Generalizing Egocentric Temporal Neighborhoods to probe for spatial correlations in temporal networks and infer their topology

TL;DR

This paper introduces a new class of motifs, that include triangles, that are not limited in their number of nodes or edges, and yet can be mined efficiently in any temporal network, and deduce an approximate formula for the distribution of the edge-centered motifs in empirical networks of social face-to-face interactions.

Abstract

Motifs are thought to be some fundamental components of social face-to-face interaction temporal networks. However, the motifs previously considered are either limited to a handful of nodes and edges, or do not include triangles, which are thought to be of critical relevance to understand the dynamics of social systems. Thus, we introduce a new class of motifs, that include these triangles, are not limited in their number of nodes or edges, and yet can be mined efficiently in any temporal network. Referring to these motifs as the edge-centered motifs, we show analytically how they subsume the Egocentric Temporal Neighborhoods motifs of the literature. We also confirm in empirical data that the edge-centered motifs bring relevant information with respect to the Egocentric motifs by using a principle of maximum entropy. Then, we show how mining for the edge-centered motifs in a network can be used to probe for spatial correlations in the underlying dynamics that have produced that network. We deduce an approximate formula for the distribution of the edge-centered motifs in empirical networks of social face-to-face interactions. In the last section of this paper, we explore how the statistics of the edge-centered motifs can be used to infer the complete topology of the network they were sampled from. This leads to the needs of mathematical development, that we inaugurate here under the name of graph tiling theory.

Figures (23)

• Figure 1: Examples of ETN instances. (a) temporal network from which the ETN instances are extracted (b) two ETN instances. Each instance is a sub-temporal graph of the network in (a), labelled by the triple (central node, starting time, depth). Here the depth is two, meaning the duration of the sub-temporal graphs is equal to two. To facilitate reading, the central node has been represented as a red square instead of a gray disk. Note that the triangle at $t=1$ cannot be part of any ETN instance, as only the interactions between an ego (the central node) and its alters (the satellites) are collected.
• Figure 2: Diagrammatic representation of ETN motifs. When we go from the instance to the motif, the identity of the nodes is lost, as well as the starting time. The diagram representation is time-ordered: from left to right, the central node is drawn together with its satellites according to time ordering. An horizontal edge is drawn between two identical nodes at different time steps. A vertical edge is drawn between a satellite and the central node if they interact.
• Figure 3: Representation of an ETN motif into a binary string. First the activity profile of each satellite is determined: a '1' means the satellite interacts with the central node. Then the profiles are sorted by increasing lexicographic order and concatenated into a single string of length $n_{s}d$, where $n_{s}$ is the number of satellites and $d$ is the ETN depth.
• Figure 4: Example of ECTN instance. The ECTN instance is extracted from the same network as Figure \ref{['fig:0']}. Here the instance considered has (0,1) as central edge, starts at time 0 and its depth equals to 3. Note then that the whole temporal network is contained in the instance.
• Figure 5: Diagram representation of an ECTN motif. We consider the same instance as in Figure \ref{['fig:3']}. From this instance, we build the diagram representing the underlying motif. Like in NCTN, horizontal edges are drawn between identical nodes at different time steps, and vertical edges indicate interactions between nodes.