# Hypergraph models for political structures

Ismar Volić  Department of Mathematics, Wellesley College, 106 Central Street,
Wellesley, MA 02481<EMAIL_ADDRESS>ivolic.wellesley.edu and  Zixu Wang
Department of Mathematics, Wellesley College, 106 Central Street, Wellesley,
MA 02481<EMAIL_ADDRESS>
###### Abstract.

Building on [18], this paper extends the modeling of political structures from
simplicial complexes to hypergraphs. This allows the analysis of more complex
political dynamics where agents who are willing to form coalitions contain
subsets that would not necessarily form coalitions themselves. We extend
topological constructions such as wedge, cone, and collapse from simplicial
complexes to hypergraphs and use them to study mergers, mediators, and power
delegation in political structures. Concepts such as agent viability and
system stability are generalized to the hypergraph context, alongside the
introduction of the notion of local viability. Additionally, we use embedded
homology of hypergraphs to analyze power concentration within political
systems. Along the way, we introduce some new notions within the hypergraph
framework that are of independent interest.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgments
  2. 2 Hypergraphs
    1. 2.1 Basic definitions
    2. 2.2 Geometric realization
    3. 2.3 Operations on hypergraphs
    4. 2.4 Hypergraph homology
  3. 3 Modeling political structures with hypergraphs
    1. 3.1 Political structures
    2. 3.2 (Local) viability and stability
    3. 3.3 Operations on political structures
    4. 3.4 Homology and political structures
    5. 3.5 Delegating power
  4. 4 Future work

##  1\. Introduction

Hypergraphs are generalizations of both graphs and simplicial complexes in
that they allow edges to connect an arbitrary number of vertices (like
simplices in a simplicial complex do), but subsets of vertices that form
hyperedges do not necessarily have to form hyperedges themselves (which is a
requirement in simplicial complexes). This flexibility makes them a convenient
tool for studying complex, non-binary relational systems, with applications in
fields ranging from computational biology to network science [2, 4, 6, 10, 12,
13, 17, 19, 21, 22, 23] (a further list of applications can be found, for
example, in [3]).

In this paper, hypergraphs model relationships among agents in a political
system. Agents are represented by vertices, and if a subset of agents is
willing to negotiate or enter a coalition, a hyperedge determined by those
vertices is formed. In [18], the same situation was modeled by a simplex in a
simplicial complex, which imposed a somewhat unnatural condition that any
subset of agents that form a coalition also must form a coalition. This, of
course, is not the case in reality since some agents might only be willing to
negotiate if other agent(s) are present. If such cohesive agents leave, the
entire coalition might collapse. The case of a mediator brought in for
coalition-building among feuding agents is an example of this. Hypergraphs do
not impose a condition of closure under taking subsets, which allows for a
more realistic model developed here.

Once the transition from simplicial complexes to hypergraphs is established,
we devote the bulk of the paper to translating and carrying over the results
from [18] to this new setting. This is sometimes straightforward and sometimes
not since the topology of hypergraphs is not as established or well studied as
that of simplicial complexes; hypergraphs are in the literature usually
treated with extensions of graph-theoretic tools rather than topological ones.
Our point of view is geometric, namely that hypergraphs are simplicial
complexes embedded in a Euclidean space but with some faces missing, and this
provides intuition for extending some basic topological notions to them: cone,
suspension, join, wedge, and (strong) collapse, among others. As in [18],
these constructions are then translated into phenomena such as mergers,
introduction of mediators, and delegation of power.

Homology is another important topological notion for us, but the standard
construction of completing a hypergraph to a simplicial complex and then
taking its homology turns out to be insufficient. This is because the closure
of a hypergraph adds faces/coalitions that are not there in the actual
political structure. Luckily, a more suitable notion of _embedded_ homology
for hypergraphs already exists [7] and this allows us to say something about
the possible viabilities, the dynamics of relationships, and concentration of
power in a political structure.

To assess the impact of different operations, we import the ideas of agent
viability and system stability from [18] into the hypergraph framework. The
results describing the interplay between these notions and various topological
operations translate to hypergraphs in a straightforward way. We also
introduce _local_ viability that captures the importance of an agent within
their framework of coalitions. As we will argue, this idea turns out to carry
richer information for hypergraphs than when restricted to simplicial
complexes.

Much like [18], this paper only initiates the study of political structures
via hypergraphs. Further avenues of investigation are suggested along the way,
and the last section is also devoted to this. We feel that hypergraphs are a
potentially fruitful approach to studying political structures with possible
exciting extensions to other topics in game theory and social choice theory.

###  1.1. Acknowledgments

Ismar Volić is grateful to the Simons Foundation for its support.

##  2\. Hypergraphs

This section introduces fundamental hypergraph concepts, with definitions that
diverge slightly from those conventionally adopted in classical graph theory
because we aim to establish a closer correspondence with simplicial complexes.
This review is by no means exhaustive; these are many resources for further
details, including, for example, [5, 8]. Some definitions, like domination and
elementary collapses, do not seem to appear in the literature, but are natural
extensions of the same notions for simplicial complexes.

###  2.1. Basic definitions

######  Definition 2.1.

A _hypergraph_ $H$ is a pair $H=(V,E)$, where

  * •

$V$ is a finite set whose elements are called _vertices_ ;

  * •

$E$ is a finite set whose elements are non-empty subsets of $V$ called
_hyperedges_. The set of hyperedges must include all singleton subsets of $V$,
i.e., every vertex $v\in V$ is also a hyperedge $\\{v\\}\in E$.

######  Remarks 2.2.

  1. (1)

In the usual definition of a hypergraph, vertices are not required to be in
the hyperedge set. In our setup, vertices will correspond to agents and
hyperedges to coalitions, and we want a single agent to be a valid coalition;
hence the inclusion of elements of $V$ in $E$.

  2. (2)

A _simplicial complex_ is a hypergraph with the additional condition that the
edge set be “downward closed,” i.e. closed under taking subsets. Thus if
$e\subset V$ is a hyperedge (or a _simplex_ in the language of simplicial
complexes), so are all subsets of $e$ (its _faces_). For a review of
simplicial complexes, see Appendix A of [18].

Since the vertex set is contained in $E$, we will often refer to a hypergraph
$H$ just by its hyperedge set $E$. For a hypergraph $H$ with $k$ vertices, we
label the vertices $v_{1},\ldots,v_{k}$, without prioritizing the order of the
enumeration.

