constructing lyapunov functions for large-scale ...jmaidens/projects/siap_li.pdf · constructing...

Constructing Lyapunov functions for large-scale hierarchical

networks of nonlinear differential equations

Michael Y. Li∗ and John Maidens†

Abstract

We present a graph theoretic framework for modeling complex hierarchical systems and a methodof systematically constructing Lyapunov functions for these systems. The effectiveness of our methodis illustrated by two applications: a new control protocol for the self-organization of swarms of mobileagents into a hierarchy of clusters, and a new result on the global stability of the endemic equilibriumfor a multi-group, multi-stage epidemic model.

Keywords. Lyapunov functions, hierarchical graphs, nonlinear differential equations on networks,Kirchhoff’s Matrix Tree Theorem, Tree-Cycle Identity, global stability.

2010 AMS mathematics subject classification. 34D23, 93A13, 92D30

1 Introduction

Many complex systems arising in science and engineering can be modeled by differential equations onnetworks [16, 35]. The network is described by a graph where each vertex is assigned a set of statevariables and a rule that describes its internal dynamics. A vertex can represent, for example, theconcentration of chemicals in a solution [21], communities of species living in discrete spacial patches[19], the velocity and position of individual vehicles in a swarm [8] or single neurons in a neural network[5]. Directed edges link the vertices, corresponding to couplings between the systems on each vertex. Inthe above contexts, the couplings can model the production of new chemical compounds, dispersal ofanimals between patches, vehicle interactions or communication between neurons.

Often the structure of the system of differential equations induces a hierarchical structure on thenetwork [27]. This can result from different dynamics on different scales such as inter- vs. intra- citymovement in global transport networks, the flow of capital within vs. between financial institutions, orthe dispersal of various species at different rates in a predator-prey system.

We are interested in using Lyapunov functions to study the asymptotic behaviour of these hierar-chically coupled systems. A Lyapunov function for a system of differential equations x = f(x) on asubset D ⊂ Rn is a map V : D → R whose derivative along trajectories of the system V (x) = ∇V · f(x)is non-positive everywhere in D. The existence of a Lyapunov function can be used to demonstrateasymptotic properties of the system, such as the stability or dichotomy of invariant sets like equilibria.We want to provide a general method of constructing Lyapunov functions for large-scale systems fromsimple Lyapunov functions Vi corresponding to each individual vertex i. Li and Shuai [16] have recentlydeveloped a method of choosing coefficients ci to construct global Lyapunov functions of the form

V (x) =

n∑i=1

ciVi(xi).

The goal of this paper is to develop a graph theoretic framework for modeling complex hierarchicalsystems and to extend the method of Li and Shuai to our new framework. To model hierarchy in networks,we label graph vertices with multiple indices and discuss how this labeling induces a multi-dimensional

∗Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada([email protected]).†Electrical & Computer Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

([email protected]).

1

Lyapunov functions for hierarchical networks 2

structure on the graph and its weight matrix. In many applications, the hierarchical information isessential to the understanding and analysis of the systems studied. Thus the methods described in thispaper can be applied in a wider range of cases than the original work of Li and Shuai.

Our construction of Lyapunov functions for these systems relies on Kirchhoff’s Matrix Tree Theoremto choose coefficients. The method is inspired by the proof of Kirchoff’s formula for calculating theeffective conductance between two nodes of a linear electrical network. For each spanning tree in thenetwork, adding a single “battery” edge between the source vertex and the sink vertex yields a graphwith a unique cycle in which the conductance can easily be found [2, 34]. We use the same idea – creatingunicyclic graphs from a spanning tree and a single edge – for the analysis of nonlinear networks. Notingthat each spanning unicyclic graph can be formed in a number of different ways gives us a method ofgrouping terms in the Lyapunov function’s derivative. This new combinatorial result is summarizedin the Tree Cycle Identity (Theorem 1). This then leads to a simple criterion for verifying that V isnon-positive (Theorem 2). Rather than considering the network as a whole, with its multiple coupledfeedback loops, Theorem 2 allows us to show that V ≤ 0 by considering only individual feedback loops.

In the second half of the paper, we demonstrate the applicability of our results by using them toconstruct two Lyapunov functions of different forms in two vastly different contexts. Section 4 outlinesa novel protocol for controlling swarms of mobile agents and gives a detailed convergence analysis basedon our methods. Section 5 describes class of models of the transmission dynamics of multi-stage diseasesin heterogeneous populations and uses the techniques we have developed to prove a new result on theuniqueness and global stability of the endemic equilibrium. In both cases, understanding the system’shierarchical structure is essential in the stability analysis.

2 Preliminaries

2.1 Terminology from graph theory

A weighted directed graph, or weighted digraph, is a graph G = (V (G), E(G)) together with a weightfunction ω : E(G) → R that assigns to each edge of G a real-valued weight. We can extend the domainof the weight function ω to the set of subgraphs of G by defining the weight of a subgraph as the productof the weights of all edges in the subgraph.

A weighted digraph can be described by its weighted adjacency matrix AG , with entries aij representing

the weight ω(~ij) of the edge ~ij directed from vertex i to vertex j. The set of neighbours of vertex i isgiven by Ni = {j ∈ V (G) : ~ji ∈ E(G)} and consists of all vertices with incoming connections to i.

A sequence of edges of G i → j → · · · → k → i that begins and ends at the same vertex is calleda directed cycle or feedback cycle if the vertices i, j, . . . , k are all distinct. In the special case that thegraph G is a signal flow graph or block diagram then the concept of a directed cycle coincides with thefamiliar notion of a feedback loop from linear control theory [7].

Given an n-vertex graph G, a subgraph of G is called a directed spanning tree rooted at vertex i if itcontains n − 1 edges and a directed path from the root vertex i to each other vertex of G. A spanningunicyclic subgraph of G is a spanning directed subgraph consisting of a collection of disjoint rooteddirected trees whose roots form a directed cycle. For a detailed discussion of these concepts, we refer thereader to [23].

