# Balanced Hodge Laplacians Optimize Consensus Dynamics over Simplicial
Complexes

Cameron Ziegler  Department of Mathematics, University at Buffalo, State
University of New York, Buffalo, NY 14260  Per Sebastian Skardal  Department
of Mathematics, Trinity College, Hartford, CT 16106  Haimonti Dutta
Management Science and Systems Department, University at Buffalo, State
University of New York, Buffalo, NY 14260  Dane Taylor<EMAIL_ADDRESS>Department of Mathematics, University at Buffalo, State University of New
York, Buffalo, NY 14260

###### Abstract

Despite the vast literature on network dynamics, we still lack basic insights
into dynamics on higher-order structures (e.g., edges, triangles, and more
generally, $k$-dimensional “simplices”) and how they are influenced through
higher-order interactions. A prime example lies in neuroscience where groups
of neurons (not individual ones) may provide the building blocks for
neurocomputation. Here, we study consensus dynamics on edges in simplicial
complexes using a type of Laplacian matrix called a Hodge Laplacian, which we
generalize to allow higher- and lower-order interactions to have different
strengths. Using techniques from algebraic topology, we study how collective
dynamics converge to a low-dimensional subspace that corresponds to the
homology space of the simplicial complex. We use the Hodge decomposition to
show that higher- and lower-order interactions can be optimally balanced to
maximally accelerate convergence, and that this optimum coincides with a
balancing of dynamics on the curl and gradient subspaces. We additionally
explore the effects of network topology, finding that consensus over edges is
accelerated when 2-simplices are well dispersed, as opposed to clustered
together.

##  I Introduction

Interest in dynamical processes occurring on and through higher-order
structures in simplicial complexes has grown rapidly in recent years, with
examples including random walks Schaub  _et al._ (2020); Mukherjee and
Steenbergen (2016); Rosenthal (2014), social contagions Iacopini  _et al._
(2019); Matamalas, Gómez, and Arenas (2020), neuronal activity Petri  _et al._
(2014), cellular networks Vergne, Decreusefond, and Martins (2014),
synchronization Gambuzza  _et al._ (2020); Skardal and Arenas (2019); Arnaudon
_et al._ (2021), and consensus Maletić and Rajković (2014); Neuhäuser, Mellor,
and Lambiotte (2020). Such dynamics are useful for applications ranging from
biological processes Arai, Brandt, and Dabaghian (2014) to the design of
machine learning Roddenberry and Segarra (2019); Bunch  _et al._ (2020) and
signal processing Schaub  _et al._ (2021). In this pursuit, a variety of
mathematical tools have been developed ranging from mean field theory Gleeson
_et al._ (2012) and probability theory Snijders, Van de Bunt, and Steglich
(2010), to persistent homology Bardin, Spreemann, and Hess (2019), algebraic
topology Hatcher (2001), and Hodge theory Muhammad and Egerstedt (2006). Thus,
new problems in modeling and understanding the dynamical systems on simplicial
complexes are revitalizing interest in analytical methods from algebraic
topology and Hodge theory.

Consensus over networks is a popular model for collective decision making in
cognitive Hinsz (1990), social Fiol (1994), and biological systems Conradt and
Roper (2005), and more recently, it has been used to investigate
interconnected human-AI decision systems Song and Taylor (2021). Notably,
consensus dynamics provide a foundation for decentralized optimization
algorithms, which strategically implement consensus using synchronous or
asynchronous gossiping between nodes Tsitsiklis, Bertsekas, and Athans (1986);
Boyd  _et al._ (2005). Such algorithms are often employed to take advantage of
distributed computing infrastructure to more efficiently train machine
learning models, such as support vector machines Bijral, Sarwate, and Srebro
(2017); Huynh, Dutta, and Taylor (2021) and deep neural networks Assran  _et
al._ (2019); Niwa  _et al._ (2020); Vogels, Karimireddy, and Jaggi (2020);
Kong  _et al._ (2021). For such systems, each node trains a local model on
local data, and at the same time, communication between nodes enables them to
reach a consensus on what the model parameter should be. Notably, much of the
literature on decentralized learning aims to understand and optimize the
converge rate to consensus, which subsequently minimizes the time and number
of computations that are required to train a predictive model.

Herein, we extend the study of consensus to simplicial complexes by assigning
states to higher-dimensional simplices and allowing interactions through both
their lower and higher-dimensional faces. Our work complements other recent
extensions of consensus to simplicial complexes Bianconi (2021); Millán,
Torres, and Bianconi (2020); Gambuzza  _et al._ (2020) in which states
assigned to each node interact through higher-dimensional simplicies.
Contrasting this prior work, we propose a model that utilizes a generalization
of the Hodge Laplacian matrix in which the relative influence of higher and
lower-order interactions can be tuned via a _balancing parameter_ $\delta$.
This leads to our definition of a _generalized Hodge Laplacian_ , and our work
is motivated by a recent similar matrix called a normalized Hodge Laplacian
Schaub  _et al._ (2020).