######  Example 2.3.

Let $H=(V,E)$ be the hypergraph with the vertex set
$V=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ and the hyperedge set

$\displaystyle E=\\{$
$\displaystyle\\{v_{1}\\},\\{v_{2}\\},\\{v_{3}\\},\\{v_{4}\\},\\{v_{5}\\},\\{v_{1},v_{2}\\},\\{v_{1},v_{5}\\},$
$\displaystyle\\{v_{2},v_{4}\\},\\{v_{3},v_{4}\\},\\{v_{1},v_{3},v_{4}\\},\\{v_{1},v_{2},v_{3},v_{4}\\}\\}.$

This can be visualized in a standard way with vertices drawn as nodes and
hyperedges depicted as closed curves containing the appropriate vertices. To
simplify the visualization, we do not draw hyperedges that consist of a single
vertex. Following this convention, the hypergraph in this example is
represented by Figure 1.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$ Figure 1. A visualization for the
hypergraph of Example 2.3.

######  Definition 2.4.

The _degree_ of a vertex $v$ in $H$ is given by

$\deg(v)=|\\{e\in E\,|\,v\in e\\}|-1.$

######  Remark 2.5.

The usual way to define the degree of $v$ is as the number of hyperedges that
contain it, but we subtract one because we do not want to include the
hyperedge consisting of $v$ alone in the count. Isolated vertices, namely
those that are not elements of any edges of size two of greater, thus have
degree 0.

######  Example 2.6.

For the hypergraph described in Example 2.3, $\text{deg}(v_{4})=4$.

######  Definition 2.7.

  * •

The _size_ of a hyperedge $e$ in a hypergraph $H$ is $|e|$, namely its
cardinality.

  * •

The _dimension_ of $H$, $\dim(H)$, is the maximum size of a hyperedge, i.e.

$\dim(H)=\max_{e\in E}|e|.$

.

  * •

The _s-vector_ of $H$ is a vector $(s_{1},s_{2},\ldots,s_{\dim(H)})$, where
$s_{i}$ denotes the number of hyperedges of size $i$.

  * •

The _s-vector of vertex $v$_, denoted by
$(s_{1}(v),s_{2}(v),\ldots,s_{m(v)}(v))$, is a vector with $s_{i}(v)$ denoting
the number of hyperedges of size $i$ in $H$ that contain the vertex $v$ and
$m(v)$ is the size of the largest hyperedge containing $v$.

######  Example 2.8.

  * •

The dimension of the hypergraph in Example 2.3 is 4 because its largest
hyperedge contains four vertices. Its $s$-vector is $(5,4,1,1)$. The
$s$-vector of $v_{1}$ is $(1,2,1,1)$.

  * •

A 2-dimensional hypergraph is an ordinary graph.

It is immediate from the definitions that $\sum_{i=1}^{\dim(H)}s_{i}=|E|$ and
$\sum_{i=1}^{m(v)}s_{i}(v)=\deg(v)+1.$

######  Definition 2.9.

The _neighborhood of a hyperedge $e$_ is the set of hyperedges in $H$ that
have $e$ as a subset:

$\operatorname{N}(e)=\\{h\in E\,|\,e\subset h\\}.$

Note that $e$ could be a vertex $v$, in which case the above defines a
neighborhood $N(v)$ of $v$. It follows that $\deg(v)=|\operatorname{N}(v)|-1$.
In case the hypergraph is a simplicial complex, the notion of a neighborhood
coincides with the _star_ of the hyperedge/face $e$, denoted by
$\operatorname{st}(e)$.

######  Definition 2.10.

Two vertices are _adjacent_ if there exists a hyperedge containing both. The
_adjacency count_ of a vertex $v$ is the number of vertices adjacent to $v$,
namely

${\operatorname{ac}}(v)=|\\{u\in V\setminus v\ |\text{ there exists }e\in
E\text{ with }v,u\in e\\}|.$

######  Definition 2.11.

A hypergraph with the set of hyperedges $E=\mathcal{P}_{0}(V)$, namely the set
of nonempty subsets of $V$, is called _complete_.

In the terminology of simplicial complexes, if $|V|=k$, then the complete
hypergraph on $V$ is a standard $(k-1)$-simplex $\Delta^{k-1}$.

######  Definition 2.12.

Given a hypergraph $H$, its _(simplicial) closure_ , denoted by $K_{H}$, the
smallest simplicial complex containing $H$.

Thus $K_{H}$ is obtained by adding all subsets of elements already in $H$ to
$H$. There is a natural inclusion map $H\to K_{H}$, providing a useful functor
from the category of hypergraphs to the category of simplicial complexes.

######  Definition 2.13.

A _maximal hyperedge_ is a hyperedge that is not contained in another
hyperedge.

######  Example 2.14.

For the hypergraph in Example 2.3, $\\{v_{1},v_{3},v_{4}\\}$ is not a maximal
hyperedge because it is a subset of the hyperedge
$\\{v_{1},v_{2},v_{3},v_{4}\\}$, while the latter is a maximal hyperedge.

######  Definition 2.15.

Let $v_{i}$ and $v_{j}$ be vertices in a hypergraph. We say that $v_{i}$ is
_dominated_ by $v_{j}$, denoted as $v_{i}\preceq v_{j}$, if the following
conditions hold:

  1. (1)

Every maximal hyperedge that contains $v_{i}$ also contains $v_{j}$.

  2. (2)

There exists a hyperedge $\\{v_{i},v_{j}\\}$.

A hypergraph without any dominated vertices is called _minimal_.

######  Remark 2.16.

In the realm of simplicial complexes, domination is defined using only the
first condition. However, for our purposes it also makes sense to impose the
second one. Namely, if an agent is dominated by another agent in a political
structure, we will want to consider the situation when the former relinquishes
their power to the latter; this is modeled by the elementary strong collapse
defined below. But for this, we want to assume that the two agents are willing
to talk to each other, namely be in a coalition of their own, and this is
represented by an edge containing them.

######  Example 2.17.

For the hypergraph in Example 2.3, $v_{3}$ is dominated by $v_{4}$, and
$v_{5}$ is dominated by $v_{1}$. However, $v_{1}$ is not dominated by $v_{4}$
because there does not exist a hyperedge that contains precisely $v_{1}$ and
$v_{4}$.

