each other,
triangle $\\{F,H,F\\}$, the edge from layer $H$ does not bridge multiple
social contexts of the ego. Bridging is characterized by cases when
$\ell_{1}\neq\ell_{2}$ and $\ell_{2}\neq\ell_{3}$, for example, in the
triangle $\\{S,N,W\\}$, where the ego’s school tie and work tie are neighbors
of each other. But in some sense, this type of closure also causes the ego’s
social circles to “overlap”, therefore, it still counts towards excess
closure, too.

The measure of _excess closure_ introduced for capturing clustering in the
opportunity structures belonging to the different network layers is related to
multiple other clustering coefficient definitions from the literature. [58]
draw attention to the fact that the multilayer nature of triadic closure has
to be taken into account when calculating clustering. Partial clustering
coefficients are introduced in [59] to capture the fraction of triangles with
edges from 1, 2, or 3 layers. The paper of [60] proposes so-called layer-
decomposed local clustering coefficients, based how many layers edges of
multilayer three-step walks come from. In their notation, our excess closure
can be written as

$c\\_{excess}=\frac{c^{(2)}+c^{(3)}}{1-c^{(1)}},$

where $c^{(i)}$ corresponds to the fraction of walks consisting of $i$
different types of inter-layer edges.

## Ethical statement

The register dataset used for creating the Dutch population-scale network is
pseudonymized before research access, and it does not contain personal
information such as address or date of birth. All data storage and analysis
was done within the secure server environments of Statistics Netherlands.
Further details about ensuring data security and privacy are described in
[49].

## Acknowledgements

We are thankful for the collaboration with Statistics Netherlands (CBS)
colleagues Jan van der Laan, Edwin de Jonge, and Gert Buiten. The project has
been funded by Platform Digitale Infrastructuur Social Sciences and Humanities
(http://www.pdi-ssh.nl). We would also like to thank the POPNET team
(https://www.popnet.io), in particular Yuliia Kazmina, for helpful suggestions
and discussions. Moreover, we are grateful for insightful comments from Márton
Karsai and Matteo Magnani on early versions of the manuscript.

### Conflicts of interest and data and code availability

The authors declare no conflicts of interest. The whole population network
file for 2018 is included in the Statistics Netherlands Microdata Catalogue as
a beta product and is available for access through the Statistics Netherlands
Microdata Services Remote Access Environment (RA); see [38]. The Python
library used as a basis for the analysis in this paper is available via
GitHub: https://github.com/popnet-io/popnet_mln.

### Author contributions

E.B., E.H., and F.T. conceptualized the study, developed the methodology, and
wrote the manuscript. E.H. and F.T. acquired funding, supervised and
administered the project. E.B. and F.T. developed the software. E.B. performed
the analysis and the investigations. E.B. visualized the results. All authors
reviewed and approved the manuscript.

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#  A new proof of a theorem of Tietze

Daniel Duverney  110, rue du chevalier français, 59000 Lille, France
<EMAIL_ADDRESS>and  Iekata Shiokawa  13-43, Fujizuka-cho, Hodogaya-
ku, Yokohama 240-0031, Japan<EMAIL_ADDRESS>
(Date: March 5, 2022)

###### Abstract.

We give a new proof of Tietze Theorem on the convergence of infinite semi-
regular continued fractions.

###### Key words and phrases:

Tietze Theorem, Convergence of semi-regular continued fractions.

###### 1991 Mathematics Subject Classification:

11A55

The purpose of this note is to give a new proof of Theorem 1 below, known as
Tietze Theorem [4]. By Theorem 1, any infinite semi-regular continued fraction
([5],[4],[2]) is convergent.

######  Theorem 1.

Assume that the infinite continued fraction

(1)
$\alpha:=b_{0}+\frac{a_{1}}{b_{1}}\genfrac{}{}{0.0pt}{}{{}}{+}\frac{a_{2}}{b_{2}}\genfrac{}{}{0.0pt}{}{{}}{+\cdots+}\frac{a_{n}}{b_{n}}\genfrac{}{}{0.0pt}{}{{}}{+\cdots}$

satisfies the following conditions:

(2) $a_{n}\in\left\\{-1,1\right\\},\qquad
b_{n}\in\left[1,+\infty\right[,\qquad b_{n}+a_{n+1}\geq 1\qquad\left(n\geq
1\right).$

Then $\alpha$ is convergent.

Tietze original proof of Theorem 1 (1911) rested on a geometrical
representation of the convergents $p_{n}/q_{n}$ as points $(p_{n},q_{n})$ in
the plane [5]. Shortly after (1913), Oskar Perron gave a different proof by
using continuants [4]. Almost 100 years later, Keith Matthews challenged Alan
Offer to find a proof of Theorem 1 avoiding the use of continuants. In [3],
Offer succeeded in doing so in the proof of Lemma 3 below, but not in the rest
of the proof. In this note we develop Offer’s ideas and give a new simple
proof of Tietze Theorem without using at all continuants.