With the help of an associated subspace decomposition called the _Hodge
Decomposition_ , we provide a convergence analysis of our model and use tools
from algebraic topology to investigate the role of _simplicial complex
homology_. We find that the collective dynamics converge to a low-dimensional
subspace lying within the homology subspace, which includes and generalizes
the notion of connected components by way of Betti numbers (i.e., connected
components characterize 0-dimensional homology). Furthermore, we show that
higher- and lower-order interactions can be optimally balanced to maximally
accelerate convergence, and that this optimum coincides with a balancing of
dynamics on the curl and gradient subspaces of a Hodge decomposition,
respectively. We additionally explore the effects of topology, finding that
consensus over edges is accelerated when 2-simplices are well dispersed, i.e.,
as opposed to clustered together.

Our findings provide new insights into how dynamical systems with higher-order
interactions can be engineered and optimized, which not only improves our
understanding of self-organizing biological and physical systems, but may also
support the development of improved consensus-based algorithms for machine
learning. Specifically, recent work Roddenberry and Segarra (2019); Bunch  _et
al._ (2020) has illustrated advantages for neural networks that use simplicial
complexes and Hodge Laplacians. Our formulation, analysis, and optimization of
consensus over simplicial complexes may help future research implement
decentralized versions of such emerging frameworks by allowing them to also
take advantage of decentralized optimization techniques that rely on
consensus.

This paper is organized as follows. We present background information in Sec.
II, our model and the role of homology in Sec. III, optimization theory in IV,
and a discussion in Sec. V.

##  II Background information

We provide here background information on the homology of simplicial complexs
(Sec. II.1) as well as Hodge Laplacians and decompositions (Sec. II.2).

###  II.1 Simplicial Complexes, Boundary Matrices and Homology

We begin with some definitions. _Simplicial complexes_ generalize undirected
graphs (which encode dyadic interactions) by allowing polyadic interactions
among sets of nodes/vertices. They consist of simplices of any number of
dimensions. For a set $\mathcal{V}$ of nodes, a $k$-dimensional simplex (or
simply, $k$-simplex) $\mathcal{S}\subset\mathcal{V}$ is a subset with $k+1$
elements. We will refer to $0$-simplices as nodes or vertices, $1$-simplices
as edges, and 2-simplices as “filled in” triangles. For a $k$-simplex
$\mathcal{S}$, any $(k-1)$-simplex that is a subset of $\mathcal{S}$ with $k$
elements is called its face. For any such face, $\mathcal{S}$ is called its
coface. For example, an edge $(i,j)$ has two faces, nodes $i$ and $j$, and
$(i,j)$ is a coface to both $i$ and $j$. Two $k$-simplices are called lower
adjacent if they share a face, and they are called upper adjacent if they
share a coface. For example, two edges are lower adjacent if they share a
common node, and the edges are upper adjacent if they are two sides of the
“boundary” of 2-simplex. (We will formally define boundaries below.)

A simplicial complex is a set of simplices in which any face of a simplex must
also be contained in the simplicial complex, and the intersection between
faces is either another simplex or an empty set. The dimension of a simplicial
complex is the maximum dimension of its simplices. For example, a
1-dimensional simplicial complex contains only nodes and edges and is
equivalent to an undirected graph. For simplicity, herein we will focus on
2-dimensional simplicial complexes; however, we emphasize that our results
naturally extend to higher dimensions.

Many of our calculations require simplicial complexes with oriented simplices.
An orientation of a $k$-simplex is an equivalence class of the ordering of its
nodes, where two classes are equivalent if they are the same up to an even
number of permuations. The most convenient way to orient a simplicial complex
is to order its vertices $x_{1},x_{2},\dots x_{n}$. From this ordering, for
all $k>1$ choose the orientation of each $k$-simplex
$\\{x_{i_{0}},x_{i_{1}},\dots x_{i_{k}}\\}$ to be $[x_{i_{0}},x_{i_{1}},\dots
x_{i_{k}}]$ such that $i_{0}<i_{1}<\dots<i_{k}$. Note that this ordering is
done only for bookkeeping purposes, and does not represent a direction to the
simplices.

Importantly, connectivity of faces and cofaces can be encoded using boundary
maps and their associated boundary matrices. Their precise definition stems
from studying functions that are defined over over $k$-simplices, and we call
such a function a $k$-chain, i.e., a linear combination of $k$-simplices. The
set of all $k$-chains for a simplicial complex is the finite-dimensional
vector space $C_{k}$. We can then define boundary maps
$\partial_{k}:C_{k}\rightarrow C_{k-1}$ by their action on a simplicial
complex:

$\partial_{k}([x_{0},x_{1},\dots,x_{k}])=\sum_{i=0}(-1)^{i}[x_{0},\dots,x_{i-1},x_{i+1},x_{k}]$
(1)

This takes a simplex to an alternating sum of its faces. Representing the map
$\partial_{k}$ gives us the boundary matrix $B_{k}$, with columns
corresponding to the $k$-simplices and rows corresponding to the
$(k-1)$-simplices. An entry $[B_{k}]_{ij}$ of $B_{k}$ is $\pm 1$ if the $i$th
$(k-1)$-simplex is a face of the $j$th $k$-simplex (where the sign indicates
orientation), and it is otherwise 0. Boundary matrices fully encode a
simplicial complex in that it can be recreated from its boundary matrices.