2.2 Hierarchical graphs

Definition 1. A graph G = (V (G), E(G)) is said to be hierarchical if its vertices are indexed by orderedn-tuples of integers (i1, . . . , in) where n ≥ 2. We call n the dimension of the hierarchical graph.

Graphs whose vertices are labeled by multiple indices have been studied in limited contexts. Multi-dimensional Manhattan networks were indexed by vectors because of their inherent spatial structure [6],and graphs resulting from the Kronecker product of smaller graphs inherit the indexing of the originalgraphs [15, 36]. These graphs both have very regular structures. The generality of our definition allowsthe framework of hierarchical graphs to be used to model many real-world networks whose structuresare often highly irregular. Examples of networks that lend themselves naturally to hierarchical mod-eling include community structures in sociology and ecology, spatially discrete systems that inherit amulti-dimensional structure from the ambient space, and distributed process control schemes.


To define certain familiar notions such as the adjacency matrix of a hierarchical graph on M vertices,we need to adopt a system of ordering the vertices. Throughout this paper we will use the lexicographicordering. We use the symbol (i) to denote the i-th vertex in the lexicographic ordering. The graph’sweighted adjacency matrix AG is then the M by M matrix whose entry in the j-th row and k-th columnis the weight ajk of the edge directed from vertex (j) to vertex (k).

We call graphs labeled by multiple indices hierarchical because a graph of dimension n can naturallybe organized into n levels of nested clusters.

Definition 2. For each l with 1 ≤ l ≤ n we define a level-l cluster

[i1, . . . , il] = {(j) = (j1, . . . , jn) ∈ V (G) : ji = i1, . . . , jl = il}.

In the case n = 2 we will simply say cluster to mean level-1 cluster.

Each level of clustering induces a partitioning of the graph’s adjacency matrix into a block matrix.Conversely, we can always define a hierarchical graph based on a block matrix, where the system ofblocks determines the clusters in the graph. The case n = 2 is illustrated in Figure 1 for a graph withM = 13. The higher-dimensional cases follow by partitioning the diagonal blocks in the same way.

(a) A 2-dimensional weighted hierarchical graph with13 vertices and 4 clusters.

(b) Vertexordering

(c) Weighted block adjacency matrix corre-sponding to the hierarchical graph in (a).

Figure 1: A hierarchical graph and the corresponding block adjacency matrix

The Laplacian matrix plays an important role in the study of dynamical systems on graphs. It isused, for example, in the study of consensus in networks of autonomous agents [24] and the study ofelectric [2] and hydraulic [20] flows. The Laplacian matrix LG is defined as LG = DG −AG where DG isthe in-degree matrix

DG = diag

∑(j)∈V (G)

aj1, . . . ,∑

(j)∈V (G)

ajM

.

Since the columns of LG all sum to zero, the cofactors corresponding to the entries of each column ofLG are all identical. Thus for each i ∈ {1, ...,M} we can define the Laplacian cofactor ci as the cofactorcorresponding to any entry of the i-th column of LG . Kirchhoff’s Matrix Tree Theorem allows us toexpress the cofactor ci as the sum of the weights of directed spanning trees rooted at vertex (i).

Proposition 1 (Kirchhoff’s Matrix-Tree Theorem).For each vertex (i) ∈ V (G) we have

ci =∑T ∈T(i)

ω(T )

where T(i) denotes the set of directed spanning trees in G rooted at vertex (i).

Proof. See [14, 23].


2.3 Differential equations on hierarchical networks

We can build a system of differential equations on an n-dimensional hierarchical graph G by assigningto each vertex (i) = (i1, . . . , in) a nonlinear equation of the form

xi = fi(xi)

where xi ∈ Rmi . We then assign to each edge directed from vertex (j) to vertex (i) a coupling functiongji (x) that represents the influence of the variable xj on xi. If there is not an edge directed from vertex

(j) to vertex (i) we set gji ≡ 0. We can then write the coupled system as

xi = fi(xi) +∑

(j)∈N(i)

gji (x). (1)

We assume that the functions fi and gji are sufficiently well-behaved to guarantee the existence anduniqueness of solutions to (1).

Our goal is to systematically construct a global Lyapunov function for the system (1) as a linearcombination of Lyapunov candidate functions V(i) corresponding to each vertex (i).

3 Main results

3.1 Tree cycle identity for hierarchical digraphs

Consider a weighted digraph G with non-negative weights aji . Assign to each edge ~ji an arbitrary edgefunction F i

j : RM → R. The vertex function for vertex (i) is then given by

F(i)(x) =∑

(j)∈N(i)

ajiFji (x).

We wish to assign weights w(i) to each vertex (i) such that the weighted sum

H(x) =∑

(i)∈V (G)

w(i)F(i)(x)

can be expressed in terms of feedback cycles in G. This will allow us to determine properties of theoverall sum H based on properties of smaller sums around each feedback cycle, which are much easierto verify in practical applications. To develop a method of choosing these vertex weights, we considerunicyclic subgraphs of G. We denote the set of all spanning unicyclic subgraphs of G by P.

An element P of P with cycle length l can be expressed in precisely l distinct ways as the union ofa directed spanning tree and a single directed edge (see Figure 2). Indeed, removing each edge ~ji fromthe cycle CP of P yields a directed spanning tree rooted at vertex (i).


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

i

j

k

(a) Spanning unicyclic graph formedfrom a directed spanning tree rootedat vertex i by adding an edge di-rected from vertex j to i

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

k

i

j k

j

i

(b) The same spanning unicyclic graph shown canbe formed in two other ways, from directed spanningtrees rooted at vertices j and k.