Let $\alpha$ be the continued fraction defined by (1). As usual, we define for
$n\geq 1$

(3)
$\left\\{\begin{array}[c]{lll}p_{-1}=1,&p_{0}=b_{0},&p_{n}=b_{n}p_{n-1}+a_{n}p_{n-2},\\\
q_{-1}=0,&q_{0}=1,&q_{n}=b_{n}q_{n-1}+a_{n}q_{n-2}.\end{array}\right.$

It is well known that

(4)
$b_{0}+\frac{a_{1}}{b_{1}}\genfrac{}{}{0.0pt}{}{{}}{+}\frac{a_{2}}{b_{2}}\genfrac{}{}{0.0pt}{}{{}}{+\cdots+}\frac{a_{n}}{b_{n}}=\frac{p_{n}}{q_{n}}\qquad\left(n\geq
1\right),$

and an easy induction using (3) shows that

(5) $p_{n}q_{n-1}-p_{n-1}q_{n}=\left(-1\right)^{n-1}a_{1}a_{2}\cdots
a_{n}\qquad\left(n\geq 1\right),$

which yields immediately

(6)
$\frac{p_{n}}{q_{n}}=b_{0}+\sum_{k=1}^{n}\frac{\left(-1\right)^{k-1}a_{1}a_{2}\cdots
a_{k}}{q_{k-1}q_{k}}.$

First we prove Theorem 1 in the following very special case, connected to
negative continued fractions [2].

######  Lemma 1.

Assume that $a_{n}=-1$ for all $n\geq 1.$ Then the continued fraction (1) is
well defined and convergent.

###### Proof.

We follow [1]. By (3) and (2) we have for $n\geq 0$

$q_{n+1}-q_{n}=\left(b_{n+1}-1\right)q_{n}-q_{n-1}\geq
q_{n}-q_{n-1}\geq\ldots\geq q_{0}-q_{-1}=1.$

Hence $q_{n}\geq n+1$ for all $n\geq 0,$ which proves that the continued
fraction (1) is well defined and convergent by (6). ∎

We now return to the general case. Since $q_{n+1}<q_{n}$ when
$\left(a_{n+1},b_{n+1}\right)=\left(-1,1\right),$ the above proof cannot be
used.

######  Lemma 2.

Let $q_{n}$ be defined by (3). We have for $n\geq 0$

(7) $q_{n}\geq 1,\quad q_{n}+a_{n+1}q_{n-1}\geq 1.$

###### Proof.

By following Offer [3, Lemma 1], we first observe that

(8) $q_{n+1}+a_{n+2}q_{n}=\left(b_{n+1}+a_{n+2}\right)q_{n}+a_{n+1}q_{n-1}\geq
q_{n}+a_{n+1}q_{n-1}\quad\left(n\geq 0\right).$

Hence by induction $q_{n}+a_{n+1}q_{n-1}\geq q_{0}+a_{1}q_{-1}=1$ for all
$n\geq 0,$ which yields

(9) $q_{n+1}=b_{n+1}q_{n}+a_{n+1}q_{n-1}\geq q_{n}+a_{n+1}q_{n-1}\geq
1\quad\left(n\geq 0\right).$

Therefore (7) holds. ∎

Lemma 2 shows that the convergents of the continued fraction $\alpha$ defined
by (1) and (2) are well defined since $q_{n}\geq 1$ for all $n\geq 0.\vskip
3.0pt plus 1.0pt minus 1.0pt$

The following lemma is proved in [4] by using continuants and in [3] without
using them. We give here a new proof, also without using continuants.

######  Lemma 3.

Let $q_{n}$ be defined by (3). Then $\lim_{n\rightarrow\infty}q_{n}=\infty.$

###### Proof.

As observed in [3, Formula (3)], it is clear by (8) and (9) that

(10) $q_{n}\geq q_{m}+a_{m+1}q_{m-1}\geq 1\qquad\left(n>m\geq 1\right).$

Assume that there exists $A\geq 1$ such that $1\leq q_{n}\leq A$ for
infinitely many $n.$ Then there exists $B\geq 1$ and an increasing sequence
$n_{k}$ such that $\lim_{k\rightarrow\infty}q_{n_{k}}=B.$ Let $C\geq 1$ be the
least real number having this property. By (10) with $n=n_{k+1}$ and
$m=n_{k}$, we have for all large $k$

$C+\frac{1}{3}\geq C-\frac{1}{3}+a_{n_{k}+1}q_{n_{k}-1}\quad\text{and}\quad
C+\frac{1}{3}+a_{n_{k}+1}q_{n_{k}-1}\geq 1.$

This yields immediately $a_{n_{k}+1}=-1,$ and therefore $q_{n_{k}-1}\leq
C-2/3$ for all large $k.$ This contradicts the minimality of $C$ and proves
Lemma 3. ∎

######  Lemma 4.

Define for $n\geq 0$ and $k\geq 1$

(11)
$x_{n,k}:=\frac{a_{n+1}}{b_{n+1}}\genfrac{}{}{0.0pt}{}{{}}{+}\frac{a_{n+2}}{b_{n+2}}\genfrac{}{}{0.0pt}{}{{}}{+\cdots+}\frac{a_{n+k}}{b_{n+k}}.$