$B_{1}=\begin{tabular}[]{ c|c c c c c c c c c c }&[1, 2]&[1, 3]&[1, 4]&[2,
3]&[2, 5]&[3, 4]&[3, 6]&[4, 6]&[5, 6]\\\ \hline\cr 1&-1&-1&-1&0&0&0&0&0&0\\\
2&1&0&0&-1&-1&0&0&0&0\\\ 3&0&1&0&1&0&-1&-1&0&0\\\ 4&0&0&1&0&0&1&0&-1&0\\\
5&0&0&0&0&1&0&0&0&-1\\\ 6&0&0&0&0&0&0&1&1&1\\\ \end{tabular}$
$B_{2}=\begin{tabular}[]{ c|c c}&[1, 3, 4]&[3, 4, 6]\\\ \hline\cr[1, 2]&0&0\\\
[1, 3]&1&0\\\ [1, 4]&-1&0\\\ [2, 3]&0&0\\\ [2, 5]&0&0\\\ [3, 4]&1&1\\\ [3,
6]&0&-1\\\ [4, 6]&0&1\\\ [5, 6]&0&0\\\ \end{tabular}$

Figure 1:  Example simplicial complex and its boundary matrices. A simplicial
complex with 6 nodes, 9 edges, and 2 triangles. The nodes are indexed from 1
to 6. The edges of the simplicial complex have arrows not because they are
directed edges, but to show their chosen orientation, where the arrow points
to the node with a larger index. Its boundary matrices are given by $B_{1}$
and $B_{2}$.

In Fig. 1, we illustrate an example simplicial complex and its boundary
matrices. Note that the edges are undirected, and arrows only indicate the
simplices’ chosen orientations. Also note that it is a 2D simplicial complex,
since there are no simplices of dimension $3$ or greater (i.e., all higher
dimensional boundary matrices are empty). Observe that each column of $B_{1}$
corresponds to an edge $(i,j)$, and all entries in a column are zero except
for the $i$-th and $j$-th entries, which contain $-1$ and $1$, respectively.
The rows and columns of $B_{2}$ correspond to edges and 2-simplices,
respectively, and values $\pm 1$ indicate incidences between faces and
cofaces. The signs indicate whether or not the edges’ orientations match that
for a triangular cycle/boundary around each 2-simplex. For example, the
orientation of edge $(1,3)$ matches that for 2-simplex $(1,3,4)$. In contrast,
the orientation of $(1,4)$ does not since the cycle would be completed by
traversing from node 4 to node 1 and not 1 to 4.

Of note, this simplicial complex has four separate cycles: two cycles are the
boundary of a 2-simplex, $\\{(1,3),(3,4),(4,1)\\}$ and
$\\{(3,4),(4,6),(6,3)\\}$; and two cycles are not such a boundary:
$\\{(1,2),(2,3),(3,1)\\}$ and $\\{(2,3),(3,6),(6,5),(5,2)\\}$.

The study of cycles (and higher-dimensional “holes” including voids)
generalizes the study of connected components in graphs and is the focus of
homology. With these boundary functions and a simplicial complex $X$, we can
define the $n$th (simplicial) homology group $H_{n}(X)$ as
Ker$(\partial_{n})/$Im$(\partial_{n+1})$. The dimension of the $H_{n}$ is
called the $n$th Betti number. These Betti numbers provide us with important
topological information about simplicial complexes. The $0$th Betti number is
equal to the number of connected components in the simplicial complex, while
the $1$st gives us the number of cycles that are not “filled-in” by triangles.
Higher-dimensional Betti numbers indicate the number of higher-dimensional
“holes” in a simplicial complex Hatcher (2001).

Figure 2:  Hodge decomposition of a real-valued function defined over
1-simplices. (A) A 1-chain ${\bf x}\in\mathbb{R}^{n^{k}}$ that is defined over
$n^{k}=9$ edges (i.e., 1-simplices). (B) Projection ${\bf x}^{(g)}$ onto the
gradient subspace. (C) Projection ${\bf x}^{(c)}$ onto the curl subspace. (D)
Projection ${\bf x}^{(h)}$ onto the harmonic subspace.  Figure 3:  Balancing
parameter $\delta$ influences convergence rate of GHL-1 consensus dynamics.
Two simulations of GHL-1 consensus were run on the simplicial complex from
Fig. 1. The top row shows the Hodge decomposition of the dynamics when
$\delta=0.6$, while the bottom row has $\delta=-0.4$. Both simulations began
with the same initial condition. Notice that the curl subspace converges
faster when $\delta$ is negative, while the gradient subspace converges faster
when $\delta$ is positive. As expected, the harmonic subspace does not change
under these dynamics, and so the total function converges to the harmonic
projection.

###  II.2 Hodge Laplacians and Hodge Decomposition

The $k$-th combinatorial Hodge Laplacian

$L_{k}=B_{k}^{T}B_{k}+B_{k+1}B_{k+1}^{T}$ (2)

is a symmetric, positive semi-definite, and square matrix of size $n_{k}$,
where $n_{k}$ is the number of $k$-simplices in the simplicial complex. From
this definition, we have that $B_{k}^{T}B_{k}$ is calculated from the lower-
adjacencies of $k$-simplices, while $B_{k+1}B_{k+1}^{T}$ is from their upper-
adjacencies. Finally, notice that $L_{0}$ is recovers the widely known
combinatorial Laplacian of a graph: $L_{0}=B_{1}B_{1}^{T}=D-A$, where $A$ is
the adjacency matrix and $D$ is a diagonal matrix that encodes node degrees,
$D_{ii}=\sum_{j}A_{ij}$.