Figure 2: Construction of spanning unicyclic graphs as the union of a directed spanning tree and a singleedge

In the following theorem, we show that by choosing vertex weights w(i) equal to the cofactors ci ofthe weighted Laplacian matrix LG we can re-organize the weighted sum H in terms of feedback cyclesin G. This result was first established for 1-dimensional networks by Li and Shuai [16]. We extend theirresult to the case of d-dimensional hierarchical networks.

Theorem 1 (Tree Cycle Identity for hierarchical graphs).Let ci be given as in Proposition 1. Then

H(x) =∑

(i)∈V (G)

ciF(i)(x) =∑P∈P

ω(P)∑

~ji∈E(CP)

F ji (x)

Proof. By Proposition 1

ci =∑T ∈T(i)

ω(T ).

Then

∑(i)∈V (G)

ciF(i)(x) =∑

(i)∈V (G)

∑T ∈T(i)

ω(T )

∑(j)∈N(i)

ajiFji (x)

=

∑(i)∈V (G)

∑T ∈T(i)

∑(j)∈N(i)

ajiω(T )F ji (x).

Now, for each triple((i), (j), T

)with (i) ∈ V (G), (j) ∈ N(i) and T ∈ T(i) there is a unique spanning

unicyclic graph P ∈ P formed from T and the edge ~ji. Conversely, for each pair(Q, ~ji

)with Q ∈ P

and ~ji ∈ E(CP) there is a unique T ∈ T(i) resulting from the removal of edge ~ji. Thus there is a direct

correspondence between triples of the form((i), (j), T

)and pairs of the form

(Q, ~ji

). Then noting that

ajiω(T ) = ω(P), we see that

H(x) =∑

(i)∈V (G)

∑T ∈T(i)

∑(j)∈N(i)

ajiω(T )F ji (x) =

∑P∈P

ω(P)∑

~ji∈E(CP)

F ji (x).

3.2 Constructing Lyapunov functions via feedback cycles

Constructing Lyapunov functions V for large-scale networked systems is difficult due to the unwieldynumber of terms in the Lyapunov derivative of V with respect to system (1). Even when each of the M


vertices is assigned only a single state variable, the number of terms in the expression

V (x) =∑

(i)∈V (G)

∂V

∂xi

fi(xi) +∑

(j)∈V (G)

gji (x)

varies as the square of M .

Theorem 1 provides us with a method of grouping terms in the Lyapunov function’s derivative interms of cycles in G. This gives us an easily-verifiable condition to guarantee that V (x) ≤ 0 for all x.

Definition 3. We say that the set of edge functions {F ji : (i), (j) ∈ V (G)} satisfies the cycle condition

if, for any directed cycle C in G, we have ∑~ji∈E(C)

F ji (x) ≤ 0.

Theorem 2. Consider a Lyapunov candidate function of the form

V (x) =∑

(i)∈V (G)

ciV(i)(x)

where each V(i) : RM → R is a function corresponding to vertex (i) and ci is given as in Proposition 1.

Suppose that there exists a set of edge functions {F ji : (i) ∈ V (G), (j) ∈ N(i)} such that

a) we haveV(i)(x) ≤ F(i)(x)

and

b) the set of edge functions satisfies the cycle condition given in Definition 3.

Then V satisfies V (x) ≤ 0. Namely, V is a Lyapunov function for the system (1).

Proof. By Theorem 1 we have

V (x) ≤∑

(i)∈V (G)

ci∑

(j)∈N(i)

ajiFji (x) =

∑P∈P

ω(P)∑

~ji∈E(CP)

F ji (x) ≤ 0.

The simplicity of this proof illustrates the power of Theorem 1. We now move on to two applicationsfrom very different contexts to demonstrate the wide range of applicability of these techniques.

4 Application: Globally stable clustering for multi-agent swarms

4.1 Background

The mathematical study of flocking behaviour in multi-agent systems has found applications in diverseareas, from the study of bird flocks and schools of fish [22, 26] to the behaviour of human crowds inemergency situation [18, 25] to engineering applications such as the control of vehicle formations [4]. Toaccurately model systems of autonomous agents we require that the motion of the agents be based ondecentralized control schemes that use only information available locally to each member of the flock. Themethod of artificial potentials [32] provides one method of stabilizing flocks based on local information.

In [28, 29], Tanner, Jadbabaie and Pappas provide a method of stabilizing swarms of agents usingartificial potentials. Their analysis uses a Lyapunov function to demonstrate that their protocol stabilizesthe formation at a local minimum of the artificial potentials and leads to the alignment of all agents.Communication networks among agents considered in [28, 29] are either complete (i.e. every agentcommunicates with every other agent) or dynamic (i.e. communication links appear and disappearbased on proximity of agents).


In the design of control protocols for multi-agent systems, it is desirable to reduce the number ofcommunication links among agents and make more efficient use of computational resources. We proposea hierarchical control scheme that is more efficient in this sense than the complete networks consideredin [28, 29]. The key steps of our hierarchical potential clustering (HPC) protocol are:

a) run a clustering algorithm based on the initial positions of the agents

b) assign leaders to each cluster

c) implement the artificial potential scheme (with complete interaction graph) within each cluster and

d) implement a velocity consensus scheme between the cluster leaders.

4.2 Model

We model each agent by specifying its position r = (x, y) ∈ R2 and velocity v = r. The dynamics ofeach agent is governed by the 4-dimensional system

r = v,

v = u,

where the thrust u ∈ R2 is the control input.Before designing the control inputs, agents are clustered based on their position using an algorithm

seeking to minimize the sum-squared distance within each cluster. See [1, 12] for a review of clusteringalgorithms. If we label the resulting clusters with integers 1, . . . , N then the clustering induces a labelingof the agents (k) = (i, j) where i is the cluster index and j ∈ {1, . . . , ni} is used to distinguish agentswithin cluster i. We use the index (i, 1) to designate the leader of cluster i. Clustering algorithms suchas the one considered in [9] provide a natural choice of leader called an exemplar. For algorithms thatdo not provide a leader, one can be assigned arbitrarily. The choice of a leader is not important for ourpurposes since we will show that stability is independent of the choice of leader. With this labeling ofvertices, we obtain a system that describes the collection of agents

rij = vij (2)

vij = uij . (3)

Now that we have clustered the agents, we need to design control inputs u to bring each cluster ofagents into formation. Within each cluster i, we implement the artificial potential control scheme. Foreach pair of agents (i, j) and (i, k) in cluster i we define an artificial potential function P ik

ij : R+ → R.

