In my previous article, titled “Time-Networks“, I introduced a new framework for analyzing social networks: time-network analysis. This article extends the concept, providing additional tools for understanding the time-network of an organization.
Before diving into the new material, here is a summary of what time-networks are:
Time-Network Basics
Time spent with others is the fundamental building block of relationships. Without it, there would be no language, no culture, no world. At the same time, time spent alone is also critical for one to have a sense of coherent self-identity. Combined, the time we spend with others and the time we spend alone makes up a large part of who we are. This fundamental structure can be described by the time-network.
The time-network is how one’s time and attention is split up and shared with others. It is comprised of each person’s self-time (time spent disconnected from others) and mutual-time (time spent in joint attention with others), such that the sum of one’s self-time and mutual-time with every other actor (person in a network) is their tenure, i.e. the total time they spent. There is no double-counting of time in the time-network; at any given moment, one is either adding to their self-time, or to their mutual-time with someone else. The time-network shows how one has chosen to spend their limited time in life or in an organization.
The time-network is represented with a matrix, where each value in the matrix represents the mutual-time between actors, where the rows and columns represent actors. The diagonal of the matrix, where the row and column indices are the same, represents each actor’s self-time. The rows (or columns) of the time-network matrix are called each actor’s time-network profile. For example, if there were three actors, actors A, B, and C, actor A’s profile would be a list of their self-time, their mutual-time with actor B, and their mutual-time with actor C. Similarly, actor B’s profile would be their mutual-time with actor A, their self-time, and their mutual-time with actor C. This is illustrated in Figure 1, which shows the time-network of actors A, B, and C, with the time-network profile of actor A circled in red.
Figure 1: Time-network basics
Time-network profiles are like the fingerprint of one’s social life. They contain the total history of how one has spent their time in life or in an organization. Comparing time-network profiles between actors, therefore, is like seeing how much they have in common. One measure for this is time-network overlap, or how much two time-network profiles intersect. For example, between actors A and B in Figure 1, their overlap would be “the minimum of actor A’s self-time and actor A & B’s mutual-time” plus “the minimum of actor A & B’s mutual-time and actor B’s self-time” plus “the minimum of actor A & C’s mutual-time and actor B & C’s mutual-time”. Overlap is constrained by the minimum of both actors’ tenures: the overlap between actors A and B is less than or equal to the tenure of A or the tenure of B, whichever is smaller.
For brevity, the time-network matrix is represented mathematically by the Greek letter \(\mathbf{\tau}\) (“tau”), and arbitrary actors in the network are represented by \(i\) and \(j\). So, the mutual-time between actors \(i\) and \(j\) is \(\tau_{ij}\), and the self-time of actor \(i\) is \(\tau_{ii}\). Tenure, which is the sum of an actor’s time-network profile, is represented by \(T_i\) and calculated according to Eq. (1) in the previous article. Time-network overlap, the sum of minimums between two actors’ time-network profiles, is represented by the Greek letter \(\Omega_{ij}\) (“Omega”) and calculated according to Eq. (2) in the previous article.
Thus the time-network concept is introduced. It is a new framework that can help one better understand relationships, organizational networks, and societal structures. It differs from traditional social network analysis in that it does not directly measure relationships; rather, it measures the underlying structure of time-investment that makes relationships possible (measurement methods and example applications are provided in the previous article).
The two main time-network analysis metrics provided thus far for measuring relationships are mutual-time and overlap. Both are measured in units of time. While these metrics are useful for describing the relative depth of relationships and the embeddedness of the network around them, the metrics are effectively unbounded (up to one’s lifespan). For networks such as corporations that have both seasoned employees and new hires, it may be difficult to interpret this data with such a large range of potential values.
So, to aid in interpretation, this article extends the time-network methodology, providing additional tools and measures for understanding the hidden structure of an organization. Specifically, the metrics of similarity, centrality, and density are introduced.