If for some $k$, we consider maps that give each $k$-simplex some real value
(i.e. real-valued $k$-chains), the space of all such maps is
$\mathbb{R}^{n_{k}}$, where $n_{k}$ is the number of $k$-simplicies in the
simplicial complex. Using the combinatorial Hodge Laplacian $L_{k}$, we can
decompose that space $\mathbb{R}^{n_{k}}$ into three mutually orthogonal
subspaces Lim (2015).

$\mathbb{R}^{n^{k}}=\text{im}(B_{k}^{T})\oplus\text{im}(B_{k+1})\oplus\text{ker}(L_{k})$
(3)

This is called the Hodge decomposition, and these three subspaces are known as
the gradient, curl, and harmonic subspaces, respectively. From this, we can
immediately see that the gradient subspace only depends on the
$(k-1)$-dimensional simplicies, the curl subspace depends on the
$(k+1)$-dimensional simplicies, and the harmonic subspace depends on both.
Importantly, the harmonic space and homology group are isomorphic vector
spaces Lim (2020), and the Betti number is the dimension of both spaces.

In Fig. 2(A), we set $k=1$ and consider a random real-valued function ${\bf
x}$ that is defined on the edges of the simplex that was shown Fig. 1. In
panels (B), (C) and (D) we depict, respectively, the projection of ${\bf x}$
onto the gradient, curl and harmonic subspaces. We denote these ${\bf
x}^{(g)}$, ${\bf x}^{(c)}$ and ${\bf x}^{(h)}$, respectively. Examining Fig.
2, we highlight several important properties of these subspaces. First, the
gradient components sum to zero along any cycles. Second, the curl components
are zero for edges that are not a 2-simplex boundary, and the entries sum to
zero around each node. Third, the harmonic component sums to zero around each
node, and it also sums to zero along each 2-simplex (but not the other cycles)
Schaub  _et al._ (2020).

##  III Consensus over Simplicial Complexes with weighted Hodge Laplacians

###  III.1 Model for Consensus

Herein, we present a new formulation for consensus dynamics over simplicial
complexes using Hodge Laplacians. However, rather than utilize combinatorial
Hodge Laplacians, we instead define and utilize Generalized Hodge Laplacians
(GHLs):

$L_{k}^{(\delta)}=(1+\delta)B_{k}^{T}B_{k}+(1-\delta)B_{k+1}B_{k+1}^{T}.$ (4)

Specifically, we introduce a balancing parameter $\delta\in[-1,1]$ to account
for the possibility that for some applications, it is reasonable to assume
that the higher- and lower-dimensional interactions should have different
strengths of influence. Notably, interactions through lower-adjacencies (e.g.,
via shared nodes when $k=1$) are more strongly weighted when $\delta>0$,
whereas interactions through higher-adjacencies (e.g., via shared 2-simplices
when $k=1$) are more strongly weighted when $\delta<0$. For the choice
$\delta=0$, the GHL $L_{k}^{(\delta)}$ recovers the combinatorial Hodge
Laplacian, i.e., $L_{k}^{(0)}=L_{k}$.

Importantly, we point out the relevance of the Hodge decomposition is exactly
the same for GHLs as for the combinatorial Hodge Laplacian. That is because
our introduction of $\delta$ in Eq. (4) is just a scalar re-weighting of the
two summands in the right-hand side of Eq. (2), and these matrices’ image
spaces are orthogonal and unchanged by $\delta$. We point out that this
“preservation” of the Hodge decomposition contrasts another recent
generalization called _normalized Hodge Laplacians_ Schaub  _et al._ (2020).
That is, normalization uses the degrees (i.e., number of neighbors) of
$k$-simplicies of different dimensions, and this can significantly alter
(i.e., rotate) the Hodge subspaces.

Given a GHL, we consider a model for consensus over a simplicial complex using
the following system of linear ordinary differential equations. We define the
linear dynamics

$\frac{d}{dt}\textbf{x}(t)=-L_{k}^{(\delta)}\textbf{x}(t)$ (5)

as Generalized Hodge Laplacian-k (GHL-k) consensus. Note that
$L_{0}^{(0)}=2L=2(D-A)$ again recovers the combinatorial graph Laplacian, and
so GHL-0 consensus with $\delta=-1$ is
$\frac{d}{dt}\textbf{x}(t)=-2L\textbf{x}(t)$, proportional to the standard
definition for consensus over a graph,
$\frac{d}{dt}\textbf{x}(t)=-L\textbf{x}(t)$ Olfati-Saber and Murray (2004). In
principle, one can define and study consensus over $k$-simplices of any
dimension. However, in this work we will reduce our scope to the case when
$k=1$ so that each edge $(i_{p},j_{p})$ is assigned a state
$x_{p}\in\mathbb{R}$, and their states collectively evolve over time. For the
remainder of this paper, we will identify fundamental insights about dynamics
evolving under Eq. (5) and the role that is played by the topology and
homology of the simplicial complex as well as the associated Hodge subspace.