The potentials P ikij can be chosen to be any positive, continuous functions of the distance d = ||rij− rik||

between agents (i, j) and (i, k) that are:

• symmetric (P ikij = P ij

ik )

• unbounded near zero

(limd→0

P ikij (d) =∞

)• and that have a unique minimum at the desired distance between agents.


P(d)

d = ||rij − rik||

Figure 3: Artificial potential function P (d) = 1d2 + log d2.

An example of an appropriate potential function is shown in Figure 3. We can then define the totalpotential for vertex (i, j) as

Pij =

ni∑k=1

P ikij .

We introduce some notation that will be useful in defining a control protocol for this system of agents.For each pair of vertices (i, j) and (i, k) from the same cluster, define the function uikij : R2 ×R2 → R as

uikij (rij , rik) = ||rij − rik||. The position derivative ∇rij of the total potential function Pij is then definedas the 2-dimensional vector

∇rijPij =

ni∑k=1

(∂(P ik

ij ◦ uikij )

∂xij,∂(P ik

ij ◦ uikij )

∂yij

).

This notation, used in [28, 29], emphasizes the fact that the control protocol given in the followingparagraph is essentially a modified gradient descent, in which each agent seeks to minimize its totalpotential using the local information available to it. The artificial potential control protocol combinesthis gradient descent with a consensus protocol in the heading θij as follows:

uij = −∇rijPij −∑k

(θij − θik)||vij ||||rij − rik||

n(ij), j 6= 1, (4)

where n(ij) is the unit vector orthogonal to vij , and θij = atan2(yij , xij) is the heading of agent (i, j).Among the leaders of each cluster we apply an additional thrust to coordinate their velocities

ui1 = −∇ri1Pi1 −∑k

(θi1 − θik)||vi1||||ri1 − rik||

n(i1) +∑h∈Ni

bih(vh1 − vi1), (5)

where bih ≥ 0 are the entries of an irreducible matrix B. This ensures that the leader communicationgraph GB that encodes the interactions among the cluster leaders is strongly connected [3].

4.3 Results

We wish to demonstrate that the system defined by Equations (2)-(5) will eventually reach a desirableequilibrium.

Definition 4. A control protocol u is said to solve the formation stabilization problem if solutions of(2)-(3) converge asymptotically to a state in which


a) the relative positions of each agent (i, j) within a cluster are such that a local minimum of the totalvertex potential Pij is achieved, and

b) for any two agents (i, j) and (h, k), their headings satisfy θij = θhk.

The following lemma is needed in the proof of our main theorem. It’s proof is the same as that forequation (7) in [29] and is therefore omitted.

Lemma 1.

θij = −ni∑k=1

θij − θik||rij − rik||2

.

Theorem 3. Given any clustering scheme, the HPC protocol solves the formation stabilization problemprovided that the leader communication graph GB is strongly connected.

Proof. For each agent indexed (i, j), define a vertex Lyapunov function

Vij =1

2

(Ni∑k=1

P ikij (||rij − rik||) + vij · vij + θ2ij

),

and let c = (c1, . . . , cN ) denote the vector of cofactors of the columns of LG , as given by Kirchhoff’sMatrix Tree Theorem. Define

V =

N∑i=1

ci

ni∑j=i

Vij .

We will show that V is a Lyapunov function for the multi-agent systems (2)-(5). Since GB is stronglyconnected, there is a tree rooted at each vertex of GB . Thus all entries of c are strictly positive andhence V is positive definite. Taking the derivative of V along trajectories of (2)-(5) we get

V =

N∑i=1

ci

ni∑j=i

Vij

=

N∑i=1

ci

ni∑j=i

1

2

[ni∑k=1

(∇rijP

ikij · rij +∇rikP

ikij · rik

)+ 2 vij · vij + 2 θij θij

]

=

N∑i=1

ci

ni∑j=i

[ni∑k=1

1

2

(∇rijP

ikij · rij +∇rijP

ijik · rij

)+ vij · vij + θij θij

].

Since we assume that the artificial potentials are symmetric, (i.e. P ikij = P ij

ik ) we can re-write this

expression for V .

V =

N∑i=1

ci

ni∑j=i

[ni∑k=1

∇rijPikij · rij + vij · vij + θij θij

]

=

N∑i=1

ci

ni∑j=i

[∇rijPij · vij + vij ·

(−∇rijPij −

∑k


n(ij)

)+ θij θij

]

+

N∑i=1

ci∑h∈Ni

bih(vh1 − vi1) · vi1

=

N∑i=1

ci

ni∑j=i

[− vij ·

∑k


n(ij) + θij θij

]+

N∑i=1

ci∑h∈Ni

bih(vh1 − vi1) · vi1.

By definition, the vector n(ij) is orthogonal to vij so we can further simplify our expression for V .


V =

N∑i=1

ci

ni∑j=i

θij θij +

N∑i=1

ci∑h∈Ni

bih(vh1 − vi1) · vi1. (6)

For each cluster i, define the cluster distance graph Gi as the complete graph on ni vertices (without

self-loops) where the edge ~jk is assigned a weight 1||rij−rik||2 corresponding to the squared distance

between agents (i, j) and (i, k). We can also define the vector θi = (θi1, . . . , θini)T ∈ Rni to be the

concatenation of the headings of all agents in group i. Using Lemma 1 we can rewrite the expression forV in terms of the weighted Laplacian matrix Li of the graph Gi as follows

V = −N∑i=1

ci θTi Liθi +

N∑i=1

ci∑h∈Ni

bih(vh1 − vi1) · vi1.

