Simplicial complexes

This work is interested in analyzing the structure of relationships between subgroups of climbers and their effect on expedition style, from an individualistic to cooperative spectrum. Relationships between climbers may result from previous joint expeditions. Such joint expeditions may involve a subgroup of individuals of size two or more. Networks, although natural to model interactions, can only capture pairwise relationships and hence fail to accurately capture higher-order interactions (interaction between more than 2 climbers in a single expedition). However, multi-node interactions can be explained by a higher-order network: a mathematical framework called simplicial complexes. For instance, a three-way interaction can be represented as a triangle, four-way as a tetrahedra etc.

Mathematically, given a set of l nodes ({n_0, n_1 ldots , n_{l-1}} in N) in a network, a (p-)simplex is a subset (sigma _p = [{n_0,n_1,ldots , n_{p-1}}]) of p nodes and a (q-)face of (sigma _p) is a set of q nodes (for (q<p)) that is a subset of the nodes of (sigma _p). A simplicial complex (Salnikov et al. 2018; Torres and Bianconi 2020) K consists of a set of simplices, that are closed under inclusion:

$$begin{aligned} tau subseteq sigma Rightarrow tau in {K} text {~for any} sigma in {K}, end{aligned}$$


where ‘(subseteq)’ denotes the subset relation between (sigma) and (tau), two subsets of the simplicial complex. When (tau subseteq sigma), we say that (tau) is a face of (sigma), which by the inclusion axiom implies every face of a simplex is again a simplex. Figure 2 shows examples of faces of a simplicial complex.

Fig. 2
figure 2

Simplices that form the faces of the simplicial complex. The number of k-simplices in the top simplicial complex are listed. The influence, or maximal simplicial dimension, (eta) for each node is listed

The dimension of a simplex equals the number of vertices in the simplex minus one; for instance 0-dimensional simplices are nodes and 1-dimensional simplices are edges. Each previous joint expedition is represented as a simplex. The dimension of a simplicial complex is the largest dimension of its simplices, where each expedition is modeled as a simplicial complex with the nodes representing climbers. By ({S}_k) we will denote the set of all simplices (sigma) with dimension k, i.e. as

$$begin{aligned} {S}_k := lbrace sigma in K : vert sigma vert = k+1rbrace , end{aligned}$$


We call the simplices in ({S}_k) the k-simplices of K and let (|S_k|) denote the number of k-simplices in the simplicial complex.

If three climbers ijk in an expedition have participated in an expedition previously, this is represented as a 2-simplex (triangle) with nodes ijk as its (0-)faces. Each climber can be a part of multiple previous expeditions, and hence be the face of multiple simplices. Let the simplices (denoted by (sigma)) that contain individual i be given by: (n_i in sigma _{k1}, sigma _{k2}, ldots , sigma _{k}) where (sigma _k) is a (k-)simplex such that their simplicial dimensions are ordered: (k1 le k2 le ldots le k). In other words, we order the simplices that a climber belongs to in order of their simplicial dimension. Explicitly, if a climber belongs to previous expeditions with 2,4 and 6 other people respectively, the node that corresponds to the climber is the face of 3 simplices ordered by degree: 2-simplex, 4-simplex, and 6-simplex respectively, where the dimension of the largest simplex is (k=6).

The influence of a climber is a measure of the size of the largest group that they belong to that have climbed together in a joint previous expedition, resulting in the formation of pre-existing group relationship. Specifically, the influence (zeta _i) of the (i)th climber is defined to be the largest number of climbers, in the current expedition, that have jointly participated in a previous expedition with the (i)th climber, i.e., dimension of its largest simplex (zeta _i = k). Each climber is associated with an influence value. For a climber who has no previous expeditions with any other climbers (zeta _i=0). The co-influence of an expedition E is given by the the average influence of all climbers participating in the expedition.

Topological data analysis

Topological Data Analysis(TDA) is a new and rapidly growing subfield in machine learning and data science for the analysis of simplicial complexes (Wasserman 2018). The central assumption of TDA is that complex and high dimensional data has an underlying shape captured by topological descriptors, which can be exploited for its analysis. Commonly used topological descriptors are simplices as well as Betti numbers, where the (k^{th}) Betti number gives the number of topological holes or (k-)dimensional cavities (e.g. an unfilled triangle is a 2-dimensional cavity) in the simplicial complex. This work derives inspiration from persistence homology (Aktas et al. 2019), the primary data analysis methodology in TDA, which attempts to extract topological descriptors (e.g. simplices) in the data that persist over various threshold values. Conventionally, topological features are recorded by creating persistence diagrams. These diagrams plot the birth time (on the x-axis), and death time (on the y-axis) of a simplex where time is measured through changing a threshold parameter. Topological features that persist for a large amount of time or across various thresholds are important in the analysis of the simplicial complex.

In a similar vein, this work assigns a weight to each simplex in an expedition, given by the number of previous expeditions of the subgroup represented by the simplex, i.e., the number of times the simplex occurs (w_{sigma } = # sigma). For example, if climbers ijk have had four previous joint expeditions together, then the simplex (sigma _2 = [{n_i,n_j,n_k}]) is a (2-)simplex with a weight (w_{sigma _2} =4). Naturally the higher the weight of the simplex, the stronger the multi-node relationships between the individuals. One can threshold the weight during generation of the simplicial complex such that only simplices with weights larger than the threshold remain, i.e., only relationships stronger than the threshold are captured. Varying the threshold over a range of values, persistent simplices (that persist across various weight thresholds (tau)). One can then study properties of the simplicial complex across (tau), such as evolution of the number, distribution and dimensionality of simplices, as well as the nature of persistent simplices on outcomes.

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