An algebraic approach to temporal network analysis based on temporal quantities

Abstract

In a temporal network, the presence and activity of nodes and links can change through time. To describe temporal networks we introduce the notion of temporal quantities. We define the addition and multiplication of temporal quantities in a way that can be used for the definition of addition and multiplication of temporal networks. The corresponding algebraic structures are semirings. The usual approach to (data) analysis of temporal networks is to transform the network into a sequence of time slices—static networks corresponding to selected time intervals and analyze each of them using standard methods to produce a sequence of results. The approach proposed in this paper enables us to compute these results directly. We developed fast algorithms for the proposed operations. They are available as an open source Python library TQ (Temporal Quantities) and a program Ianus. The proposed approach enables us to treat as temporal quantities also other network characteristics such as degrees, connectivity components, centrality measures, Pathfinder skeleton, etc. To illustrate the developed tools we present some results from the analysis of Franzosi’s violence network and Corman’s Reuters terror news network.

Appendix: Algorithms

1.1 Clustering coefficient

Algorithm 4 presents an algorithm for computing different types of temporal clustering coefficient. The function $\textit{nRows}(\mathbf {A})$ returns the size (number of rows) of matrix $\mathbf {A}$. The function $\textit{VecConst}(n,v)$ constructs a temporal vector of size n filled with the temporal quantity v. The function $\textit{MatBin}(\mathbf {A})$ transforms all values in the triples in the matrix $\mathbf {A}$ to 1. The function $\textit{MatSetDiag}(\mathbf {A},c)$ sets all the diagonal entries of the matrix $\mathbf {A}$ to the temporal quantity c. The function $\textit{MatSym}(\mathbf {A})$ makes the transformation $\mathbf {S} = \mathbf {A}\oplus \mathbf {A}^T$.

Functions $\textit{VecSum}$ and $\textit{VecProd}$ implement a component wise composition of temporal vectors:

$$\begin{aligned} \textit{VecSum}(a,b) = [a_i \oplus b_i,\ i = 1, \ldots , n] \end{aligned}$$

and

$$\begin{aligned} \textit{VecProd}(a,b) = [a_i \odot b_i,\ i = 1, \ldots , n]. \end{aligned}$$

Similarly $\textit{VecInv}(a) = [\textit{invert}(a_i),\ i = 1, \ldots , n]$ in the combinatorial semiring; where

$$\begin{aligned} \textit{invert}(a) = [ (s,f,1/v)\, \mathbf{for}\, (s,f,v) \in a ]. \end{aligned}$$

The function $\textit{MatProd}(\mathbf {A},\mathbf {B})$ determines the product $\mathbf {A \odot B}$. Since we need only the diagonal values of the matrix $\mathbf {SAS}$ we applied a special function $\textit{MatProdDiag}$ that determines only the diagonal vector of the product $\mathbf {A \odot B}$. Afterward, to get the clustering coefficient, we have to normalize the obtained counts. The number of neighbors of the node v is determined as its degree in the corresponding undirected temporal skeleton graph (in which an edge $e=(v:u)$ exists iff there is at least one arc between the nodes v and u). The maximum number of neighbors $\Delta$ can be considered either for a selected time point ($\textit{type}=2$) or for the complete time window ($\textit{type}=3$). Note that to determine the temporal $\Delta$ we used summing of temporal degrees over the maxmin semiring $(\mathbb {R},\max ,\min ,-\infty ,\infty )$.

1.2 Equivalences

The transformation of the temporal equivalence matrix $\mathbf {E}$ into the corresponding temporal partition $\mathbf {p}$ is implemented as a procedure $\textit{eqMat2Part}(\mathbf {E})$ (see Algorithm 5). Maybe in the future implementations we shall add a loop with the check of the injectivity of this mapping. The classes of the obtained temporal partition are finally renumbered with consecutive numbers using the function renumPart(p) (see Algorithm 6). The variable C in the description of the function renumPart is a dictionary (data structure).

1.3 Temporal closeness

To compute the vector of closeness coefficients of nodes we have to sum the temporal distances to other nodes over the combinatorial semiring, see Algorithm 7. While summing, we replace gaps (inactivity intervals inside $\mathcal{T}$) with time intervals with the value infinity, using the procedure $\textit{fillGaps}$.

1.4 Temporal PathFinder

The scheme of Pathfinder is implemented (see Algorithm 8) as the function $\textit{pathFinder}$. The temporal version of the statement

if $\mathbf {W}^{(q)}[u,v] = \mathbf {W}[u,v]$ then $\mathcal {L}_{PF} := \mathcal {L}_{PF} \cup \{ e \}$

is implemented in the function $\textit{PFcheck}$ (Algoritm 9) using the merging scheme.

The function $\textit{MatPower}(A,k)$ computes the kth power of the matrix $\mathbf {A}$.