Then for $n\geq 0$ and $k\geq 1$

(12) $0<x_{n,k}\leq 1\text{ if }a_{n+1}=1,\qquad-1\leq x_{n,k}<0\text{ if
}a_{n+1}=-1.$

###### Proof.

The proof follows [2, Proposition 3] and is by induction on $k.$ If $k=1,$
then $x_{n,1}=a_{n+1}/b_{n+1}$ and so (2) implies (12) for $k=1$ and any
$n\geq 0.$ Assume that (12) holds for some $k\geq 1$ and any $n\geq 0.$ Then
by (11) we have

(13) $x_{n,k+1}=\frac{a_{n+1}}{b_{n+1}+x_{n+1,k}},$

where for any $n\geq 0$

(14) $0<x_{n+1,k}\leq 1\text{ if }a_{n+2}=1,\qquad-1\leq x_{n+1,k}<0\text{ if
}a_{n+2}=-1.$

If $b_{n+1}\geq 2,$ then by (14)

(15) $b_{n+1}+x_{n+1,k}\geq 1.$

Otherwise, we have $b_{n+1}<2$ and therefore $a_{n+2}=1$ by (2), and so we
find again (15) by the first inequality in (14). Hence (13) and (15) imply

$0<x_{n,k+1}\leq 1\text{ if }a_{n+1}=1,\qquad-1\leq x_{n,k+1}<0\text{ if
}a_{n+1}=-1,$

which proves (12) by induction. ∎

######  Lemma 5.

Let $x_{n,k}$ be defined by (11). Then

(16)
$\frac{p_{n+k}}{q_{n+k}}=\frac{p_{n}+x_{n,k}p_{n-1}}{q_{n}+x_{n,k}q_{n-1}}\qquad\left(n\geq
0,k\geq 1\right).$

###### Proof.

By induction on $n.$ Clearly (16) is true for $n=0$ and every $k\geq 1$ since

$\frac{p_{0}+x_{0,k}p_{-1}}{q_{0}+x_{0,k}q_{-1}}=b_{0}+\frac{a_{1}}{b_{1}}\genfrac{}{}{0.0pt}{}{{}}{+}\frac{a_{2}}{b_{2}}\genfrac{}{}{0.0pt}{}{{}}{+\cdots+}\frac{a_{k}}{b_{k}}=\frac{p_{k}}{q_{k}}.$

Assume that (16) is true for a given $n\geq 0$ and all $k\geq 1.$ By (13), we
see that

$\frac{p_{n+1+k}}{q_{n+1+k}}=\frac{p_{n+k+1}}{q_{n+k+1}}=\frac{p_{n}+x_{n,k+1}p_{n-1}}{q_{n}+x_{n,k+1}q_{n-1}}=\frac{p_{n}\left(b_{n+1}+x_{n+1,k}\right)+a_{n+1}p_{n-1}}{q_{n}\left(b_{n+1}+x_{n+1,k}\right)+a_{n+1}q_{n-1}},$

which proves (16) by induction.∎

From Lemmas 5 and 4 we deduce immediately that

(17)
$\left|\frac{p_{n+k}}{q_{n+k}}-\frac{p_{n}}{q_{n}}\right|\leq\frac{1}{q_{n}\left|q_{n}+x_{n,k}q_{n-1}\right|}\qquad\left(n\geq
0,k\geq 1\right).$

Now we prove Tietze Theorem. By Lemma 1, we can assume that there exist
infinitely many $m$ such that $a_{m+1}=1.$ For every large $n,$ let $m=m(n)$
be the greatest integer less than $n$ such that $a_{m+1}=1.$ Clearly $m$ tends
to infinity with $n.$ For every large $n$ and every $k\geq 1,$ we have by (17)

$\displaystyle\left|\frac{p_{n+k}}{q_{n+k}}-\frac{p_{n}}{q_{n}}\right|$
$\displaystyle\leq\left|\frac{p_{n+k}}{q_{n+k}}-\frac{p_{m}}{q_{m}}\right|+\left|\frac{p_{n}}{q_{n}}-\frac{p_{m}}{q_{m}}\right|$
$\displaystyle\leq\frac{1}{q_{m}\left|q_{m}+x_{m,n+k-m}q_{m-1}\right|}+\frac{1}{q_{m}\left|q_{m}+x_{m,n-m}q_{m-1}\right|}.$

However $x_{m,n+k-m}>0$ and $x_{m,n-m}>0$ by Lemma 4. Hence

$\left|\frac{p_{n+k}}{q_{n+k}}-\frac{p_{n}}{q_{n}}\right|\leq\frac{2}{q_{m\left(n\right)}^{2}}$

for all large $n$ and all $k\geq 1.$ As
$\lim_{n\rightarrow\infty}q_{m\left(n\right)}=\infty$ by Lemma 3,
$p_{n}/q_{n}$ is a Cauchy sequence, which proves Tietze Theorem.

## References

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