Time-Network Similarity
The previous article provided two measures that describe a relationship between two actors: mutual-time and overlap. Mutual-time is the portion of each actors’ time that is attributed to the other, while overlap is how much their time-network profiles intersect. These measures give an idea of the depth of investment and embeddedness that exists between two people. However, depth by itself does not tell the whole story: 100 hours of overlap between new hires is much more significant than 100 hours of overlap between seasoned employees. Overlap relative to tenure is what gives it its significance, and this significance can be described with a similarity measure.
A similarity measure is a value, typically between 0 and 1, that describes how similar two things are.1 There are a large variety of similarity metrics to choose from, but the one that is the focus for this article is the Jaccard index.
The Jaccard index is a commonly used similarity measure for comparing two sets of data.2 It is best explained with Venn diagrams: it is the size of the intersection divided by the size of the union, as shown in Figure 2.
Time-network overlap, as explained in the main article, is like the intersection of two time-network profiles. Thus, the Jaccard index between actors \(i\) and \(j\) can be calculated as shown in Eq. \eqref{Jaccard} below:
$$J_{ij} = \frac{\Omega_{ij}}{T_i + T_j – \Omega_{ij}} \tag{1} \label{Jaccard}$$
The Jaccard index is constrained by the ratio of the smallest tenure to the largest tenure between two actors, as in \(J_{ij} \le \frac{\min(T_i,T_j)}{\max(T_i,T_j)}\). So, it will not only measure the degree to which actors have mutual connections, but it will also reflect the difference in age/tenure between them.
For example, if actor A had 100 hours of overlap with actor B, actor A had 200 total hours, and actor B had 300 total hours, the ratio between their tenures would be 2/3, or 67%, while their Jaccard index would be 100/(200 + 300 – 100) = 1/4, or 25%.
Excel Application
To calculate the Jaccard index in Excel, apply the following LAMBDA function to the time-network and overlap matrices:
=LAMBDA(tau,Omega,Omega/(BYROW(tau,SUM)+BYCOL(tau,SUM)-Omega))UCINET Application
In UCINET, there are two ways that the Jaccard index can be calculated. One way is to use the built-in function “Tools → Similarities & Distances → Valued Jaccard” with “Diagonal values are valid” checked, as shown in the screenshot below:
The other method is to run the following code in the command-line interface:
J = divby(Omega,add(outer(rowsums(tau),sum),neg(Omega)))Using the example dataset in the previous article, the Jaccard index between all actors is as shown in Table 1 below, and the similarity scores of actor (or in this case employee) E1 is shown in Figure 3.
Table 1: Jaccard index
Notice that employees E1 and E7, which were compared in the previous article and shown to be quite disconnected from each other, have a corresponding low Jaccard index of 0.16, or 16%. Employees E1 and E3, on the other hand, which were shown to have very similar time-network profiles, have a Jaccard index of 68%. Also note that the diagonal of the matrix, each actor’s self-similarity, is always 1.00, or 100%. Any actors with a similarity measure of 100% would have identical time-network profiles, and thus would be, in terms of the time-network, structurally equivalent.
The Jaccard index normalizes the time-network overlap matrix; while the values in the time-network and overlap matrices can vary wildly, the Jaccard index only ranges from zero to one. A Jaccard index of 0% corresponds to zero time-network overlap, while a Jaccard index of 100% corresponds to time-network structural equivalence. A large Jaccard index between two actors means that overlap comprises a large part of both actors’ tenures. Therefore, it can be interpreted as an estimate of whether a mutual relationship exists.
Time-Network Centrality
So far, all the time-network measures that have been introduced have been dyadic in scope: mutual-time, overlap, and similarity are all attributes of a relationship between two actors. While these are useful for understanding the interpersonal dynamics within an organization, it is not as useful for things like performance evaluations and incentive programs. Paychecks go to individuals, not to the bond that exists between people.
So, if the time-network is to be incorporated into performance evaluations and incentive systems, a measure is needed that ranks actors in terms of total connectivity. In network science, this is called a centrality measure.