###  III.2 Decoupling of Dynamics within Hodge Subspaces

We first show that the Hodge subspaces are invariant subspaces for the
dynamics evolving under Eq. (5). Consider a solution
$\textbf{x}(t)=\textbf{x}^{(g)}(t)+\textbf{x}^{(c)}(t)+\textbf{x}^{(h)}(t)$ to
Eq. (5), where $\textbf{x}^{(g)}(t)$, $\textbf{x}^{(c)}(t)$, and
$\textbf{x}^{(h)}(t)$, respectively, give the gradient, curl and harmonic
components (i.e., projections onto the respective subspaces). We substitute
this form into Eq. (5) to obtain

$\displaystyle\frac{d}{dt}\left[\textbf{x}^{(g)}(t)+\textbf{x}^{(c)}(t)+\textbf{x}^{(h)}(t)\right]$
$\displaystyle=-[L_{k}^{(\delta)}\textbf{x}^{(g)}(t)]-[L_{k}^{(\delta)}\textbf{x}^{(c)}(t)]-[L_{k}^{(\delta)}\textbf{x}^{(h)}(t)]$
$\displaystyle=-[(1+\delta)B_{1}^{T}B_{1}\textbf{x}^{(g)}(t)]-[(1-\delta)B_{2}B_{2}^{T}\textbf{x}^{(c)}(t)]-[0].$
(6)

These simplifications use

  * •

$B_{2}B_{2}^{T}\textbf{x}^{(g)}(t)=0$ since
$\textbf{x}^{(g)}(t)\in\text{ker}(B_{2}B_{2}^{T})$,

  * •

$B_{1}^{T}B_{1}\textbf{x}^{(c)}(t)=0$ since
$\textbf{x}^{(c)}(t)\in\text{ker}(B_{1}^{T}B_{1})$,

  * •

and $L_{1}^{(\delta)}\textbf{x}^{(h)}(t)=0$ since
$\textbf{x}^{(h)}(t)\in\text{ker}(L_{1}^{(\delta)})$.

Due to the orthogonality of these subspaces, we obtain the following decoupled
set of linear ODEs:

$\displaystyle\frac{d}{dt}\textbf{x}^{(g)}(t)$
$\displaystyle=-(1+\delta)B_{1}^{T}B_{1}\textbf{x}^{(g)}(t),$
$\displaystyle\frac{d}{dt}\textbf{x}^{(c)}(t)$
$\displaystyle=-(1-\delta)B_{2}B_{2}^{T}\textbf{x}^{(c)}(t),$
$\displaystyle\frac{d}{dt}\textbf{x}^{(h)}(t)$ $\displaystyle=0,$ (7)

which have the solutions

$\displaystyle\textbf{x}^{(g)}(t)$
$\displaystyle=e^{-t(1+\delta)B_{1}^{T}B_{1}}\textbf{x}^{(g)}(0),$
$\displaystyle\textbf{x}^{(c)}(t)$
$\displaystyle=e^{-t(1-\delta)B_{2}B_{2}^{T}}\textbf{x}^{(c)}(0),$
$\displaystyle\textbf{x}^{(h)}(t)$ $\displaystyle=\textbf{x}^{(h)}(0).$ (8)

In other words, the dynamics evolve separately within each Hodge subspace.
Because the matrices $B_{1}^{T}B_{1}$ and $B_{2}B_{2}^{T}$ are both positive
semi-definite, the gradient and curl components converge to zero with
increasing time, i.e., $\lim_{t\to\infty}{\bf x}^{(g)}(t)=0$ and
$\lim_{t\to\infty}{\bf x}^{(c)}(t)=0$. On the other hand, the harmonic
component remains fixed in time, and so the full state converges to
$\lim_{t\to\infty}{\bf x}(t)={\bf x}^{(h)}(0)$, which we will call the
_consensus vector_.

Of particular interest is the _asymptotic convergence rate_ $\mu(\delta)$ for
Eq. (5), which is the convergence rate for the slower subspace dynamics. That
is, each subspace in the Hodge decomposition also converges, but not all at
the same rate. Let $\lambda_{2}^{(1)}$ and $\lambda_{2}^{(2)}$, be the
smallest nonzero eigenvalues of $B_{1}^{T}B_{1}$ and $B_{2}B_{2}^{T}$,
respectively. The gradient subspace is dependent on the lower-dimensional
interactions and has convergence rate $(1+\delta)\lambda_{2}^{(1)}$. The
convergence curl subspace is similarly determined by the higher-dimensional
interactions, and so its convergence rate is $(1-\delta)\lambda_{2}^{(2)}$.
Notably, the balancing parameter $\delta$ tunes the relative convergence rates
for the invariant dynamics in the gradient and curl subspace. It also follows
that
$\mu(\delta)=\min\\{(1+\delta)\lambda_{2}^{(1)},(1-\delta)\lambda_{2}^{(2)}\\}$.