For each group i, the Laplacian Li is symmetric and positive semidefinite. Since Gi is strongly connected,the nullspace of Li is spanned by the consensus vector 1 = (1, . . . , 1)T [24]. Therefore we have

V ≤N∑i=1

ci∑h∈Ni


with equality if and only if we have heading consensus θij = θik for all agents within each group i. Wenow have

V ≤N∑i=1

ci∑h∈Ni


=

N∑i=1

ci∑h∈Ni

bih(xh1xi1 − x2i1 + yh1yi1 − y2i1)

≤N∑i=1

ci∑h∈Ni

bih(x2h1 + x2i1

2− x2i1 +

y2h1 + y2i12

− y2i1)

=

N∑i=1

ci∑h∈Ni

bih1

2(x2h1 − x2i1 + y2h1 − y2i1).

It is easy to verify that the set of edge functions{Fhi = 1

2 (x2h1 − x2i1 + y2h1 − y2i1) : i, h ∈ V (GB)}

satisfies the cycle condition. Thus by Theorem 2 we have V ≤ 0. We also have V = 0 if and only ifθij = θik for all agents within each group i and the velocities of the leaders of each group are identical(vi1 = vh1 for all group indices i, h ∈ {1, ..., N}). Thus the system’s configuration will converge to thelargest invariant set M contained in {(r, v) : θ11 = · · · = θNnN

and v11 = · · · = vN1}. So we see thatpart (b) of Definition 4 is satisfied.

To see that part (a) is satisfied, consider the derivative of Pij along trajectories of the system (2)-(5).In M , we have

Pij = ∇rijPij · vij = ∇rijPij · (−∇rijPij) ≤ 0.

Thus trajectories converge to a configuration corresponding to a local minimum of Pij .

4.4 Simulations

To demonstrate our results numerically, we begin by randomly assigning initial positions and velocitiesto a set of 13 agents. Positions are assigned within a 10 by 10 area using a uniform distribution, andvertical and horizontal velocities are assigned uniformly in the range [−1, 1] (Figure 4a).

To group agents into clusters, we perform a k-means clustering [12] with the parameter k set to 4(Figure 4b). Each cluster is assigned a leader at random and any pair of distinct agents within a clusteris linked by the artificial potential function P (x) = 1

x2 + log x2 that has a global minimum at x = 1. The


resulting formation is shown in Figure 4d. The distances between agents within each group converge toa value of approximately 1 which minimizes the potential functions. Although it cannot be seen fromthe diagrams, the agent headings also converge.

(a) Randomly generated positions of 13 agents at t = 0 (b) Agents are grouped using a k-means clustering

(c) Position at time t = 1 (d) Position at time t = 6

Figure 4

5 Application: Global stability of the endemic equilibrium formulti-stage diseases in heterogeneous populations

5.1 Background

Compartmental models have been used extensively to model the spread of disease in large populations[10, 33]. The Kermack-McKendrick SIR model [13] was introduced in 1927 and provided some of thefirst mathematical insights into how disease spreads through a population then disappears. To modelinfections in a heterogeneous population more accurately, the basic SIR models are extended by dividingthe population into homogeneous groups based on factors including age, size, spatial position, socialgrouping and gender. This leads to a networked multi-group model coupled by the spread of diseasebetween groups [17, 30, 31].

Many diseases such as HIV/AIDS progress through a long period of time and exhibit distinctivedisease stages. At different stages, individuals may have varying levels of both infectivity and mortality.Thus we require a compartmental model structured based on both disease stage and population hetero-geneity. In [11], Jacques et al. present a multi-stage multi-group model of HIV/AIDS. They demonstratethat when an endemic exists, it is unique and locally asymptotically stable, and they conjecture that


it is in fact globally stable. In this section, we present a model similar to the Jacques model and useTheorem 2 to construct a global Lyapunov function to show that the endemic equilibrium for this modelis globally asymptotically stable whenever it exists. While most multi-stage models in the literature onlyconsiders forward progression of the disease through stages, the network topology of our model is moregeneral than that of Jacquez et al. in that we allow the disease to progress or ameliorate between anytwo stages.

5.2 Model

The dynamics of an infection in a population structured into N groups, in which an infected individualfrom the i-th group might progress through some number ni of stages can be described by a system ofdifferential equations in double-indexed variables of the form xij , the index i to denote the group andthe index j to denote the stage of infection. Such a system is described by the multi-group, multi-stageepidemic model given by the following differential equations:

dxi1dt

= Λi − di1xi1 −N∑

h=1

nh∑k=2

βhki xhkxi1

dxi2dt

=

N∑h=1

nh∑k=2

βhki xhkxi1 +

ni∑l=2

δili2xil −(di2 +

ni∑l=2

δi2il

)xi2

dxijdt

=

ni∑l=1

δilijxil −(dij +

ni∑l=2

δijil

)xij , if j 6∈ {1, 2},

(7)

where

• xi1 denotes the population of the susceptible compartment of group i

• xij denotes the population of the j-th infectious stage of group i where j ranges from 2 to ni

• Λi denotes the birth rate into group i

• dij denotes the rate of removal from xij (the sum of the death rate and rate of progression to theterminal stage)

• δikij denotes the rate of movement from infectious stage k to stage j in group i

• βhki is the transmission coefficient for infection of susceptible individuals in group i by individuals

in group h at infectious stage k.

Figure 5: Flow between disease stages corresponding to group i.


Figure 6: Routes of disease transmission.