One of the most commonly used centrality measures is degree centrality, or how many ties/relationships a single actor has.3 Degree centrality ranges from \(0\) to \(N-1\), where a score of zero represents a completely isolated individual, and a score of \(N-1\) represents an individual who is connected to everyone. The time-network equivalent is as follows:
Let \(C_{\tau,i}\) represent the time-network centrality of actor \(i\), calculated as shown in Eq. \eqref{centrality} below:
$$ C_{\tau,i} = \frac{1}{\mu_T} \left(\sum_{j=1}^N \Omega_{ij} – T_i\right) \tag{2} \label{centrality} $$
where \(\mu_T\) is the average tenure of all actors in the network, calculated as shown in Eq. \eqref{age} below:
$$ \mu_T = \frac{1}{N} \sum_{i=1}^N T_i \tag{3} \label{age} $$
So, an actor’s time-network centrality, \(C_{\tau,i}\), is the sum of their overlap with everyone (except themselves), divided by the average tenure.4 It ranges from \(0\) to \(N-1\), the same as degree centrality (see Appendix for proof).
Excel Application
To calculate time-network centrality measures in Excel, apply the following LAMBDA function to the time-network and overlap matrices:
=LAMBDA(tau,Omega,BYROW(Omega-tau,SUM)/AVERAGE(BYROW(tau,SUM)))UCINET Application
To calculate time-network centrality measures in UCINET,5 run the following code in the command-line interface, where “tau” is the time-network matrix and “Omega” is the overlap matrix.
C_tau = divby(rowsums(add(Omega,neg(tau))),wavg(rowsums(tau)))For the example in the previous article, with seven actors (and four events), the centrality scores for each actor (compared to tenure) are as shown in Figure 4 below:
Note that the actors with the most tenure are not necessarily the most central individuals. This is because while time-network overlap is constrained by one’s total number of hours, it is also dependent on whether that time was spent alone or with others. The most central actors in the time-network are those who have not only spent a lot of time in the group, but who have also spent that time in a way that maximizes overlap with everyone.
Time-Network Density
Another common metric for assessing a network is network density. Network density involves counting up the number of edges (relationships) in a network and dividing by the total number of possible edges. For an undirected graph (as in, the relationships are reciprocated) of \(N\) nodes and a total of \(E\) edges, density is calculated as shown in Eq. \eqref{density}:
$$ D = \frac{2 E}{N (N-1)} \tag{4} \label{density} $$
This ranges from 0 to 100%, where 0% density represents a completely sparse graph with no edges at all, while a density of 100% represents a completely dense graph, where everyone is connected to everyone. See Figure 5 for what networks at various density levels look like.
Network density is a good metric for understanding, at a high-level, what the network is like. Low density networks are likely to be quite fragmented and disconnected, while high density networks may be overly redundant and chaotic. There is no “right” level of network density, every network is unique after all, but typically, a moderately dense network is most efficient.6
So what is the time-network equivalent of network density? Consider Eq. \eqref{centrality}. It ranges from \(0\) to \(N-1\), and applies to all \(N\) actors. Therefore, the sum of all time-network centrality measures ranges from \(0\) to \(N(N-1)\), the same as the denominator in Eq. \eqref{density}. So, let time-network density be defined as follows in Eq. \eqref{Dtau}:
$$ D_\tau = \frac{\sum_{i=1}^N C_{\tau,i}}{N (N-1)} \tag{5} \label{Dtau} $$
This density measure ranges from 0 to 100%, the same as the traditional network density measure. A density of 0% represents a network of actors who have no overlap with each other at all, and a density of 100% represents a network of actors who have equal time spent with everyone.
Excel Application
To calculate time-network density in Excel, apply the following LAMBDA function to the column of time-network centrality values:
=LAMBDA(C_tau,SUM(C_tau)/(ROWS(C_tau)*(ROWS(C_tau)-1)))UCINET Application
To calculate time-network density in UCINET, run the following code in the command-line interface, where “C_tau” is the vector of time-network centrality values and “tau” is the time-network matrix.