In Fig. 3, we see illustrate timeseries ${\bf x}(t)$ for simulations of Eq.
(5) as well as the projections onto the three Hodge subspaces. The top and
bottom rows indicate the same initial condition ${\bf x}(0)$ but two different
choices for the balancing parameter: (top) $\delta=0.6$; and (bottom)
$\delta=-0.4$. Each curve indicates a trajectory $x_{p}(t)$ for some edge
$(i_{p},j_{p})$ in the simplicial complex that was shown in the previous
figures. Observe in both rows that the gradient and curl components converge
to zero, whereas the harmonic components remain fixed in time. In fact, for
both values of $\delta$ the system states converge to the same consensus
vector ${\bf x}^{(h)}(0)$—that is because the initial conditions are the same.
That is, $\delta$ has no effect on the final state. In contrast, $\delta$ has
a significant impact on the convergence rates for the invariant dynamics
within the gradient and curl subspaces. Observe for significantly positive
$\delta$ (i.e., when the lower-dimensional interactions are prioritized) that
the gradient component converges much more quickly than the curl component.
The reverse is true when $\delta$ is sufficiently negative.

Figure 4:  Consensus dynamics converge to the harmonic subspace that is
isomorphic to the homology space. The zero and first dimensional homology
groups for a few example simplicial complexes. The homology groups are
illustrated as vector spaces. We plot the convergence of GHL-0 (A) and GHL-1
(B) consensus with 3 different random initial conditions starting from the
white circles. Notice that they all converge to the harmonic subspace. Within
each panel (A) and (B), the initial conditions are the same. While the
harmonic subspaces are plotted in full, the converging flows are projected
down into 3 dimensions to simplify this visualization.

###  III.3 Uncovering the Role of Homology

In the previous subsection, we saw that the consensus dynamics remains
unchanged in the harmonic subspace. That is to say, the consensus vector is
the harmonic component of the initial condition. It’s therefore important to
take a look at properties of the harmonic subspace and their dependence on the
topology and homology of the simplicial complex. Importantly, the harmonic
subspace and homology group are isomorphic vector spaces, and so the space of
possible solutions to Eq. (5) is determined by the homology of the simplicial
complex. In particular, the dimension, i.e., Betti number, gives the dimension
of the solution space.

In Fig 4, we show several example simplicial complexes, their associated
harmonic subspaces of $L_{k}$ (which are identical to those of
$L_{k}^{(\delta)}$), and the convergence of GHL-k consensus to the harmonic
space with either (A) $k=0$ or (B) $k=1$.

Starting with Fig 4(A), in the first column we depict 1-dimensional simplicial
complexes such that Betti-0 (i.e., the dimension of both the harmonic and
homology spaces for $L_{0}$) increases as one considers lower rows (see second
column). Note that Betti-0 is simply the number of connected components. In
this case, the harmonic subspace of $L_{0}$ is spanned by vectors that have a
constant value for entries associated with nodes in a given component and all
other entries are zeros. If there are $\beta_{0}$ such components, then there
are $\beta_{0}$ such vectors that span $\text{Ker}(L_{0})$. The third column
illustrates that GHL-0 consensus—which is equivalent to the traditional
definition of consensus over an undirected graph—converges to a consensus
value in this harmonic subspace. For practical purposes, one typically assumes
that the graph is strongly connected so that the harmonic space is a
1-dimensional line and the consensus value gives the mean value for the
initial condition $\\{x_{p}(0)\\}$.

Turning our attention to Fig 4(B), in the first column we depict 2-dimensional
simplicial complexes such that Betti-1 (i.e., the dimension of both the
harmonic and homology spaces for $L_{1}$) increases as one considers lower
rows (see second column). Observe that Betti-1 corresponds to the number of
cycles that are not the boundary of a 2-simplex, i.e., “unfilled triangles” in
these three examples. The third column illustrates that GHL-1 consensus
converges to a consensus value in the harmonic subspace $\text{Ker}(L_{1})$.
For some applications, we predict that it is beneficial for ${\bf x}(t)\to 0$.
This can be achieved by either choosing the initial condition to be orthogonal
to the harmonic subspace, ${\bf x}^{(h)}(0)=0$, or ensuring that the
simplicial complex has a zero-dimensional harmonic subspace, i.e., no homology
groups (i.e., unfilled cycles) in dimension $k$. For other applications—in
particular, multiscale dynamical systems, it may also be beneficial for the
dynamics to instead converge to a low-dimensional subspace, within which other
dynamics could occur (e.g., oscillations and synchronization). Thus, although
we call our proposed model GHL-k “consensus”, we expect the findings that we
present herein to also be relevant to much more diverse types of dynamics (see
e.g., Millán, Torres, and Bianconi (2020); Schaub  _et al._ (2020)).

##  IV Optimizing Consensus Convergence Rate

Figure 5:  Balancing invariant subspace dynamics maximally accelerates
convergence. Working with the simplicial complex from Fig. 1, plot (A) shows
the convergence rate of the total value as well as the gradient and curl
subspaces in GHL-1 consensus simulations for different values of $\delta$. The
orange and purple lines show the expected values for these convergence rates
calculated using the eigenvalues of the component matrices of the generalized
Hodge Laplacian as explained in Section III.2. For this simplicial complex,
the balancing parameter that optimizes convergence rate is
$\delta^{*}\approx.08$. In (B), we plot the log normed error, $\log{(||{\bf
x}(t)-\bf{x}^{(h)}||_{2})}$, again for the total as well as the gradient and
curl subspaces. Remember that the consensus here converges to the harmonic
projection of the initial condition.