To study the stability of an equilibrium

x∗ = (x∗11, x∗12, . . . , x

∗NnN

),

we define the equilibrium flow graph G as the hierarchical graph with block adjacency matrix

AG = [ahkij ] =

D1 B12 B13 · · · B1N

B21 D2 B23 · · · B2N

B31 B32 D3 · · · B3N

......

.... . .

...BN1 BN2 BN3 . . . DN

,

where the diagonal blocks Di encode the progression and transmission of disease within each group oncethe system has reached equilibrium

Di =

0

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 0 · · · 0

βi2i x∗i2x∗i1 0 δi2i3x

∗i2 · · · δi2ini

x∗i2βi3i x∗i3x∗i1 δi3i2x

∗i3 0 · · · δi3ini

x∗i3...

......

. . ....

βinii x∗ini

x∗i1 δinii2 x

∗ini

δinii3 x

∗ini

· · · 0

,

and the other blocks each have a single nonzero column that encodes the transmission of disease betweengroups

Bhi =

0 0 0 · · · 0

βh2i x∗h2x

∗i1 0 0 · · · 0

βh3i x∗h3x

∗i1 0 0 · · · 0

......

.... . .

...

βhnhi x∗hnh

x∗i1 0 0 · · · 0

.The equilibrium flow graph is shown in Figure 7.


Figure 7: The equilibrium flow graph G. Edge weights are not labeled to avoid clutter.

5.3 Results

The following two lemmas are needed for the proof of the main theorem of this section. Lemma 2establishes a property of the Laplacian cofactors of graph G, and Lemma 3 gives a general property ofthe Laplacian cofactors that will be useful in the proof of the main theorem.

Lemma 2. For each i ∈ {1, . . . , N} the Laplacian cofactors ci1 and ci2 are identical.

Proof. For each i ∈ {1, ..., N} the (i, 1)-th row of LG contains only two nonzero entries:

• the diagonal entry

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 corresponding to the in-degree of vertex (i, 1) and

• the entry in the ((i, 1) + 1)-th column: −N∑

h=1

nh∑k=2

βhki x∗hkx

∗i1

(the negative of the (1, 2) entry of Di)

The Laplacian matrix is always singular. So performing a cofactor expansion along the (i, 1)-th row weget

0 = det(LG) = ci1

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 − ci2

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

and thus ci1 = ci2.

Lemma 3. The vector c = (c11, ..., cNnN) of Laplacian cofactors is a right eigenvector of LG correspond-

ing to the eigenvalue 0.

Proof. Let (LG)i denote the i-th row of LG . Performing a cofactor expansion along the i-th row we seethat 0 = det(LG) = (LG)i · c for all i and hence LGc = 0.


Let K denote the closed positive orthant of the state space. An equilibrium of the epidemiologicalsystem (7) is called endemic if it lies in the interior K of K. Biologically, points in the interior of Kcorrespond to states in which disease persists throughout the population. The following theorem showsthat whenever an endemic equilibrium exists it is necessarily uniqe. Furthermore, any disease introducedinto the population will persist for all time, and the solutions will converge asymptotically to the endemicequilibrium.

Theorem 4. If the graph G is strongly connected and there exists an endemic equilibriumx∗ = (x∗11, x

∗12, . . . , x

∗NnN

) of the system (7) then x∗ is unique and globally stable in K.

Proof. Consider the Lyapunov function

V (x) =

N∑i=1

ni∑j=1

cij

[xij − x∗ij − x∗ij ln

(xijx∗ij

)],

where cij is the cofactor corresponding to the (i, j)-th column of LG . Since G is strongly conneted, all thecofactors are positive and therefore V is globally positive definite and radially unbounded. Its derivativealong trajectories of the systems is given by

V (x) =

N∑i=1

ni∑j=1

cij

(1−

x∗ijxij

)dxijdt

=

N∑i=1

[ci1

(1− x∗i1

xi1

)(Λi − di1xi1 −

N∑h=1

nh∑k=1

βhki xhkxi1

)

+ci2

(1− x∗i2

xi2

)( N∑h=1

nh∑k=1

βhki xhkxi1 +

ni∑l=1

δili2xil −(di2 +

ni∑l=1

δi2il

)xi2

)

+

ni∑j=3

cij

(1−

x∗ijxij

)( ni∑l=1

δilijxil −(dij +

ni∑l=2

δijil

)xij

) .We can then use the equilibrium equations

Λi = di1x∗i1 +

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1,(

di2 +

ni∑l=2

δi2il

)x∗i2 =

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 +

ni∑l=2

δili2x∗il,(

dij +

ni∑l=2

δijil

)x∗ij =

ni∑l=1

δilijx∗il, if j 6∈ {1, 2},

to rewrite our expression for V :

V (x) =

N∑i=1

[ci1

(1− x∗i1

xi1

)(di1x

∗i1 +

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 − di1xi1 −

N∑h=1

nh∑k=1

βhki xhkxi1

)

+ci2

(1− x∗i2

xi2

)( N∑h=1

nh∑k=1

βhki xhkxi1 +

ni∑l=1

δili2xil −(

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 +

ni∑l=2

δili2x∗il

)xi2x∗i2

)

+

ni∑j=3

cij

(1−

x∗ijxij

)( ni∑l=1

δilijxil −ni∑l=1

δilijx∗il

xijx∗ij

) .


We can now expand these expressions, and group terms with a single variable in the numerator separately.

V (x) =

N∑i=1

[ci1

N∑h=1

nh∑k=2

βhki xhkx

∗i1 + (ci2 − ci1)

N∑h=1

nh∑k=2

βhki xhkxi1 + ci2

ni∑l=1

δili2xil

−ci2(

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 −

ni∑l=2

δili2x∗il

)xi2x∗i2

+

ni∑j=3

cij

(ni∑l=1

δilijxil −ni∑l=1

δilijx∗il

xijx∗ij

)+

N∑i=1

[ci1 di1x

∗i1

(2− xi1

x∗i1− x∗i1xi1

)+ ci1

(1− x∗i1

xi1

) N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

+ci2

(N∑

h=1

nh∑k=2

βhki x∗hkx

∗i1 −

x∗i2xi2

N∑h=1

nh∑k=1

βhki xhkxi1

)

+

ni∑j=2

cij

(ni∑l=1

δilijx∗il −

x∗ijxij

ni∑l=1

δilijxil

)].