D_tau = divby(total(C_tau),total(dichot(tau GE 0)))The density of the network provided in the previous article, using the centrality scores in Figure 4, is approximately 50%, as demonstrated in the example calculation below:
$$ D_\tau = \frac{3.4 + 2.6 + 2.7 + 2.4 + 3.7 + 3.8 + 2.6}{7 (7-1)} $$
$$ D_\tau \approx \frac{21}{42} = 50\% $$
For convenience, the example is provided in an Excel file at the bottom of this article.
Conclusion
Time is the most precious resource that we have to spend, and who we spend it with says a lot about who we are. This obvious fact of life is formalized in the structure of the time-network, which shows how one’s total time in life or in an organization is divided up and shared with others.
The time-network is a matrix, where the rows represent each actor’s time-network profile. Each actor’s profile is comprised of their self-time alone and their mutual-time with others, such that A’s mutual-time with B is equal to B’s mutual-time with A. Additionally, the sum of each actor’s profile equals their total time, or tenure.
The previous article introduced this concept, elucidated its theoretical underpinnings, and provided a measurement method appropriate for HR professionals. This article extended the methodology by providing additional measures for analyzing and interpreting time-networks: similarity, centrality, and density.
Appendix
The limits to time-network centrality, defined in Eq. \eqref{centrality}, are \(0 \le C_{\tau,i} \le N-1\). While the lower bound is trivial (an actor with no overlap with other actors will have a centrality of zero), the upper limit is less obvious. The proof is as follows:
Let actors be sorted in order of increasing tenure, such that \(T_1 \le T_2 \le … \le T_N\). Then, because time-network overlap is constrained by the minimum tenure between both actors (i.e. \(\Omega_{ij} \le \min(T_i,T_j)\)), for every \(\Omega_{ij}\) where \(i < j\), \(\Omega_{ij} \le T_i\), and for every \(\Omega_{ij}\) where \(i > j\), \(\Omega_{ij} \le T_j\). Therefore, for any given actor \(i\), their time-network centrality \(C_{\tau,i}\) is bounded by the following:
$$ C_{\tau,i} \le \frac{\sum_{j=1}^{i-1} T_j + (N-i) T_i}{\mu_T} $$
Furthermore, because of the selected order of actors in increasing tenure, the following inequality holds:
$$ \frac{\sum_{j=1}^{i-1} T_j + (N-i) T_i}{\mu_T} \le \frac{\sum_{j=1}^{N-1} T_j}{\mu_T} $$
Additionally, because \(\mu_T\) is greater than or equal to the average of the \(N-1\) smallest tenures, the following is true:
$$ \frac{\sum_{j=1}^{N-1} T_j}{\mu_T} \le \frac{(N-1) \mu_T}{\mu_T} $$
Therefore, \(C_{\tau,i} \le N-1\).
Q.E.D.
- Avivit Levy et al., “A Guide to Similarity Measures,” 2408.07706v1, preprint, arxiv, August 7, 2024, https://arxiv.org/html/2408.07706v1. ↩︎
- Joshua Noble, “What Is Jaccard Similarity?,” IBM Think, n.d., https://www.ibm.com/think/topics/jaccard-similarity. ↩︎
- Stephen P. Borgatti et al., Analyzing Social Networks, 3rd ed. (Sage Publications, 2024), 163. ↩︎
- A measure with the same bounds could be trivially obtained by dividing an actor’s total overlap by their own tenure, but this would make it possible for a brand new employee to be the ranked as the most central individual on day one. Dividing by the average tenure ensures that the centrality score reflects not just how connected one is, but how long one has been involved. ↩︎
- Stephen P. Borgatti et al., Ucinet 6 for Windows: Software for Social Network Analysis., Analytic Technologies, released 2002, Windows. ↩︎
- Alex Derr, “Using Network Density to Evaluate and Optimize Collaboration Intensity,” Visible Network Labs, November 13, 2024, https://visiblenetworklabs.com/2024/11/13/using-network-density-to-evaluate-and-optimize-collaboration-intensity/. ↩︎