###  IV.1 Convergence Rate is Maximized by Optimal Balancing of Subspace
Dynamics

Recall the convergence rate of Eq. (5) is given by
$\mu(\delta)=\min\\{(1+\delta)\lambda_{2}^{(1)},(1-\delta)\lambda_{2}^{(2)}\\}$
and that convergence occurs faster when $\mu$ is larger. We therefore seek to
maximize the convergence rate by solving the following optimization problem

$\delta^{*}=\text{argmax}_{\delta}~{}\mu(\delta)=\frac{\lambda_{2}^{2}-\lambda_{2}^{1}}{\lambda_{2}^{2}+\lambda_{2}^{1}}.$
(9)

In this case, the optimum coincides with the intersection of two lines and is
found by equating $(1+\delta)\lambda_{2}^{(1)}=(1-\delta)\lambda_{2}^{(2)}$.
In other words, the fastest total convergence requires that we choose the
balancing parameter $\delta$ so that the decoupled dynamics with the gradient
and curl subspaces have the same convergence rate. Such an optimum may be
beneficial to engineering applications and the optimization of naturally
occurring systems.

In Fig. 5, we provide numerical validation for such an optimum. The solid
orange and dashed pink lines lines Fig. 5(A) indicate our theoretical
predictions of convergence rate within the gradient and curl subspaces,
respectively,. These are in excellent agreement with empirically observed
convergence rates for these two spaces, which are indicated by the orange
triangles and pink squares, respectively. Black dots indicate empirically
observed convergence rates for the full system, and note that they align the
minimum of the two subspace convergence rates.

Vertical dotted lines in Fig. 5(A) highlight three values of $\delta$ that are
chosen to be below, at, and above the optimum $\delta^{*}\approx 0.08$. In
Fig. 5(B), we plot empirically observed values of the error $||{\bf x}(t)-{\bf
x}(\infty)||$, in log scale, for the total system as well as the gradient and
curl components. Considering the left and right subpanels of Fig. 5(B),
observe that the gradient component converges more slowly than the curl
component when $\delta<\delta^{*}$, and the opposite occurs when
$\delta>\delta^{*}$. In either case, the slower dynamics slows the full
system’s convergence. As shown in the center panel of Fig. 5(B), the system
converges fastest at the optimum $\delta^{*}$, where the curl and gradient
components have the same convergence rate. Under this optimal choice for the
balancing parameter, we say that $L_{1}^{(\delta^{*})}$ is _balanced_.

Figure 6: 2-simplex arrangement affects convergence rate of curl component.
(A) Simplicial complexes with different 2-simplices. Since they have identical
nodes and edges, they have the same gradient subspaces but different curl
subspace. (B) Convergence rate $\mu(\delta)$ versus $\delta$ for GHL-1
consensus on these three simplicial complexes. The three dots highlight how
the optimum $\delta^{*}$ differs across the three simplicial complexes.
Notably, convergence is fastest for SC 3 (i.e., where the 2-simplices are most
dispersed), and the optimum occurs at a larger value of $\delta$.

###  IV.2 Dispersed Higher-order Interactions Accelerate Consensus

It is natural to wonder how the optimum $\delta^{*}$ is influenced by the
topology and homology of a simplicial complex. To this end, we created three
example simplicial complexes that have the same underlying $1$-skeleton (the
same nodes and edges), but different 2-simplices. Specifically, each
simplicial complex has the same number of triangles, but they have different
placements, as shown in Fig. 6(A). Because they share the same nodes and
edges, all three simplicial complexes have the same $B_{1}$ boundary matrix
and associated gradient subspace. In contrast, their $B_{2}$ boundary matrices
differ as well as their curl subspaces. In Fig 6(B), we plot the convergence
rates of the gradient and curl components for the three simplical complexes.
First, observe that their gradient subspaces’ convergence rates are all the
same (see black line). In contrast, their curl subspaces have different
convergence rates (see dashed lines). The three dots highlight the optimum
$\delta^{*}$ for each simplicial complex. Observe that these dots move up and
to the right as one considers simplicial complex 1, 2 and 3. That is,
convergence occurs fastest for simplicial complex 3 and slowest for simplicial
complex 1. It follows that convergence is faster for the curl subspace (and
the total system) when the 2-simplices are spaced apart, i.e., rather than
clustered together.

Another interesting property is that while the optimal convergence rate
$\mu(\delta^{*})$ increases with each subsequent simplicial complexes (i.e., 1
to 3), the associated optimum $\delta^{*}$ also increases. Recall for our
definition for the weighted Hodge Laplacian $L^{(\delta)}_{k}$ given by Eq.
(4) that larger $\delta$ implies stronger coupling via lower-dimensional
interactions (i.e., adjacency via shared nodes). In other words, because the
total coupling weight is conserved [i.e., the weights are $(1+\delta)$ and
$(1-\delta)$], and because the curl subspace (which arises due to higher-
dimensional interactions) converges faster for simplicial complex 3, this
allows the coupling weight to be increased for the lower-dimensional
interactions. We find this to be somewhat counter intuitive—that is, as
convergence through higher-dimensional interactions becomes faster, the
optimal system relies more strongly on the lower-dimensional interactions.