######  Definition 2.18.

The _deletion of a vertex $v$_ in a hypergraph $H$, denoted by $H\setminus v$,
is the subhypergraph of $H$ obtained by removing the neighborhood of $v$ from
it, i.e. retaining all hyperedges in $H$ that do not contain $v$:

$H\setminus v=\\{e\in E\,|\,v\notin e\\}.$

######  Definition 2.19.

  * •

The deletion of a dominated vertex $v$ from a hypergraph $H$ is called an
_elementary strong collapse_ , denoted by
$H\,{\searrow\\!\\!\\!\searrow}\,H\setminus v$.

  * •

A sequence of elementary strong collapses is a _strong collapse_.

  * •

The inverse of a strong collapse is a _strong expansion_.

  * •

If $\operatorname{H}_{1}$ and $H_{2}$ are related by a sequence of strong
collapses and expansions, then they have the same _strong homotopy type_.

  * •

A hypergraph $H$ is _strongly collapsible_ if it has the strong homotopy type
of a hypergrph consisting of a single vertex.

######  Example 2.20.

Figure 2 below gives an example of a strong collapse.

In a minimal hypergraph, no elementary strong collapses are possible. Each
hypergraphs admits a unique minimal subhypergraph via a strong collapse called
the _core of $H$_ and denoted by $H^{c}$. For example, picture (E) of Figure 2
(single vertex) is the core of all the hypergraphs before it.

######  Definition 2.21.

A _hypergraph map_ $\phi\colon\operatorname{H}_{1}\to H_{2}$ between
hypergraphs $\operatorname{H}_{1}$ to $H_{2}$ is a function that sends
vertices to vertices and hyperedges to hyperedges. A bijective hypergraph map
whose inverse is also a hypergraph map is an _isomorphism_.

###  2.2. Geometric realization

The standard way to visualize and topologize a simplicial complex $K$ is to
associate to it its geometric realization $|K|$. Each simplex of $K$ of
cardinality $n+1$ corresponds to the standard geometric $n$-simplex in
Euclidean space (and topologized as its subset), and the simplices are glued
along common faces. The same procedure can be performed for hypergraphs. One
way to think about the realization $|H|$ of $H$ is to first take the geometric
realization $|K_{H}|$ of the simplicial closure of $H$, and then remove faces
corresponding to elements of $K_{H}\setminus H$.

######  Example 2.22.

The geometric realization of the hypergraph in Example 2.3 is given in picture
(A) of Figure 2. Dashed lines indicate that the corresponding 2-vertex
hyperedges do not exist. The filled-in triangle indicates that the hypergraph
contains the hyperedge $\\{v_{1},v_{3},v_{4}\\}$. Uncolored triangles indicate
that the 3-vertex hyperedges corresponding to those faces do not exist.
Additionally, there is a tetrahedron determined by vertices
$v_{1},v_{2},v_{3},v_{4}$ because there is a hyperedge
$\\{v_{1},v_{2},v_{3},v_{4}\\}$. We refrained from trying to color in the
tetrahedron in order to make the picture simpler.

The rest of Figure 2 gives an illustration of a strong collapse. Since the
last pictures is that of a single vertex, the hypergraph is strongly
collapsible.

$v_{1}$$v_{4}$$v_{3}$$v_{2}$$v_{5}$ (a)

$v_{1}$$v_{4}$$v_{3}$$v_{2}$ (b)

$v_{1}$$v_{4}$$v_{2}$ (c)

$v_{1}$$v_{2}$ (d)

$v_{1}$ (e)

Figure 2. Illustration of a strong collapse on a sequence of geometric
realizations.

From now on, we will use $H$ for a hypergraph $H$ and its realization $|H|$
and will not distinguish between the two.

###  2.3. Operations on hypergraphs

There are several standard constructions that can be performed on simplicial
complexes, and they have straightforward generalizations to hypergraphs.

######  Definition 2.23.

Let $\operatorname{H}_{1}$ and $H_{2}$ be hypergraphs with distinguished
vertices $v$ and $w$. Define the _wedge_ of $\operatorname{H}_{1}$ and
$H_{2}$, denoted by $\operatorname{H}_{1}\vee H_{2}$, to be the the union of
$\operatorname{H}_{1}$ and $H_{2}$, except that one of $v$ or $w$ is removed
(and the remaining one is relabeled if necessary).

Geometrically, $\operatorname{H}_{1}\vee H_{2}$ is obtained as a quotient
space

$\operatorname{H}_{1}\vee H_{2}=(\operatorname{H}_{1}\sqcup H_{2})/(v\sim w),$

where $\sqcup$ denotes disjoint union. Thus $\operatorname{H}_{1}\vee H_{2}$
is simply the space obtained by identifying $v$ and $w$, i.e. “attaching”
$\operatorname{H}_{1}$ and $H_{2}$ along these points.

One can also attach hypergraphs along common subhypergraphs in a more general
construction called the _pushout_.

Another general construction is the _join_ , obtained by taking the union of
$\operatorname{H}_{1}$ and $H_{2}$ as well as the unions of the elemens of all
the possible pairs of hyperedges from $\operatorname{H}_{1}$ and $H_{2}$. A
special case is

######  Definition 2.24.

Suppose $\\{c\\}$ is a hypergraph consisting of a single vertex. The _cone_ on
a hypergraph $H$ is the hypegraph $CH$ given by

$CH=H\sqcup\\{c\\}\sqcup\\{e\cup c\colon e\in H\\}.$

Vertex $c$ is the _cone vertex_ or the _cone point_.

In terms of realizations, the cone is the quotient $(H\times
I)/(H\times\\{1\\})$, where $I=[0,1]$.

######  Example 2.25.

Figure 3 gives an illustration of the cone construction. All three vertices
from $H$ are connected to the cone point $c$ by the edges $\\{v_{1},c\\}$,
$\\{v_{2},c\\}$, and $\\{v_{3},c\\}$. The edges $\\{v_{1},v_{2}\\}$ and
$\\{v_{1},v_{3}\\}$ of $H$ form hyperedges $\\{v_{1},v_{2},c\\}$ and
$\\{v_{1},v_{3},c\\}$ in $CH$. Since the edge $\\{v_{2},v_{3}\\}$ is missing
in $H$, the face $\\{v_{2},v_{3},c\\}$ is also absent. The hyperedge
$\\{v_{1},v_{2},v_{3}\\}$ of $H$ leads to the hyperedge
$\\{v_{1},v_{2},v_{3},v_{4}\\}$ in $CH$, and as a result, the tetrahedron is
solid.

$v_{1}$$v_{2}$$v_{3}$

$H$

$v_{1}$$C$$v_{3}$$v_{2}$

$CH$

Figure 3. Illustration of the cone operation.