We can then apply Lemma 2 to replace the second term in the first sum and rewrite the other terms toobtain an expression for V that can be expressed in terms of the Laplacian matrix:

V (x) =

N∑i=1

[ci1

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

xhkx∗hk

+ (ci2 − ci1)

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

xi1x∗i1

+ ci2

ni∑l=1

δili2x∗il

xi1x∗i1

−ci2(

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1 −

ni∑l=2

δili2x∗il

)xi2x∗i2

+

ni∑j=3

cij

(ni∑l=1

δilijx∗il

xilx∗il−

ni∑l=1

δilijx∗il

xijx∗ij

)+

N∑i=1

[ci1 di1x

∗i1

(2− xi1

x∗i1− x∗i1xi1

)+ ci1

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i1

xi1

)

+ci2

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i2

xi2

xhkx∗hk

xi1x∗i1

)

+

ni∑j=2

cij

ni∑l=1

δilijx∗il

(1−

x∗ijxij

xilx∗il

)]

=

[x11x∗11

· · · xNnN

x∗Nnn

] LG

c11

...cNnN

+

N∑i=1

[ci1 di1x

∗i1

(2− xi1

x∗i1− x∗i1xi1

)+ ci1

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i1

xi1

)

+ci2

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i2

xi2

xhkx∗hk

xi1x∗i1

)

+

ni∑j=2

cij

ni∑l=1

δilijx∗il

(1−

x∗ijxij

xilx∗il

)].

Using Lemma 3 we see that the term involving the Laplacian is always zero. We also have

2− xi1x∗i1− x∗i1xi1≤ 0


for all positive xi1, with equality if and only if xi1 = x∗i1. So our equation becomes

V (x) ≤N∑i=1

[ci2

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i1

xi1

)

+ci1

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

(1− x∗i2

xi2

xhkx∗hk

xi1x∗i1

)

+

ni∑j=2

cij

ni∑l=1

δilijx∗il

(1−

x∗ijxij

xilx∗il

)].

Table 1

Edge Type Connects Vertices Edge Function

Infection edge (h, k)→ (i, 1) (k 6= 1) Fhki1 (xi1, xi2, xhk) =

(1− x∗i2

xi2

xhkx∗hk

xi1x∗i1

)Disease acquisition edge (i, 1)→ (i, 2) F i1

i2 (xi1) =

(1− x∗i1

xi1

)Disease progression edge (i, l)→ (i, j) (j, l 6= 1) F il

ij(xij , xil) =

(1−

x∗ijxij

xilx∗il

)

Defining edge functions Fhkij as in Table 1 and noting that

ai1i2 =

N∑h=1

nh∑k=2

βhki x∗hkx

∗i1

ahki1 = βhki x∗hkx

∗i1 (k 6= 1)

ailij = δilijx∗il (i, l 6= 1)

we see that

V (x) ≤N∑i=1

[ci2a

i1i2F

i1i2 (xi1)

+ci1

N∑h=1

nh∑k=2

ahki1 Fhki1 (xi1, xi2, xhk)

+

ni∑j=2

cij

ni∑l=1

ailij Filij(xij , xil)

]=

∑(i,j)∈V (G)

cij∑

(h,k)∈V (G)

ahkij Fhkij (x).

Thus hypothesis (a) of Theorem 2 is satisfied. To verify hypothesis (b), consider a cycle C of length lin the graph GA (shown in Figure 7). By construction, an infection edge (h, k) → (i, 1) appears in thecycle if and only if it is followed by the disease acquisition edge (i, 1)→ (i, 2). Thus we have(

1− Fhki1 (x)

)(1− F i1

i2 (x))

=(x∗i2xi2

xhkx∗hk

xi1x∗i1

)(x∗i1xi1

)=x∗i1xi1

xhkx∗hk· x∗i2

xi2

xi1x∗i1

.


Since the geometric mean of l positive numbers is bounded by their arithmetic mean, we now have∑((h,k)→(i,j)

)∈E(C)

Fhkij (x) = l −

∑((h,k)→(i,j)

)∈E(C)

(1− Fhkij (x))

≤ l − l

∏((h,k)→(i,j)

)∈E(C)

(1− Fhkij (x))

1/l

= l − l

∏((h,k)→(i,j)

)∈E(C)

x∗ijxij

xhkx∗hk

1/l

= 0.

So all the hypotheses of Theorem 2 are satisfied. Thus we have V (x) ≤ 0 for all x ∈ K. To have theequality V (x) = 0, we require that xi1 = x∗i1 for all i ∈ {1, . . . , N} and that (1− Fhk

ij ) = (1− Fh′k′

i′j′ ) forany pair of edges (h, k) → (i, j) and (h′, k′) → (i′, j′) in the graph. It is easy to see that this occurs ifand only if for all (i, j) ∈ V (G) we have xij = x∗ij .

So V (x) = 0 if and only if x = x∗. Therefore the endemic equilibrium x∗ = (x∗11, x∗12, . . . , x

∗NnN

)is globally stable in K. The uniqueness of x∗ follows from its global stability.

Acknowledgments

This research is supported in part by grants from the Natural Sciences and Engineering Research Councilof Canada (NSERC) and Canada Foundation for Innovation (CFI). MYL acknowledges the financialsupport from the NCE-MITACS. JM acknowledges the financial support of an NSERC USRA awardedat the University of Alberta.