Figure 7:  2-simplex placement changes optimal convergence rate. Visualization
of the optimal convergence rate $\mu(\delta^{*})$ and the optimal balancing
parameter $\delta^{*}$ for different simplicial complexes that have the same
nodes and edges but different locations and numbers of 2-simplices.
Interestingly, we do not identify a clear relation between the number of
2-simplices and $\mu(\delta^{*})$ or $\delta^{*}$. However, we again find that
convergence tends to be faster when the 2-simplices are more dispersed.

In Fig. 7, we further study the the optimal convergence rate $\mu(\delta^{*})$
and optimal coupling $\delta^{*}$ for several additional simplicial complexes
that have the same nodes and edges but different arrangements of 2-simplices.
We also now allow them to contain different numbers of 2-simplices, but we do
not identify an obvious relation between that number and $\mu(\delta^{*})$ or
$\delta^{*}$. That said, similar to Fig. 6 we observe that faster convergence
is associated with larger $\delta^{*}$. However, it is important to stress
that in general, this need not be the case. This phenomenon occurs here
because we restricted our attention to simplicial complexes that have the same
nodes and edges, and thus the same gradient subspaces.

##  V Discussion

We have presented here a model for consensus dynamics over simplicial
complexes that uses a generalization of the Hodge Laplacian that we call
_generalized Hodge Laplacians_. It is likely that in many real-world
applications of simplicial complex dynamics, higher- and lower-dimensional
interactions between simplices are not equally weighted. In these
applications, the generalized Hodge Laplacian could prove a useful tool in
modeling this phenomenon.

Our work identified a decoupling of consensus dynamics given by Eq. (5) onto
the Hodge subspaces, and we showed that the collective dynamics converge to
the harmonic space (which is isomorphic to the homology group of the
simplicial complex). These findings open several new direction of research in
working with dynamics, consensus, and homology of simplicial complexes. The
literature on consensus and other dynamical processes over networks (e.g.,
diffusion) has been restricted to studying convergence toward a 1D subspace.
Understanding how systems converge toward higher-dimensional subspaces
represents an important new direction that may be foundational to more
complicated dynamical systems. In particular, biological neurosystems are also
likely to organize along low-dimensional subspaces and/or manifolds McIntosh
and Jirsa (2019), and it has been noted that deep neural networks are non-
convergent and may even approach limit cycles Bock and Weiß (2019) that are
within subspaces of dimension greater than 1.

On the practical side, we also developed theory to optimally balance higher-
and lower-order interactions to maximally accelerate convergence.
Interestingly, the optimum coincides with a balancing of dynamics on the
gradient and curl subspaces of the Hodge decomposition. We additionally
explored the effects of topological structure, finding that consensus over
edges is accelerated when 2-simplices are well dispersed. There is much to
learn about the specific quantitative reasons for this behavior. Research may
also be done into the optimal balancing parameter and convergence rate for
randomly generated simplicial complexes, or those that satisfy certain
criteria specific to neuroscience, machine learning, or social network
applications.

Other possible future directions can be found by revisiting our modeling
choices. While GHL consensus is defined for higher dimensions, we restricted
our simulations to the $k=1$ case for simplicity. It is worth investigating
not only GHL-$k$ consensus for $k>1$, but also properties of the gradient,
curl, and harmonic subspaces for these higher dimensions. At the same time, we
introduced a balancing based on the combinatorial Hodge Laplacian, and it
would also be beneficial to balance normalized Hodge Laplacians Schaub  _et
al._ (2020) (which were recently introduced to model random walks on
simplicial complexes). Our generalized Hodge Laplacian may be useful for other
dynamics as well, and we are currently investigating the relation between our
work and explosive synchronization on simplicial complexes Millán, Torres, and
Bianconi (2020). Another interesting direction would be to consider adaptive
dynamics in which the balancing parameter is allowed to change with time,
i.e., $\delta(t)$. Alternatively, we could introduce a learning rate
$\beta(t)$ and replace Eq. (5) by $\frac{d}{dt}{\bf
x}(t)=-\beta(t)L^{\delta}_{k}{\bf x}(t)$. We could allow the learning rate to
change over time, which is commonly done in decentralized machine learning.

Finally, we conclude by highlighting how our proposed model and theory may be
useful to help support the development of decentralized algorithms for machine
learning and artificial intelligence (AI). In this application, data is
dispersed across a network of computer nodes (e.g., separate CPUs that are
either organized into high-performance computer cluster or are spatially
dispersed through a network of remote sensors). Such algorithms already rely
on various forms of consensus in which the states of nodes (i.e., 1-simplices)
coordinate to learn a single model for data. In this context, most research
focuses on designing algorithms that converge as fast and efficient as
possible (see review Nokleby, Raja, and Bajwa (2020)). Our findings here
suggest that it may be feasible to generalize such algorithms to also
coordinate the states of 2-simplices and higher-dimensional simplices. This
potential new direction for decentralized learning is especially relevant,
given this community’s growing interest Roddenberry and Segarra (2019); Bunch
_et al._ (2020) to utilize simplicial complexes and Hodge Laplacians to design
deep neural networks.

Python code that implements GHL-1 consensus on simplicial complexes and
reproduces our experiments can be found at Ziegler (2021).

###### Acknowledgements.

CZ, HD, and DT were supported in part by the National Science Foundation
(EDT-1551069). DT also acknowledges NSF grant DMS-2052720 and the Simons
Foundation (grant #578333). PSS acknowledges NSF grant MCB-2126177.

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