###  2.4. Hypergraph homology

The standard way to arrive at the homology of a hypergraph $H$ is to regard it
as a poset (ordered by inclusion) and then construct the nerve of that poset.
The homology of the resulting space is the homology of the $H$. It turns out
that this homology is isomorphic to the standard homology of the simplicial
complex $K_{H}$ obtained by the closure of the hypergraph.

However, we find this definition insufficient for our purposes because we wish
for homology to “remember” which faces are missing. Instead, a more relevant
notion of the homology of hypergraphs is that of _embedded homology_ [7]
(which generalizes _path homology_ [15]). In brief, the chain complex that
computes the embedded homology of $H$ is the smallest subcomplex of the
singular chain complex for $K_{H}$ that contains the graded group generated by
the hyperedges of $H$. In case $H$ is a simplicial complex, this produces the
usual homology. But $H$ and $K_{H}$ can in general have different homology
groups, as illustrated in the example below. We will denote the $n$th embedded
homology o a hypergraph $H$ by $\operatorname{H}^{emb}_{n}(H)$.

######  Example 2.26.

Figure 4 shows a hypergraph $H$ and its closure $K_{H}$. It turns out that
$\operatorname{H}^{emb}_{0}(H)=\mathbb{Z}\oplus\mathbb{Z}$ while
$\operatorname{H}_{0}(K_{H})=\mathbb{Z}$ (see [7] for details). Adding one of
the two missing edges to $H$ would change its 0th homology to ${\mathbb{Z}}$
(as calculated in [14, Example 3.7]). Intuitively, this makes sense since
contracting the single edge in $H$ would produce two vertices, i.e. two
componenents, while both $H$ with an added edge and $K_{H}$ are contractible
and hence consist of one component.

$v_{1}$$v_{2}$$v_{3}$

$H$

$v_{1}$$v_{2}$$v_{3}$

$K_{H}$

Figure 4.

##  3\. Modeling political structures with hypergraphs

The premise of [18] is that, given a collection of _agents_ in a political
system (voters, political parties, members of a board of directors, etc.), we
can regard those as vertices of a simplicial complex and simplices as
potential coalitions. The presence of a simplex indicates that the agents
represented by its vertices are willing to negotiate or vote the same way on a
resolution. Various topological ideas and constructions on simplicial
complexes then have interpretations and consequences for the functioning of
the political system.

In this section, we extend much of what was done in [18] to hypergraphs. One
restrictive assumption in that paper is that any subset of agents in a viable
configuration also forms a viable configuration. This had to be true as
simplicial complexes require closure under subsets. Current extension to
hypergraphs eliminates this assumption, as it should since there is no
guarantee a coalition will stay intact after one or more agents leave it. If
an agent departs a coalition, other agents might do so as well or the entire
coalition might fall apart. Dropping the requirement that every subset of a
hyperedge is also a hyperedge allows for this possibility.

###  3.1. Political structures

This section is a straightforward extension of the basic dictionary set up in
[18]. We follow the terminology from that paper.

We make the following definition after [1, Definition 1] and [18, Definition
3.1].

######  Definition 3.1.

A _political structure_ $P$ is an ordered pair $P=(A,\mathcal{C})$, where

  * •

$A$ is a finite set whose elements are called _agents_ ;

  * •

$\mathcal{C}$ is a collection of non-empty subsets of $A$ called _viable
configurations_ which includes all elements of $A$, i.e., every agent $a\in A$
forms a viable configuration $\\{a\\}$ on their own.

A correspondence between political structures and hypergraphs should be clear;
vertices are agents and hyperedges are viable configurations, which should be
thought of a coalitions. From now on, we will blur the disctinction between
the two and will use the terms political structure and hypergraph
interchangeably.

A _map_ $P\to Q$ between political structures is simply a hypergraph map of
underlying hypergraphs. Such a map might capture a change, represent a
transformation of the system after an event, a consolidation of agents, or an
embedding of one structure into another. In particular, agents are sent to
agents, but not necessarily in an injective of surjective way, meaning that
multiple agents might have been consolidated or merged, or that new agents
have been introduced. Since hyperedges map to hyperedges, viable
configurations are mapped to viable configuration, indicating that, if agents
were compatible before, they are still compatible after the event modelled by
the map.

If there are $k$ agents and all configurations are viable, then we get the
complete hypergraph or a $(k-1)$-simplex $\Delta^{k-1}$. We say the structure
is in this case _fully viable_. If no coalitions are possible, we get a
0-dimensional hypergraph consisting only of $k$ vertices and no other
hyperedges. If all viable configurations are of size 2, the structure is
modeled by an ordinary graph.

We can interpret various other definitions from Section 2 in the context of
political structures.

######  Definition 3.2.

  * •

The _dimension_ of a political structure is the size of its largest viable
configuration.

  * •

A viable configuration is _maximal_ if it is not contained in any other
configuration.

  * •

An agent $a_{j}$ is _more central_ than agent $a_{i}$ if $a_{i}$ is dominated
by $a_{j}$ in the corresponding hypergraph.

###  3.2. (Local) viability and stability

One of the main goals of modeling political structures by hypergraphs (or
simplicial complexes) is that their topology, geometry, and combinatorics
allow us to define and quantify how stable the system is and how well-
positioned its agents are individually or in coalitions. This can then in turn
be used for deciding how the structure can be further stabilized or where the
most urgent interventions are.

To make this precise, we provide the following three definitions. The notion
of local viability is new, while the other two appear in [18, Definition 3.8]
and are modified appropriately to hypergraphs here.

######  Definition 3.3.

Given a political structure $P=(A,\mathcal{C})$ with the set of agents
$A=\\{a_{1},...,a_{k}\\}$, $k\geq 2$, and the set of hyperedges $\mathcal{C}$,
define the

  * •

_viability of $a_{i}$_ to be

$\operatorname{via}(a_{i})=\frac{|\operatorname{N}(a_{i})|-1}{2^{k-1}-1};$

  * •

_local viability of $a_{i}$_, where $a_{i}$ is not an isolated agent (i.e.
$|\operatorname{N}(a_{i})|>1$) to be

$\operatorname{lvia}(a_{i})=\frac{|\operatorname{N}(a_{i})|-1}{2^{{\operatorname{ac}}(a_{i})}-1};$

In case $a_{i}$ is isolated, define $\operatorname{lvia}(a_{i})=0$.