References

[1] W. A. Barbakh, Y. Wu, and C. Fyfe, Review of clustering algorithms, in Non-Standard Pa-rameter Adaptation for Exploratory Data Analysis, Springer, 2009, pp. 7–28.

[2] R. Brooks, C. Smith, A. Stone, and W. Tutte, Determinants and current flows in electricnetworks, Discrete Math., 100 (1992), pp. 291–301.

[3] R. A. Brualdi, Combinatorial Matrix Classes, Encyclopedia Math. Appl. no. 108, CambridgeUniversity Press, 2006.

[4] Y.-L. Chuang, Y. R. Huang, M. R. D’Orsogna, and A. L. Bertozzi, Multi-vehicle flocking:scalability of cooperative control algorithms using pairwise potentials, in IEEE Int. Conf. Robot.,2007, pp. 2292–2299.

[5] M. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memorystorage by competitive neural networks, IEEE T. Syst. Man. Cyb., 13 (1983), pp. 815–826.

[6] C. Dalf, F. Comellas, and M. Fiol, The multidimensional manhattan network, Electron. NotesDiscrete Math., 29 (2007), pp. 383–387.

[7] R. C. Dorf and R. H. Bishop, Modern Control Systems, Prentice Hall, 2008.

[8] M. Egerstedt and X. Hu, Formation constrained multi-agent control, IEEE T. Robotic. Autom.,17 (2001), pp. 947–951.


[9] B. J. Frey and D. Dueck, Clustering by passing messages between data points, Science, 315(2007), pp. 972–976.

[10] H. W. Hethcote, A thousand and one epidemic models, in Frontiers in Mathematical Biology,no. 100 in Lecture Notes in Biomathematics, vol. 100, Springer, 1994, pp. 504–515.

[11] J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel, and T. Perry, Modeling andanalyzing HIV transmission: the effect of contact patterns, Mathematical Biosciences, 92 (1988),pp. 119–199.

[12] A. Jain, K. Murty, N. M, and P. J. Flynn, Data clustering: a review, ACM Comput. Surv.,31 (1999), pp. 264–323.

[13] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epi-demics, Proc. R. Soc. of Lond. Ser. A, 115 (1927), pp. 700–721.

[14] D. E. Knuth, Fundamental Algorithms, The Art of Computer Programming, vol, 1, Addison-Wesley, Massachusetts, 1997.

[15] J. Leskovec, D. Chakrabarti, J. Kleinberg, C. Faloutsos, and Z. Ghahramani, Kro-necker graphs: An approach to modeling networks, J. Mach. Learn. Res., 11 (2010), pp. 985–1042.

[16] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations onnetworks, J. Differential Equations, 248 (2010), pp. 1–20.

[17] X. Lin and J. W.-H. So, Global stability of the endemic equilibrium and uniform persistence inepidemic models with subpopulations, J. Aust. Math. Soc. B, 34 (1993), pp. 282–295.

[18] D. J. Low, Following the crowd, Nature, 407 (2000), pp. 465–466.

[19] Z. Lu and Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion systems,J. Math. Biol., 32 (1993), pp. 67–77.

[20] C. Maas, Transportation in graphs and the admittance spectrum, Discrete Appl. Math., 16 (1987),pp. 31–49.

[21] M. Mincheva and D. Siegel, Stability of mass action reaction-diffusion systems, Nonlinear Anal.,56 (2004), pp. 1105–1131.

[22] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros, Mutual interactions, potentials,and individual distance in a social aggregation, J. Theoret. Biol., 47 (2003), pp. 353–389.

[23] J. Moon, Counting Labelled Trees, Canadian Mathematical Monographs, vol. 1, Canadian Mathe-matical Congress, Montreal, 1970.

[24] R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switchingtopology and time-delays, IEEE T. Automat. Contr., 49 (2004), pp. 1520–1533.

[25] X. Pan, C. S. Han, K. Dauber, and K. H. Law, A multi-agent based framework for the sim-ulation of human and social behaviors during emergency evacuations, AI and Society, 22 (2007),pp. 113–132.

[26] C. Reynolds, Flocks, herds and schools: A distributed behavioral model, Comp. Graph., 21 (1987),pp. 25–34.

[27] M. Sales-Pardo, R. Guimer, A. A. Moreira, and L. A. N. Amaral, Extracting the hierar-chical organization of complex systems, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 15224–15229.

[28] H. G. Tanner, A. Jabdabaie, and G. J. Pappas, Stable flocking of mobile agents, Part I: Fixedtopology, in IEEE Decis. Contr. P., 2003, pp. 2010–2015.

[29] H. G. Tanner, A. Jadbabaie, and G. J. Pappas, Stability of flocking motion. Technical Report,University of Pennsylvania, 2003.


[30] H. R. Thieme, Local stability in epidemic models for heterogeneous populations, in Mathematicsin Biology and Medicine, V. Capasso, E. Grosso, and S. Paveri-Fontana, eds., Lecture Notes inBiomathematics 57, Springer, 1985, pp. 185–189.

[31] , Mathematics in Population Biology, Princeton University Press, 2003.

[32] R. B. Tilove, Local obstacle avoidance for mobile robots based on the method of artificial potentials.General Motors Research Laboratories, Research Publication GMR-6650, 1989.

[33] P. van den Driessche, Deterministic compartmental models: Extensions of basic models, in Math-ematical Epidemiology, Lecture Notes in Mathematics, Springer, 2008, pp. 147–157.

[34] J. H. van Lint and R. M. Wilson, A course in combinatorics, Cambridge University Press, 1992.

[35] A. I. Vol’pert, Differential equations on graphs, Sb. Math., 17 (1972), pp. 571–582.

[36] P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc., 13 (1962), pp. 47–52.

constructing lyapunov functions for large-scale ...jmaidens/projects/siap_li.pdf · constructing...

Documents