  * •

_stability of $P$_ to be

$\operatorname{stab}(P)=\frac{|\mathcal{C}|-k}{2^{k}-k-1}.$

The viability of $a_{i}$ is determined by the number of hyperedges that
contain $a_{i}$ as a vertex, normalized to range between 0 and 1. A higher
value of $\operatorname{via}(a_{i})$ means that $a_{i}$ has many coalition
options and is well-connected or well-trusted in the system. An isolated agent
has viability 0. On the other hand, an agent who forms every possible
coalition with other agents has viability 1. The minimal hypergraph for which
this is possible is the the $(k-1)$-simplex $\Delta^{k-1}$ with the face
$\\{a_{1},...,\widehat{a_{i}},...,a_{k}\\}$ removed (where $\widehat{a_{i}}$
indicates the omission of $a_{i}$). Thus if $P$ is represented by
$\Delta^{k-1}$, i.e. it is fully viable, every agent has viability 1.

The local viability of $a_{i}$ again counts the number of hyperedges that
include $a_{i}$ as a vertex, but now normalized relative to the number of
vertices in its neighborhood. Since there is no dependence on $k$, this
provides a more local metric compared to viability. Low local viability
indicates that an agent prefers or is only able to form small coalitions. For
example, if an agent only forms coalitions with one other agent at a time (so
its neighborhood looks like a graph), local viability is 0. Higher local
viability indicates an agent’s willingness to form many coalitions with its
adjacencies, demonstrating more flexibility. As an extreme, we have

######  Proposition 3.4.

The local viability of an agent $a$ is 1 if and only if the agent forms all
possible coalitions with its adjacencies.

###### Proof.

Local viability of an agent $a$ is 1 if and only if
$|N(a)|=2^{|{\operatorname{ac}}(a)|}$, and this is true if an only if $a$
forms a hyperedge with every subset of its adjancencies. ∎

One way to think about the above is that, if local viability is 1, then $a$
and its adjacencies form a $\Delta^{|{\operatorname{ac}}(a)|}$, possibly with
the face determined by the adjacencies alone (the face “opposite” $a$) and
some or all of its subfaces missing. If any of the other faces are missing,
$|N(a)|$ decreases and local viability is strictly less than 1. In the case
when $H$ is a simplicial complex, this means that an agent has local viability
1 if they belong to exactly one maximal coalition.

We will make some further comments about local viability and its relation to
viability after working out some examples.

The stability of $P$ is the total number of hyperedges of $P$, again
normalized. This is essentially the sum of the viabilities of all the agents,
but taking into account the overcount of hyperedges. Greater
$\operatorname{stab}(P)$ indicates more compatibilities among agents and more
willingness to form coalitions. As one would expect, a fully viable system
(simplex) has stability 1 while the system with no coalitions has viability 0.

######  Example 3.5.

Consider political structures $P_{1},P_{2},P_{3},P_{4}$, each containing five
agents $a_{1},a_{2},a_{3},a_{4},a_{5}$. In each, the neighborhood $N(a_{1})$
of $a_{1}$ has 4 elements:

$\displaystyle\text{In }P_{1}:$ $\displaystyle\
N(a_{1})=\\{\\{a_{1}\\},\\{a_{1},a_{2}\\},\\{a_{1},a_{3}\\},\\{a_{1},a_{2},a_{3}\\}\\}$
$\displaystyle\text{In }P_{2}:$ $\displaystyle\
N(a_{1})=\\{\\{a_{1}\\},\\{a_{1},a_{2}\\},\\{a_{1},a_{2},a_{3}\\},\\{a_{1},a_{4}\\}\\}$
$\displaystyle\text{In }P_{3}:$ $\displaystyle\
N(a_{1})=\\{\\{a_{1}\\},\\{a_{1},a_{2},a_{3}\\},\\{a_{1},a_{4},a_{5}\\},\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}\\}$
$\displaystyle\text{In }P_{4}:$ $\displaystyle\
N(a_{1})=\\{\\{a_{1}\\},\\{a_{1},a_{2}\\},\\{a_{1},a_{3}\\},\\{a_{1},a_{4},a_{5}\\}\\}$

The viability of $a_{1}$ is the same in all four, since, in each structure,

$\operatorname{via}(a_{1})=\frac{4-1}{{2^{5-1}-1}}=\frac{1}{5}.$

However, the local viability of $a_{1}$ differs:

In $P_{1}$ : ${\operatorname{ac}}(a_{1})=2$, and $\operatorname{lvia}(a_{i})$
is 1. This makes sense in light of Proposition 3.4 because the neighborhood of
$a_{1}$ consists of all possible hyperedges that contain one or both of
$a_{2}$ and $a_{3}$.

In $P_{2}$ : ${\operatorname{ac}}(a_{1})=3$, and $\operatorname{lvia}(a_{1})$
is $3/7$.

In $P_{3}$ : ${\operatorname{ac}}(a_{1})=4$, and $\operatorname{lvia}(a_{1})$
is $1/5$. Note that this value is the same as $a_{1}$’s viability. This is
because $a_{1}$ is connected to all other agents so the local and global
measures coincide.

In $P_{4}$ : ${\operatorname{ac}}(a_{1})=4$, and $\operatorname{lvia}(a_{1})$
is $1/5$.

While local viability offers more insight than viability in this example, it
still cannot distinguish between the cases of $P_{3}$ and $P_{4}$. To further
detect differences between $P_{3}$ and $P_{4}$, one might consider bringing in
the $s$-vector of a vertex.

The following example illustrates that viability, with its global view of the
position of an agent, can also provide a more nuanced view than local
viability.

######  Example 3.6.

Consider these political structures:

$\displaystyle P_{1}$
$\displaystyle=\\{\\{a_{1}\\},\\{a_{2}\\},\\{a_{3}\\},\\{a_{1},a_{3}\\},\\{a_{1},a_{2},a_{3}\\}\\}$
$\displaystyle P_{2}$
$\displaystyle=\\{\\{a_{1}\\},\\{a_{2}\\},\\{a_{3}\\},\\{a_{4}\\},\\{a_{5}\\},\\{a_{1},a_{3}\\},\\{a_{1},a_{2},a_{3}\\},\\{a_{3},a_{4}\\},\\{a_{3},a_{5}\\},\\{a_{4},a_{5}\\},\\{a_{2},a_{3},a_{5}\\}\\}$

The local viability of $a_{1}$ is equal in the two structures since the
neighborhoods of $a_{1}$ are the same.

However, their viabilities are:

In $P_{1}$: $\operatorname{via}(a_{1})=\frac{{3-1}}{{2^{3-1}-1}}=\frac{2}{3}$.

In $P_{2}$:
$\operatorname{via}(a_{1})=\frac{{3-1}}{{2^{5-1}-1}}=\frac{2}{15}$.

Small viability indicates a low overall level of interaction between an agent
and the other agents. If local viability is simultaneously high, this agent
forms many coalitions, but with a small subset of agents. This might point to
the existance of a clique or a faction within a political structure with the
agent in question as the ringleader.

######  Example 3.7.

For a complete hypergraph $\Delta^{k-1}$, $|\operatorname{N}(a_{i})|=2^{k-1}$
for all $a_{i}$ so the viability of each agent is 1. The adjacency count of
each agent is $k-1$, so the local viability of each agent is also 1. Since the
number of hyperedges in a complete hypergraph is $2^{k}-1$, the stability is
also 1.

At the other extreme is the situation where the structure consists of $k$
isolated vertices so no agents are compatible. Now
$|\operatorname{deg}(a_{i})|=0$ for all $a_{i}$, and the viability of each
agent is 0. For any agent, $|\operatorname{N}(a_{i})|=1$, as the hyperedge
containing the vertex itself is the sole hyperedge in its neighborhood.
Consequently, the local viability of each agent is 0. Since the only
hyperedges are those consisting of a single vertex, $|P|=|V(P)|=k$ and so the
stability of $P$ is 0.

###  3.3. Operations on political structures

Hypergraph constructions from Section 2.3 carry over into the setting of
political structures in a straightforward way. This section explains how and
establishes some results about the interplay between the notions of the
previous section and these constructions.

Suppose a political structure $P$ with $k\geq 2$ agents consists of two
distinct path-connected components, $P_{1}$ and $P_{2}$, so $P=P_{1}\amalg
P_{2}$. Suppose an agent $a_{0}\in P_{1}$ and an agent $b_{0}\in P_{2}$ decide
to merge or join forces, effectively becoming a single agent. If we rename the
newly formed agent by $a$, we then have a situation that corresponds to the
wedge of hypergraphs $P_{1}\vee P_{2}$ (see Definition 2.23).

######  Proposition 3.8.

If agents are merged from two path-connected components of a political
structure, the viability of the newly formed agent is greater than that of
each agent individually. In fact,

$\operatorname{via}(a)>\operatorname{via}(a_{0})+\operatorname{via}(b_{0}).$

###### Proof.

From the definition of viability,
$|N(a_{0})|=(2^{k-1}-1)\cdot\operatorname{via}(a_{0})+1$. Similarly for
$b_{0}$. By definition of the wedge,
$|\operatorname{N}(a)|=|\operatorname{N}(a_{0})|+|\operatorname{N}(b_{0})|-1$,
where the subtraction of 1 is because the hyperedge containing only the wedge
vertex itself should be counted only once. Additionally, the number of agents
decreases by $1$ after the wedge operation.

Hence

$\displaystyle\operatorname{via}(a)$
$\displaystyle=\frac{1}{2^{k-2}-1}\cdot(|\operatorname{N}(a_{0})|+|\operatorname{N}(b_{0})|-2)$
$\displaystyle=\frac{1}{2^{k-2}-1}\cdot\left((2^{k-1}-1)\cdot\operatorname{via}(a_{0})+1+(2^{k-1}-1)\cdot\operatorname{via}(b_{0})+1-2\right)$
$\displaystyle=\frac{1}{2^{k-2}-1}\cdot\left((2^{k-1}-1)\cdot(\operatorname{via}(a_{0})+\operatorname{via}(b_{0}))\right)$
$\displaystyle=\frac{2^{k-1}-1}{2^{k-2}-1}\cdot(\operatorname{via}(a_{0})+\operatorname{via}(b_{0}))$

Since $\frac{2^{k-1}-1}{2^{k-2}-1}>1$ when $k>2$, it follows that
$\operatorname{via}(a)>\operatorname{via}(a_{0})+\operatorname{via}(b_{0})$. ∎

For local viability, the situation is different. The discrepacy comes from the
fact that $k$ is the same for both $a_{0}$ and $b_{0}$ in the above, while the
adjacency count for the two can be different.

######  Proposition 3.9.

If agents are merged from two path-connected components of a political
structure, the local viability satisfies

$\operatorname{lvia}(a)=\frac{2^{{\operatorname{ac}}(a_{0})}-1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\operatorname{lvia}(a_{0})+\frac{2^{{\operatorname{ac}}(b_{0})}-1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\operatorname{lvia}(b_{0}).$

In particular, since the two fractions are less than 1 for
${\operatorname{ac}}(a_{0}),{\operatorname{ac}}(b_{0})\geq 1$, we have

$\operatorname{lvia}(a)<\operatorname{lvia}(a_{0})+\operatorname{lvia}(b_{0}).$

###### Proof.

As in the above proof, we have
$|\operatorname{N}(a_{0})|=(2^{{\operatorname{ac}}(a_{0})}-1)\operatorname{lvia}(a_{0})+1$
and similarly for $b_{0}$. Since
${\operatorname{ac}}(a)={\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})$,

$\displaystyle\operatorname{lvia}(a)$
$\displaystyle=\frac{1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\left(|\operatorname{N}(a_{0})|+|\operatorname{N}(b_{0})|-2\right)$
$\displaystyle=\frac{1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\left((2^{{\operatorname{ac}}(a_{0})}-1)\cdot\operatorname{lvia}(a_{0})+1+(2^{{\operatorname{ac}}(b_{0})}-1)\cdot\operatorname{lvia}(b_{0})+1-2\right)$
$\displaystyle=\frac{2^{{\operatorname{ac}}(a_{0})}-1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\operatorname{lvia}(a_{0})+\frac{2^{{\operatorname{ac}}(b_{0})}-1}{2^{{\operatorname{ac}}(a_{0})+{\operatorname{ac}}(b_{0})}-1}\cdot\operatorname{lvia}(b_{0})$

∎

######  Proposition 3.10.

Merging agents from different path-connected components of a political
structure increases stability:

$\operatorname{stab}(P_{1}\amalg P_{2})<\operatorname{stab}(P_{1}\vee P_{2}).$

###### Proof.

The stability before the merging of the agents is given by

$\operatorname{stab}(P_{1}\amalg P_{2})=\frac{1}{2^{k}-1-k}\cdot(|P_{1}\amalg
P_{2}|-k)$

where by $|\cdot|$ we as usual mean the number of hyperedges of the structure.
After the merging, there is one fewer agent and hence one fewer hyperedge.
Therefore, the stability after the merging is

$\operatorname{stab}(P_{1}\vee
P_{2})=\frac{1}{2^{k-1}-1-(k-1)}\cdot(|P_{1}\amalg P_{2}|-1-(k-1))$

or

$\operatorname{stab}(P_{1}\vee P_{2})=\frac{1}{2^{k-1}-k}\cdot(|P_{1}\amalg
P_{2}|-k)$

Since $2^{k}-1-k>2^{k-1}-k$ when $k>2$,
$\frac{1}{2^{k}-1-k}<\frac{1}{2^{k-1}-k}$. Therefore,
$\operatorname{stab}(P_{1}\amalg P_{2})<\operatorname{stab}(P_{1}\vee P_{2})$.
∎

Two political structures can also merge along two matching substructures
(subhypergraphs). This is known as the _pushout_ and it would be interesting
to generalize Proposition 3.10 to that situation.

A similar yet distinct situation that can also be modeled using the wedge
model is that, instead of being two components of one political structure,
$P_{1}$ and $P_{2}$ are initially two separate political structures and are
then merged along two agents. There seems to be no easy relationship between
the stability of the individual structures and the merged one.

For example, if

$\displaystyle P_{1}$
$\displaystyle=\\{\\{v_{1}\\},\\{v_{2}\\},\\{v_{3}\\},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\},\\{v_{1},v_{2},v_{3}\\}\\}$
$\displaystyle P_{2}$
$\displaystyle=\\{\\{w_{1}\\},\\{w_{2}\\},\\{w_{3}\\},\\{w_{1},w_{3}\\},\\{w_{1},w_{2},w_{3}\\}\\}$

then merging along $v_{1}$ and $w_{1}$ produces a structure whose stability is
lower than that of $P_{1}$ and $P_{2}$ individually. On the other hand, if
$P_{1}$ consists of ten agents and a single viable configuration containing
all of them while $P_{2}$ is a 5-simplex, then the merged structure (along any
two vertices) has stability greater than $P_{1}$ but less than $P_{2}$.

Sometimes we want to introduce a _mediator_ into a political structure $P$.
This is a new agent who is willing to negotiate or enter coalitions with any
existing agents of viable configurations. This is modeled by $CP$, the cone on
$P$. It is easy to show that the stability in this case increases, and in fact
the proof of the same result for simplicial complexes, [18, Proposition 3.14],
applies here.

The question of introducing a mediator into a substructure of $P$ is less
straightforward (and more interesting). Such a scenario could arise, for
instance, when certain agents are deeply divided, necessitating separate
mediation to facilitate their participation in broader discussions. This
situation can be modeled by taking a cone on a subhypergraph of $P$. In this
case, the stability of the political structure does not necessarily increase,
which may seem counterintuitive. The weakening occurs as the mediator
strenghtens the viable configuration structure among some agents but may have
the effect of isolating them from the rest of the system. For details, see
Proposition 3.15 in [18] whose proof applies when simplicial complexes are
replaced by hypergraphs.

######  Remark 3.11.

In [18], an extension to _weighted_ political structures is considered. This
is the model for the scenario where agents are deciding their position on more
than one issue and each simplex is labeled by how many issues the agents
forming that coalition agree on. Weighted versions of viability and stability
for hypergraphs can be defined the same way they are for simplicial complexes
and all the results about weighted political structures from [18] carry over
to the setting of hypergraphs in the same way.

###  3.4. Homology and political structures

In this section, we initiate a study of how embedded homology can be used to
extract information about the dynamics in a political structure. One
observation is that all the results of [18, Section 3.5] hold in the case of
hypergraphs and embedded homology since this homology defines cycles the same
way as standard homology (the difference is that there are fewer of them in
embedded homology). Thus in particular, a non-zero Betti number (rank of a
homology group) points to non-viabilities among certain subsets of agents.

However, it is also helpful to play embedded and standard homology off each
other. Neither is robust enough to be sufficiently useful on its own; homology
is homotopy invariant, and in much of what we do, we want to keep track of
structures that are not preserved by homotopy equivalences. For example, if we
know $\operatorname{H}^{emb}_{0}(H)=\mathbb{Z}\oplus\mathbb{Z}$, this means
that the agents are partitioned into two subsets with no hyperedges of size 2
between them. If there are $k$ agents, we can say that $H$ is missing at least
$k-1$ hyperedges from a complete hypergraphs, but that is about all the
information we can extract.

But if we know both embedded and singular homology of a hypergraph, we can
extract more information about the possible viable configurations in the
structure. For example, for the left image in Figure 4,
$\operatorname{H}^{emb}_{0}(H)=\mathbb{Z}\oplus\mathbb{Z}$. However, this
homology would be the same even if the viable configuration
$\\{v_{1},v_{2},v_{3}\\}$ was not there. On the other hand, the standard
homology of the closure of that figure (the right side of Figure 4) is
$\operatorname{H}_{0}(K_{H})=\mathbb{Z}$. But if the hyperedge
$\\{v_{1},v_{2},v_{3}\\}$ was not there, then
$\operatorname{H}_{0}(K_{H})=\mathbb{Z}\oplus\mathbb{Z}$. In fact, it is not
hard to see that zeroth and first embedded and singular homology groups
together completely determine the political structure on three agents.

Here is a sample result of what we believe could be a fruitful line of
investigation. The statement is about the 0th homology (i.e. path-connected
components), but a generalization to higher homology groups is also possible.
Let $\beta^{emb}_{0}(H)$ and $\beta_{0}(K_{H})$ be the 0th Betti numbers, i.e.
ranks of $\operatorname{H}^{emb}_{0}(H)$ and $\operatorname{H}_{0}(K_{H})$,
respectively.

######  Proposition 3.12.

For any hypergraph $H$,

  * •

$\beta^{emb}_{0}(H)\geq\beta_{0}(K_{H})$.

  * •

If the above is an equality, then the structure is path connected, i.e. there
is a path through hyperedges of cardinality 2 from any agent to any other
agent.

  * •

If the inequality is strict, then there are hyperedges of size greater than
two whose union contains vertices from $\beta^{emb}_{0}(H)-\beta_{0}(K_{H})$
different components of the hypergraph.

###### Proof.

Zeroth homology in general count the number of path-connected components of a
space, i.e. it keeps track of paths of edges of cardinality 2. Taking the
closure of a hypergraph might introduce such edges as faces of other
hyperedges. Thus $K_{H}$ might contain more paths than $H$ itself and
introduce edges of size 2 between distinct path-components of $H$, thereby
decreasing their count. The only way this can happen is if there are
hyperedges of size 3 or greater containing vertices from different components.
These hyperedges provide new paths once the closure is taken. ∎

If the difference in Betti numbers is small, this indicates a fragmentation in
the structure with lots of clusters of agents but few coalitions between the
clusters. If the number is high, this means that the coalitions between
clusters exist, but they are potentially large. Note the extreme case when
there is a single large coalition containing all agents. Then
$\beta_{0}(K_{H})=1$.

###  3.5. Delegating power

In political structures, agents often delegate or cede authority to other,
more favorably positioned agents. In [1], the better-posioned agent is said to
be _more central_ and is modeled in the simplicial complex setting by a more
dominant vertex. An elementary strong collapse is then precisely the the model
for relinquishing authority. This is called a _friendly delegation_ [1]. It
was shown in [18] that the notion of viability and the process of delegating
power interface in a compatible way, namely the viability of an agent
increases if another agent delegates power to them, as one would expect.

We extend this result below to the setting of hypergraphs. We could quote the
proof of [18, Theorem 4.1(1)] and say that it goes through the same way here,
but instead we offer a more streamlined version.

######  Proposition 3.13.

Suppose $P$ is a political structure with $k$ agents and agent $b$ is
dominated by agent $a$. Relabel agent $a$ as $a^{\prime}$ after a friendly
delegation from $b$ to $a$. Then

$\operatorname{via}(a^{\prime})\geq\operatorname{via}(a).$

###### Proof.

The viability of agent $a$ is initially

$\operatorname{via}(a)=\frac{|\operatorname{N}(a)|-1}{2^{k-1}-1}.$

After the friendly delegation from $b$ to $a$, the number of agents decreases
by 1. There is also a decrease in $|\operatorname{N}(a)|$; denote the
difference by $n$, so that

$\operatorname{via}(a^{\prime})=\frac{|\operatorname{N}(a)|-n-1}{2^{k-2}-1}.$

Since $n$ is the number of edges involving both $a$ and $b$ before the
friendly delegation, $n\leq 2^{k-2}$. We also have $|\operatorname{N}(a)|\leq
2^{k-1}$. Using these two inequalities, it is easy to deduce that
$\operatorname{via}(a^{\prime})\geq\operatorname{via}(a)$. ∎

Equality is achieved when $|\operatorname{N}(a)|=2^{k-1}$ and $n=2^{k-2}$,
resulting in $\operatorname{via}(a)=1$. I.e. equality only occurs when $a$ is
fully viable to begin with.

The situation with local viability is not as straighforward. For example,
consider the political structure

$\displaystyle P=\\{$
$\displaystyle\\{a\\},\\{b\\},\\{c\\},\\{d\\},\\{a,b\\},\\{b,c\\},\\{b,d\\},\\{c,d\\},\\{b,c,d\\},\\{a,b,c,d\\}\\}.$

Here $a$ is more central than $b$, but after $b$ delegates to $a$, the
resulting structure is $\\{\\{a\\},\\{b,c\\}\\}$. Now $a$ is an isolated agent
with its local viability going down to 0 from $2/7$. But if we amend $P$ by
adding in the configuration $\\{\\{a,c\\}\\}$ and then delete $b$, producing
the structure $\\{\\{a\\},\\{a,c\\},\\{b,c\\}\\}$, $a$’s viability increases
from $3/7$ to 1.

The difference between these two situations is that, in the first, agent $a$
had no connections to agents $c$ or $d$, so when $b$ delegated power to $a$,
both $a$ and the pair $\\{c,d\\}$ became isolated. In a sense, the structure
of the system was damaged because of $b$’s poor judgement to delegate to $a$.
This does not happen in the second example since $a$ and $c$ form a coalition
prior to delegation. Another measure of this is that the stability of $P$ in
both examples decreases, but the decrease is smaller in the second situation.

Note that, in the first example, $a$ is also dominated by $b$, so that $a$
could also delegate to $b$. This would increase the local viability of $b$,
indicating that this is a better move since the coalition structure would be
stronger after this delegation. This provides a potential delegation strategy
– delegation should only occur if the local viability of the agent being
delegated to increases.

##  4\. Future work

This paper sets up the basic framework of modeling political structures by
hypergraphs and there are still many potential directions of further
investigation. For example, one could take any of the standard notions and
constructions from the theory of hypergraphs (and simplicial complexes) and
try to translate them into this setting. One such is the _clustering
coefficient_ [11, 16] which, in its various forms, can be defined to measure
both local hypergraph structure or provide aggregated infromation about the
entire hypergraph (some even result in approximations as necessitated by
computational complexity). Our notion of local viability could in particular
be upgraded to some kind of a clustering coefficient which might be a more
powerful tool of gauging the power of an agent. Especially interesting would
be a connection to the game-theoretic approach to hypergraph clustering [20]
since the work in this paper borders game theory and social choice theory.

We would also like to extend the Banzhaf and Shapley-Shubik power indices from
simplicial complexes, as was done in [9], to hypergraphs. This would supply a
refined notion of power distribution in weighted voting systems where certain
coalitions are unfeasible or forbidden. In addition, each hyperedge is
geometrically a simplex emebedded in some Euclidean space, so one could
interpret the position in the simplex as influence of each agent in the
corresponding coalition. This information could be added to the calculation of
power indices.

Our elementary foray into (embedded) homology deserves to be explored further.
Higher homology groups should be incorporated into the picture. The existence
of a homology cycle indicates incompatibilities among a subset of agents.
While this was relatively easy to describe in the case of simplicial
complexes, hypergraphs are a different story because of the many ways subsets
of agents forming a cycle may or may not form (sub)coalitions. What is likely
required is a consideration of all homology groups at the same time,
suggesting a combinatorial difficulty but also a structural richness and
potential for a powerful tool.

We have not examined maps between hypergraphs, but they could play an
important role. A hypergraph map sends hyperedges to hyperedges (and faces to
faces) and as such captures a change in a political system that causes agents
or substructures to be consolidated or introduces new ones. The collection of
hypergraphs with hypergraph maps forms a category, which means that political
structures inherit a category structure as well. This could provide a useful
framework and an organizing mechanism to study political structures.
Hypergraphs are are category-theoretically related to simple games from social
choice theory and political structures can also easily be brought into that
framework.

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